Applied Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Neuenfeld, Dominik
2019-07-31T15:51:03Z
2019
Doctor of Philosophy - PhD
University of British Columbia
In this thesis we discuss applications of quantum information theoretic concepts to
quantum gravity and the low-energy regime of quantum field theories.
The first part of this thesis is concerned with how quantum information spreads
in four-dimensional scattering experiments for theories coupled to quantum electrodynamics
or perturbative quantum gravity. In these cases, every scattering process
is accompanied by the emission of an infinite number of soft photons or gravitons,
which cause infrared divergences in the calculation of scattering probabilities.
There are two methods to deal with IR divergences: the inclusive and dressed
formalisms. We demonstrate that in the late-time limit, independent of the method,
the hard outgoing particles are entangled with soft particles in such a way that the
reduced density matrix of the hard particles is essentially completely decohered.
Furthermore, we show that the inclusive formalism is ill-suited to describe scattering
of wavepackets, requiring the use of the dressed formalism. We construct the
Hilbert space for QED in the dressed formalism as a representation of the canonical
commutation relations of the photon creation/annihilation algebra, and argue that it
splits into superselection sectors which correspond to eigenspaces of the generators
of large gauge transformations.
In the second part of this thesis, we turn to applications of quantum information
theoretic concepts in the AdS/CFT correspondence. In pure AdS, we find an
explicit formula for the Ryu-Takayanagi (RT) surface for special subregions in the
dual conformal field theory, whose entangling surface lie on a light cone. The
explicit form of the RT surface is used to give a holographic proof of Markovicity
of the CFT vacuum on a light cone. Relative entropy of a state on such special
subregions is dual to a novel measure of energy associated with a timelike vector flow between the causal and entanglement wedge. Positivity and monotonicity of
relative entropy imply positivity and monotonicity of this energy, which yields a
consistency conditions for solutions to quantum gravity.
https://circle.library.ubc.ca/rest/handle/2429/71167?expand=metadata
Aspects of quantum information in quantum field theoryand quantum gravitybyDominik Neuenfelda thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Physics)The University of British Columbia(Vancouver)July 2019© Dominik Neuenfeld, 2019The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Aspects of quantum information in quantum field theory and quantumgravitysubmitted by Dominik Neuenfeld in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in Physics.Examining Committee:Gordon Semenoff, PhysicsCo-supervisorIan Affleck, PhysicsUniversity ExaminerJoel Feldman, MathUniversity ExaminerAlison Lister, PhysicsSupervisory Committee MemberRobert Raussendorf, PhysicsSupervisory Committee MemberAdditional Supervisory Committee Members:Mark Van Raamsdonk, PhysicsCo-supervisoriiAbstractIn this thesis we discuss applications of quantum information theoretic concepts toquantum gravity and the low-energy regime of quantum field theories.The first part of this thesis is concerned with how quantum information spreadsin four-dimensional scattering experiments for theories coupled to quantum electro-dynamics or perturbative quantum gravity. In these cases, every scattering processis accompanied by the emission of an infinite number of soft photons or gravi-tons, which cause infrared divergences in the calculation of scattering probabilities.There are two methods to deal with IR divergences: the inclusive and dressedformalisms. We demonstrate that in the late-time limit, independent of the method,the hard outgoing particles are entangled with soft particles in such a way that thereduced density matrix of the hard particles is essentially completely decohered.Furthermore, we show that the inclusive formalism is ill-suited to describe scatter-ing of wavepackets, requiring the use of the dressed formalism. We construct theHilbert space for QED in the dressed formalism as a representation of the canonicalcommutation relations of the photon creation/annihilation algebra, and argue that itsplits into superselection sectors which correspond to eigenspaces of the generatorsof large gauge transformations.In the second part of this thesis, we turn to applications of quantum informationtheoretic concepts in the AdS/CFT correspondence. In pure AdS, we find anexplicit formula for the Ryu-Takayanagi (RT) surface for special subregions in thedual conformal field theory, whose entangling surface lie on a light cone. Theexplicit form of the RT surface is used to give a holographic proof of Markovicityof the CFT vacuum on a light cone. Relative entropy of a state on such specialsubregions is dual to a novel measure of energy associated with a timelike vectoriiiflow between the causal and entanglement wedge. Positivity and monotonicity ofrelative entropy imply positivity and monotonicity of this energy, which yields aconsistency conditions for solutions to quantum gravity.ivLay SummaryQuantum information theory, the theory of how information is processed in quantumsystems, plays an important role in deepening our understanding of quantumgravity,a theory which seeks to unify quantum and gravitational physics. In this thesis weapply quantum information theoretic concepts in two contexts.First, we investigate the quantum information carried away by radiation pro-duced after particles interact gravitationally or through the electromagnetic inter-action. In such interactions, an infinite number of very low-energy particles areproduced; these particles carry away a large amount of information about the parti-cles undergoing the interaction. We formulate methods of calculation which allowinvestigation of the information spread due to the production of these low-energyparticles.Second, we translate quantum information theoretic inequalities into inequal-ities in quantum gravity. This supplements the equations of gravitational physicswith additional constraints that must be obeyed in a consistent theory of quantumgravity.vPrefaceA large part of the body of this thesis has been published elsewhere and is includedverbatim. The ordering of author names is alphabetical.Most of chapter 4 is an adapted version of D. Carney, L. Chaurette, D. Neuenfeldand G. Semenoff, Infrared quantum information, Phys.Rev.Lett. 119 (2017) no.18,180502 [1]. Like the two following papers, this publication is a result of manydiscussions and close collaboration between all authors. My main contributionswere towards the identification of the currents and the the proof of their relation tothe decoherence condition. The manuscript was drafted by D. Carney and editedby all authors. Chapter 4.5 is unpublished, original work. I thank L. Chaurette fordiscussions at an early stage.A version of chapter 5 has appeared as D. Carney, L. Chaurette, D. Neuenfeldand G. Semenoff, Dressed infrared quantum information, Phys.Rev. D97 (2018)no.2, 025007 [2]. The calculation which lead to equation (5.9) was carried outby D. Carney and L. Chaurette. The generalization to multi-particle states andthe proof of the finiteness of the reduced density matrix was joint work betweenall authors. Furthermore I contributed to chapters 5.4 and 5.5 which discuss thephysical interpretation of dressed states and the relation to black hole information.A first draft of the manuscript was prepared by D. Carney and L. Chaurette andedited by all authors.Chapter 6 contains a version of D. Carney, L. Chaurette, D. Neuenfeld andG. Semenoff, On the need for soft dressing, J. High Energ. Phys. (2018) 2018:121[3]. Most of the preliminary calculations were work shared between L. Chauretteand myself. I contributed the findings on the inconsistency of scattering of normal-ized wave packets in the inclusive formalism, chapter 6.4, and a first draft of thevimanuscript, which was edited by all authors. Versions of chapters 4 - 6 have alsoappeared in [4].A version of chapter 7 was uploaded to the Arxiv as Infrared-safe scatteringwithout photon vacuum transitions and time-dependent decoherence [5]. I amthe sole author of this work, which has greatly benefited from discussions withD. Carney, L. Chaurette and G. Semenoff.Chapter 9 has been published as D. Neuenfeld, K. Saraswat andM. Van Raams-donk, Positive gravitational subsystem energies from CFT cone relative entropies,J. High Energ. Phys. (2018) 2018:50, [6]. The paper is a result of close collab-oration between the authors. Calcuations were shared work between K. Saraswatand myself, while drafting the manuscript was shared work between all authors.Related material also appeared in [7].viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Quantum information in fundamental physics . . . . . . . . . . . . 11.1 Black hole entropy and the quest for quantum gravity . . . . . . . 11.2 Quantum information theory in fundamental physics . . . . . . . 21.3 The roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A very short introduction to quantum information . . . . . . . . . . 52.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Relative entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Markovicity of quantum states . . . . . . . . . . . . . . . . . . . 82.5 Quantum information in quantum field theories . . . . . . . . . . 9viiiI Quantum information in the infrared . . . . . . . . . . . . . . 113 Infrared divergences in quantum field theory . . . . . . . . . . . . . 123.1 Scattering and the asymptotic Hilbert space . . . . . . . . . . . . 143.2 Infrared divergences in S-matrix scattering . . . . . . . . . . . . . 163.3 A semiclassical analysis . . . . . . . . . . . . . . . . . . . . . . . 213.4 Dealing with infrared divergences . . . . . . . . . . . . . . . . . 243.4.1 The inclusive formalism . . . . . . . . . . . . . . . . . . 253.4.2 Dressed formalisms . . . . . . . . . . . . . . . . . . . . . 293.5 An infinity of conserved charges . . . . . . . . . . . . . . . . . . 333.5.1 Anti-podal matching and conserved charges . . . . . . . . 333.5.2 Hard and soft charges . . . . . . . . . . . . . . . . . . . . 353.5.3 Weinberg’s soft theorems . . . . . . . . . . . . . . . . . . 364 Infrared quantum information . . . . . . . . . . . . . . . . . . . . . 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Decoherence of the hard particles . . . . . . . . . . . . . . . . . 384.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Entropy of the soft bosons . . . . . . . . . . . . . . . . . . . . . 434.5 Relation to large gauge symmetries . . . . . . . . . . . . . . . . . 434.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Dressed infrared quantum information . . . . . . . . . . . . . . . . 485.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 IR-safe S-matrix formalism . . . . . . . . . . . . . . . . . . . . . 495.3 Soft radiation and decoherence . . . . . . . . . . . . . . . . . . . 505.4 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . 545.5 Black hole information . . . . . . . . . . . . . . . . . . . . . . . 555.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 On the need for soft dressing . . . . . . . . . . . . . . . . . . . . . . 576.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Scattering of discrete superpositions . . . . . . . . . . . . . . . . 596.2.1 Inclusive formalism . . . . . . . . . . . . . . . . . . . . . 60ix6.2.2 Dressed formalism . . . . . . . . . . . . . . . . . . . . . 626.3 Wavepackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3.1 Inclusive formalism . . . . . . . . . . . . . . . . . . . . . 646.3.2 Dressed wavepackets . . . . . . . . . . . . . . . . . . . . 656.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4.1 Physical interpretation . . . . . . . . . . . . . . . . . . . 666.4.2 Allowed dressings . . . . . . . . . . . . . . . . . . . . . 676.4.3 Selection sectors . . . . . . . . . . . . . . . . . . . . . . 716.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 An infrared-safe Hilbert space for QED . . . . . . . . . . . . . . . . 737.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.1.1 Summary of results . . . . . . . . . . . . . . . . . . . . . 757.2 Representations of the canonical commutation relations . . . . . . 787.2.1 Inequivalent CCR representations . . . . . . . . . . . . . 787.2.2 Von Neumann space . . . . . . . . . . . . . . . . . . . . 797.2.3 Unitarily inequivalent representations on IDPS . . . . . . 807.3 Asymptotic time-evolution and definition of the S-matrix . . . . . 837.3.1 The naive S-matrix . . . . . . . . . . . . . . . . . . . . . 837.3.2 The asymptotic Hamiltonian . . . . . . . . . . . . . . . . 847.3.3 The dressed S-matrix . . . . . . . . . . . . . . . . . . . . 867.4 Construction of the asymptotic Hilbert space . . . . . . . . . . . . 887.4.1 The asymptotic Hilbert space . . . . . . . . . . . . . . . 887.4.2 Multiple particles and classical radiation backgrounds . . 927.4.3 Comments on the Hilbert space . . . . . . . . . . . . . . 937.5 Unitarity of the S-matrix . . . . . . . . . . . . . . . . . . . . . . 947.6 Example: Classical current . . . . . . . . . . . . . . . . . . . . . 967.6.1 Calculation of the dressed S-matrix . . . . . . . . . . . . 967.6.2 Tracing out long-wavelength modes . . . . . . . . . . . . 987.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102xII Quantum information in quantum gravity . . . . . . . . . . . 1048 The AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . . 1058.1 Holography in string theory . . . . . . . . . . . . . . . . . . . . . 1058.1.1 AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.1.2 The dictionary . . . . . . . . . . . . . . . . . . . . . . . 1078.1.3 Holographic entanglement entropy . . . . . . . . . . . . . 1088.1.4 Causal wedge vs entanglement wedge . . . . . . . . . . . 1109 Positive gravitational subsystem energies from CFT cone relativeentropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169.2.1 Relative entropy in conformal field theories . . . . . . . . 1169.2.2 Gravity background . . . . . . . . . . . . . . . . . . . . . 1189.3 Bulk interpretation of relative entropy for general regions boundedon a lightcone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.4 Perturbative expansion of the holographic dual to relative entropy 1249.4.1 Light cone coordinates for AdS . . . . . . . . . . . . . . 1249.4.2 HRRT surface in pure AdS . . . . . . . . . . . . . . . . . 1259.4.3 The bulk vector field . . . . . . . . . . . . . . . . . . . . 1289.4.4 Perturbative formulae for ∆Hξ . . . . . . . . . . . . . . . 1299.5 Holographic proof of the Markov property of the vacuum state . . 1339.5.1 The Markov property for states on the null-plane . . . . . 1339.5.2 The Markov property for states on the lightcone . . . . . . 1359.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.1 Infrared quantum information . . . . . . . . . . . . . . . . . . . . 13810.2 Quantum information and holography . . . . . . . . . . . . . . . 140Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A Infrared quantum information . . . . . . . . . . . . . . . . . . . . . 152xiB Dressed soft factorization . . . . . . . . . . . . . . . . . . . . . . . . 156C On the need for soft dressing . . . . . . . . . . . . . . . . . . . . . . 158C.1 Proof of positivity of ∆A,∆B . . . . . . . . . . . . . . . . . . . . 158C.2 The out-density matrix of wavepacket scattering . . . . . . . . . . 159C.2.1 Contributions to the out-density matrix . . . . . . . . . . 159C.2.2 Taking the cutoff λ→ 0 vs. integration . . . . . . . . . . 162D Cone Relative Entropies . . . . . . . . . . . . . . . . . . . . . . . . . 164D.1 Equivalence of Hξ on the boundary and the modular Hamiltonian 164D.2 The HRRT surface ending on the null-plane . . . . . . . . . . . . 165D.3 Calculation of the binormal . . . . . . . . . . . . . . . . . . . . . 167D.4 Hollands-Wald gauge condition . . . . . . . . . . . . . . . . . . . 168xiiList of FiguresFigure 3.1 Feynman diagram for an electron scattering off of a potential . 16Figure 3.2 Construction of loops on external legs . . . . . . . . . . . . . 17Figure 3.3 Penrose diagram for Minkowski space . . . . . . . . . . . . . 34Figure 6.1 Scattering of plane waves through a single slit and productionof radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 7.1 The dressed S-matrix . . . . . . . . . . . . . . . . . . . . . . 75Figure 8.1 The Ryu-Takayanagi prescription . . . . . . . . . . . . . . . . 108Figure 8.2 Phase transition of the Ryu-Takayanagi surface . . . . . . . . 109Figure 9.1 A region with boundary on a past lightcone . . . . . . . . . . 114Figure 9.2 The surfaces A, A˜ and Aˆ . . . . . . . . . . . . . . . . . . . . 115Figure B.1 Emission of radiation in the dressed formalism . . . . . . . . 156xiiiAcknowledgmentsI would like to thank my supervisors Gordon Semenoff and Mark Van Raamsdonkfor their support throughout this thesis. Moreover, I thank my collaborators DanCarney, Laurent Chaurette, and Krishan Saraswat, as well as the additional mem-bers of my supervisory committee Alison Lister and Robert Raussendorf. It is apleasure to also thank all other members of the String theory group at UBC and allother physics-enthusiasts I had the chance of meeting for interesting and insightfulconversations about science and otherwise.My co-authors and I thank Scott Aaronson, Tim Cox, Aidan Chatwin-Davis,Colby Delisle, William Donnelly, Wojciech Dybalski, Bart Horn, Raphael Flauger,John Preskill, Alex May, Duff Neill, Wyatt Reeves, Mohammad Sheikh-Jabbari,Philip Stamp, Andy Strominger, Bill Unruh, Jordan Wilson-Gerow, and ChrisWaddell for discussions and comments.This thesis would not have been possible without the understanding and supportof my friends on both sides of the Atlantic, my family, and – above all – my partner,Birthe Lente.I amgrateful for financial support provided by theUniversity ofBritishColumbiathrough a Four Year Doctoral Fellowship, the Simons Foundation, the Natural Sci-ences and Engineering Research Council of Canada, and Green College throughthe Norman Benson award.xivChapter 1Quantum information infundamental physics1.1 Black hole entropy and the quest for quantum gravityBased on the requirement that the second law of thermodynamics should hold evenin the presence of black holes, Bekenstein [8] conjectured that black holes shouldpossess entropy. If this were not the case, one could drop a system with non-zeroentropy into a black hole and thus – at least operationally – violate the secondlaw of thermodynamics. Bekenstein conjectured the entropy of a black hole to beproportional to the area of its event horizon, ABH, divided by Newton’s constantGN . Consequently, to save the second law of thermodynamics, the concept ofentropy should be replaced with a generalized entropySgen =ABH4GN+ Sout, (1.1)which does not decrease; here, Sout denotes the entropy of matter outside the blackhole horizon. The conjecture that black holes have entropy and thus should beseen as thermodynamical systems was subsequently supported by Hawking [9],who demonstrated that black holes radiate at a temperature proportional to theirsurface gravity. The results were in line with the predicted scaling of entropy withhorizon area, andmade black hole thermodynamics consistent. In thermodynamics,1statistical physics, and information theory, entropy is a measure of the lack ofknowledge about the microstate of a system, assuming we know its macroscopicproperties. At least in string theory, this interpretation also applies to the entropyof certain black holes, as can be shown by microstate counting.The relation between area and entropy indicates that certain quantities in quan-tum gravity can be understood in information theoretic terms. If, as is widelybelieved, quantum gravity is a true quantum theory, it thus seems reasonable thatprogress can be made by using concepts from quantum information theory in thestudy of quantum gravity.1.2 Quantum information theory in fundamental physicsIn the past decades, the application of quantum information theory has been atthe center of various important discoveries in fundamental physics. One of themost renowned discoveries is the black hole information paradox. The radiationemitted by black holes, as calculated by Hawking, was found to be completelyrandom. If this were to remain true in the full quantum theory, the evaporationof a black hole would erase all information about what has fallen into it, therebyviolating the basic premise of quantum theory that time-evolution is unitary, i.e.,information conserving [9]. A version of the black hole information paradox [10]can be explained in terms of quantum information theoretic quantities. Excitedmodes close to the black hole horizon have to be strongly entangled with modesbehind the horizon in order to give a smooth geometry and thus allow for theequivalence principle of general relativity to hold, which states that a freely fallingobserver should not note anything out of the ordinary when they cross the horizon.On the other hand, at least at late times, modes close to the horizon must also bestrongly entangled with early-time modes of the Hawking radiation if unitarity is tobe preserved [11]. A property referred to as monogamy of entanglement prohibitsstrong entanglement with two disparate subsystems, thus posing a paradox: undercertain additional physically-motivated assumptions, either the geometry at thehorizon is not smooth and the equivalence principle fails, or black hole evaporationis not unitary.Concepts from quantum information theory have also played an important role2in understanding how spacetime emerges in the AdS/CFT correspondence. In itssimplest form, the AdS/CFT correspondence [12]1 is a proposed duality between asuperconformal field theory in d dimensions and string theory in an asymptoticallyanti-de Sitter spacetime in d + 1 dimensions. The conformal field theory can bethought of as living at the conformal boundary of the anti-de Sitter spacetime.The entanglement entropy of a subregion of the field theory can be computed inthe gravitational theory as the area of a special surface anchored on this boundary[13, 14]. This suggests that in a holographic theory, spacetime in the gravitationalpicture is intimately linked to entropy in the field theory [15, 16].The use of quantum information theoretic quantities has lead to new conjecturesand proofs in semiclassical gravity and quantum field theory; see e.g., [17–19].Moreover, concepts from quantum information theory have been used to obtaina better understanding of the dynamics of black holes [20], and to find discretetoy models [21] and explain properties of the AdS/CFT correspondence such assubregion duality [22].The success of quantum information theoretic ideas in black hole physics andquantum gravitymotivates furthering those investigations, and applying thesemeth-ods to other problems such as scattering theory [23, 24].1.3 The roadmapThe first part of this thesis analyzes the impact of infrared (IR) divergences onquantum information theoretic quantities. We will investigate how the presenceof IR divergences affects the information carried away by unobserved particles inscattering. The surprising result is that QED and perturbative quantum gravity bothpredict that unobservable radiation carries away an essentially maximal amount ofinformation and leaves the observed particles in a mixed state; this is independent ofwhich method is used to render IR divergences finite. However, closer investigationshows that the typical prescription for removing IR divergences, while applicableto the scattering of momentum eigenstates, cannot be used to study the scatteringof wavepackets. This hints at a rich structure of the Hilbert spaces of QED andperturbative quantum gravity, which split into superselection sectors corresponding1see also chapter 83to representations of the canonical commutation relations. The so-called dressedformalism takes into account this structure, and can be used to define approximatefinite-time scattering amplitudes, which allow for the calculation of decoherencerates.There has also been a recent resurgence of interest in the infrared structure ofgauge theories and gravity coming from a seemingly different perspective.2 It hasbeen shown that certain theorems involving soft bosons can be understood as Wardidentities of asymptotic symmetries; they can be thought of as gauge transformationsthat extend to infinity [27–29]. This has lead to speculations about how black holesstore information [30–33]. We will see below that in four dimensions, infrareddivergences, decoherence and large gauge transformations are intimately linked.We will use this relation to comment on the role of the infrared in solutions to theblack hole information paradox.In the second part of this thesis, we will briefly introduce the AdS/CFT cor-respondence. There exists a large body of work which links information-theoreticinequalities in the CFT to geometric constraints in gravitational theories, i.e., [34–39]. Here, we extend results that posit an equivalence between the relative entropyof ball-shaped subregions of a holographic CFT and a measure of energy defined ona subregion of its holographic dual, broadening these results to a more general classof subregions. We obtain explicit expressions for extremal surfaces in pure AdSand use them to give straightforward holographic proofs of the Markov propertyfor the vacuum state of a ball-shaped region.In the next chapter, we give a brief review of concepts fromquantum informationtheory which are relevant for the rest of this thesis. Reviews of infrared divergencesand ways of dealing with them, as well as concepts relevant to the AdS/CFTcorrespondence, can be found in the introductions to parts I and II of this thesis,respectively.2For a review, see [25]. For earlier work, see [26].4Chapter 2A very short introduction toquantum informationThis chapter will give a brief introduction to the quantum information theoreticquantities which appear in this thesis. Sections 2.1 and 2.2 are relevant for bothparts of the thesis, whereas sections 2.3 to 2.5 are only relevant for the second part.More detailed introductions can be found in [40, 41].2.1 Quantum mechanicsIn quantum mechanics, the state of a physical system is described by a unit nor-malized vector in Hilbert space, up to a phase. Given two physical systems A andB with Hilbert spaces HA and HB, any state of the joint system is described by avector in the product Hilbert space HA∪B = HA ⊗ HB. In particular, the systemcan be in a superposition|ψ〉 = cos(α) |0〉A ⊗ |0〉B + sin(α) |1〉A ⊗ |1〉B . (2.1)Here, the states |0〉A/B and |1〉A/B are two orthogonal states of the Hilbert spaceHA/B. For generic values of α, it is impossible to write the total state as the productof the states of the systems A and B,|ψ〉 , |φ〉A ⊗ |η〉B , (2.2)5and thus the states of subsystems A and B are correlated.Measurements on quantum systems are represented as Hermitean operators,O† = O. The real eigenvalues of O give the allowed measurement outcomes, andthe average outcome of a measurement of O in the state |ψ〉 is given by the innerproduct 〈ψ |O |ψ〉. We can equivalently describe the state |ψ〉 by a density matrixρ = |ψ〉 〈ψ | , (2.3)such that the expectation value is given by〈ψ |O |ψ〉 = tr(Oρ). (2.4)If ρ is constructed from a state as shown in equation (2.3), it is called pure.In the case of a multi-partite system HA∪B, we can imagine operations whichonly act on one subsystem, say subsystem A. Such measurements are representedby operators OA = O˜A ⊗ 1B. If the multipartite system is in a product state, forexample |ψ〉 = |1〉 ⊗ |0〉, the expectation value of OA (and all composite operators)can be calculated by ignoring |0〉B,(〈1| ⊗ 〈0|)(O˜A ⊗ 1B)(|1〉 ⊗ |0〉) = 〈1| O˜A |1〉 〈0|0〉B = 〈1| O˜A |1〉 . (2.5)It can be shown that for operations which only act on subsystem A, |1〉A is acomplete description.However, if the system is in an entangled state, such as equation (2.1), adescription of A in terms of a state in the Hilbert spaceHA is not available anymore.Instead, a complete description of the quantum state for operators which only acton the A subsystem is given by the reduced density matrixρA = trB(ρ), (2.6)where ρ is the density matrix which describes the system and trB traces over allstates in HB. Unless the system was in a product state, the trace will turn a6previously pure state into a mixed one withρmixed =N∑ici |ηi〉 〈ηi | . (2.7)The quantum system described by this density matrix describes a classical ensembleof pure states |ηi〉. The probability to find the system in the state |ηi〉 is ci.2.2 Entanglement entropyTo quantify the lack of knowledge of how the reduced densitymatrix ρAwas purifiedby subsystem B we can use the von Neumann entropy of the reduced density matrix[42],S(ρA) = −tr(ρA log ρA). (2.8)The von Neumann entropy S(ρ) vanishes if ρ is a pure state and is maximal if ρis maximally mixed, i.e., proportional to the identity matrix. The von Neumannentropy of a reduced density matrix is oftentimes called entanglement entropy,which indicates that non-zero von Neumann entropy can result from entanglementwith another system. However, note that a non-zero von Neumann entropy alsomeasures classical uncertainty, for example if the whole system is described by athermal ensemble. The reason is that if ρ is not pure, S(ρ) also obtains a contributiondue to the ci in equation (2.7); this counts the statistical entropy of the ensemble ofpure states. In the following, we will use the terms entanglement entropy and vonNeumann entropy interchangeably.2.3 Relative entropyWe can define a measure for the distinguishability of two states of our quantumsystem, called relative entropy,S(ρ‖σ) = tr(ρ log ρ) − tr(ρ logσ). (2.9)7Relative entropy is positive definite, i.e., it is positive (or infinite) except whenρ = σ, for which it vanishes. It is also monotonic, meaningS(ρA‖σA) ≤ S(ρAB‖σAB). (2.10)Furthermore, positivity of relative entropy can be used to show certain propertiesof entanglement entropy, such as subadditivity, S(ρA) + S(ρB) ≥ S(ρAB).We can get some intuition for relative entropy by considering the case whereρ and σ are simultaneously diagonalizable, such that the trace in (2.9) reduces toa sum over eigenvalues ρi and σi. If ρ and σ describe orthogonal pure states,then tr(ρ logσ) = ∑i ρi logσi contains a term where ρi is positive but σi vanishes.Thus S(ρ‖σ) = ∞ and the two states are perfectly distinguishable. This is also trueif ρ is maximally mixed and σ is pure. On the other hand, if ρ is pure and σ ismaximally mixed, the relative entropy will be finite. Thus, relative entropy is notsymmetric. Roughly speaking, relative entropy measures how easy it is to disprovethe hypothesis that a system is described by σ, given that its actual state is given byρ.2.4 Markovicity of quantum statesIf three randomvariables X,Y, Z have conditional probabilities that satisfy p(X |Y, Z) =p(X |Y ), they are said to form a Markov chain. Using the definition of conditionalprobability, p(A|B) ≡ p(A,B)/p(B), one can show that this is equivalent toS˜(XYZ) + S˜(Y ) = S˜(XY ) + S˜(YZ), (2.11)where S˜ = −∑i pi log pi is the Shannon entropy. A “quantum version” of thisequation isS(A ∪ B) + S(A ∩ B) = S(A) + S(B), (2.12)which uses the von Neumann entropy and where we have identified subsystem Awith the random variables X and Y and subsystem B with the variables Y and Z .A state of the joint system A ∪ B which obeys equation (2.12) is called a Markov8state. In fact, obeying the Markov condition is equivalent to saturating strongsubadditivity,S(A ∪ B) + S(A ∩ B) ≤ S(A) + S(B), (2.13)which generally holds for entanglement entropies.2.5 Quantum information in quantum field theoriesIn this thesis, we see the above concepts applied to states in quantum field theories.The subsystems under consideration will either be subsystems in momentum spaceor position space. For a discussion in momentum space we want to define the traceon Hilbert space. If the Hilbert space is non-separable, it is in general not clear howsuch a definition would look like and we thus want to require that our Hilbert spaceis separable. This is generally the case in free field theories with massive particles,and we will see that this requirement has implications for the Hilbert space structureof theories with IR divergences.If we are interested in subsystems in position space, the situation is morecomplicated, since the Hilbert space does not factorize into a product of Hilbertspaces of subregions. Instead of considering the Hilbert space of a subregion, weshould consider the von Neumann algebra of operators associated with a subregion.It is then possible to define relative entropy in terms of the von Neumann algebra.Entanglement entropy of a subregion is an ill-defined concept since it is alwaysdivergent due to an infinite amount of entanglement in high energy modes acrossthe boundary of that region. Nonetheless, if suitably regularized, the naive treatmentof entanglement entropy works for all practical purposes and thus in this thesis wewill be taking on this naive picture. Alternatively, oftentimes one can study UVdivergence-free quantum information theoretic quantities such as relative entropy.For a more detailed review see, e.g., [43]. Defining the modular Hamiltonianassociated to a subregion as the negative logarithm of the reduced density matrixon that subregion,Hρ = − log ρ, (2.14)9we can bring relative entropy into a form which will be useful in chapter 9,S(ρ| |σ) = tr(ρ log ρ) − tr(ρ logσ)= tr(ρ log ρ) − tr(σ logσ) + tr(σ logσ) − tr(ρ logσ)= ∆S − ∆ 〈Hσ〉 ,(2.15)where ∆S is the difference in von Neumann entropies of the states ρ and σ, and∆ 〈Hσ〉 is the difference of expectation values of Hσ in those states. The modularHamiltonian generally is a non-local operator. However, in some special cases, Hρcan be written as an integral over local operators. For example, this is the case ifρ is the reduced vacuum density matrix of a half-space in a local quantum fieldtheory, or if ρ is the reduced density matrix of the vacuum state on a ball-shapedregion in a conformal field theory.10Part IQuantum information in theinfrared11Chapter 3Infrared divergences in quantumfield theoryIn chapter 1.1 we briefly reviewed how quantum information theory has lead toimportant insights in fundamental physics. In the cases discussed, the applicationsof concepts from quantum information theory to quantum gravity are mostly basedon an analysis in position space. For example, quantities of interest are relativeor entanglement entropies of subregions. A natural extension of these ideas isto investigate whether quantum information theory in momentum space can yieldequally interesting insights. The question which motivate the research in this partof the thesis is: How does quantum information spread in scattering?In the following we will investigate quantum information theoretic aspects ofscattering in four dimensions in the presence of long range forces such as gravityand electromagnetism. In such situations, scattering amplitudes are plagued byinfrared divergences (IR divergences), which occur beyond leading order in thecalculations of Feynman diagrams. Their appearance sets almost all scatteringamplitudes to zero. We will be concerned with how to define information theoreticquantities in the presence of IR divergences and what IR divergences teach us aboutthe Hilbert space structure. This will lay the foundation of a framework in whichthe spread of entanglement in scattering can be determined, even in the presence ofIR divergences.There are many more motivations to better understand information theory and12dynamics in the infrared. Apart from the importance of infrared physics for the un-derstanding of confinement, long wavelength modes seem to play an important rolein the quest for a theory of quantum gravity. They are important for understandingnon-locality [44], soft hair is proposed to capture black hole microstates [33]3 andseveral solutions of the black hole information paradox [10, 47] which are based onlow-energy physics have been proposed, e.g., [48–51]. Since black hole formationand evaporation can be understood as a scattering problem, it seems worthwhileinvestigating the fate of information in scattering. Lastly, if the lessons learned sofar from the AdS/CFT correspondence are correct, the bulk and the boundary theoryshould share the same Hilbert space and a better understanding of the Hilbert spaceof flat-space perturbative quantum gravity might yield hints towards the structureof the correct dual theory.More concretely, themethods developed here are useful for investigating variousquestions related to quantum gravity. It has been argued [30–33] that informationabout what has fallen into a black hole can be stored in and retrieved from low-energy or soft field modes. A detailed understanding of the spread of informationin scattering would enable us to quantify how much information can be carriedby different parts of the spectrum. This question is also potentially relevant forexperiments testing quantum mechanics or quantum gravity in the laboratory. Aswe will see below, almost all processes are accompanied by the emission of softradiation which potentially destroys quantum coherence. While the tools developedin this thesis enable a thorough analysis of the above questions, answering them isbeyond the scope of this thesis and will be deferred to possible future work.The present chapter gives a review of infrared divergences in quantum fieldtheory, before we give the main results in the subsequent chapters. Most parts ofchapters 4 to 6 are heavily based on work which previously appeared in [1–3], andchapter 7 is a redacted version of a preprint [5]. Section 4.5 is original work whichhas not been published before.3See also the older proposals [30, 31] and criticism thereof [45, 46].133.1 Scattering and the asymptotic Hilbert spaceTo begin, let us briefly review the standard method of how scattering amplitudes arecalculated in quantum field theory (see, e.g., [52, 53]). Physical states of a quantumfield theory are represented as vectors in a Hilbert space H. Time evolution isimplemented by a unitary operator e−iHt which acts on states in the Hilbert space(Schrödinger picture) or evolves operators in time (Heisenberg picture).To motivate the definition of the S-matrix we imagine an idealized experiment.An experimentalist sets up a set of well-separated particles at some early time andis interested in the amplitude4 with which the system turns into some set of well-separated particles at very late times. The S-matrix captures this information, andif we express the states in the Heisenberg picture, it is defined asSβ,α = out,H〈β|α〉in,H, (3.1)where |α〉in,H and |β〉out,H are Heisenberg states which correspond to well-separatedparticles if measured at early or late times, respectively. In order to calculatequantum information theoretic quantities before and after scattering, we need thedensity matrices which describe incoming and outgoing states,ρin(α) = |α〉in 〈α | , ρout(β) = |β〉out 〈β| . (3.2)Since the particle content of |α〉in,H , |β〉out,H as measured at early and late times,respectively, is well-separated, the particles can be described as approximately non-interacting. This means that at early and late times we should be able to describethe system by a free theory with a Hamiltonian H0 with the same spectrum as thefull Hamiltonian H. In other words, if we use Schrödinger picture state |α, ti〉in,S tomake the time-dependence explicit, there are states |α, ti〉in,0 which evolve with thefree Hamiltonian and approximate the Schrödinger picture states at early times ti,ti − t < 0,e−iH(t−ti ) |α, ti〉in,S ∼ e−iH0(t−ti ) |α, ti〉in,0 , (3.3)4Technically, she is interested in the probability which can be obtained from the amplitude.14and similarly for |β〉out at late times. These states are called asymptotic states.Consequently, we can write the S-matrix in the Schrödinger picture asSβ,α = limt′/t′′→∓∞ out〈β| eiH0(t′′−t f )e−iH(t′′−t′)e−iH0(t′−ti ) |α〉in . (3.4)In this expression, we have dropped the zero subscript and will do so for therest of this thesis. We have furthermore defined fixed times ti/ f at which thestates |α〉in /|β〉out in the Heisenberg and Schrödinger picture agree. H0 is the freeHamiltonian inwhich themass parameter takes its physical value. At amathematicallevel, the role of the terms including H0 is that they ensure convergence of the aboveexpression [54]. We could remove the dependence on ti/ f by redefining the S-matrixS → eiH0(t f −ti )S.5Going to the interaction picture in which operators evolve with the free Hamil-tonian, while states evolve with the interaction Hamiltonian Hint allows us to rewritethe S-matrix in the well-known form [53],S = Te−i∫ ∞−∞ dt Hint(t). (3.5)The time-dependence in the interaction Hamiltonian comes from the interactionpicture fields and possibly some explicit time dependence. The interaction Hamil-tonian is controlled by a small coupling constant which typically allows us to expandthe S-matrix order by order in the coupling.The Hilbert space of asymptotic states is usually constructed by expandingthe fields of the theory in terms of creation and annihilation operators a†(k), a(k)and constructing the Fock representation of the canonical commutation relations[a(k),a†(k′)] = (2pi)32Ekδ(3)(k−k′). The Fock representation are all normalizablestates which can be constructed by acting with creation operators which are convo-lutedwith square-integrable functions on a vacuum state |0〉, which is annihilated byall annihilation operators, i.e., a(k′) |0〉 = 0. We will see below that this choice ofrepresentation for asymptotic scattering states is problematic, if long-range forcesare present. The reason is that in this case, even at very early or late times, the fields5Oftentimes one chooses the convention that t f = ti = T , i.e., the incoming and outgoing particlesare defined on the same, arbitrary timeslice.15pp′(a)pp′(b)Figure 3.1: (a) Tree-level diagram for a fermion scattering off of a potentialrepresented by the shaded blob. (b)The IR divergent one-loop correctionto the process (a).cannot be treated as approximately free.3.2 Infrared divergences in S-matrix scatteringFollowing the standard prescription for calculating scattering amplitudes in theorieswith massless bosons in four dimensions between Fock space states, we encounterinfrared divergences. For example, consider a scattering process inQED inwhich anelectronwithmomentump scatters off of a potential while transferring amomentumq = p − p′, figure 3.1. The correction to the tree-level diagram in Feynman gaugeis given by(ie)3(−i)3∫d4k(2pi)4[γν(−/p′ − /k + m)γµ(−/p − /k + m)γν]((p′ + k)2 + m2 − i)((p + k)2 + m2 − i)(k2 − i), (3.6)where we have, like in the rest of this chapter, followed the notation of [55].If the fermion propagators are almost on-shell, the integrand scales like 1k . Thissuggests a logarithmic divergence as |k|, k0 → 0. This expectation is indeed correctand is a general feature of any non-trivial scattering process in four-dimensionalelectrodynamics [56, 57].For a treatment of the general case, let us consider a matrix element M ofan arbitrary scattering process, figure 3.2a. As argued above, IR divergences are16pn. . .p1p′m. . .p′2p′1(a)pn. . .p1p′m. . .p′2p′1(b)pn. . .p1p′m. . .p′2p′1(c)Figure 3.2: Construction of loops on the external legs. (a) An arbitraryscattering process which involves n incoming and m outgoing particles.(b) A vertex can be added to any external leg which emits a soft photon.(c) Multiple vertices can be connected by photon propagators to yield(soft) photon loops.expected to appear as propagators go on-shell while emitting or absorbing a virtualphoton of long wavelength. This can happen if incoming or outgoing legs emit orabsorb virtual photons. The emission of a (virtual) soft photon from an outgoingleg requires us to add a vertex to the amplitude, for example, figure 3.2b,us(p)M(p) → us(p)(ieγµ)(−i)(−/p − /k + m)(p + k)2 + m2 − i ×M(p + k). (3.7)Performing a similar replacement on a different leg, connecting the vertices with aphoton propagator and integrating over the loopmomentum gives us an IR divergentloop correction to the amplitude. To extract the divergence we split the loop integralinto an integral over soft (0 < λ ≤ |k| < Λ) and hard (Λ < |k|) momenta,∫d4k(2pi)4 →∫ Λλd4k(2pi)4 +∫ ∞Λd4k(2pi)4 . (3.8)The integral with |k| > Λ is UV divergent and needs to be renormalized. Wewill implicitly include contributions from hard momenta intoM and writeMΛ toindicate that these contributions depend on Λ. The scale Λ should be much smallerthan the electron mass and other relevant energy scales. We have also introduced17a cutoff λ to regulate the IR divergence. At the end of the calculation λ has to betaken to zero.Only the divergent parts of the integral with |k| ≤ Λ are relevant to the discus-sion of IR effects, which is what we will be focussing on. Using the explicit formof the spinors [55] equation (3.7) can be written asM = us(p)M˜(p) →(epµp · k − i)× us(p)M˜(p) + (non-divergent). (3.9)In the general case, a similar argument shows that leading order in the inverse bosonmomentum, a vertex that emits a (virtual) photon of momentum kµ is added bymultiplying the matrix element withηnenpµnpn · k − iηn + O(1). (3.10)The factors pn and en are the momentum and charge carried along the n-th leg.ηn takes values +1 or −1 if the n-th leg is outgoing or incoming, respectively. Toleading order the matrix elementM is independent of k and the contributions fromsoft photon loops factorize. The one-loop contribution coming from a soft loopbetween the n-th and m-th leg is enemηnηmJmn withJmn ≡ (−i)∫ Λλd4k(2pi)4pn · pm(k2 − i)(pn · k − iηn)(−pm · k − iηm) . (3.11)The k0 and |k| integrals can be performed and evaluate toJmn = − 12(2pi)3∫dΩvn · vm(1 − kˆ · vn)(1 − kˆ · vm)log(Λλ)− i4piβnm(1 + ηnηm)2log(Λλ),(3.12)with vµ = pµ/p0. The remaining integral over the unit vector kˆ yieldsJmn =18pi21βnmlog(1 + βnm1 − βnm)log(Λλ)− i4piβnm(1 + ηnηm)2log(Λλ). (3.13)We will postpone the physical interpretation of these terms to section 3.3. The18imaginary part of Jmn, called Coulomb phase, only contributes to loops whichconnect two outgoing or two incoming legs. The factor of βnm is the relativevelocity of particles n and m in either rest frame and is given byβnm ≡√1 − m2nm2m(pn · pm)2 . (3.14)As we take the IR cutoff λ to zero we see that equation (3.13) – and thus theone loop correction which is proportional to Jmn – diverges. This indicates thatfor small enough IR cutoff perturbation theory breaks down and we need to resumthe result to all orders. Luckily, the structure of IR divergent contributions in theinfrared is simple enough to do this.If we resum the contribution from soft loops to all orders, we need an expressionwhich takes multiple photon emissions per leg into account. At leading order,adding a second vertex which emits momentum k1 to the n-th leg, which alreadyemits momentum k2, yields a factor of(ηnenpµnpn · k1 − iηn) (ηnenpνnp · (k1 + k2) − iηn)(3.15)in front of the matrix element, which corresponds to the case where k2 is emittedbefore k1. In addition, there will be a term which is obtained by swapping k1 andk2 corresponding to the case where k1 is emitted before k2,(ηnenpµnpn · k2 − iηn) (ηnenpνnp · (k1 + k2) − iηn). (3.16)Summing both terms, we obtain(ηnenpµnp · k1 − iηn) (ηnenpνnp · k2 − iηn). (3.17)Note that one would, in principle, expect a contribution of O(k−1) that could alsobe IR divergent. However, it turns out that all the divergences factorize [58]. Thissuggests the following rule, which can be proved by induction: To leading order inlow momenta, we can account for the emission of M (virtual) soft bosons from the19n-th leg of a Feynman diagram by multiplying the amplitude withM→(M∏iηnenpµinpn · ki − iηn)×M. (3.18)These can be connected by photon propagators as before.Since the soft contributions to loop integrals factorize, the leading part of theN-th order correction is proportional to the N-th power of equation (3.11). Moreprecisely, for the scattering between states |α〉 and |β〉 it isMΛ ×∑N12NN!( ∑n,m∈α,βηnηmenemJmn)N=MΛ ×(λΛ)Aβ,α/2, (3.19)withAβ,α = −∑n,m∈α,βenemηnηm8pi2β−1nm log(1 + βnm1 − βnm)+ (phase factor). (3.20)The phase factor is given in (3.13). The factor of 2−N in equation (3.19) makessure we do not count twice diagrams which only differ by the orientation of thephoton line, while the factor of N! corrects for overcounting different permutationsof the photon lines. The function Aβ,α is positive. Trivial terms in the S-matrixhave Aα,α = 0 and thus can also be multiplied be the prefactor(λΛ)Aβ,α/2. Thus,the factor which multiplies the whole S-matrix is the same as that for the matrixelement. The scattering probability isp(α→ β) =(λΛ)Aβ,α SΛβ,α2 , (3.21)where the superscript on the S-matrix reminds us that loop diagrams are calculatedwith a cutoff Λ.The prefactor damps the amplitude such that it vanishes in the limit λ→ 0. Aswewill see in the remainder of this thesis, this is not merely some technical problem.Quantum electrodynamics and perturbative quantumgravity in fact correctly predictthat transition amplitudes between Fock space states vanish.20As can be seen from equation (3.11), the occurrence of IR divergences is tiedto the number of spacetime dimensions and the structure of the propagators of theinvolved particles, most notably the absence of a regulating mass term in the bosonpropagator. It is thus not surprising that analogous divergences appear in fourdimensions whenever massless bosons are exchanged. One example of particularimportance is gravity. A similar argument to the discussion above shows that softgraviton loops contribute an infrared divergence of the form [56]M =MΛ ×(λΛ)Bβ,α/2, (3.22)with the positive coefficientBβ,α =∑n,m∈α,βmnmmηnηm16pi2M2p1 + β2nmβnm√1 − β2nmlog(1 + βnm1 − βnm). (3.23)Another large class of theories with IR divergences are four-dimensional Yang-Mills theories. While in a non-perturbative treatment confinement might make surethat no IR divergences appear, in perturbative calculations they do appear in thefashion outlined above as soft divergences and need to be treated as well.It turns out that the preceding discussion does not cover all possibilities forhow IR divergences can appear in quantum field theory. Another source of IRdivergences are collinear emissions which appear when massless particles emitother massless particles along their direction of propagation. Apart from Yang-Mills theory, this effect also appears in massless QED and gravity at high energies.In this thesis, we will not be interested in effects of collinear divergences, and onlyrefer the reader to the existing literature [59, 60].3.3 A semiclassical analysisIn order to make predictions that can be compared to experiment, one needs toeliminate infrared divergences. Approaches which accomplish this are based onthe observation that, during a scattering process, Bremsstrahlung is produced. Theproduced radiation carries a finite amount of energy at arbitrarily long wavelengths,21such that the number of photons of arbitrarily long wavelengths must diverge, sincelimω→0N(ω) = limω→0E(ω)~ω→∞. (3.24)The Fock space representation does not allow for infinite occupation numbers,which explains why generically, Fock space states cannot be used as scatteringout-states, and consequently why the S-matrix elements between those states mustvanish. In this section, we will use a semiclassical argument to derive the form ofthe expected photon out-state.Assume we have a charged particle with momentum pµ, which is scattered atthe origin. After scattering, it has momentum p′µ. The current for this particle isgiven byjµ(x) = e∫ ∞0dτp′µmδ(4)(xµ − p′µmτ)+ e∫ 0−∞dτpµmδ(4)(xµ − pµmτ). (3.25)We now want to investigate the corresponding classical radiation field. To this end,we Fourier transform the above expression after introducing convergence factors i ,jµ(k) =∫d4xe−ikx jµt0(x) = −ie(p′µp′ · k − i −pµp · k + i). (3.26)In Lorenz gauge, the solution to Maxwell’s equations can be written asAµ(x) = −ie∫d4k(2pi)4 eikx 1k2(p′µp′ · k − i −pµp · k + i). (3.27)The term 1k2is the Green’s function for the vector potential in Lorenz gauge.To obtain the radiation produced in the scattering, we need to choose the Green’sfunction to be the difference between the retarded and advanced Green’s function.6The outgoing radiation can then be obtained by closing the k0 contour in the lower6In order to describe the full outgoing vector potential wewould consider only the retardedGreen’sfunction. However, the conclusion we will be drawing is the same in both cases.22half-plane and is given byAµcl,out(x) = e∫d3k2|k|(2pi)3(p′µk · p′ −pµk · p) (eik ·x + e−ik ·x) k0= |k |. (3.28)The quantum field theoretical description of a classical field is given in termsof a coherent state. We can formally write the coherent state which corresponds toequation (3.28) as a state in the photon Fock space,|Aµcl,out〉 = N exp(∫d3k(2pi)32|k| fµ(p,p′,k)a†µ(k))|0〉 , (3.29)withf µ(p,p′,k) = e(p′µp′ · k −pµp · k), (3.30)and the normalizationN = exp(−12∫d3k(2pi)32|k| | f (p,p′,k)|2). (3.31)The expectation value of the number operator in a thin shell in momentum spacearound momentum k, Nk = a†µ(k)aµ(k)d3k, is〈Aµcl,out |Nk |Aµcl,out〉 ∼d3k2|k|(v′µk · v′ −vµk · v)2∼ d |k||k| . (3.32)This clearly shows that in the quantummechanical description the infrared containsan infinite number of photons. The logarithm of the normalizationN of the coherentstate |Aµcl,out〉 is also proportional to equation (3.32) and thus divergent, and we seethat the state which represents the classical Bremsstrahlung is not part of the Fockspace represention.The above argument explains why all S-matrix elements vanish. Quantumelectrodynamics should reproduce classical physics at long distances. However, aswe have seen, the expected out-state has a vanishing norm. Moreover, the overlap ofFock states with coherent states of the above form vanishes, which implies that theS-matrix maps Fock space states into a vector space orthogonal to Fock space. In23conclusion, the IR divergences coming from loop corrections should be understoodas a physical prediction. It sets all scattering probabilities between different Fockstates to zero, simply because an infinite number of soft modes will be created.The terms which appear in the normalization are reminiscent of the real partof equations (3.11) and (3.13). However, as noted around equation (3.13), thereis also a divergent phase factor. It appears if more than one particle is present inthe in- or out-state. Physically, it comes from the potential energy of a chargedparticle in the field created by a second particle, which can be seen by consideringthe non-relativistic case. The energy of a non-relativistic outgoing particle in thefield of a different outgoing particle is given byE(t) = m + m2v21 +q1q24pi(r0 + v1t) . (3.33)Thus, at very late times, the phase of the corresponding state goes like−i∫ tdt ′E(t ′) ∼ −iE0t − i q1q24piv1 log(t). (3.34)The treatment of outgoing particles as free Fock states only accounts for the firstterm, −iE0t, and the mismatch between the time evolution as a free state andequation (3.33) gives rise to the divergent phase factor in equation (3.13).3.4 Dealing with infrared divergencesThe prescriptions used to cancel the IR divergences can be classified into inclusiveand dressed formalisms. The philosophy behind the inclusive formalisms is thatone should not ask questions one cannot experimentally answer. We cannot build adetector that measures photons of arbitrarily low energies and therefore we shouldnot ask how likely it is to scatter from a certain in-state to an out-state withoutany additional photons that might have escaped detection. Instead, we should askfor inclusive probabilities, i.e., the probability to scatter from |α〉 to |β〉 plus anypossible configuration of photons which escape detection. This implies a treatmentin which amplitudes are regulator-dependent and cannot be assigned a physicalinterpretation. In light of more fundamental questions, however, this approach isunsatisfactory. For example, in the AdS/CFT correspondence, gravity is a quantum24theory whose states live in a Hilbert space. If anything like this should be true in flator de Sitter space, there must be a way of assigning regulator-independent quantumstates to the out-state of a scattering experiment. Moreover, questions about theunitarity of time evolution can only be answered at the level of amplitudes.A different approach is followed by dressed formalisms. These formalismsare built on the assumption that asymptotic states are not correctly modeled byFock space states. Instead, physical states, such as electrons or generally massiveparticles, are accompanied by a certain photon/graviton field configuration calleddressing. These dressings resemble the coherent states of the previous subsection.Amplitudes between dressed fields are finite, since the excitations contained withinthe dressing cancel IR divergences order by order.7 In the following we will brieflysummarize the inclusive and dressed formalisms.3.4.1 The inclusive formalismThe objects of interest in scattering calculations are typically not scattering am-plitudes, but scattering probabilities or scattering cross-sections, since these arephysical and can be determined in experiment. As we have seen above, non-trivial scattering processes produce asymptotic states with an infinite number ofsoft bosons. Due to the limited volume of any apparatus and the limited durationof any experiment, it is clear that some of these bosons will escape undetected.To predict a detector response, we need to sum over all possible outcomes of ourexperiment which are consistent with our measurement, i.e., we need to sum overall possible soft boson emissions, where the energy of the boson is below somedetection threshold. Consequently, the probability to scatter a state |α〉 to a finalstate |β〉 should be calculated aspincl(α→ β) =∑unobs.b| 〈β, b|α〉 |2, (3.35)where the sum indicates that we consider the addition of unobservable soft bosonsb in the final state. The so-obtained probability pincl(α→ β) is called the inclusive7Sometimes dressing is used to describe the process of adding photon/graviton field excitations toa state in order to make it gauge invariant, an idea pioneered in [61]. The IR part of such a dressingcan in principle also be chosen to cancel IR divergences, but a priori both concepts are independent.25transition probability between states |α〉 and |β〉.Calculating inclusive probabilities as opposed to the naive probabilities p =| 〈β |α〉 |2 is the textbook way of dealing with IR divergences [53, 57, 58], firstestablished for QED [58, 62] and subsequently expanded to include the case of softgravitons [56].In section 3.2 we already discussed that the emission of soft (virtual) photonsfrom incoming or outgoing legs requires multiplying the matrix element by a softfactor, equation (3.10). If a real photon is emitted, the same soft factor appears,the momentum k has to be put on-shell and the free index of the vertex needs tobe contracted with a polarization vector, ε`(k) for outgoing and ε∗`(k) for incomingphotons. For on-shell photons, the notion of soft is controlled by an additionalthreshold energy scale ET which is smaller than all relevant energy scales of theexperiment. This includes scales associated with the experiment’s dimensions.The addition of on-shell soft factors directly leads to Weinberg’s soft theorems.To leading order in the inverse photon momentum, the S-matrix element for scat-tering between two asymptotic states |α,a〉in, out 〈β, b|, with hard particles α, β andsoft bosons a, b can be written asSβb,αa =∏i∈a,b( ∑n∈α,βηnenε`iµ (ki)pnµpn · ki)× Sβ,α, (3.36)where ki and `i are the momentum and helicity of the i-th photon and k0 = |k|.The momentum ki is taken to be outgoing from the vertex. This formula and theequivalent formula for soft gravitonsSβb,αa =∏i∈a,b( ∑n∈α,β1Mpηn`iµν(ki)pµnpνnpn · ki)× Sβ,α (3.37)are known asWeinberg’s soft theorems. Here the index n runs over all the incomingand outgoing hard particles, i runs over the outgoing soft bosons; ηn = −1 for anincoming and +1 for an outgoing hard particle. The en are electric charges andMp = (8piGN )−1/2 is the Planck mass, and the ’s are polarization vectors or tensorsfor outgoing soft photons and gravitons, respectively. Recently, it was shownthat these soft theorems can be understood as the Ward identities of asymptotic26symmetries [25, 29, 63]. We will briefly discuss this in section 3.5.We can use the soft theorems to show that inclusive transition probabilities arefinite. Consider equation (3.35). The sum over soft bosons is implemented byintegrating the momenta of all possible soft photon emissions up to some scaleEi and summing over all photon helicities. The sum of all photon energies isconstrained to be less that ET . For the leading order contribution, the emission ofa single soft photon with unknown helicity we obtainpincl,(1)(α→ β) =∑`=±∫ ETλd3k2|k|(2pi)3 Sβk,αS∗βk,α = C × Sβ,αS∗β,α, (3.38)where again the boson’s momentum is on-shell andC = −∑n,m∈α,β∑`=±∫ ETλd3k(2pi)32|k|(ηnenε`µ(k)pµnpn · k) (ηmemε`ν(k)pνmpm · k). (3.39)We can simplify this expression by using that∑`=±ε`∗µ (k)ε`ν(k) = gµν − kµcν − cµkν,kµ = |k|(1kˆ)µ, cµ =12|k|(−1kˆ)µ.(3.40)The terms proportional to kµ vanish upon contraction in equation (3.39). Theintegral over angles is precisely the same integral we have already encountered inequation (3.13) and we are left with( ∑n,m∈α,βηnηmenem∫ Eλd |k|(2pi)32|k|∫dΩ(vn · vm(1 − vn · kˆ)(1 − vm · kˆ)))= Aα,β∫ Eλd |k||k| ,(3.41)where Aα,β was given in equation (3.20). For the emission of N bosons with totalenergy below the threshold, we consider N factors of the form (3.39) whose totalenergy is constrained to be less than ET . Summing over all possible emissions27yields∞∑N=0ANN!N∏i=1(∫ Eiλd3ki|ki |)θ(ET −∑iEi). (3.42)Here, we havemade sure not to overcount identical photon emissions by introducinga factor of 1N ! . The Heaviside theta function can be rewritten asθ(ET −∑iEi) = 1pi∫ ∞−∞sin(ETu)uexp(iu∑iEi). (3.43)In the following we will assume that all Ei = E . With this, the inclusive probabilitybecomespincl(α→ β) = F(E/ET , Aα,β)(Eλ)Aα,βp(α→ β), (3.44)where F comes from evaluating the integral in equation (3.43) and p(α→ β) is thehard scattering probability. The function F is given byF(x, A) = 1pi∫ ∞−∞dusin(u)uexp(A∫ x0dωω(eiωu − 1)). (3.45)The parameters E and ET can be chosen such that if A 1, F is close to one, e.g.,F(1, A) ≈ 1 − 112pi2A2. Note that, due to the positivity of Aα,β , the prefactor inequation (3.44) diverges in the limit λ → ∞. The dependence on λ is just right tocancel against the λ dependence which makes the loop-corrected amplitude vanish,equation (3.21). Thus, the inclusive scattering probability is free of λ dependencesand IR finite,pincl(α→ β) =(EΛ)Aα,β SΛβ,α2 F(E/ET , Aα,β). (3.46)For gravity, the situation seems more complicated. Since gravitons are them-selves a source of stress-energy, they can set off a cascade of softer gravitons whichmight spoil the simple form of the expression for soft photon emission. However,we are fortunate as the coupling to gravitons is proportional to the energy of a par-28ticle. Consequently, such terms are subleading in momentum and do not contributeto divergences. Similarly, this also explains why loop-corrected graviton loops donot play any role: the above argument goes through and we end up with the sameexpression, equation (3.46) with Aα,β replaced by Bα,β .For Yang-Mills theories this argument does not work, since the coupling is notproportional to the momenta of the involved particles. Furthermore, apart fromthe soft divergences discussed, the appearance of collinear divergences causes ad-ditional problems. However, in these cases the KLN theorem [59, 60] guaranteesthat a modified prescription also produces scattering probabilities free of IR di-vergences. The modification consists of also including a sum over incoming softparticles.3.4.2 Dressed formalismsThe inclusive formalism outlined above gives up the notion of scattering amplitudes.Dressed formalisms are an alternative approach with which finite amplitudes canbe calculated. The underlying idea is to add additional soft radiation to incomingand outgoing states whose emission and absorption cancels IR divergences. Theadded radiation takes the form of the coherent states of section 3.3. In this sectionwe will give a rough outline of the idea, following early work by Chung [64], whichis sufficient until chapter 7. There, we will take a closer look at the more elaboratedressed formalisms of Faddeev and Kulish [65] and investigate the Hilbert spacestructure.Dressed formalisms propose to replacemomentumeigenstates by dressed states,|p1, . . . ,pn〉 → ‖p1, . . . ,pn〉〉. (3.47)In the case of a one-particle momentum eigenstate, the corresponding dressed stateis defined as‖p〉〉 = Wλ[ f`(p,k)] |p〉 , (3.48)29whereWλ[ f`(p,k)] is an operator that creates a coherent stateWλ[ f`] ≡ exp(∫ Eλd3k(2pi)32|k|∑`=±(f`(p,k)a†`(k) − h.c.))(3.49)andf`(p,k) = −e p · `p · k φ(p,k). (3.50)Here, p, k are on-shell four-vectors, and φ(p,k) can be any function that goes to 0as |k| → 0. The dressed state depends on an IR cutoff λ through the coherent stateoperator. This IR cutoff ensures that the normalization of the state created byW[ f`]is finite, compare to the discussion around equation (3.31). The extension to themulti-particle case is straight forward and will be discussed in section 5.3. Thesestates can be used to calculate scattering amplitudes,Sβ,α ≡ 〈〈β‖S‖α〉〉 = 〈β|W†βSWα |α〉 , (3.51)which are finite as λ→ 0. We call S the dressed S-matrix.To see how the dressing removes IR divergences, consider the scattering of adressed electron with momentum p to a dressed electron with momentum p′. Thecancellation of IR divergences takes place order by order, and we will show thefirst non-trivial order, O(e2). We need to replace |p〉 by ‖p〉〉 which at leading orderreads‖p〉〉 =(1 − 12∑`=±∫ Eλd3k(2pi)32|k| | f`(p,k)|2)×(1 +∑`=±∫ Eλd3k(2pi)32|k| f`(p,k)a†`(k))|p〉 ,(3.52)where f ∼ O(e). Note that the dressing only needs to be expanded to order e, sincethe absorption or emission of a photon from the dressing is also of order e, yieldinga term of order e2. Dressed S-matrix elements Sp′,p equal bare S-matrix elements,30Sp′,p, multiplied by a correction,(1 +∫ Eλd3k(2pi)32|k|∑`=±(f ∗` (p′,k) f`(p,k) −12| f`(p,k)|2 − 12 | f`(p′,k)|2))Sp′,p+∫ Eλd3k(2pi)32|k|∑`=±(f`(p,k)Sp′,pk + f ∗` (p′,k)Sp′k,p).(3.53)The first line comes from the process where the photon does not interact with thescattered particles at all and the change in normalization of the in- and out-goingstate. The second line consists of terms which appear since the dressing of theincoming and outgoing state interacts with the scattering process.The second line can be rewritten using the soft theorem (3.36) as∫ Eλd3k(2pi)32|k|∑`=±©« f`(p,k)∑n∈{p,p′ }ηn f ∗` (pn,k) − f ∗` (p′,k)∑n∈{p,p′ }ηn f`(pn,k)ª®¬ Sp′,p.(3.54)In summary, the total correction isSp′,p =Sp′,p(1 +∫ Eλd3k(2pi)32|k|∑`=±(− f ∗` (p′,k) f`(p,k) +12| f`(p,k)|2 + 12 | f`(p′,k)|2)).(3.55)Now recall from equations (3.12) and (3.19) that we can split off the IR divergencecoming from loops in the calculation of the S-matrix asSp′,p = SΛp′,p(1 − 12∑`=±∑n,m∫ Λλd3k(2pi)32|k| (ηnηm f`(pn,k) f`(pn,k))). (3.56)Using equation (3.40) it is easy to show that the corrections coming from thedressing, equation (3.55), and soft loops, equation (3.56), exactly cancel to orderO(e2). This argument can be extended to all orders [64].The reason this procedure works can be understood from the semi-classicalanalysis in section 3.3. We can see from equation (3.28) that the IR radiation31produced in a scattering process consists of two terms, one which depends on theincoming and one which depends on the outgoing momenta. The dressed statesdiscussed here correspond to a case where we send fine-tuned radiation into thescattering region which only depends on the incoming hard particles and cancelsthe part of the outgoing radiation which depends on the incoming hard momenta.We will see in chapter 6 how this can be generalized.The proposal reviewed here has several shortcomings. First, although we havewell-defined amplitudes, we have to introduce an IR regulator into the states. As wesend it to zero, the states become non-normalizable. As a consequence of this, thestructure of the Hilbert space is unclear in the limit of vanishing regulator. Second,in the form presented here, it is not clear whether the divergence associated withthe Coulomb phase, equation (3.34), still persists.Steps to ameliorate these problems were taken in a series of papers by Kibble,who modified the procedure to take into account the divergent Coulomb phase andproposed to use a von Neumann space as the Hilbert space of dressed states withoutIR cutoff [66–69]. The proposed Hilbert space is non-separable, i.e., it does nothave a countable basis, and the S-matrix maps states between different separablesubspaces. This proposal was developed further by Faddeev and Kulish [65] whogave a derivation of the dressing from first principles and identified a subspaceof Kibble’s Hilbert space which is separable and stable under the action of theS-matrix. Their derivation of the dressing from first principles will be reviewed inchapter 7.Another slightly different dressed formalism was proposed by Bagan, Lavelleand McMullan [70, 71]. However, the only difference between their approach andthat of Faddeev-Kulish is that instead of dressing asymptotic states, they dressoperators. For example the dressed operator Aµ creates modes on top of a clas-sical radiation background, equation (3.28). Thus, with slight modifications, allstatements made in this thesis also apply to their dressed formalism.In [72] a dressed formalism for gravity was proposed. In this case, the dressingis again given by (3.49). This time, however, a and a† are the graviton annihilationand creation operators and the functions f (k,p) depend on the polarization of the32graviton µν,f gr`(k,p) = pµµν`pνk · p φ(k,p). (3.57)3.5 An infinity of conserved chargesRecently, interest in the IR behavior of gauge theories and gravity was revivedfrom a different perspective. The work initiated in [28, 63, 73] demonstratedthat soft theorems and asymptotic symmetries can be understood in a unified way.Moreover, these findingswere used to suggest newways of how black holes can storeinformation [30, 31]. Dressed states also arise naturally in the recent discussionsof asymptotic gauge symmetries [25, 28–30, 74, 75], which imply the existence ofselection sectors [76–79]. See also [80, 81] for work on soft charges and dressing inholography. Throughout this thesis we will comment on the relation of our findingsto asymptotic symmetries: large gauge transformations [73] in the case of QED andBMS transformations [82] in the case of gravity. The next subsections reviews therelevant aspects of the connection between asymptotic symmetries and Weinberg’ssoft theorems in the case of QED. A more complete review, also covering the caseof non-abelian gauge theories and gravity can be found in [25].3.5.1 Anti-podal matching and conserved chargesAt leading order, solutions to Maxwell’s equations obey an anti-podal matchingcondition at light-like infinity I±, c.f. figure 3.3. This is easy to see for the Liénard-Wiechert field of a point particle with charge e moving at constant velocity v,Frt (x, t) = e4piγv(r − txˆ · v)|γ2v(t − r xˆ · v)2 − t2 + r2 |3/2. (3.58)Here, xˆ · r is the three vector at which the field is evaluated at time t and γv is therelativistic gamma factor. We are interested in the electric field at light-like infinityI±. To obtain an expression on I+ we change coordinates to (u = t − r,r, xˆ) and take33i+i0i−i0I+I+I−I−uvFigure 3.3: This figure shows the Penrose diagram of Minkowski space. In-finity is conformally mapped to a finite distance, thus distances are notfaithfully represented, however, the causal structure is. Light runs at45◦ angles. Lightlike future and past infinity, I±, are a good Cauchyslices for massless particles, while massive particles start and end at i±.Spacelike infinity is denoted by i0.the limit of r →∞. while keeping u and xˆ constant. The result isFrt (x, t)I+=e4pir21γ2v(1 − xˆ · v)2. (3.59)Using coordinates (v = t + r,r, xˆ) we can take the limit of Frt to I− and findFrt (x, t)I−=e4pir21γ2v(1 + xˆ · v)2. (3.60)Equations (3.59) and (3.60) are related to each other by xˆ→ −xˆ.Light-like infinity has the topology of a cylinderR×S2, whereR is parametrizedby u or v and xˆ parametrizes the S2. To make the resulting equations simpler, oneconventionally changes coordinates on the S2 to complex coordinates (z, z) such thatthe coordinates on the sphere of future infinity I+ are related to those on the sphereof past light-like infinity I− by (z, z) → (−z,−z). This way, a light ray which enterson I− through the point (z, z) exits at I+ at an angle given by the same coordinates.With these conventions, the field strength tensor Fµν obeys the matching con-34ditionF(2)ru (z, z)I+−= F(2)rv (z, z)I−+, (3.61)where F(n)µν denotes the coefficient of the r−n term in a large-r expansion of Fµν.Since equations (3.59) and (3.60) are u and v independent, we have decided toevaluate F on I+− and I−+. The two-sphere I+− is located on I+ at u → −∞ andsimilarly I−+ is located on I− at v → ∞. Thanks to the matching condition, thereexists an infinite number of trivially conserved chargesQ+ε ≡∫I+−ε(z, z)? F =∫I−+ε(z, z)? F ≡ Q−ε, (3.62)where ? is the Hodge star operator. This expression is true for any function ε(z, z)defined such that ε(z, z)|I+− = ε(z, z)|I−+ . For constant ε the conserved charge issimply the electric charge.These charges are the generators of large gauge transformations, i.e., gaugetransformations which do no vanish at infinity but reduce to transformations whichare only functions of the coordinates z and z at infinity.3.5.2 Hard and soft chargesUsing Maxwell’s equationsd ? F = ?j, (3.63)the charges can be rewritten asQ+ε =∫I+−ε ? F =∫I+d(ε ? F) +∫I++ε ? F=∫I+dε ∧?F︸ ︷︷ ︸Q+ε,S+∫I++ε ? F︸ ︷︷ ︸Q+ε,H, (3.64)where we have used that∫I+ε ? j = 0, since in QED there are no charges leavingMinkowski space through light-like infinity. The first term in equation (3.64) only35depends on the behavior of the transverse electric field at future light-like infinityand is called the soft charge, Q+ε,S . The second term depends on the longitudinalpart of the electric field, weighted by ε, and is called the hard charge, Q+ε,H . Thesoft charge is a measure of soft radiation, while the hard charge is a measure ofthe long-wavelength part of the longitudinal fields of charged matter particles. Thesame argument can be used to show that Q−ε also splits into a soft and hard part.It can be shown [29, 74] that, independent of their photon content, out-states ofdefinite momentum are eigenstates ofQ±ε,H . Similarly, dressed states are eigenstatesof Q±ε,S . Their eigenvalue is proportional to an integral, whose integrand dependson the residue of the dressing function fh(p,k) as |k| goes to zero.3.5.3 Weinberg’s soft theoremsConservation of Q± implies that the operator commutes with the Hamiltonian andthus in particular with the S-matrix,0 = 〈β | [Qε,S] |α〉 = 〈β| (Q+εS − SQ−ε ) |α〉 . (3.65)The presence of IR divergences can be related to the conservation of the chargesQ [29]. Calling the eigenvalues with respect to the soft charges Q±ε,S , Nout and Nin,respectively, we find(Nout − Nin) 〈β| S |α〉 =∑n∈α,β√21 + zzenηn+ · pnpn · k 〈β| S |α〉 . (3.66)For the states in the Fock space representation one can check explicitly that theeigenvalue of the soft charge operator is zero, i.e., the left hand side of equation(3.66) vanishes. Hence, the only way equation (3.66) can hold is if the factor thatmultiplies the amplitude on the right-hand side vanishes or the amplitude itselfis zero. For any non-trival scattering process, the prefactor is non-zero, so theamplitude must vanish. Moreover, it can be shown that equation (3.66) is simply acoordinate-transformed version of Weinberg’s soft theorem [25].The case of gravity is completely analogous, with the electric field in equation(3.62) replaced by the gravitational field.36Chapter 4Infrared quantum informationThis chapter is a redacted version of [1].4.1 IntroductionWe have seen in the previous chapter that in the standard treatment of scattering theS-matrix becomes ill-defined due to divergences coming from low-energy virtualbosons. The usual solution to this problem is to use the inclusive formalism, i.e.,to argue that an infinite number of low-energy bosons are radiated away during ascattering event; this leads to divergences which cancels the divergences from thevirtual states, and physical predictions in terms of infrared-finite inclusive transitionprobabilities.In this chapter, we study quantum information-theoretic aspects of this proposal.Since each photon and graviton has two polarization states and three momentumdegrees of freedom, one might suspect that the low-energy radiation producedduring scattering could carry a huge amount of information. Here we demonstratethat, according to the methodology of [56, 58, 62], which was summarized insection 3.4.1, if the initial state is an incoming n-particle momentum eigenstate,the soft bosonic divergences can lead to complete decoherence of the outgoinghard particles, with the momentum eigenstates as the pointer basis [83]. Thisdecoherence is avoided only for superpositions of pairs of outgoing states forwhich an infinite set of angle-dependent currents match, see equation (4.9). In37simple examples like QED, this will be enough to get complete decoherence of allmomentum superpositions. In less simple cases, one is still left with an extremelysparse density matrix dominated by its diagonal elements. See [84–86] for relatedwork.Having traced the radiation in this fashion, we obtain an infrared-finite, mixedreduced density matrix for the hard particles. In the simple cases when we get acompletely diagonal matrix, we compute the entanglement entropy carried by thesoft gauge bosons. The answer is finite and scales like the logarithm of the energyresolution E of a hypothetical soft boson detector.While the tracing out of the soft radiation can be viewed as a physical statementabout the energy resolution of a real detector, in this formalism, the trace is alsoforced on us by mathematical consistency: it is the only way to get well-definedtransition probabilities from the infrared-divergent S-matrix.Recently, the infrared structure of gauge theories has become a topic of muchinterest due to the proposal that soft radiation may encode information about thehistory of formation of a black hole [30, 31, 45]. We also hope that this workcan make the discussion more quantitatively grounded; we comment on blackholes at the end of this chapter. More generally, it is of interest to understand theinformation-theoretic nature of the infrared sector of quantum field theories, andthis work is intended to make some first steps in this direction.4.2 Decoherence of the hard particlesFix a single-particle energy resolution E . We define soft bosons as those withenergy less than E , and hard particles as anything else. Consider an incoming state|α〉in consisting of hard particles, charged or otherwise, of definite momenta.8 TheS-matrix evolves this into a coherent superposition of states with hard particles βand soft bosons b = γ, h (photons γ and gravitons h),|α〉in =∑βbSβb,α |βb〉out . (4.1)8Labels like α, β, bmean a list of free-particle quantum numbers, e.g., |α〉in = |p1`1, . . .〉in listingmomenta and spin of the incoming particles.38Hereafter we drop the subscript on kets, which will always be out-states. Tracingout the bosons |b〉, the reduced density matrix for the outgoing hard particles isρ =∑ββ′bSβb,αS∗β′b,α |β〉 〈β′ | . (4.2)Using the usual soft factorization theorems, equations (3.36) and (3.37), we canwrite the amplitudes in terms of the amplitudes for α → β multiplied by softfactors, one for each boson. By an argument identical to the one employed in thelast chapter, and assuming we can neglect the total lost energy ET compared to theenergy of the hard particles, we can use this factorization to perform the sum oversoft bosons in (4.2), and we find that∑bSβb,αS∗β′b,α = Sβ,αS∗β′,α(Eλ) A˜ββ′ ,α (Eλ) B˜ββ′ ,α× F(EET, A˜ββ′,α)F(EET, B˜ββ′,α).(4.3)Here λ E is an infrared regulator used to cut off momentum integrals which wewill send to zero later; one can think of λ as a mass for the photon and graviton.The exponents areA˜ββ′,α = −∑n∈α,βn′∈α,β′enen′ηnηn′8pi2β−1nn′ log[1 + βnn′1 − βnn′]B˜ββ′,α =∑n∈α,βn′∈α,β′mnmn′ηnηn′16pi2M2p1 + β2nn′βnn′√1 − β2nn′log[1 + βnn′1 − βnn′],(4.4)and F is given in equation (3.45). In these formulas, βnn′ is the relative velocitybetween particles n and n′, given in (3.14). For future use, we note that 0 ≤ β ≤ 1,and both of the dimensionless functions of β appearing in (4.4) run over [2,∞) asβ runs from 0 to 1. We have βnm = 0 if and only if pn = pm.The divergences as λ→ 0 in (4.3) come from summing over an infinite numberof radiated, on-shell bosons. There are also infrared divergences inherent to thetransition amplitude Sβ,α itself coming from virtual bosons. We can add these39divergences up, and we have thatSβ,α = SΛβ,α(λΛ)Aβ,α/2 ( λΛ)Bβ,α/2, (4.5)where now SΛβ,α means the amplitude computed using only virtual bosons of energyabove Λ, and Aβ,α and Bβ,α were given in equations (3.20) and (3.23) and arerepeated here for convenience,Aβ,α = −∑n,m∈α,βenemηnηm8pi2β−1nm log[1 + βnm1 − βnm]Bβ,α =∑n,m∈α,βmnmmηnηm16pi2M2p1 + β2nmβnm√1 − β2nmlog[1 + βnm1 − βnm].(4.6)The infrared-divergent Coulomb phase from equation (3.20) is suppressed in (4.5).We will see shortly that this phase cancels out of all the relevant density matrixelements.Putting the above results together, we find that the reduced density matrixcoefficient for |β〉 〈β′ | is given byρββ′ = SΛβ,αSΛ∗β′,α(Eλ) A˜α,ββ′ ( λΛ)Aβ,α/2+Aβ′ ,α/2×(Eλ) B˜α,ββ′ ( λΛ)Bβ,α/2+Bβ′ ,α/2F(A˜ββ′,α)F(B˜ββ′,α).(4.7)The question is how this behaves in the limit of vanishing infrared regulator, λ→ 0.The coefficient scales as λ∆A+∆B, where∆Aββ′,α =Aβ,α2+Aβ′,α2− A˜ββ′,α∆Bββ′,α =Bβ,α2+Bβ′,α2− B˜ββ′,α.(4.8)In appendixA,weprove that both of these exponents are positive-definite,∆Aββ′,α ≥0 and ∆Bββ′,α ≥ 0. The density matrix components (4.7) which survive as the reg-ulator λ → 0 are those for which ∆A = ∆B = 0; all other density matrix elements40will vanish.To give necessary and sufficient conditions for ∆A = ∆B = 0, we define twocurrents for each spatial velocity vector v. We assume for simplicity that only mas-sive particles carry electric charge. For massive particles, there are electromagneticand gravitational currents defined asjemv =∑ieiai†(pi(v))ai(pi(v)),jgrv =∑iEi(v)ai†(pi(v))ai(pi(v)).(4.9)Here i labels particle species, ei their charges and mi their masses; the kinematicquantities pi(v) = miv/√1 − v2 and Ei(v) = mi/√1 − v2 are the momentum andenergy of species i when it has velocity v. For lightlike particles we have toseparately define the gravitational current, since a velocity and species does notuniquely determine a momentum:jgr,m=0v =∑i∫ ∞0dωωai†(ωv)ai(ωv). (4.10)Momentum eigenstates of any number of particles are obviously eigenstates of thesecurrents and we denote their eigenvalues jv |α〉 = jv(α) |α〉.The photonic exponent ∆Aββ′,α is zero if and only if the charged currents inβ are the same as those in β′; the gravitational exponent ∆Bββ′,α is zero if andonly if all the hard gravitational currents in β are the same as those in β′. This isdemonstrated in detail in appendix A. For any such pair of outgoing states |β〉 , |β′〉,(4.7) becomes independent of the IR regulator λ and is thus finite as λ→ 0,ρββ′ = SΛ∗β′,αSΛβ,αGβα (E,ET ,Λ) , (4.11)whereGβα = F(EET, Aβ,α)F(EET,Bβ,α) (EΛ)Aβα+Bβα. (4.12)This is the case in particular for diagonal density matrix elements β = β′, for which41we obtain the standard transition probabilitiesρββ =SΛβ,α2 Gβα (E,ET ,Λ) . (4.13)On the other hand, if there is even a single v for which one of the currents (4.9)or (4.10) does not have the same eigenvalue in |β〉 and |β′〉, then the densitymatrix coefficient decays as λ∆A+∆B → 0 as the regulator λ → 0. We see that theunobserved soft bosons have almost completely decohered the momentum state ofthe hard particles. Only a very sparse subset of superpositions survive, in whichthe currents agree for all velocities v,jv(β) = jv(β′). (4.14)4.3 ExamplesTo get a feel for the results presented in the previous section, we consider a fewexamples. First, consider any scattering with a single incoming and outgoingcharged particle, like potential or single particle Compton scattering. Let theincoming momentum be α = p and the outgoing momenta of the two branchesβ = q, β′ = q′. We have either directly from the definition (4.8) or the theorem(A.1) that∆Aqq′,p = − e28pi2[2 − γqq′], (4.15)where γqq′ = β−1qq′ log(1 + βqq′)/(1 − βqq′). This ∆A is easily seen to equal zeroif and only if q = q′. Thus other than the spin degree of freedom, the resultingdensity matrix for the charge is exactly diagonal in momentum space.To see an example where the current-matching condition is non-trivially ful-filled, consider a theory with two charged particle species of charge e and e/2 andthe same mass. Then we can get an outgoing superposition of a state β = (e,q) andone with two half-charges β′ = (e/2,q′1) + (e/2, iq′2). The differential exponent forsuch a superposition is∆Aββ′,p = − e28pi2[3 +12γq1q2 − γqq1 − γqq2], (4.16)42which is zero if q = q1 = q2. In other words, the currents (4.9) cannot distinguishbetween a full charge of momentum q and two half-charges of the samemomentum.4.4 Entropy of the soft bosonsWe have seen that the reduced density matrix for the outgoing hard particles isvery nearly diagonal in the momentum basis. In a simple example like a theorywith various scalar fields φi of different, non-zero masses mi, the soft gravitonemission causes complete decoherence into a diagonal momentum-space reduceddensity matrix for the hard particles. More generally, we may have a sparse setof superpositions, and in any case spin and other internal degrees of freedom areunaffected by the soft emission.In a simple example with a purely diagonal reduced density matrix, it is straight-forward to compute the entanglement entropy of the soft emitted bosons. The totalhard + soft system is in a bipartite pure state, with the partition being between thehard particles and soft bosons, so the entanglement entropy of the bosons is thesame as that of the hard particles. Following the calculation in [23, 24, 87], we cansimply write down the entropy:S =∑βSΛβ,α2 Gβα log [SΛβ,α2 Gβα] . (4.17)This sum is infrared-finite; again, G is given in (4.12), and the superscript Λ meansthe naive S-matrix computed with virtual bosons only of energies greater than Λ.Given the explicit form of G, we see that the entropy scales like the log of theinfrared detector resolution E .4.5 Relation to large gauge symmetriesThe decoherence condition (4.14) can be rephrased in the language of large gaugetransformations. The condition that given two momentum eigenstates |β〉 and |β′〉,the density matrix element ρββ′ vanishes unless the same amount of charge iscarried with the same velocity vector in both states, is equivalent to the conditionthat the hard charges Q+ε,H agree on |β〉 and |β′〉 for all ε(z, z).43To prove this, we start by showing that if condition (4.14) holds for momentumeigenstates |β〉 and |β′〉, it follows that the eigenvalues of Q+ε,H also agree. Let|β〉, |β′〉 be two momentum eigenstates which contain a finite number of chargedparticles which carry electric chargeQ(v) (andQ′(v), respectively) with velocity v,alongside with a number of uncharged particles which we will ignore. For example,if two different particles carry charge e along v = v0, then Q(v0) = 2e. If equation(4.14) holds,thenQ(v) = Q′(v) (4.18)for every v. The eigenvalues of the out-states with respect to the hard charges aregiven byQ+ε,H |β〉 =∫I++d2z√γε(z, z)F(2),βrt (z, z) |β〉 (4.19)whereF(2),βrt (z, z) =∑i14piγ2iQ(vi)(1 − xˆ · vi)2 (4.20)for |β〉 and the sum runs over the (finite) number of velocity vectors along whichcharge is carried. For |β′〉 we get the same expression where we have to replaceQ → Q′. However, since Q(v) = Q′(v), the same amount of charge is carriedwith the same velocity and F(2)rt (z, z) is the same on |β〉 and |β′〉. Therefore theeigenvalues of the hard charges agree.Conversely, we will now show that equal eigenvalues with respect to Q+ε,H fortwomomentum eigenstates |β〉, |β′〉 imply that the same amount of charge is carriedalong the same velocity in both states. That is for either state we can constructfunctions Q(v) and Q′(v), respectively, which represents the charge carried alongvelocity vectors v, with Q(v) = Q′(v). Since we know that these functions are inone-to-one correspondence with eigenvalues of the operators jv we can concludethat also the eigenvalues jv(β) and jv(β′) agree. Consider two states |β〉, |β′〉 withQ+ε,H eigenvalues qεβ and qεβ′. We assume that the eigenvalues agree for any choice44of ε and in particular we can chooseε ∝ δ(2)(z − w). (4.21)Then the condition qεβ = qεβ′ translates to a pointwise equality for the functionsF(2),βrt and F(2),β′rt ,∑n∈β14piγ2nen(1 − xˆ · vn)2 =∑m∈β′14piγ′2mem(1 − xˆ · v′m)2. (4.22)It is clear that by combining terms this can be rewritten as∑i∈V14piγ2iQ(vi)(1 − xˆ · vi)2 =∑i∈V14piγ2iQ′(vi)(1 − xˆ · vi)2 . (4.23)The set V contains all velocities along which charge is carried in either β or β′. Wenow assume that equation (4.23) holds but Q(vi) disagrees with Q′(vi) and showthat this leads to a contradiction. We solve for one of the terms in disagreement,whose associated velocity we denote by v0. This leaves us withQ(v0) −Q′(v0)γ20(1 − xˆ · v0)2=∑i∈V\{0}Q′(vi) −Q(vi)γ2n(1 − xˆ · vi)2. (4.24)The sum on the right hand side runs over all velocities except v0. Multiplying byall denominators and defining ∆Q(vi) = Q(vi) −Q′(vi) we find∏i∈V\{0}(1 − xˆ · vi)2 = −γ20(1 − xˆ · v0)2∆Q(v0)©«∑i∈V\{0}∆Q(vi)γ2i∏j∈V\{0,i }(1 − xˆ · vj)2ª®¬ .(4.25)Treated as functions of xˆ, both sides are polynomials on S2. Since the ring ofpolynomials with real coefficients on the sphere is a unique factorization domain,the factorization of both sides in factors of the form (1+ v1x + v2y + v3z) is uniquewith two factors being identical if and only if all vi agree. Since the right hand sidecontains a factor of (1− xˆ · v0)2 it follows that such a factor must also appear on theleft hand side of the equation, but we have assumed that a term containing v0 is not45included in the product. This contradicts our initial assumption and hence we haveshown that Q(vi) = Q′(vi). It then also follows that the eigenvalues of the currentsjv acting on |β〉 and |β′〉 must agree.4.6 DiscussionAccording to the solution of the infrared catastrophe advocated in [56, 58, 62],an infinite number of very low-energy photons and gravitons are produced duringscattering events. We have shown that if taken seriously, considering this radiationas lost to the environment completely decoheres almost any momentum state ofthe outgoing hard particles. The basic idea is simple: the radiation is essentiallyclassical, so any two scattering events are easy to distinguish by their radiation.The physical content of this result is somewhat unclear. A conservative viewis that the methodology of [56, 58, 62] is ill-suited to finding outgoing densitymatrices. As remarked earlier, in this formalism, one must trace the radiation to getwell-defined transition probabilities. An alternative would be to use the infrared-finite S-matrix program [64–69, 72], in which no trace over radiation is neededat all. But then we need to understand where the physical low-energy radiationis within that formalism–since after all, a photon that is lost to the environmentcertainly does decohere the system. We will turn to this in the next chapters.The decoherence found here is for the momentum states of the particles: atlowest order in their momenta, soft bosons do not lead to decoherence of spindegrees of freedom. However, the sub-leading soft theorems [27, 88, 89] do involvethe spin of the hard particles, so going to the next order in the soft particles would beinteresting.9 We would also like to understand to what extent our answers dependon the infinite-time approximation used in the S-matrix approach.To end, we comment on potential applications to the black hole informationparadox. The idea advocated in [30, 31] is that correlations between the hard andsoft particles mean that information about the black hole state can be encoded intosoft radiation. In [45, 46, 76], the dressed-state formalism and soft factorizationhas been used to argue that the soft particles simply factor out of the S-matrix andthus contain no such information. In the approach used here, it is manifest that9We understand that Strominger has confirmed this. (Private communication)46the outgoing hard state and outgoing soft state are highly correlated, leading to thedecoherence of the hard state. The outgoing density matrix for the hard particles,while not completely thermal, has been mixed in momentum as much as possiblewhile retaining consistency with standard QED/perturbative gravity predictions.It is tempting to conjecture that this generalizes to all asymptotically measurablequantum numbers.47Chapter 5Dressed infrared quantuminformationThis chapter is a redacted version of [2].5.1 IntroductionIn the inclusive formalism, one is forced to trace out soft photons to get finite an-swers. In the previous chapter, we have seen that this leads to an almost completelydecohered densitymatrix for the outgoing state after a scattering event. This chapteranalyses the situation in dressed state formalisms, in which no trace over IR photonsis needed to obtain a finite outgoing state. However, consider the measurement ofan observable sensitive only to electronic and high-energy photonic degrees of free-dom. We show that for such observables, there will be a loss of coherence identicalto that obtained in the inclusive probability method. Quantum information is lostto the low-energy bremsstrahlung photons created in the scattering process.The primary goal of this chapter is to give concrete calculations exhibiting thedressed formalism and how it leads to decoherence. To this end, we work with theformulas from the papers of Chung [64] and Faddeev-Kulish [65]. The result ofthis calculation should carry over identically to any of the existing refinements ofChung’s formalism. In section 5.4, we make a number of remarks on possible re-finements to the basic dressing formalism, give an expanded physical interpretation48of our results, and relate our work to literature in mathematical physics on QEDsuperselection rules. In section 5.5 we make remarks on how this work fits intothe recent literature on the black hole information paradox; in brief, we believe thatour results are consistent with the recent proposal of Strominger [32], but not theoriginal proposal of Hawking, Perry and Strominger [30, 31].5.2 IR-safe S-matrix formalismFollowing Chung, we study an electron with incoming momentum p scattering off aweak external potential. This 1→ 1 process is simple and sufficient to understandthe basic point; at the end of the next section, we show how to generalize ourresults to n-particle scattering. The electron spin will be unimportant for us and wesupress it in what follows. The standard free-field Fock state |p〉 for the electron ispromoted to a dressed state ‖p〉〉 as discussed in section 3.4.2,‖p〉〉 = Wp |p〉 ≡ Wλ[ f`(k,p)]. (5.1)This consists of the electron and a coherent state of on-shell, transversely-polarizedphotons.We introduce an IR regulator (“photon mass”) λ and an upper infrared cutoffE > λ, which can be thought of as the energy resolution of a single-photon detectorin our experiment. Here and in the following all momentum-space integrals areevaluated in the shell λ < |k| < E .Consider now an incoming dressed electron scattering into a superpositionof outgoing dressed electron states. The outgoing state is, to lowest order inperturbation theory in the electric charge,|ψ〉 =∫d3qSqp‖q〉〉. (5.2)At higher orders there will be additional photons in the outgoing state; as explainedin the next section, these will not affect the infrared behavior studied here, so weignore them for now. Here the S-matrix is just the standard Feynman-Dyson time49evolution operator, evaluated between dressed states. That is,Sqp = 〈〈q‖S‖p〉〉, (5.3)with S = ‘T exp(−i ∫ ∞−∞V(t)dt) as usual. As calculated by Chung, the dressed1 → 1 elements of this matrix are independent of the IR regulator λ and thusinfrared-finite as we send λ→ 0. We can write the matrix elementSqp =(EΛ)ASΛqp (5.4)whereA = − e28pi2β−1 log[1 + β1 − β], β =√1 − m4(p · q)2 . (5.5)As discussed in section 3.4, the undressed S-matrix element on the right side meansthe amplitude computed by Feynman diagrams with photon loops evaluated onlywith photon energies aboveΛ and evaluated between undressed electron states, thatis, with no external soft photons. By definition, this quantity is infrared-finite andthe dependence on the scale Λ cancels between the prefactor and SΛ.5.3 Soft radiation and decoherenceThe state (5.2) is a coherent superposition of states, each containing a bare electronand its corresponding photonic dressing. The presence of hard photons in theoutgoing state will not change our conclusions below, so for simplicity we ignorethem. In particular, the density matrix formed from this state has off-diagonalelements of the formS∗q′pSqp‖q〉〉〈〈q′‖. (5.6)These states have highly non-trivial photon content. However, if one is doing ameasurement involving only the electron degree of freedom, then these photonsare unobserved, and we can make predictions with the reduced density matrix ofthe electron, obtained by tracing the photons out. The resulting electron densitymatrix has coefficients damped by a factor involving the overlap of the photon states,50namelyρelectron =∫d3qd3q′S∗q′pSqpDqq′ |q〉 〈q′ | (5.7)where the dampening factor is given by the photon-vacuum expectation valueDqq′ = 〈0|W†q′Wq |0〉 . (5.8)Straightforward computation gives this factor asDqq′ = exp{−e22∑`=±∫d3k(2pi)32|k| f`(q) − f ∗` (q′)2}= exp{−e2∫d3k(2pi)32|k|(q − q′)2(q · k)(q′ · k)}.(5.9)In this integrand, since q and q′ are two timelike vectors with the same temporalcomponent, we have that the numerator is positive definite and the denominator ispositive. It is therefore manifest that we have D = 1 if q = q′ and D = 0 otherwise,since the integral over d3k diverges in its lower limit. Thus, tracing the photonsleads to an electron density matrix that is completely diagonalized in momentumspace.It is noteworthy that the factor (5.9) depends only on properties of the outgoingsuperposition; it has no dependence on the initial state. This may seem surprisingsince we are tracing over outgoing radiation, the production of which dependson both the initial and final electron state. The point is that the damping factormeasures the distinguishability of the radiation fields given the processes p → qand p→ q′. The radiation field for a scattering process consists of two pieces addedtogether: a term Aµ ∼ pµ/p · k peaked in the direction of the incoming electronand a term Aµ ∼ qµ/q · k peaked in the direction of the outgoing electron. Theoutgoing radiation fields with outgoing electrons q,q′ are then only distinguishableby the second terms here, since both radiation fields will have the same pole in theincoming direction.The damping factor (5.9) is precisely what was found in the previous chap-ter, reduced to the problem of 1 → 1 scattering. The mechanism is the same:physical, low-energy photon bremsstrahlung is emitted in the scattering. These51photons are highly correlated with the electron state and thus, if one does not ob-serve them jointly with the electron, one will measure a highly-decohered electrondensity matrix. The only difference is bookkeeping: in the dressed formalism,the bremsstrahlung photons are folded into the dressed electron states (the in-coming/outgoing parts of the bremsstrahlung in the incoming/outgoing dressing,respectively). However, referring to “an electron” as a state including these soft pho-tons is an abuse of semantics. In an actual measurement of the electron momentum,one does not measure these soft photons.The dressed states are not energy eigenstates, and in fact contain states of arbi-trarily high total energy. This should be contrasted with the inclusive-probabilitytreatment used byWeinberg, which has a cutoff on both the single-photon energy Eand the total outgoing energy contained by all the photons ET ≥ E in the outgoingstate [56]. This additional parameter, however, appears only in the ratio ET/E inWeinberg’s probability formulas, and one finds that the dependence on ET van-ishes as ET → ∞. This can be understood because what is important is the verylow-energy behavior of the photons, so moving an upper cutoff has limited impact.We note that (5.2) does not include effects from the bremsstrahlung of additionalsoft photons beyond those in the dressing. There is no kinematic reason to excludesuch photons, so the outgoing state should properly be written as|ψ〉 =∞∑n=0∑{` }∫d3qd3nkSq{k` };p‖q〉〉. (5.10)Here {k`} = {k1`1, . . . ,kn`n} is a list of n photon momenta and polarizations. Bythe dressed version of the soft photon factorization theorem (see appendix B), wehave thatSqk`;p = Sqp × eO(|k|0), (5.11)or in other words lim |k |→0 |k|Sqk`;p = 0. Thus, when we take a trace over n-photondressed states in (5.10), we obtain a sum of additional decoherence factors of the52formDnmqq′ = en+mO(|k|0)×∑`1,...,`n∑`′1,...,`′m∫d3nkd3mk′〈0|a`′m (k′m) · · · a`′1(k′1)W†q′Wqa†`1(k1) · · · a†`n (kn)|0〉 .(5.12)Evaluating the inner product one findsDnmqq′ ∼[∑`=±∫d3k Re ( f`(q) − f`(q′))]n+m, (5.13)which is infrared-finite. Summing these contributions, which exponentiate, will notchange the conclusion that (5.9) leads to vanishing off-diagonal electron densitymatrix elements.Finally, we explain the generalization to n-electron states. We will find that thesame decoherence is found in the dressed formalism as in the inclusive formalism.Following Faddeev-Kulish [65], we define the multi-particle dressing operator byreplacingf`(p,k) →∫d3p(2pi)3 f`(p,k)ρ(p), (5.14)in the definition of Wλ[ f`]. Here, we have introduced an operator which countscharged particles with momentum p.ρ(p) =∑s(b†p,sbp,s − d†p,sdp,s), (5.15)and the b and d are electron and positron operators, respectively.10 As in theone-particle case, additional photons do not affect the IR behaviour of scatteringamplitudes. Hence, we will ignore them and only consider the case where theout-state is a linear superposition of dressed electron states. In that case we haveto replace the outgoing momentum by list of momenta, q→ β = {q1,q2, . . .} and10Note that in the multi-particle case there is an infinite phase factor which needs to be included inthe definition of the S-matrix. Since this phase factor does not affect our discussion, we ignore it inthe following.53similarly q′→ β′ = {q′1,q′2, . . .}. This results in a replacement in (5.9) off`(q) →∑n∈βf`(qn)f ∗` (q′) →∑m∈β′f ∗` (q′m).(5.16)Using the explicit form of F in the limit k→ 0, the damping factor (5.9) then thenbecomesDββ′ = exp[−e2∫d3k(2pi)32|k|∑m,n∈β,β′ηmηnpm · qn(qm · k)(qn · k)]. (5.17)In this equation the labels m,n both run over the full set β ∪ β′, and ηn = +1 ifn ∈ β while ηn = −1 if n ∈ β′. This is precisely the quantity ∆Aββ′,α defined inthe previous section, so we see that the results carry over to the dressed formalismsused here.5.4 Physical interpretationDressed-state formalisms are engineered to provide infrared-finite transition ampli-tudes, as opposed to inclusive probabilities constructed in the traditional approachstudied in the previous section. In the dressed formalism, the outgoing state (5.2) isa coherent superposition of states ‖p〉〉 consisting of electrons plus dressing photons.However, if one does a measurement of an observable sensitive only to the electronstate, the measurement will exhibit decoherence because the unobserved dressingphotons are highly correlated with the electron state. We have given a concretecalculation showing that the damping factor (5.17) is identical in either the dressedor undressed formalism.The physical relevance of this calculation rests on the idea that the basic observ-able is a simple electron operator in Fock space. What would be much better wouldbe to use a dressed LSZ reduction formula to understand the asymptotic limits ofelectron correlation functions [90, 91]. Nevertheless, the basic physical pictureseems clear: in a scattering experiment, one does not measure an electron plus afinely-tuned shockwave of outgoing bremsstrahlung photons, just the electron on54its own. This is responsible for well-measured phenomena like radiation damping.QED has a complicated asymptotic Hilbert space structure which is still some-what poorly understood. For example, although Faddeev-Kulish try to define asingle, separable Hilbert space Has [65, 91] other authors have argued that oneneeds an uncountable set of separable Hilbert spaces [66, 90]. Formally, this isrelated to the fact that the dressing operator does not converge on the usual Fockspace. We will discuss this in chapter 7. A related idea is that one can argue thatQED has an infinite set of superselection rules based on the asymptotic chargesQ(Ω) = limr→∞ r2Er (r,Ω) (5.18)defined by the radial electric field at infinity [92, 93]. Webelieve that the calculationspresented here and in chapter 4 demonstrate the physical mechanism for enforcingsuch a superselection rule. The charges (5.18), the currents defined in the previouschapter, and the large-U(1) charges defined in [29, 74] are presumably closelyrelated, and working out the precise relations is an interesting line of inquiry.5.5 Black hole informationLet us again comment on the proposal of Hawking, Perry and Strominger sug-gesting that information apparently lost in the process of black hole formation andevolution could be encoded in soft radiation [30, 31]. The original proposal wasthat there are symmetries which relate hard scattering (like the black hole forma-tion or evaporation process) to soft scattering and thus led to constraints on theS-matrix. As emphasized by a number of authors, this is not true in the dressedstate approach [45, 46, 76, 94]. As we review in appendix B, soft modes de-couple from the dressed hard scattering event at lowest order, in the sense thatlimω→0[aω,Sdressed] = 0. Dropping a soft boson into the black hole will not yieldany information about the black hole formation and evaporation process.However, a more recent proposal due to Strominger is to simply posit thatoutgoing soft radiation purifies the outgoing Hawking radiation [32]. That is, thestate after the black hole has evaporated is of the form |ψ〉 = ∑a ca |a〉Hawking |a〉soft,such that the Hawking radiation is described by a thermal density matrix, i.e.,ρHawking = trsoft |ψ〉 〈ψ | ≈ ρthermal. We believe that both the results presented55here and those in our previous work are consistent with this proposal. In either theinclusive or dressed formalism, the final state of any scattering process contains softradiation which is highly correlated with the hard particles because the radiation iscreated due to accelerations in the hard process. The open issue is to explain whythe hard density matrix coefficients behave thermally, which likely relies on detailsof the black hole S-matrix.5.6 ConclusionsWhen charged particles scatter, they experience acceleration, causing them to ra-diate low-energy photons. If one waits an infinitely long time (as mandated byan S-matrix description), these photons cause severe decoherence of the chargedparticle momentum state. This was demonstrated in the preceding chapter in thestandard formulation of QED involving IR-finite inclusive cross section, and herewe have shown the same conclusion holds in IR-safe, dressed formalisms of QED;they should carry over in a simple way to perturbative quantum gravity. Theseresults constitute a sharp and robust connection between the infrared catastropheand quantum information theory, and should provide guidance in problems relatedto the infrared structure of gauge theories.56Chapter 6On the need for soft dressingThis chapter is a redacted version of [3].6.1 IntroductionBoth, the dressed and inclusive formalisms, are designed to give the same predic-tions for the probability of scattering from an incoming set of momenta p1, . . . ,pninto an outgoing set of momenta p′1, . . . ,p′m. The measurement of observableswhich only depend on the hard particles should be predictable from the reduceddensity matrix obtained by tracing over soft bosons, which are invisible to a finitesize detector. Given an incoming momentum eigenstate, we have argued in theprevious two chapters that the two formalisms agree. Thus, one might naively thinkfor calculating cross-sections it does not matter which formalism one chooses. Weshow in this chapter that this is not the case: the two approaches differ in theirtreatment of incoming superpositions.Consider a simple superposition of two momentum eigenstates for a singlecharged particle|ψ〉 = 1√2(|p〉 + |q〉), (6.1)scattering off of a classical potential. We expect the out-state to be described by a57density matrix of the formρ =12S (|p〉 〈p| + |p〉 〈q| + |q〉 〈p| + |q〉 〈q|) S†. (6.2)Here S is the scattering operator and we have performed a trace over the softradiation, hence ρ is the density matrix for the hard particles. If |p〉 , |q〉 are dressedstates, this expectation is indeed correct. In the inclusive formalism, however, where|p〉, |q〉 are Fock space momentum eigenstates, there is no interference betweenthe different momenta as opposed to the diagonal terms of (6.2). We find that thediagonal entries of the density matrix which encode the cross-sections are of theformσψ→out ∝ 〈out| ρincl |out〉 = 12 〈out| S (|p〉 〈p| + |q〉 〈q|) S† |out〉 . (6.3)In other words, the cross-section behaves as if we had started with a classicalensemble of states withmomenta p and q. The entire scattering history is decoheredby the loss of the soft radiation. This appears to contrast starkly with any realisticexperiment.Moreover, as we will show, repeating the analysis for wavepackets, e.g., |ψ〉 =∫dp f (p) |p〉, leads to the nonsensical conclusion that a wave-packet is not observedto scatter at all. However, in the dressed state formalism of Faddeev-Kulish theinterference appears as in equation (6.2). This strongly suggests that scatteringtheory in quantum electrodynamics and perturbative quantum gravity should reallynot be formulated in terms of standard Fock states of charged particles. Formulatingthe theories using dressed states seems to be a good alternative.Our findings have a nice interpretation in the language of asymptotic symme-tries: only superpositions of states within the same selection sector, defined usingthe charges that generate the symmetries, can interfere. This explains the failureof the undressed approach. In the inclusive formalism, essentially any pair of mo-mentum eigenstates live in different charge sectors. In contrast, the Faddeev-Kulishformalism is designed so that all of the dressed states live within the same chargesector.Our results can also be viewed in the context of the black hole information prob-58lem [10, 47]. In particular, Hawking, Perry, and Strominger [30] and Strominger[32] have recently suggested that black hole information may be encoded in softradiation. In black hole thought experiments, one typically imagines preparing aninitial state of wavepackets organized to scatter with high probability to form anintermediate black hole. Our results suggest then that one needs to use dressedinitial states to study this problem. See also [45, 46] for some remarks on the useof dressed or inclusive formalisms for studying black hole information.The rest of the chapter is organized as follows. We start by presenting thecalculations showing that the dressed and undressed formalisms disagree in section6.2 for discrete superpositions and in section 6.3 for wavepackets. The discussionand interpretation of the results takes place in section 6.4. There, we will arguewhy our findings imply that dressed states are better suited to describe scatteringthan the inclusive Fock-space formalism. We will give a new very short argumentfor the known result of [78] that the dressing operators and the S-matrix weaklycommute and argue for a more general form of dressing beyond Faddeev-Kulish.We will then interpret our results in terms of asymptotic symmetries and selectionsectors before concluding in section 6.5. Appendix C contains proofs of certainstatements in sections 6.2 and 6.3.6.2 Scattering of discrete superpositionsIn this and the next section we generalize the results of chapters 4 and 5 to the caseof incoming superpositions of momentum eigenstates. We begin in this sectionby studying discrete superpositions |ψ〉 = |α1〉 + · · · + |αN 〉 of states with variousmomenta α = p1,p2, . . .. We will see that the dressed and inclusive formalisms givevastly different predictions for the probability distribution of the outgoingmomenta:dressed states will exhibit interference between the αi whereas undressed states donot.596.2.1 Inclusive formalismConsider scattering of an incoming superposition of chargedmomentum eigenstates|in〉 =N∑ifi |αi〉 , (6.4)with∑i | fi |2 = 1. The outgoing density matrix vanishes due to IR divergences invirtual photon loops. However, as before, we can obtain a finite result if we traceover outgoing radiation [1, 56, 58, 62]. The resulting reduced density matrix of thehard particles takes the formρ =∑bN∑i, j∬dβ dβ′ fi f ∗j Sβb,αi S∗β′b,αj |β〉 〈β′ | , (6.5)where β and β′ are lists of the momenta of hard particles in the outgoing state, andthe sum over b denotes the trace over soft bosons. We will be interested in the effectof infrared divergences on this expression.The sum over external soft boson states b produces IR divergences whichcancel those coming from virtual boson loops. We can regulate these divergencesby introducing an IR cutoff (e.g., a soft boson mass λ). Following the standardsoft photon resummation techniques [56], one finds that the total effect of thesedivergences yields reduced density matrix elements of the formρββ′ =N∑i, jfi f ∗j SΛβ,αiSΛ∗β′,αjλ∆Aββ′ ,αiα j +∆Bββ′ ,αiα j Gββ′,αiαj (E,ET ,Λ). (6.6)Here we have introduced “UV” cutoffs Λ,E on the virtual and real soft bosonenergies, so SΛ are S-matrix elements with the soft boson loops cut off belowΛ andwe only trace over outgoing bosonswith individual energies up to E and total energyET . The explicit form of the Sudakov rescaling function G defined analogously to(4.12). What concerns us here is the behavior of this expression in the limit where60we remove the IR regulator λ→ 0, which is controlled by the exponents∆Aββ′,αα′ = −12∑n,n′∈α,α¯′,β,β¯′enen′ηnηn′8pi2β−1nn′ log[1 + βnn′1 − βnn′],∆Bββ′,αα′ = −12∑n,n′∈α,α¯′,β,β¯′mnmn′ηnηn′16pi2M2pβ−1nn′1 + β2nn′√1 − β2nn′log[1 + βnn′1 − βnn′].(6.7)The factor ηn is defined as +1 (−1) if particle n is incoming (outgoing). Thequantities βnn′ are the relative velocities between pairs of particles given in equation(3.14) and a bar interchanges incoming states for outgoing and vice versa. Theexpressions for ∆A and ∆B come from contributions of soft photons and gravitons,respectively. The question now is which terms survive.The special case of no superposition, αi = αj = α, was discussed in chapter4. There it was shown that ∆Aββ′,αα ≥ 0 and ∆Bββ′,αα ≥ 0, so that in the limitλ → 0, all of the terms in the sum except those with ∆A = ∆B = 0 will vanish.The equality holds if and only if the out states β and β′ contain particles such thatthe amount of electrical charge and mass carried with any choice of velocity agreesfor β and β′. This can be phrased in terms of an infinite set of operators whichmeasure charges flowing along a velocity v, defined in equations (4.9) and (4.10).Momentum eigenstates are eigenstates of these operators. Using them, the equalityof currents readsjv |β〉 ∼ jv |β′〉 , (6.8)where the tildemeans that the eigenvalues of the states are the same on both sides forall velocities. In appendix C.1, we show that the more general exponents ∆Aββ′,αα′and ∆Bββ′,αα′ are positive. Similarly to the argument in the previous chapters, onecan show that ∆A and ∆B are non-zero if and only ifjv |αi〉 + jv |β′〉 ∼ jv |αj〉 + jv |β〉 , (6.9)that is if the list of hard currents in states |α〉 and |β′〉 is the same as the list of hardcurrents in states |α′〉 and |β〉. An easy way to understand the form of equation(6.9) is by looking at equation (6.7). There, the bar over α′ (which corresponds to61αj) indicates that it should be treated as an outgoing particle, i.e., similarly to β.On the other hand β¯′ should be treated similarly to α. Hence, we obtain equation(6.9) from (6.8) by replacing β′→ αi + β′ and β→ αj + β. On the other hand it isclear that in the case of |αi〉 = |αj〉 = |α〉 equation (6.9) reduces to equation (6.8).Armedwith these results, we can calculate the cross-sections given an incomingsuperposition. These are proportional to the diagonal elements β = β′ of the densitymatrix; for simplicity we ignore forward scattering terms. The diagonal terms ofthe density matrix (6.6) are proportional to λ∆A+∆B. This factor reduces to unity ifjv |αi〉 ∼ jv |αj〉 for all of the currents (4.9) and (4.10) and is zero otherwise. Fora generic superposition, this implies that only terms with i = j contribute and wefindσin→β ∝ ρββ =N∑i, jfi f ∗j Gββ,αiαj SΛβαiSΛ∗βαj δαiαj =N∑i| fi |2 |SΛβ,αi |2Gββ,αiαi .(6.10)As we see, no interference terms between incoming states are present. Instead, thetotal cross-section is calculated as if the incoming states were part of a classicalensemble with probabilities | fi |2. The reason is that in the inclusive approach theinformation about the interference is carried away by unobservable soft radiation. Todefine the scattering cross-section, however, we need to trace out the soft radiationand we obtain the above prediction, which is at odds with the naive expectation,equation (6.2).6.2.2 Dressed formalismThe calculation above was done using the usual, undressed Fock states of hardcharges, which required to calculate inclusive cross-sections. The alternative ap-proach we will now turn to is to consider transitions between dressed states. Forconcreteness, we will follow the dressing approach of Chung and Faddeev-Kulish11,which contains charged particles accompanied by a cloud of real bosons which radi-ate out to lightlike infinity [64, 65, 72]. For a given set of momenta α = p1,p2, . . .,11Recently, a generalization of Faddeev-Kulish states was suggested [77]. We will extend ourdiscussion to those states in section 6.4.62we write the dressed state as‖α〉〉 ≡ Wα |α〉 ≡ Wλ[ f`(k, α)], (6.11)where multi-particle dressed states are introduced as discussed in the previouschapter,f`(k, α) =∑p∈α` · pk · p φ(k,p) (6.12)The operatorWα equips the state |α〉 with a cloud of photons/gravitons. For QED,Wα with a finite cutoff λ is a unitary operator LettingWα act on Fock space statesfor λ = 0 gives states with vanishing normalization, hence in the strict λ→ 0 limitWα is no good operator on Fock space. Thus, as before, we will do calculationswith finite λ and only at the end will we take λ→ 0.12Consider now an incoming state consisting of a discrete superposition of suchdressed states,‖in〉〉 =∑ifi ‖αi〉〉. (6.13)The outgoing density matrix is thenρ =∑i, j∬dβdβ′ fi f ∗j SβαiS∗β′αj ‖β〉〉〈〈β′‖. (6.14)However, every experiment should be able to ignore soft radiation. Followingchapter 5, we treat the soft modes as unobservable and trace them out. This yieldsthe reduced density matrix for the outgoing hard particles,ρhardββ′ =∑i, jfi f ∗j SβαiS∗β′αj 〈0|W†βWβ′ |0〉 . (6.15)The last term is the photon vacuum expectation value of the out-state dressingoperators. This factor reduces to one or zero as shown in chapter 4 and 5; one12Note that as argued in [65], a proper definition ofW in the limit λ → 0 should be possible on avon Neumann space.63if j(β) ∼ j(β′) and zero otherwise. This is responsible for the decay of mostoff-diagonal elements in (6.15). However, if we are interested in the cross-sectionfor a particular outgoing state β, this is again given by a diagonal density matrixelement,σin→β ∝ ρββ =∑i, jfi f ∗j Sβ,αiS∗β,αj. (6.16)In stark contrast to the result obtained in the inclusive formalism, equation (6.10),this cross-section exhibits the usual interference between the various incomingstates, like expected in equation (6.2). The reason for this is that in the dressedformalism, the outgoing radiation is described by the dressing which only dependson the out-state and not on the in-state. We will discuss this in more detail in section6.4. This establishes that the inclusive and dressed formalism are not equivalentbut yield different predictions for cross-sections of finite superpositions.6.3 WavepacketsWe will now proceed to look at scattering of wavepackets and find that the resultis even more disturbing. After tracing out infrared radiation in the undressedformalism, no indication of scattering is left in the hard system. On the contrary,once again we will see that with dressed states, one gets the expected scatteringout-state.6.3.1 Inclusive formalismWe consider incoming wavepackets of the form|in〉 =∫dα f (α) |α〉 , (6.17)normalized such that∫dα | f (α)|2 = 1. The full analysis of the preceding sectionstill applies, provided we replace∑αi →∫dα, αi → α, fi → f (α) and similarlyfor aj → α′. The only notable exception is the calculation of single matrix elements64as in equation (6.10), which now readsρββ =∬dαdα′ f (α) f ∗(α′)SΛβ,αSΛ∗β,α′δαα′Gββ,αα′(E,ET ,Λ). (6.18)Note that here, by the same argument as before, the λ-dependent factor is turnedinto a Kronecker delta, which now reduces the integrand to a measure zero subseton the domain of integration. The only term that survives the integration is theinitial state, which is acted on with the usual Dirac delta δ(α − β), i.e., the “1” termin S = 1 − 2piiM. The detailed argument can be found in appendix C.2. Thus weconclude thatρoutββ′ = f (β) f ∗(β′) = ρinββ′ . (6.19)The hard particles show no sign of a scattering event.6.3.2 Dressed wavepacketsThe dressed formalism has perfectly reasonable scattering behavior. Considerwavepackets built from dressed states‖in〉〉 =∫dα f (α)‖α〉〉, (6.20)with ‖α〉〉 a dressed state in the same notation as in equation (6.11). The S-matrixapplied on dressed states is infrared-finite and the outgoing density matrix can beexpressed asρ =∬dβdβ′∬dαdα′ f (α) f ∗(α′)Sβ,αS∗β′,α′ ‖β〉〉〈〈β′‖. (6.21)Tracing over soft modes, we findρββ′ =∬dαdα′ f (α) f ∗(α′)Sβ,αS∗β′,α′ 〈W†βWβ′〉 . (6.22)Again the expectation value is taken in the photon vacuum. The crucial point hereis that this factor is independent of the initial states α. Upon sending the IR cutoffλ to zero, the expectation value for W†W takes only the values 1 or 0, leading to65decoherence in the outgoing state, but the cross-sections still exhibit all the usualinterference between components of the incoming wavefunction,ρββ =∬dαdα′ f (α) f ∗(α′)Sβ,αS∗β,α′, (6.23)unlike in the inclusive formalism.6.4 ImplicationsIn this section we will discuss the implications of our results and generalize andre-interpret our findings in particular in view of asymptotic gauge symmetries inQED and perturbative quantum gravity.6.4.1 Physical interpretationThe reason for the different predictions of the inclusive and dressed formalism isthe IR radiation produced in the scattering process. The key idea is that acceleratedcharges produce radiation fields made from soft bosons. In the far infrared, theradiation spectrum has poles as the photon frequency k0 → 0 of the form pi/pi · k,where pi are the hard momenta. These poles reflect the fact that the radiationstates are essentially classical and are completely distinguishable for different setsof asymptotic currents jv.In the inclusive formalism, we imagine incoming states with no radiation, andso the outgoing radiation state has poles from both the incoming hard particles αand the outgoing hard particles β. In the dressed formalism, the incoming part of theradiation is instead folded into the dressed state ‖α〉〉, which is designed preciselyso that the outgoing radiation field only includes the poles from the outgoinghard particles. Thus if we scatter undressed Fock space states, a measurementof the radiation field at late times would determine the dynamical history at longwavelengths of the process α → β, leading to the classical answer (6.10). If weinstead scatter dressed states, the outgoing radiation has incomplete informationabout the incoming charged state, which is why the various incoming states stillinterfere in (6.16). Given that this type of interference is observed all the time innature, this seems to strongly suggest that the dressed formalism is correct for any66problem involving incoming superpositions of momenta.Based on the result of section 6.2, one might argue that equation (6.10) perhapsis the correct answer and one would have to test experimentally whether or notinterference terms appear if we give a scattering process enough time so that thedecoherence becomes sizable. After all, the inclusive and dressed approach tocalculating cross-sections are at least in principle distinguishable, although maybenot in practice due to very long decoherence times. However, we have demonstratedin section 6.3 that the inclusive formalism predicts an even more problematic resultfor continuous superpositions, namely that no scattering is observed at all. We thuspropose that using the dressed formalism is the most conservative and physicallysensible solution to the problem of vanishing interference presented in this chapter.6.4.2 Allowed dressingsDressing operators weakly commute with the S-matrixIt was conjectured in [77] and proven in [78] that the far IR part of the dressingweakly commutes with the S-matrix to leading order in the energy of the bosonscontained in the dressing. In particular, this means that the amplitudes〈β |W†βSWα |α〉 ∼ 〈β|W†βWαS |α〉 ∼ 〈β | SW†βWα |α〉 (6.24)are all IR finite, while they might differ by a finite amount. A short proof of thisin QED, complementary to [78], can be given as follows (the gravitational casefollows analogously). Recall that Weinberg’s soft theorem for QED states that tolowest order in the soft photon momentum q of outgoing soft photons〈`1a`1q1 . . . `N a`NqN S〉 ∼N∏i=1(M∑jηjej`i · pjqi · pj)〈S〉 . (6.25)67A similar argument holds for incoming photons. For incoming photons with mo-mentum q we find that〈S∗`1a`1†q1 . . . ∗`N a`N †qN 〉 ∼N∏i=1(−M∑jηjej∗`i · pjqi · pj)〈S〉 . (6.26)The reason for the relative minus sign is that incoming photons add energy-momentum to lines in the diagram instead of removing it. That means that themomentum in the denominator of the propagator changes (p−q)2+m2 → (p+q)+m2and vice versa. For small momentum, the denominator becomes −2pq → 2pq.From this it directly follows that for general dressings at leading order in the IRdivergences,〈SW〉 = 〈Se∫d3k( f` (k)a`†k − f ∗` (k)a`k )〉 ∼ N 〈Se∫d3k f` (k)a`†k 〉∼ N 〈e−∫d3k f ∗` (k)a`k S〉∼ 〈e∫d3k( f` (k)a`†k − f ∗` (k)a`k )S〉 = 〈WS〉 .(6.27)Here, we have suppressed a factor of ((2pi)32|k|)−1 and the sum over polarizationsin the integrals. In the first and third step we have split the exponential using theBaker-Campbell-Hausdorff formula (N is the normalization which is finite for finiteλ) and in the second equality we have used Weinberg’s soft theorem for outgoingand incoming particles.Dressings cannot be arbitrarily moved between in- and out-statesThis opens up the question about the most general structure of a consistent Faddeev-Kulish-like dressing. For example, one could ask whether one can consistentlydefine S-matrix elements with the dressing only acting on the out-state. To answerthis question, we assume that the dressing of the out-state has the same IR structureas equation (3.49), but is more general in that it may also include the momenta of(some) particles of the in-state, i.e., Wβ → WβWα˜ or any other momenta whichmight not even appear in the process, WβWα˜ → WβWα˜Wζ . The IR structure ofthe in-dressing is then fixed by the requirement that the S-matrix element is finite.In addition to the requirement of IR-finiteness we ask that the so-defined S-matrix68elements give rise to the correct rules for superposition and the correct scatteringfor wavepackets, even after tracing out soft radiation.Applying the logic of the previous sections and 5, one finds that tracing overthe soft bosons yields for a diagonal matrix element ρββρhardββ =∑i, jfi f ∗j SβαiS∗β′α′j〈0|W†α˜′Wα˜ |0〉 (6.28)andρhardββ =∬dαdα′ f (α) f ∗(α′)SβαS∗β′α′ 〈0|W†α˜′Wα˜ |0〉 (6.29)for finite and continuous superpositions, respectively. Here, we have used that〈W†α˜′W†β′WβWα˜〉β=β′= 〈W†α˜′Wα˜〉 . (6.30)The expectation value is taken in the soft boson Fock space. The expression in thecase of α˜ = α and α˜′ = α′ was already encountered in sections 6.2 and 6.3 in thecontext of inclusive calculations, where it was responsible for the unphysical formof the cross-sections. By the same logic it follows that even in the case where α˜ is aproper subset of α, we will obtain a Kronecker delta which sets α˜ = α˜′ and we againdo not obtain the expected form of the cross-section. Instead, particles from thesubset α˜ will cease to interfere. We thus conclude that the dressing of the out-statesmust be independent of the in-states and it is not consistent to build superposition ofstates which are dressed differently. This means that building superpositions fromhard and charged Fock space states is not meaningful. In particular, we cannot useundressed states to span the in-state space by simply moving all dressings to theout-state.Generalized Faddeev-Kulish statesHowever, it would be consistent to define dressed states by acting with a constantdressing operatorWζ for fixed ζ on states ‖α〉〉,‖α〉〉ζ ≡ W†ζWα |α〉 . (6.31)69(a)Σ(b) (c)Figure 6.1: (a) A plane wave goes through a single slit and emerges as alocalizedwavepacket. The scattering of the incomingwavepacket resultsin the production of Bremsstrahlung. (b) We can also define someCauchy slice Σ and create the state by an appropriate initial condition.(c) Evolving this state backwards in time while forgetting about the slitresults in an incoming localized particle which is accompanied by aradiation shockwave.Physically this corresponds to defining all asymptotic states on a fixed, coherentsoft boson background, defined by some momenta ζ . This state does not affect thephysics since soft modes decouple from Faddeev-Kulish amplitudes [45] and thusthis additional cloud of soft photons will just pass through the scattering process.The difference between the Faddeev-Kulish dressed state ‖α〉〉 and the generalizedstates of the form ‖α〉〉ζ is that the state ‖ζ〉〉ζ = W†ζWζ |ζ〉 = |ζ〉 does not containadditional photons. This also explains why QED calculations using momentumeigenstates without any additional dressing give the correct cross-sections once wetrace over soft radiation. Such a calculation can be interpreted as happening in aset of dressed states defined by‖α〉〉in = W†inWα |α〉 , (6.32)such that the in-state ‖in〉〉in does not contain photons and looks like a standardFock-space state.Localized particles are accompanied by radiationWe also conclude from the previous sections that there are no charged, normalizablestates which do not contain radiation. The reason is that within each selection sector70there is at most one delta-function normalizable state which does not contain radi-ation. Thus building a superposition to obtain a normalizable state will necessarilyinclude dressed states which by definition contain soft bosons. A nice argumentwhich makes this behavior plausible was given by Gervais and Zwanziger [92], seefigure 6.1.6.4.3 Selection sectorsEverything said so far has a nice interpretation in terms of the charges Q±ε oflarge gauge transformations (LGT) for QED and supertranslations for perturbativequantum gravity.It turns out that also our generalized version of Faddeev-Kulish states ‖α〉〉ζ ,equation (6.31), are eigenstates of the generatorsQ±ε with eigenvalues which dependon ζ . To see this note that [76][Q±ε,W†ζ ] = [Q±ε,S,W†ζ ] ∝∫S2d2z√γζ2ζ · qˆ ε(z, z), (6.33)and similarly for gravity [78]. Thus the generalized Faddeev-Kulish states span aspace of states which splits into selection sectors parametrized by ζ . The statementthat we can build physically reasonable superpositions using generalized Faddeev-Kulish states translates into the statement that superpositions can be taken within aselection sector of the LGT and supertranslation charges Q±ε .6.5 ConclusionsCalculating cross-sections in standardQEDand perturbative quantumgravity forcesus to deal with IR divergences. Tracing out unobservable soft modes seems to bea physically well-motivated approach which has successfully been employed forplane-wave scattering. However, as we have shown this approach fails in moregeneric examples. For finite superpositions it does not reproduce interferenceterms which are expected; for wavepackets it predicts that no scattering is observed.We have demonstrated in this chapter that dressed states à la Faddeev-Kulish (andcertain generalizations) resolve this issue, although it is not clear if the inclusive anddressed formalism are the only possible resolutions. Importantly, we have shown71that predictions of different resolutions can disagree, making them distinguishable.Superpositions must be taken within a set of states with most of the statesdressed by soft bosons. The corresponding dressing operators are only well-definedon Fock space if we use an IR-regulator which we only remove at the end ofthe day. In the strict λ → 0 limit, the states are not in Fock space but ratherin the much larger von Neumann space which allows for any photon content,including uncountable sets of photons [66, 90]. This suggests an interesting picturewhich seems worth investigating. The Hilbert space of QED is non-separable buthas separable subspaces which are stable under action of the S-matrix and formselection sectors. These subspaces are not the usual Fock spaces but look likethe state spaces defined by Faddeev and Kulish [65], in which almost all chargedstates are accompanied by soft radiation. In the next chapter, we will make thesestatements more precise.Our results also raise doubt on whether physical observables exist which cantake a state from one selection sector into another. If they did we could use them tocreate a superpositions of states from different sectors. But as we have seen above,in this case interference would not happen, which is in conflict with basic postulatesof quantum mechanics.Our results may have implications for the black hole information loss problem.Virtually all discussions of information loss in the black hole context rely on thepossibility of localizing particles – from throwing a particle into a black holeto keeping information localized. We argued above that normalizable (and inparticular localized) states are necessarily accompanied by soft radiation. It is wellknown that the absorption cross-section of radiation with frequency ω vanishes asω→ 0 and therefore it seems plausible that, whenever a localized particle is throwninto a black hole, the soft part of its state which is strongly correlated with the hardpart remains outside the black hole. If this is true a black hole geometry is alwaysin a mixed state which is purified by radiation outside the horizon.72Chapter 7An infrared-safe Hilbert space forQEDThis chapter is a redacted version of [5].7.1 IntroductionThe dressed formalisms discussed previously remove the IR divergences by includ-ing the radiation as coherent states in incoming and/or outgoing states. However,due to the infinite number of soft-modes, the dressed states are not Fock space states.Instead, as we will discuss in section 7.2, they live in representations of the photoncanonical commutation relations (CCR) which are different from the standard Fockrepresentation. Physically speaking, one could either say that states in differentCCR representations differ by an infinite number of low-energy excitations, or thatthey represent states which are expanded around classical backgrounds which differat arbitrarily long wavelengths. Since the radiation produced in scattering dependson the momenta of incoming and outgoing charges, a state which contains a chargedparticle with momentum p will generally be in a different CCR representation thana state containing a charged particle with momentum q , p.In this chapter we will restrict our attention to the case of QED. The infraredstructure of perturbative quantum gravity shares many qualitative features with thestructure of QED at low energies. Thus, a first step towards a detailed analysis of73IR physics of gravity can be taken by investigating the IR dynamics and kinematicsof QED.The fact that generic out-states consist of superpositions of states in differ-ent CCR representations becomes an issue if one wants to ask questions about theinformation content or the dynamics of low energy modes, since a meaningful com-parison of the photon content between different states in different representationsis impossible. A related problem recently mentioned in [95] is that the entirety ofdressed states is non-separable [65], i.e., they do not have a countable basis, andthus existing dressed formalisms do not allow for the definition of a trace. Andin fact, when using an IR cutoff to make the trace over IR modes well-defined,the reduced density matrix of the hard modes again essentially complete decoheresonce the cutoff is removed, see chapter 5.The soft photon production which is responsible for the IR divergences iswell approximated by a classical process, but a classical analysis suggests thenumber of zero-modes should stay constant: although the radiation fields which areclassically produced during scattering modify the vector potential at arbitrarily longwavelengths, this change is compensated by the change of the Liénard-Wiechertpotentials sourced by the charges. Hence, taking the off-shell modes of the classicalfield into account, the dynamics of the zero-modes become completely trivial andin the deep IR, the field remains constant in all physical processes.In this chapter we will see that this picture is accurate even at the quantum level.We develop a new dressed formalism for QED in which the asymptotic Hilbertspaces carry only a single representation of the canonical commutation relations.In other words, all relevant photon states only differ by a finite amount of excitedmodes. Moreover, the representations for in- and out-states are unitarily equivalent.This implies that the S-matrix is a manifestly unitary operator. Our proposal is amodification of the dressed state formalism of [65]. In addition to coherent statesdescribing radiation, we also incorporate off-shell modes into the definition of statesand approximate the time-evolution at late times. The outgoing density matrix ofany scattering is IR finite and tracing-out IR modes of the field is well-definedand does not completely decohere the density matrix at finite times. This allowsfor an IR safe investigation of scattering at late but finite times and enables us todiscuss information theoretic properties of quantum states, e.g., time evolution of74Hin Houtti t fTe−i∫ −∞tidtHas (t)S = Te−i∫ ∞−∞ dtHTe−i∫ t f∞ dtHas (t)scattering regionasymptotic in-region asymptotic out-regionFigure 7.1: The asymptotic Hilbert spaces Hin/out are defined at finite timesti and t f . We assume the particles to be well-separated before andafter ti and t f , respectively (shaded regions). The time evolution oftheories with long range forces is not given by the free HamiltonianH0, but approximated by the asymptotic Hamiltonian Has which takesthe coupling to very long wavelength modes of the gauge field intoaccount. Charged eigenstates of the free Hamiltonian are replaced bystates dressed with transverse off-shell photons which reproduce thecorrect Liénard-Wiechert potential at long wavelengths. The dressedS-matrix S evolves a state from t = ti to t = −∞ with the asymptoticHamiltonian, which removes the off-shell modes. It is then evolvedby the standard S-matrix S to t = ∞ and mapped onto Hout by anotherasymptotic time-evolution, dressing it with the correct Liénard-Wiechertmodes. The statesHin/out are related by a unitary transformation.entanglement.7.1.1 Summary of resultsAt times earlier than some initial time ti or later than some final time t f , wellseparated states of the full theory are well approximated by states in an asymptoticHilbert space. The dynamics relevant at long wavelengths are captured by time-evolution with an asymptotic Hamiltonian, which differs from the free Hamiltonian.This is summarized in figure 7.1. The asymptotic Hilbert spaces of QED are of the75formHin/out = Hm ⊗H⊗( f`), (7.1)whereHm is the free fermion Fock space andH⊗( f`) is an incomplete direct productspace (IDPS) (which despite the name is a Hilbert space and in particular complete)with a single representation of the photon canonical commutation relations. Theprecise definition is discussed in section 7.4. The choice of representation dependson a function f` , which generally is different for different incoming particles.H⊗( f`) can be understood as the image of Fock space under a coherent stateoperator and the function f` as specifying the low energy modes of the classicalbackground. States in this Hilbert space are dressed and take the form‖p,k〉〉 { f˜` } = |p〉 ⊗W[ f˜`(p, . . . )] |k〉 , (7.2)whereW[ f˜`] are operator valued functionals which create coherent states of trans-verse modes whose wavefunction is given by f˜` with polarization `. The constrainton f˜` is that for small photon momenta it agrees with f` appearing in equation(7.1).13 This guarantees that it is a state in H⊗( f`). The coherent state generallycontains transverse off-shell excitations which ensure that at low energies, the ex-pectation value of the photon field agrees with the classical expectation value. Itcontains additional on-shell radiation which makes sure that the bosonic part of thedressed state lives inH⊗( f`). The dressed S-matrix is defined asS =(Te−i∫ t f∞ dtHas (t))S(Te−i∫ ti−∞ dtHas (t))†(7.3)and is a unitary operator on H⊗( f`) for any f` . The first and last terms in thedefinition of the S-matrix remove off-shell modes from the states. This leaves statesdressed with on-shell photons which are scattered by the standard S-matrix, similarto the proposal of [65].This framework can be used to investigate the correlation between chargedparticles and IR modes. EachH⊗( f`) inherits the trace operation from Fock space.13Note that, unlike in [65], the IR profile of soft modes in the state ‖p,k〉〉α does not depend on pbut only on α.76Tracing the density matrix of a superposition of dressed states over soft modeswith wavelengths above some scale Λ yields time-dependent decoherence in themomentum eigenbasis. At late times, off-diagonal density matrix elements areproportional toρreducedoff-diagonal ∝ (tΛ)−A1eA2(t ,Λ). (7.4)The precise form of the exponents is discussed around equation (7.87). The ex-ponents are proportional to a dimensionless coupling and depend on the relativevelocities of the charged matter. The factor A1 is the same one found in [56] andwhose role for decoherence was discussed in chapter 4. The dependence on timeand energy scale has been found in [95] through a heuristic argument. The newfactor A2 suppresses decoherence relative to (tΛ)−A1 . The only information storedin the zero-momentum modes is the information about the CCR representation anddecoherence is caused by modes with non-zero momentum. As time passes, thesemodes become strongly entangled with the hard charges.In the following, we assume that QED is quantized in Coulomb gauge, sincethis makes the physical interpretation of our construction more obvious. Section7.2 reviews the construction of different representations of the CCR which areimportant for our purposes. Section 7.3 derives the asymptotic Hamiltonian andthe dressed S-matrix in Coulomb gauge. The construction of the asymptotic Hilbertspace is explained in section 7.4. Section 7.5 contains a proof of the unitarity ofthe S-matrix. In section 7.6 we explicitly calculate the S-matrix in the presence ofa classical current and investigate the correlation between IR modes and chargedparticles. The density matrix of superpositions of the fields of classical currents,reduced over IR modes, decoheres with time. The conclusions comment on furtherdirections.777.2 Representations of the canonical commutationrelations7.2.1 Inequivalent CCR representationsTheories with massless particles allow for different representations of the CCRalgebra which are not unitarily equivalent. This can easily be seen in a toy model[96]. Consider the HamiltonianH =∫d3k(2pi)32|k| |k|a†(k)a(k) −∫d3k(2pi)32|k| j(k, t)(a†(k) + a(−k)), (7.5)where j(x) is a real source. The Hamiltonian can be diagonalized using a canonicaltransformationa(k) → b(k) = a(k) + j(k)|k| a†(k) → b†(k) = a†(k) + j∗(k)|k| , (7.6)so that the commutation relations agree for b(k), b†(k) and a(k),a†(k). The diago-nalized Hamiltonian is given byH˜ =∫d3k(2pi)32|k| |k|b†(k)b(k) + 12∫d3k(2pi)3| j(k)|2|k|2 . (7.7)We will assume that lim |k |→0 j(k) = O(1). In this case and with appropriatefalloff conditions at large momenta, H˜ is bounded from below. We will assumethis in the following. The formally unitary transformation which implements thetransformation in equation (7.6) takes the formW ≡ eF = exp(∫d3k(2pi)32|k|(j(−k)|k| a†(k) − h.c.)). (7.8)However, W is not a good operator on the representation of the a(k),a†(k) CCR,since for example‖F |0〉 ‖2 =∫d3k(2pi)32|k|3 | j(k)|2 = ∞. (7.9)78This argument shows that generally, representations of the CCR of a massless fieldin 3+1 dimensions coupled to different currentswill be unitarily inequivalent, whichis exactly the problem discussed in the introduction. The choice of representation ofthe commutation relations of the photon field will generally depend on the presenceof charged particles. Before we discuss how to deal with this in the case of QED,we first need to develop some formalism.7.2.2 Von Neumann spaceFormally unitary operators like the one in (7.8) can be given a meaning as operatorson a complete direct product space [97], henceforth von Neumann space H⊗.The non-separable von Neumann space splits into an infinite number of separableincomplete direct product spaces (IDPS) on each of which one can define anirreducible representation of the canonical commutation relations [98]. Let usreview this construction in this and the next subsection.Given a countably infinite set of separable Hilbert spaces Hn, we define theinfinite tensor product spaceH′⊗ asH′⊗ ≡⊗nHn. (7.10)Vectors |ψ〉 ∈ H′⊗ of this space are product vectors built from sequences |ψn〉 ofnormalized vectors inHn,|φ〉 =⊗n|ψn〉 . (7.11)Two such vectors are called equivalent, |ψ〉 ∼ |φ〉, if and only if∑n|1 − 〈ψn |φn〉 | < ∞. (7.12)If the vectors are equivalent their inner product is defined via〈ψ |φ〉 =∏n〈ψn |φn〉 . (7.13)If two vectors are inequivalent, their inner product is set to zero by definition. The79von Neumann space H⊗ is then defined as the space obtained by extending thedefinition to all finite linear combinations of the vectors in H′⊗ and subsequentcompletion of the resulting space. In order to make the inner product definite,we also require that two states are equal if their difference has zero inner productwith any state in H⊗. The so-obtained space is non-separable, but splits intoseparable Hilbert spaces H⊗(ψ) called incomplete direct product spaces (IDPS).H⊗(ψ) consists of all vectors equivalent to some |ψ〉.Given a unitary operator Un on each Hn we can define a unitary operator U⊗onH⊗ throughU⊗⊗n|ψn〉 ≡⊗nUn |ψn〉 (7.14)and extend its definition to all states in H⊗ by linearity. Clearly, this is not theset of all possible unitary operators on H⊗. Multiplication and inverse of suchoperators is defined through multiplication and inverse of the Un. It can thenbe shown that these unitary operators map different IDPS onto each other, i.e.,U⊗H⊗(ψ) ∼ H⊗(ψ ′) with U⊗ |ψ〉 = |ψ ′〉. An operator U⊗ is a unitary operator onH⊗(ψ) if U⊗ |ψ〉 ∼ |ψ〉.In a quantum mechanical Hilbert space physical states are only identified withvectors up to a phase. In order to make this precise in a von Neumann spacewe define a generalized phase. Given a set of real numbers λ = {λ1, λ2, . . . } wedefine the generalized phase operator V⊗(λ) as a unitary operator with Vn = eiλn .If∑n λn converges absolutely, V⊗(λ) = ei∑n λn . Two vectors which differ bya generalized phase represent the same physical state. States are called weaklyequivalent |ψ〉 ∼w |φ〉, if and only if there exists a V⊗(λ) such thatV⊗(λ) |ψ〉 ∼ |φ〉 . (7.15)7.2.3 Unitarily inequivalent representations on IDPSGiven the notion of a unitary operator on a von Neumann space, we can findrepresentations of the photon CCR [66]. Let us define the Hilbert space Hγ of80photon wavefunctions f`(k) which obey∑`=±∫d3k(2pi)32|k| | f`(k)|2 < ∞. (7.16)The inner product is given by〈g | f 〉 =∑`=±∫d3k(2pi)32|k| g∗`(k) f`(k). (7.17)We are only interested in a special class of CCR representations discussed in [66].We define the coherent state operator14W[ f`] ≡ exp(∫d3k(2pi)32|k|[∑`=±f`(. . . ,k, t)a†`(k) − h.c.])(7.18)which formally obeysW[ f`]W[g`] = exp(∑`=±∫d3k(2pi)32|k|(g∗` f` − f ∗` g`))W[g`]W[ f`]. (7.19)Note that this is the same definition as equation (3.49), but with vanishing IR cutoff,λ → 0. By functionally differentiating this equation with respect to f` and g∗` atf` = g∗` = 0 we see that the operators a†`(k) and a`(k) obey the standard CCR. Iff`,g` are elements of Hγ the integrals in equation (7.19) converge and we obtaina representation on H⊗(0) which consists of all states equivalent to the photonvacuum |0〉 = ⊗n |0n〉. This is the standard Fock representation. It is clear thatany operator of the formW[h`] with h` ∈ Hγ is a unitary operator on Fock space.To obtain other representations we need to find operators which obey equation(7.19) on an IDPSH⊗(ψ) which is not weakly equivalent to Fock spaceH⊗(0). (Itwas shown in [98] that commutation relation representations on weakly equivalent14To make contact with the previous definition in terms of modes n, we need to expand f` in a basisen of the space of wavefunctions and define an ∼∫d3ken(k)a`(k) to be the annihilation operator onHn.81IDPS are unitarily equivalent.) Consider the space of functions Aγ defined by∑`∫ d3k(2pi)32|k||k| + 1|k| | f`(k)|2 < ∞. (7.20)Functions which obey this inequality are still dense in Hγ. The dual vector spaceA∗γ, taken with respect to the inner product, equation (7.17), consists of functionsfor which ∑`∫ d3k(2pi)32|k||k||k| + 1 | f`(k)|2 < ∞ (7.21)and 〈g | f 〉 is well defined for all g ∈ A∗γ and f ∈ Aγ. Let us define the state|h〉 = W[h`] |0〉, where h` lies inA∗γ, but not inAγ. SinceW[h`] formally diverges,the state |h〉 is inequivalent to the photon vacuum |0〉 (even weakly). This time,operatorsW[ f`] with f` ∈ Hγ do not yield a representation of the CCR onH⊗(h),since〈h|W[ f`] |h〉 = exp(−12∫d3k(2pi)32|k| | f` |2)exp(∫d3k(2pi)32|k|(h∗` f` − f ∗` h`) )(7.22)and the integral in the argument of the second exponential will generally diverge.Here, we left the sum over ` implicit. However, if we choose f` ∈ Aγ, the phaseconverges and we obtain a representation, this time on the separable space H⊗(h)which can be obtained from Fock space by the formally unitary operator W[h].These are the representations we will need in the following.827.3 Asymptotic time-evolution and definition of theS-matrix7.3.1 The naive S-matrixIn the standard treatment of scattering in quantum field theory, one defines theS-matrix asSβ,α ' limt′/t′′→∓∞〈β | e−iH(t′′−t′) |α〉 . (7.23)However, already in free theory it is clear that the limits t ′ → −∞ and t ′′ → ∞do not exist due to the oscillating phase at large times. More carefully we take thestates |α〉in /|β〉out at some fixed times ti/ f and define the S-matrix asSβ,α = limt′/t′′→∓∞ out〈β| eiH0(t′′−t f )e−iH(t′′−t′)e−iH0(t′−ti ) |α〉in . (7.24)H0 is the free Hamiltonian in which the mass parameter takes its physical value. Attimes later (earlier) than t f (ti) we assume that all particles are well separated suchthat their time-evolution can approximately be described by the free Hamiltonian.The contribution to phase factors coming from the renormalized Hamiltonian H =H0 + Hint cancels the one coming from the free evolution as t ′, t ′′→ ∓∞.However, it is well known that the free-field approximation is not valid for QEDeven at late times, since the interaction falls off too slowly. Mathematically, theproblem is that the expression for the S-matrix, equation (7.24), does not converge[54]. Physically, the issue is that massless bosons given rise to a conserved charge(e.g., electric charge in QED or ADM mass in gravity) which can be measured atinfinity as an integral over the long range fields. Turning off the coupling completelyat early and late times, no field is created. In this chapter we use canonicallyquantized QED in Coulomb gauge. One might argue that the conserved charge isalready taken into account by the solution to the constraint equation, which createsa Coulomb field around the source. However, for all but stationary particles, thisis not the correct field configuration. Well-separated particles with non-vanishingvelocity should be accompanied by the correct Liénard-Wiechert field which differsfrom the Coulomb field by transverse off-shell modes. Again, these modes can only83be excited if the coupling is not turned off completely.7.3.2 The asymptotic HamiltonianIn order to understand which terms of the full Hamiltonian remain important atearly and late times, let us approximate how the states evolve if they do not in-teract strongly for a long time. We ignore all UV issues, which are dealt with byusing renormalization, and consider the normal ordered version of the interactionHamiltonian,Hint ∼ −e∫d3x : ψ¯γiψ : (x) · Ai(x) +∬d3xd3y : ψ†ψ(x)ψ†ψ(y) :4pi |x − y| . (7.25)In the asymptotic regions it is then assumed that the fields, masses and couplingstake their physical values instead of the bare ones. In [65] it was shown that atlate times coupling to long-wavelength photon modes still remain important. Herewe will take a slightly different route to arrive at the exact same expression for theasymptotic Hamiltonian, i.e., the Hamiltonian which approximates time evolutionat very early and late times.The normal ordered current in the interaction picture in momentum space isgiven by: jµ(x) :∼ e∑s,t∬d3pd3q(2pi)64EpEq(b†s(p)bt (q)us(p)γµut (q)e−i(p−q)x−d†t (q)ds(p)vs(p)γµvt (q)ei(p−q)x + . . .),(7.26)where we have omitted terms proportional to b†s(p)d†t (q) and bt (q)ds(p). Theycorrespond to pair creation or annihilation with the emission or absorption of a highenergetic photons. In the asymptotic regions it should be a reasonable assumptionto ignore these effects. Generally, we do not want external momenta to stronglycouple to the current. Thus we restrict the integral over q to a small shell aroundp and set p = q everywhere except in the phases. After a Fourier transform and84keeping only leading order terms in |k| we obtain the asymptotic current,: jµas(k, t) : ∼ e∑s∫d3p(2pi)32EppµEp(b†s(p)bs(p) − d†s (p)ds(p))e−ivpkt∼ e∫d3p(2pi)32EppµEpρ(p)e−ivpkt,(7.27)where we have defined ρ(p) = ∑s (b†s(p)bs(p) − d†s (p)ds(p)) and vp = p/Ep. Atlate and early times, the free Hamiltonian in equation (7.24) should thus be replacedby the time-dependent asymptotic Hamiltonian,Has(t) = H0 + Vas(t), (7.28)which is obtained by replacing the current with the asymptotic current. The inter-action potential Vas(t) which replaces the interaction Hamiltonian is given in theinteraction picture byVas(t) = −∫IRd3k(2pi)3(: ji(−k, t) : Ai(k, t) − 12|k|2 : j0(k, t) j0(−k, t) :). (7.29)The domain of integration is restricted to soft momenta. The first term describesthe coupling of transverse photon degrees of freedom to the transverse current,V (1)as (t) = −∫IRd3k(2pi)32|k| ji(k, t)[ε∗i` (−k)a`(−k)e−i |k |t + εi`(k)a†`(k)ei |k |t],(7.30)with a sum over the spatial directions i implied. The second term,V (2)as (t) = e22∫d3p(2pi)32Ep∫IRd3k(2pi)31|k|2 : ρ(p) j0(q, t) : e−ivpkt, (7.31)gives the energy of a charge in a Coulomb field created by a second charge.857.3.3 The dressed S-matrixIn the spirit of equation (7.24) we define the dressed S-matrix as an operator whichmaps the asymptotic Hilbert space of incoming statesHin to the asymptotic Hilbertspace of outgoing statesHout,S = limt′/t′′→∓∞Te−i∫ t ft′′ dtHas(t)e−iH(t′′−t′)Te−i∫ t′tidtHas(t), (7.32)where T denotes time-ordering. It seems plausible that in the case of QED thisexpression has improved convergence over equation (7.23), since Has takes intoaccount the asymptotic behavior of H.15 In order to simplify the expressionfor the S-matrix and relate it to the standard expression, we insert the identity,1 = e−iH0(t′′−t f )eiH0(t′′−t f ) and 1 = e−iH0(t′−ti )eiH0(t′−ti ), between the time orderedexponentials and the full time evolution. We then obtainS = limt′/t′′→∓∞U(t f , t ′′) S U(t ′, ti), (7.33)where S = eiH0(t′′−t f )e−iH(t′′−t′)e−iH0(t′−ti ) reduces to the usual S-matrix in non-dressed formalisms once the limits are taken. The unitaries U(t1, t0) obey thedifferental equationi∂∂t1U(t1, t0) = Vas(t1)U(t1, t0), (7.34)where Vas is in the interaction picture and given by equation (7.29). The solution tothis is standard16U(t1, t0) = Te−i∫ t1t0dtVas(t). (7.35)15It has been conjectured in [91] that a similar expression in the context of the Nelson modelconverges. However, other work [99] indicates that there might be subleading divergences comingfrom current-current interactions.16See, e.g., chapter 4.2 of [53].86We can bring this into an even more convenient form [65] by splitting U(t1, t0) inthe following way,U(ti, t0) = Te−i(∫ titi− +· · ·+∫ t0+t0)dtVas(t)= Te−i∫ titi− dtVas(t) . . . e−i∫ t0+t0dtVas(t)= Te−i∫ titi− dtVas(t) . . . Te−i∫ t0+t0dtVas(t).(7.36)In the limit → 0 we can remove the time-ordering symbols. Since [Vas(t),Vas(t ′)]only depends on ρ(p) which commutes with all operators we can use the Baker-Campbell-Hausdorff formula eAeB = eA+Be1/2[A,B] to combine the exponentalsintoU(ti, t0) = e−i∫ tit0dtVas(t)e−12∫ tit0dt∫ tt0dt′[Vas(t),Vas(t′)]. (7.37)The first factor couples currents to the transverse electromagnetic potential and alsocontains the charge-charge interaction given in equation (7.31). The second factormakes sure that U(t2, t1)U(t1, t0) = U(t2, t0). We are interested in the limit wheret0 → −∞. In this case the second factor can be calculated as follows. Since thedensity ρ(p) commutes with all operators present in the asymptotic potential, theonly relevant contributions to the commutator come from the photon annihilationand creation operators. The unequal-time commutator of the asymptotic potentialwith itself is given by[Vas(t),Vas(t ′)] =∫IRd3k(2pi)32|k| j⊥as(−k, t)j⊥as(k, t ′)(ei |k |(t′−t) − e−i |k |(t−t′)), (7.38)with the transverse current j⊥,i(k, t) = ∑` εi∗` (k)ε j`(k)jj(k, t). We can now performthe integral over t ′ and drop the boundary conditions as t = −∞ knowing that inany final calculation they will be canceled by the corresponding term coming from87the full Hamiltonian. The result isH⊥c (t) = −12∫ t−∞dt ′[V(t),V(t ′)]=i2∫IRd3k(2pi)32|k|∫d3p(2pi)32Epvp − k(k·vp)|k |2|k| − k · vp×[: ρ(p)jas(−k, t) : e−ikvpt + h.c.],(7.39)where we have used that ρ(p) : jas(−k, t) :=: ρ(p) jas(−k, t) : up to terms thatare renormalized away [65]. This corrects the phase due to the Coulomb energy,equation (7.31), toeiΦ(t) ≡ ei∫ t−∞ dt′(Hc (t′)+H⊥c (t′)), (7.40)which gives the phase due to the energy of a charge in the Liénard-Wiechert fieldof another charge. The total asymptotic time evolution takes the formU(−∞, ti) = eiΦ(t)ei∫ t−∞ dt′V (1)as (t′). (7.41)An analogous expression follows forU(t f ,∞), where we have to drop the boundaryterms at t = ∞.7.4 Construction of the asymptotic Hilbert space7.4.1 The asymptotic Hilbert spaceWe can finally discuss the asymptotic Hilbert space. For now, we will ignorefree photons and moreover focus on a single particle. The generalization to manyparticles and the inclusion of free photons is straight forward and will be donelater. We require that our asymptotic states evolve with the asymptotic Hamiltonianinstead of the free one. Naively, we might be tempted to think that our asymptoticparticle agrees with a free field excitation at some time t. However, as discussed inthe previous section, if our field couples to a massless boson this will generally not88be correct. Given a charged excitation of momentum p we define‖p〉〉inp ≡ U(ti,−∞)(|p〉in ⊗ |0〉)≡ |p〉in ⊗W[ f in` (p,k, t)] |0〉 .(7.42)The state |p〉in is a free field fermion Fock space state defined at time ti and |0〉 isthe photon Fock space vacuum. U(ti,−∞) was given in equation (7.41) and doesnot change the matter component of the state. We can therefore write its actionas an operator on the photon Hilbert space, W[ f in` ], with W[ · ] given in equation(7.18). In (7.41), we have dropped the boundary term at −∞. This is analogousto the standard procedure one uses to get the electric field of a current at a timet from the retarded correlator. The subscript in equation (7.42) indicates that theasymptotic Hilbert space containing the state ‖p〉〉inp isHas = Hm ⊗H⊗( f in` (p,k, ti)), (7.43)whereHm is the standard free fermion Fock space and H⊗( f in` (p,k, ti)) is an incom-plete direct product space which carries a representation of the canonical commu-tation relations for the photon as explained in the previous subsection. Performingthe integral in U(ti,−∞), we can determine f in` (p,k, ti) to bef in` (p,k, t) = −ep · `(k)p · k θ(kmax − |k|)e−iv ·kti . (7.44)Here, pµ and kµ are on-shell and vµ = pµ/Ep. The Heaviside function makes surethat only modes with wave number smaller than kmax are contained in the dressing.Analogously, we can construct out-states as‖p〉〉outp ≡ U(t f ,∞)(|p〉out ⊗ |0〉)≡ |p〉out ⊗W[ f out` (p,k, t)] |0〉 ,(7.45)andf out` (p,k, t f ) = −ep · `(k)p · k θ(kmax − |k|)e−iv ·kt f = f in` (p,k, t f ). (7.46)89In the following, we will leave the sum over ` and the dependence of f`(p,k, t) onk and t implicit. It can be checked by power counting that the exponent of〈0|W[ f in` (p)] |0〉 = exp(−12∑`∫ d3k(2pi)32|k| | fin` (p)|2)(7.47)is IR divergent, so W[ f in` (p)] is not a unitary operator on Fock space. It can alsobe checked that W[ f in` (p)] obeys equation (7.21) so that the commutation relationrepresentation is inequivalent to the Fock space representation. On the other handW[ f out` (p) − f in` (p)] is a unitary operator on any representation since its argumentis inAγ, defined through equation (7.20). This operator maps in-states to out-statesand it follows thatH⊗( f (p)out` ) = H⊗( f (p)in` ). Since the Hilbert spaces are relatedby unitary time-evolution using the asymptotic Hamiltonian, in the following wewill oftentimes drop the in and out labels on the states. Equivalently we can setti = t f = T without affecting any argument in the following.The coherent state of transverse modes in equation (7.42) which accompaniesthe matter field |p〉in is not a cloud of on-shell photons. The reason is that thetime-dependence of f in` (p) modifies the dispersion relation of the modes createdby this coherent state from Ek = |k| to Ek = kv. To understand the role of thesemodes consider the expectation values of the four-potential in such a dressed state,〈〈p‖A0‖p〉〉 =∫d3k(2pi)31|k|2 〈〈p‖ j0(k, t)‖p〉〉eikx, (7.48)〈〈p‖A‖p〉〉 = e∫ kmax0d3k(2pi)32|k|vp − k(k·vp)|k |2|k| − k · vp[eik(x−vpt) + h.c.]〈〈p‖p〉〉. (7.49)The expectation value of A agrees with the classical 3-vector potential of a pointcharge moving in a straight line with velocity vp at long wavelength which passesthrough x = 0 at t = 0,jµ(k, t) = evµe−ivpkt . (7.50)In otherwords, the dressed state constructed above obeysEhrenfest’s theoremat longwavelengths. If we had not dressed the state, we would have found 〈〈p‖A‖p〉〉 = 090and the corresponding electric field would have been only the Coulomb field of astatic charge.17Given two momenta p , q, the Hilbert spacesH⊗( f in` (p)) andH⊗( f out` (q)) areinequivalent. To see this, note that W˜ ≡ W[ f out` (q)]W†[ f in` (p)]mapsH⊗( f in` (p)) toH⊗( f in` (q)) and up to a phase equals W˜ = W[ f in` (q) − f in` (p)]. If the Hilbert spaceswere equivalent, W˜ would have to be a unitary operator on H⊗( f in` (p)). However,it is easy to see that f in` (q) − f in` (p) does not obey (7.20) and thus the two Hilbertspaces cannot be equivalent.Since we have started with the claim, that we want all in- and out-states to beelements of the Hilbert space (7.43), it seems our program has failed. However,this is too naive. Assume we scatter an initial state ‖p〉〉p off of a classical potential.Our outgoing state will be a superposition of different momentum eigenstates.However, the state ‖q〉〉q will not be part of this superposition. A scattering processproduces an infinite number of long-wavelength photons as bremsstrahlung, but‖q〉〉q contains no such radiation. The IR part of the classical radiation fieldproduced during scattering from momentum p to q is created by a coherent stateoperatorR(p, q¯) ≡ W[ f rad` (p,k, t) − f rad` (q,k, t)]= W[ f rad` (p,k, t)]W†[ f rad` (q,k, t)](7.51)withf rad` (p,k) =ep · `(k)p · k g(|k|) ≈ − fin` (p,k,0). (7.52)The bar in the definition of R(p, q¯) denotes that the terms containing q come witha relative minus sign. Here, g(|k|) is a function which goes to 1 as |k| → 0 andcan be chosen at will otherwise. Thus the state which is obtained by scattering anexcitation with momentum p into an excitation with momentum q plus the longwavelength part of the corresponding bremsstrahlung is given by‖q〉〉p ≡ |q〉 ⊗W[ f in` (q)]R(q, p¯) |0〉 (7.53)17In the case of a plane wave the charge distribution is smeared over all of space.91up to a finite number of photons. This state contains the field of the state ‖q〉〉q aswell as the radiation produced by scattering the state ‖p〉〉p to momentum q at longwavelengths.The operator W[ f in(q)]R(p, q¯) again is not a unitary operator on any CCRrepresentation. However, the combinationW[ f out` (q)]R(q, p¯)W†[ f in` (p)] (7.54)converges on Fock space. The convergence up to phase is easy to see since up toa phase, equation (7.54) equals W[ f out` (q) + f rad` (q) − f in` (p) − f rad` (p)] and sincethe function in the argument vanishes as |k| → 0 it clearly satisfies equation (7.20).It is an easy exercise to prove that the phase is also convergent. We will give anexample below. This shows that the states ‖p〉〉p and ‖q〉〉p live in the same subspaceH⊗( f in` (p)). Moreover, all states which are physically accessible from ‖p〉〉p mustcontain radiation. States of the form ‖q〉〉p are constructed to precisely contain the IRtail of the classical radiation. Hence, all single fermion states which are physicallyaccessible take the form of equation (7.53) up to a finite number of photons and thuslive in the same separable IDPS. With the appropriate dressing, also multi-fermionstates and thus all physically accessible states live in this subspace. Note that thisstructure is different to existing constructions [65, 66, 95], where an out-state isgenerally a superposition of vectors from inequivalent subspaces ofH⊗.7.4.2 Multiple particles and classical radiation backgroundsThe generalization to multiple particles is straight forward. Given a state whichcontains multiple charges with momenta p1,p2, . . . , the operatorU†(ti,−∞) acts onthe photon state as18W[∑if in` (pi)](7.55)and maps the Fock space vacuum into a a different separable Hilbert space,H⊗(∑i f in` (pi)), which acts as our asymptotic photon Hilbert space. Similarly,18In the case of multiple particle species with different charges, we should replace e → ei in thedefinition of f in`(p).92we can define a coherent state operatorR(p1,p2, . . . ;q1,q2, . . . ) (7.56)which lets us define states‖q1,q2, . . .〉〉 {p1,p2,... } ∈ H⊗(∑if in` (pi)), (7.57)which contain particles with momenta q1,q2, . . . and the appropriate bremsstrah-lung produced by scattering charged particles of momenta {p1,p2, . . . } to chargedparticles of momenta {q1,q2, . . . }. Up to a finite number of additional photons allout states will be of this form.We can also incorporate classical background radiation described byA0 = 0 (7.58)A =∫d3k(2pi)32|k|[h`(k)eikx + h.c.](7.59)with lim |k |→0 |k|h`(k) = O(1), i.e., backgrounds which contain an infinite numberof additional infrared photons. In the presence of charged particles with momentap1,p2, . . . the corresponding asymptotic Hilbert space isH⊗(h` +∑i f in` (pi)).7.4.3 Comments on the Hilbert spaceThe construction presented in this chapter has a number of properties which areknown to be realized in theories with long range forces in 3 + 1 dimensions.Existence of selection sectorsThe existence of selection sectors in four-dimensional QED and gravity is wellestablished [26, 100] and has recently been rediscovered [29]. In the presentconstruction, the choice of selection sector corresponds to a choice of representationof the canonical commutation relations on a separable Hilbert spaceH⊗(ψ) ⊂ H⊗.That these are indeed selection sectors will be shown in the next section where weprove that S is unitary.93Charged particles as infraparticlesIt was shown in [96, 100, 101] that there are no states in QED (or more generally intheories with long range forces), which sit exactly on the mass-shell p2 = −m2. Ourconstruction reproduces this behavior. Although P · P‖p〉〉p = −m2‖p〉〉p, the stateis not non-normalizable.19 A normalizable state must be built from a superpositionof different states ‖q〉〉p. However, any other state in H⊗( f in` (p)) contains extraphotons and thus cannot be on the mass-shell p2 = −m2. Also note that in [3] itwas argued that consistent scattering of wavepackets in theories with long rangeforces in four dimensions requires to take superpositions of particle states includingphotons.Spontaneous breaking of Lorentz invarianceThe spontaneous breaking of Lorentz invariance in QED has already been notedin [100, 102] (see also [103]). In our construction, there is an infinite number ofpossible H⊗(ψ) one can choose from. This choice spontaneously breaks Lorentzinvariance. The states ‖p〉〉p and ‖q〉〉q describe boosted versions of the same config-uration, namely a charged particle in the absence of radiation. However, as shownabove they live in inequivalent representations. Thus, a Lorentz transformation can-not be implemented as a unitary operator on H⊗( f in` (p)). An analogous argumentapplies for any configuration of charged particles p1,p2, . . . .7.5 Unitarity of the S-matrixThe form of the S-matrix follows from equation (7.32),S = U(t f ,∞) SU†(ti,−∞), (7.60)with U(t1, t0) given in equation (7.37). The operator S is the textbook S-matrix.Comparing to equation (7.42) we see that the role of the operators U(t f ,∞),U†(ti,−∞) is to remove the part of the dressing which corresponds to the clas-sical field. Thus, the off-shell dressingU(ti,−∞) in the definition of the asymptoticstates, equation (7.53), can be ignored whenever we are calculating S-matrix ele-19P is the 4-momentum operator.94ments.Consider the action of the dressed S-matrix on ‖p1,p2, . . .〉〉 { f` } ∈ H⊗( f`). Weestablish unitarity on H⊗( f`) by showing that dressed S-matrix elements betweenstates with given f` are finite, as well as that dressed S-matrix elements betweenstates of different separable subspaces, i.e., various f`, f˜` with different IR asymp-totics vanish. Unitarity then follows from unitarity ofU in the von Neumann spacesense and unitarity of S.For the sake of clarity we will neglect the possibility of a classical backgroundradiation field in the following. Taking this possibility into account corresponds toacting with some coherent state operator R˜ on the Fock space vacuum and does notaffect the proof. We take an otherwise arbitrary, dressed in-state‖in〉〉 = |p1, . . .〉 ⊗W[ f in` (p1) + . . . ]R(p1, . . . ;q1, . . . ) |k1, . . .〉 (7.61)and similarly define a general out-state‖out〉〉 = |p′1, . . .〉 ⊗W[ f out` (p′1) + . . . ]R(p′1, . . . ;q1, . . . ) |k′1, . . .〉 . (7.62)Both states are elements of H⊗(∑i f`(qi)). For ease of notation, we will omit theellipses . . . and indices in the following. The S-matrix elements take the formSout,in = 〈〈out‖U(t f ,∞)SU†(ti,−∞)‖in〉〉=(〈p′ | ⊗ 〈k′ | R†(p;q))S(|p〉 ⊗ R(p;q) |k1〉).(7.63)It was conjectured in [77] and shown in [78] (see also [3]) that we can movedressings through the S-matrix without jeopardizing the IR-finiteness. We cantherefore move all qi dependent terms on one side and obtain〈〈out‖R†(p′;q)SR(p;q)‖in〉〉 = 〈〈out‖R(q;p′)SR(p;q)‖in〉〉= 〈〈out‖R(0;p′)SR(p; 0)‖in〉〉 + (finite).(7.64)95Hence, the divergence structure of the matrix element is the same as the one ofSout,in ∼(〈p′1, . . .| ⊗ 〈k′1, . . .| R†(p′1, . . . ; 0))S(|p1, . . .〉 ⊗ R(p1, . . . ; 0) |k1, . . .〉).(7.65)However, these are just Faddeev-Kulish amplitudes which are known to be IR finite[65].Let us now show that if ‖p1, . . .〉〉q1,... and ‖p′1, . . .〉〉q′1,... live in inequivalentrepresentations, the matrix element vanishes. We again omit the ellipses andindices. ConsiderSout′,in = 〈〈out‖U(t f ,∞)SU†(ti,−∞)‖in〉〉 (7.66)=(〈p′ | ⊗ 〈k′ | R†(p′;q′))S(|p〉 ⊗ R(p;q) |k〉). (7.67)Moving the dressing through the S-matrix, we find that up to finite termsSout′,in ∼ 〈out′ | R(q′,q)R†(p′; 0)SR(p; 0) |in〉 . (7.68)The previous proof showed that R†(p′; 0)SR(p; 0) is a unitary operator on Fockspace. Further, it can be shown that R(q′,q) vanishes on Fock space if q1, · · · ,q′1, . . . [1]. Therefore we can conclude that the S-matrix element vanishes and haveshown that the S-matrix is a stabilizer of the asymptotic Hilbert spaces defined insection 7.4.7.6 Example: Classical current7.6.1 Calculation of the dressed S-matrixThe formalism devised in the preceding sections can be used to investigate the timedependence of decoherence in scattering processes. A simple example can be givenby considering QED coupled to a classical current jµ(x). The current enters with96momentum p and at xµ = xµ0 is deflected to a momentum p′,jµ(x) = e∫ ∞0dτp′µmδ(4)(xµ − xµ0 −p′µmτ)+ e∫ 0−∞dτpµmδ(4)(xµ − xµ0 −pµmτ).(7.69)We assume that initially no radiation is present and the current is carried by aninfinitely heavy particle. The initial state of the transverse field excitations is notthe Fock vacuum but ‖in〉〉 = W[ f in` (p)] |0〉, which is the vacuum of the CCRrepresentation H⊗( f in` (p)). This state represents a situation in which the classicalfield of the current jµ is present at wavelengths longer than the inverse mass. Sincewe deal with an infinitely massiv source, the integrals are taken over all of valuesof k. The IR divergent Fock space S-matrix in the presence of a current can becalculated explicitly, see e.g., [52], and is given byS = R(q,p) = W[ f rad` (q,k) − f rad` (p,k)]. (7.70)According to our prescription, the dressed S-matrix is given byS = W[ f out` (q,k, t f )] S W†[ f in` (p,k, ti)]. (7.71)The out state is given by ‖out〉〉 = S ‖in〉〉 and contains the radiation field producedby the acceleration as well as a correction to the Coulomb field which depends onthe outgoing current. Combining everything, the dressed S-matrix becomesS = W[ f S` (p,q,k, ti, t f )] exp(ie2∫d3k(2pi)32|k|Φ(k,q,p))(7.72)97withf S` (p,q,k, t) =e(q · ε`(k)q · k (1 − eivq ·kt f ) − p · ε`(k)p · k (1 − eivp ·kti )),Φ(k,q,p) =(q⊥q · k −p⊥p · k) (q⊥q · k sin(vq · kt f ) +p⊥p · k sin(vp · kti))+q⊥q · kp⊥p · k sin((t f vq − tivp) · k )(7.73)The superscripts on the momentum vectors p⊥ ≡ P⊥(kˆ)p denote the part of pwhich is perpendicular to k. The projection operator P⊥(kˆ) arises from the sumover polarizations, P⊥(kˆ) = ∑`=± ε∗`(k)ε`(k). From here it is easy to see that as|k| → 0, f S`has no poles and Φ only goes like |k|−1. Therefore, S is a well definedunitary operator.7.6.2 Tracing out long-wavelength modesA big advantage of formulating scattering in terms of the dressed states introducedabove is that it allows an IR divergence free definition of the trace operation the onasymptotic Hilbert space. The trace operation is inherited from Fock space. Forexample, a basis for the Hilbert space of photon excitations inH⊗( f in` (p)) is givenbyW[− f rad` (p)] |0〉 ,W[− f rad` (p)]a†`′(k) |0〉 ,. . . ,W[− f rad` (p)]1√n!(a†`′(k))n |0〉 (7.74)Wecould have chosen any other f˜`(p,k, t) in place of f rad` as long as limk→0 |k| f in` (p,k, ti) =limk→0 |k| f˜`(p,k, t). For example we could have chosen f˜`(p,k, t) = f out` (p,k, t f ),since the trace is invariant under a change of basis.As an example, let us consider a superposition of fields created by classicalcurrents, i.e., the outgoing state is‖out〉〉 = 1√2N(Wq1 +Wq2) |0〉 , (7.75)98whereWqi ≡ W[ f out` (qi,k, t)] W[ f rad` (qi,k) − f rad` (p,k)] (7.76)and N is given byN = 1 + Re(〈0|W†q1Wq2 |0〉). (7.77)In order to calculate the reduced density matrix we split the dressingWqi = W IRqi +WUVqi into a part we will trace over (IR) and the complement (UV). The “IR” partcontains all modes with wavelength longer than some cutoff Λ, which is smallerthan kmax. The reduced density matrix obtained by tracing over “IR” then becomesρUV =1N(WUVq1 |0〉 〈0|WUV†q1 + 〈0|W IR†q2 W IRq1 |0〉 WUVq1 |0〉 〈0|WUV†q2 (7.78)+ (q1 ↔ q2)). (7.79)We see that the off-diagonal elements are multiplied by a factor of 〈0|W IR†q2 W IRq1 |0〉which is responsible for decoherence. A similar dampening factor already appearedin chapter 5. There, the calculation was done for Faddeev-Kulish dressed states andit was shown that the dampening factor has an IR divergence in its exponent whichmakes it vanish, unless q1 = q2. As we will see, using the dressing devised in thischapter, the dampening factor is IR finite for finite times.The magnitude of the dampening factor is simply the normal-ordering constantofW IR†q2 WIRq1 which is given byexp(−12∫d3k(2pi)32|k|∑`=±| f 1` − f 2` |2)(7.80)withf i` (qi,k, t) = eqi · ε`(k)qi · k (1 − e−iv ·kt ). (7.81)We can rearrange the terms proportional to | f i |2. We go to spherical polar coordi-99nates and separate the |k| integral to find∫d3k2|k|∑`=±| f i` |2 = e2∫d2Ωq⊥i q⊥i(qi · k)2∫ Λ0d |k||k| sin2(|k| (−vi · kˆ)2t)(7.82)The |k| integral can be performed and the result can be expressed in terms oflogarithms and cosine integral functions Ci(x).∫ Λ0d |k||k| sin2(|k| (−vi · kˆ)2t)=12(log(Λt) + γ + log(|vi · kˆ |) − Ci(Λt |vi · kˆ |)).(7.83)Here, γ is the Euler-Mascheroni constant. Using Ci(x) ∼ γ + log(x) + O(x2) forsmall x, we see that at Λ, t = 0 the exponent vanishes. The |k| integral for thecross-term involving f 1` and f2` is only slightly more complicated and can also beperformed. One finds∫d3k2|k|∑`=±Re( f 1∗` f 2` )= 2e2∫d2Ωq⊥1 q⊥2(q1 · kˆ)(q2 · kˆ)∫ Λ0d |k||k| sin(|k| (−v1 · kˆ)2t)×sin(|k| (−v2 · kˆ)2t)cos(|k| (−(v1 − v2) · kˆ)2t) (7.84)The integral evaluates to14(2 log(Λt) + γ + log(|v1 · kˆ |) + log(|v2 · kˆ |) − log(Λt |(v1 − v2) · kˆ |)−Ci(Λt |v1 · kˆ |) − Ci(Λt |v2 · kˆ |) + Ci(Λt |(v1 − v2) · kˆ |)).(7.85)Clearly, as t → 0 the dampening factor becomes zero and no decoherence takesplace. This is sensible is the example at hand, where we have assumed that thecurrent changes direction at t = 0. Different to the situation in [1], the densitymatrix is well defined even without an IR cutoff. In any real experiment wemeasure the field at very late times after the scattering process has happened and100all wavelengths shorter than those that will be traced out had enough time to beproduced, i.e., Λt 1. In this limit, the integrals are dominated by the logarithms.Furthermore, we need to keep the term which contains Ci(Λt |(v1 − v2) · kˆ |) − γ,since the cosine integral diverges as v1 → v2 and kˆ ⊥ v1,v2.Similarly, the phases of the off-diagonal terms in the density matrix can becalculated. Since we only have a single charge present, the Coulomb interactionsHc + H⊥c does not contribute anything to the phase. The only contributions comefrom the normal ordering of the coherent state operators. After some cancellationsand performing the integration over |k| we obtainexp(ie22(2pi)3∫d2Ωq⊥1 q⊥2(q1 · kˆ)(q2 · kˆ)Si(Λt(v1 − v2)kˆ)). (7.86)Thus, at late times, the dampening factor becomes〈0|W IR†q2 W IRq1 |0〉 = (Λt)−A1eA2(Λ,t) (7.87)withA1 =e22(2pi)3∫d2Ω( q⊥1q1 · kˆ− q⊥2q2 · kˆ) ( q⊥1q1 · kˆ− q⊥2q2 · kˆ)(7.88)A2(t,Λ) = − e22(2pi)3∫d2Ωq⊥1 q⊥2(q1 · kˆ)(q2 · kˆ)(Ci(Λt |(v1 − v2) · kˆ |)− iSi(Λt(v1 − v2) · kˆ) − γ − log(Λt |(v1 − v2) · kˆ |)).(7.89)This is consistent with earlier results obtained in [85, 95]. The appearance ofthe factor A2 makes the decoherence rate for particles milder than suggested by theterm which only depends on A1. The qualitative behavior at infinite times, however,reproduces exactly what has been found before based on calculations which onlytake the emitted radiation into account, namely that any reduced density matrixdecoheres in the infinite time limit.1017.7 ConclusionsIn this chapter we presented a construction of an infinite class of asymptotic Hilbertspaces which are stable under S-matrix scattering with a unitary, dressed S-matrix.The major improvement over existing work is that all asymptotic states live in thesame separable Hilbert space with a single representation of the photon canonicalcommutation relations. Our construction relied on the fact that transverse IRmodes of the Liénard-Wiechert field are included in the definition of the asymptoticstates. This should be a good approximation if the included wavelengths are smallerthan any other scale in the problem. The construction enables an analysis of theinformation content of IR modes in the late-time density matrix. As an example, westudied a density matrix which describes a superposition of the field of two classicalcurrents. The reduced density matrix decoheres as a power law with time. Theincrease of decoherence with time shows that the entanglement of charged particleswith infrared modes increases over time. The physical reason for the decoherence isthat at times t ∼ 1Λwe can tell apart on- and off-shell modes with wavelengths largerthan λ ∼ 1Λ. Since charged matter is accompanied by a cloud of off-shell modescreating the correct momentum dependend electric field, this allows to identifythe momenta of the involved particles. One might argue that this is incompatiblewith the picture of conserved charges from large gauge transformations (LGT)(for a recent review see [25]). There it is argued that a photon vacuum transitionmust happen since the soft charge generally changes during a scattering process.However, in our approachwe take into account off-shell excitationswhich contributeto the hard charge. The increase of decoherence with time can be understood aslearning to tell apart soft and hard charges as time goes on. Hence, in flat spacescattering, no information is stored in the LGT charges, but in the way the chargesplits between the hard and soft part.This work leaves open some interesting questions. We have seen that near-zeroenergymodes decohere the outgoing density matrix in the momentum basis. Unlikein chapter 4, this decoherence happens although the scattering is fundamentally IRfinite. Furthermore, the decoherence cannot be avoided by chosing an appropriatedressing, since we can only add radiation, i.e., on-shell modes, as additional dress-ing. At zero energy there is no difference between on and off-shell modes, however,102at finite times those can be distinguished which leads to decoherence. This opensup the possibility that a similar mechanism at subleading order in the asymptoticcurrent could also decohere additional quantum numbers like spin. Moreover, al-though we have constructed dressed states, we have not discussed how they can beobtained by an LSZ-like formalism from operators. Due to the presence of longwavelength modes of classical fields and radiation, the correct operators must benon-local. Presumably there should be an infinite family of operators, similar tothe situation in [70, 71], for each Hilbert space which must contain radiative modesin their definition. Filling in the details is left for future work.Lastly, as motivated in the introduction, an extension of the presented ideas togravity would be desirable for a variety of reasons. While one might expect that ageneralization to linearized gravity should be fairly straight forward, an extensionbeyond linear order will presumably more difficult. The discussion in the contextof gravity could be interesting in the context of the black hole information paradox:We have seen that in our construction no information is stored in the zero-energyexcitations. This agrees with statements made in [45, 46]. However, bywaiting longenough, charged matter can be arbitrarily strong correlated with near zero-energymodes and those modes might store information. Tracing out the matter thus leavesone with a completely mixed density matrix of soft modes, which might be relatedto the ideas presented in [33]. The fact that “softness” is an observer-dependentnotion might aid arguments in favor of complementarity. Clearly, more work isrequired to make these arguments more precise.103Part IIQuantum information inquantum gravity104Chapter 8The AdS/CFT correspondence8.1 Holography in string theoryWhile a complete understanding of quantum gravity in a four-dimensional de Sitteruniverse, such as the one we live in, does not seem to be in reach, considerableprogress has been made in quantum gravity in anti-de Sitter spacetime. Basedon the early developments outlined in section 1.1, it was proposed that gravity isholographic [104, 105], i.e., that the true degrees of freedom within a volume ofspacetime can be thought of as being encoded on a hypersurface of one dimensionless. The AdS/CFT correspondence is a duality between a gravitational theory in ananti-de Sitter (AdS) spacetime and a conformal field theory (CFT) in one dimensionless, and therefore a concrete realization of the holographic principle.8.1.1 AdS/CFTIn its generic form, the AdS/CFT duality relates a d-dimensional conformal fieldtheory CFTd to a gravitational theory on AdSd+1.20 One of the most prominentexamples of this duality is the conjecture that N = 4 Super Yang-Mills (SYM)theory on four-dimensionalMinkowski space is dual to String theory on anAdS5×S5background. This can be motivated by the following argument first posited in [12].20The stringy origin of the gravitational side enables one to rewrite the gravitational theory as atheory on AdSd+1× X where X is some compact internal space. The dimensions of X and the anti-deSitter space add up to 10 or 11.105Ten dimensional type IIB string theory contains non-perturbative, 3+ 1 dimen-sional hypersurfaces called D-branes. At weak coupling, the dynamics of theseobjects can be described by open strings, whose ends are restricted to lie on thebranes. Consider a stack of N such D-branes. At low energies, the dynamics of thissystem are described by two sectors: supergravity in ten-dimensional flat space andfour-dimensional N = 4 SYM theory with gauge group SU(N), which describesthe brane dynamics. At very low energies, these two sectors decouple.At strong coupling, the branes backreact on the geometry and their low energydescription is a p-brane solution of ten-dimensional supergravity. In the low energylimit, the dynamics of string theory on that background split into a sector away fromthe branes which effectively lives in flat space and string theory close to the horizonof the backreacted solution. Again, these sectors decouple.The AdS/CFT conjecture identifies the theories at strong and weak coupling.Both theories contain a decoupled sector of low energy supergravity in flat space.The non-trivial statement is that the other sectors, namely N = 4 SYM theory andstring theory in the near-horizon region of the p-branes, should also be identified.They are different descriptions of the same theory at different couplings.The regime of validity of either description can be extracted from the aboveargument. The Yang-Mills coupling constant in the gauge theory is given in termsof the string coupling by g2YM = 2pigs, while Newton’s constant on the AdS side isgiven by GN = α′4g2s . The constant α′ is related to the string length via α′ = l2s .Lastly, the tension of the brane stack is given by Ngsα′2. The characteristic lengthscale we can build from these quantities is R4 = Ngsα′2, which is proportional tothe curvature scale of the p-brane background, R4AdS = 4piNgsα′2.The gravity description should be a good approximation at low curvature, i.e.,if the radius of curvature is much bigger than the string length, 4piNgsα′2 α′2.Moreover, the description of the gravitational theory as stringsmoving onAdS spacerequires that the string scale is bigger than the Planck scale, i.e., l2p ≡ gsl2s < l2s .This tells us that the gravitational description should be valid ifN > Ngs 1. (8.1)The gauge theory description is valid in the opposite regime, Ngs < 1.1068.1.2 The dictionaryMore precisely, the duality states that the partition function of string theory on anegatively curved space equals the partition function of a conformal theory in lowerdimensions which can be thought of as being located on the conformal boundaryof the AdS space [106, 107]. The partition functions of both theories agree,〈exp(∫φ0O)〉CFT = ZGrav(φ0). (8.2)Here, φ0 schematically denote the asymptotic value of fields φ in the gravity theorywith partition function ZGrav(φ0). The left-hand side is the partition function ofthe dual conformal theory in which the φ0 play the role of sources for operatorsO. The operator O which multiplies φ0 is called the dual operator to the field φ.The equivalence of the partition functions allows one to translate quantities in theconformal field theory to quantities in the dual gravitational theory. By takingfunctional derivaties with respect to the sources we can express CFT correlationfunctions in terms of derivatives of the gravity partition function.21 Alternatively,we can choose the following prescription, known as the extrapolate dictionary.Close to the boundary of AdS, a scalar field can be expanded asφ = azd−∆ + . . . + bz∆ + . . . , (8.3)where the boundary sits at z = 0. The value of φ0 is given by the coefficienta, which defines a non-normalizable solution to the Klein-Gordon equation. Thenormalizable solution with highest power z∆ has a leading coefficient b which isrelated to the expectation value of the CFT operator dual to φ.Similar arguments can be made for fields of higher spin. For example, the CFTstress-energy tensor is the operator dual to the metric. One can choose coordinatesclose to the boundary such that the metric takes the Fefferman-Graham form,ds2 =1z2(−dt2 + dz2 + dxµdxµ + zdΓ(d)µν (x, z)dxµdxν). (8.4)21In order to obtain well-defined expressions free of divergences, holographic renormalization[108] needs to be employed.107AA˜ΣFigure 8.1: The relation between extremal surface and boundary region. Thisimage shows a time-slice of AdS3. The dual conformal field theory canbe thought of as living on the boundary (the black circle). The orangepart of the boundary is the subregion A of the CFT. Its entanglemententropy is dual to the length of an extremal codimension two surfaceA˜ (blue line) in the gravitational theory. The region Σ between theboundary and the extremal surface (shaded orange) is a slice of theentanglement wedge.In this gauge, the extrapolate dictionary gives the expectation value of the stress-energy tensor as〈Tµν〉 = d16piGN Γ(d)µν (x,0). (8.5)8.1.3 Holographic entanglement entropyThe entanglement entropy of a subsystem A, equation (2.8), equals the area of anextremal bulk A˜ surface which is homologous to the subsystem A [13, 14, 109–111].Roughly speaking, this means that it can be smoothly deformed onto A. Moreprecisely, the Ryu-Takayanagi formula or its covariant formulation, the Hubeny-Rangamani-Takayanagi prescription, states that the von Neumann entropy of thereduced density matrix on A is proportional to, at leading order in N , the area ofthe extremal surface A˜,S =Area(A˜)4GN, (8.6)108AAEA = CAEA = CA(a)AACA ⊂ EACA ⊂ EAEA(b)Figure 8.2: a) A disconnected region on the boundary of a timeslice of AdS3(orange) is dual to a a disconnected region in the bulk bounded by twoseparate RT surfaces (blue). b) If the boundary region is bigger thanhalf of the total boundary, the dual region becomes connected. It is stillbounded by the corresponding RT surface; however, not all points in thebulk region can causally communicate with DA. The boundary of thecausal wedge is indicated by dashed lines. For a detailed discussion,refer to the main text.see figure 8.1. In the rest of this theses we will use the acronym HRRT for thisexpression. If there are many extremal surfaces A˜, the one with smallest area mustbe chosen.As mentioned in section 2.5, entanglement entropy of subregions is an ill-defined concept in the quantum field theory because of UV divergences close tothe boundary of that region. The divergences can be regulated using a UV cutoff.Similarly, the extremal surface defined above has divergent area. The divergencearises from the fact that the boundary of AdS is infinitely far away from everypoint in the interior. This is an example of the UV/IR connection of AdS/CFT: UVdivergences in the CFT are related to divergences which appear as a consequenceof the infinite size of the AdS spacetime [112].1098.1.4 Causal wedge vs entanglement wedgeThere are two natural subregions in the bulk one can construct given a boundarysubregion. On the conformal boundary, a spatial subregion A has an associatedcausal diamond or domain of dependence, DA. A point to the future of A lieswithin DA if all past-directed timelike curves through p intersect A. Similarly, apoint to the past of A lies withinDA if all future-directed curves through that pointintersect A. In other words, DA is the spacetime region whose time evolution isuniquely determined by specifying initial conditions on A.The causal wedge CA is the intersection of the causal future and past of DA inthe bulk, i.e., all points which can send and receive lightlike signals from and toDA. The information contained in the reduced density matrix ρA associated withsubregion A captures the physics at least in the causal diamond CA. If this wasnot the case, we could e.g., place a small mirror inside the causal wedge withoutchanging the density matrix and thus change the boundary conditions of fields inthe bulk. Via the AdS/CFT dictionary, this would affect expectation values in theCFT, which leads to a contradiction [113].The entanglement wedge EA is the domain of dependence of a bulk Cauchy slicebounded by the boundary and the bulk extremal surface. Figure 8.2a shows a t = 0slice of pure AdS3 with two boundary regions and their associated RT surfaces.Both the causal and the entanglement wedge agree and are bounded by the HRRTsurface.However, the entanglement wedge and the causal wedge are generally not thesame. Figure 8.2b shows slightly bigger regions and their RT surfaces, which haveundergone a phase transition. The boundary of the causal wedge are given bythe dashed lines, while the entanglement wedge corresponds to the shaded region.Thus we see that in this case the causal and entanglement wedge are different. Thatthe causal and entanglement wedge are different is the generic situation away fromAdS vacuum, even if the boundary region is connected.While it seems natural that the boundary subregion contains information aboutthe associated causal wedge, the HRRT formula shows that in fact it must containinformation about the entanglement wedge. Since the area of the HRRT surfacecan be computed from the reduced density matrix, the density matrix must contain110information about the geometry close to the HRRT surface. An in fact, the densitymatrix of a subregion of a holographic CFT can be used to reconstruct bulk physicsin the associated entanglement wedge [22].111Chapter 9Positive gravitational subsystemenergies from CFT cone relativeentropiesThis chapter is a redacted version of [6].9.1 IntroductionVia the AdS/CFT correspondence, it is believed that any consistent quantum theoryof gravity defined for asymptotically AdS spacetimes with some fixed boundarygeometry B corresponds to a dual conformal field theory defined on B. Recently,it has been understood that many natural quantum information theoretic quantitiesin the CFT correspond to natural gravitational observables (see, for example [13],or [114, 115] for a review). Through this correspondence, properties which holdtrue for the quantum information theoretic quantities can be translated to statementsabout gravitational physics. In this way, we can obtain an alternative/deeper under-standing of some known properties of gravitational systems, but also discover novelproperties that must hold in consistent theories of gravity. A particularly interestingquantum information theoretic quantity to consider is relative entropy [116]. Aswe have seen in chapter 2, for a general state |Ψ〉 of the CFT, we can associate a re-duced density matrix ρA to a spatial region A by tracing out the degrees of freedom112outside of A and relative entropy S(ρA| |ρ0A) quantifies how different this state isfrom the vacuum density matrix ρ0Areduced on the same region. Relative entropyis typically UV-finite, always positive, and has the property that it increases as weincrease the size of the region A (known as the monotonicity property). Accordingto the AdS/CFT correspondence, this should correspond to some quantity in thegravitational theory which also obeys these positivity and monotonicity properties.As we review in section 9.2, by making use of the holographic formula re-lating CFT entanglement entropies to bulk extremal surface areas (the “HRRTformula” [13, 14]), it is possible to explicitly write down the gravitational quan-tity corresponding to relative entropy as long as the vacuum modular Hamiltonian(H0A= − log ρ0A) for the region A is local, that is, it can be written as a linearcombination of local operators in the CFT. Until recently, such a local form wasonly known for the modular Hamiltonian of ball-shaped regions [110]. For theseregions, relative entropy has been shown to correspond to an energy that can beassociated with the bulk entanglement wedge corresponding to this ball [37, 117].The positivity of relative entropy then implies an infinite family of positive energyconstraints (reviewed below) [39].Ball-shaped regions (ofMinkowski space) have the property that their boundarylies on the past lightcone of a point p and the future lightcone of some other point q.In the recent work [118], it has been shown that the vacuum modular Hamiltonianfor a region A has a local expression so long as the boundary ∂A of A lies on thepast lightcone of a point p or the future lightcone of a point q.22 Thus, we have amuch more general class of regions for which the relative entropy and its propertiescan be interpreted gravitationally. The main goal of the present chapter is to explainthis interpretation.In the general case, we denote by Aˆ the region of the lightcone bounded by ∂A,as shown in figure 9.1. The modular Hamiltonian can then be written asH0A =∫AˆζµA(x)Tµν(x)ν , (9.1)where Tµν is the CFT stress-energy tensor, µ is a volume form defined in section22The existence of such a region depends on the relativistic nature of the theory under consideration,which guarantees the existence of a codimension-0 domain of dependence.113pA∂AAˆFigure 9.1: Subregion A of the CFT whose boundary ∂A is on the past light-cone of the point p. Aˆ denotes the surface of the cone bounded by∂A.9.2, and ζµA(x) is a vector field on Aˆ directed towards the tip of the cone andvanishing at the tip of the cone and on ∂A.To describe the gravitational interpretation of the relative entropy for region A,we consider any codimension one spacelike surface Σ in the dual geometry suchthat Σ intersects the AdS boundary at Aˆ and is bounded in the bulk by the HRRTsurface A˜ (the minimal area extremal surface homologous to A). This is illustratedin figure 9.2. Next, we define a timelike vector field ξ in a neighborhood of Σwith the properties that ξ approaches ζA at the AdS boundary and behaves near theextremal surface A˜ like a Killing vector associated with the local Rindler horizon atA˜. The timelike vector field ξ represents a particular choice of time on the surfaceΣ and we can define an energy Hξ associated with this. While generally there aremany choices for the surface Σ and the vector field ξ, we can show that all of themlead to the same value for the energy Hξ . It is this quantity that corresponds to theCFT relative entropy S(ρA| |ρ0A).23The independence of Hξ on the surface Σ used to define it can be understoodas a bulk conservation law for this notion of energy. In the case of a ball-shaped23In this work, we focus on the leading contribution to the CFT relative entropy at large N andmake use of the classical HRRT formula. More generally, we expect that the bulk quantity will becorrected by a term −∆SΣ measuring the vacuum-subtracted bulk entanglement of the region Σ.114AA˜ΣAˆFigure 9.2: CFT relative entropy associated with boundary region A corre-sponds to a certain energy associated with a gravitational subsystemdefined by the domain of dependence of any spatial region Σ boundedby cone region Aˆ with ∂ Aˆ = ∂A and extremal surface A˜ with ∂ A˜ = ∂A.region [39], the energy Hξ is conserved in a stronger sense (or a bigger volume),since there we are also free to vary the boundary surface Aˆ to be any spatial surfaceA′ homologous to A in the domain of dependenceDA of A. In that case, the vectorfield ζA can be defined everywhere in DA such that the expression (9.1) for themodular Hamiltonian gives the same result for any surface A′. The bulk vectorfield ξ can be defined on the full entanglement wedge for A, i.e., the union ofspacelike surfaces ending on A˜ and on any A′ inDA, so we can think of the energyHξ as being associated with the entire entanglement wedge. In the more generalcase considered here, the collection of allowed surfaces Σ generally still define acodimension zero regionWA of the bulk spacetime (equivalent to the bulk domainof dependence of any particular Σ), but this region intersects the boundary only onthe lightlike surface Aˆ rather than the whole domain of dependence region DA.In section 9.4, we consider the limit where the geometry is a small deformationaway from pure AdS. For pure AdS, we show that the extremal surface A˜ associatedwith a region A whose boundary lies on the lightcone of p always lies on the bulk115lightcone of p. Thus, in a limit where perturbations to AdS become small, thewedge WA collapses to the portion Aˆbulk of this lightcone between p and A˜. Wepresent an analytic expression for the extremal surface A˜ and a canonical choice forthe vector field ξ on Aˆbulk. In terms of these, we can write an explicit expressionfor the leading perturbative contribution to the energy Hξ , which takes the form ofan integral over Aˆbulk quadratic in the bulk field perturbations.In section 9.5, we point out that the explicit form of the extremal surface A˜ inthe pure AdS case (in particular, the fact that it lies on the bulk lightcone) leadsimmediately to a holographic proof of the Markov property for subregions of a CFTin its vacuum state, namely that for two regions A and B the strong subadditivityinequalityS(A) + S(B) − S(A ∩ B) − S(A ∪ B) ≥ 0, (9.2)is saturated if their boundaries lie on the past or future lightcone of the same pointp. This was shown for general CFTs in [118], so it had to hold in this holographiccase. The holographic proof extends easily to cases where the field theory isLorentz-invariant but non-conformal, for example a CFT deformed by a relevantperturbation. In this case, the statement holds for subregions A, Bwhose boundarieslie on a null-plane.We conclude in section 9.6 with a discussion of some possible future directions.9.2 Background9.2.1 Relative entropy in conformal field theoriesRecall from section 2.3 that we can rewrite the expression for relative entropy as[116]S(ρ‖σ) = ∆〈Hσ〉 − ∆S (9.3)where ∆ indicates a quantity calculated in the state ρ minus the same quantitycalculated in the reference state σ.For a conformal field theory in the vacuum state, the modular Hamiltonian ofa ball-shaped region takes a simple form [110]. For a ball B of radius R centeredat x0 in the spatial slice perpendicular to the unit timelike vector uµ, the modular116Hamiltonian isHB =∫B′ζµBTµνν, (9.4)where ν = 1(d−1)!νµ1 · · ·µd−1dxµ1 ∧ · · · ∧ dxµd−1 is a volume form and ζB is theconformal Killing vectorζµB =piR{[R2 − (x − x0)2]uµ + [2uν(x − x0)ν](x − x0)µ} , (9.5)with some four-velocity uµ. The modular Hamiltonian is the same for any surfaceB′ with the same domain of dependence as B.Using the expression (9.4) in (9.3), the relative entropy for a state ρ comparedwith the vacuum state may be expressed entirely in terms of the entanglement en-tropy and the stress tensor expectation value. For a holographic theory in a statewith a classical gravity dual, these quantities can be translated into gravitationallanguage using the HRRT formula (which also implies the usual holographic re-lation between the CFT stress-energy tensor expectation value and the asymptoticbulk metric [119]). Thus, the CFT relative entropy for a ball-shaped region corre-sponds to some geometrical quantity in the gravitational theory with positivity andmonotonicity properties. In [117] and [39], this quantity was shown to have theinterpretation of an energy associated with the gravitational subsystem associatedwith the interior of the entanglement wedge associated with the ball.Recently, Casini, Testé, and Torroba have provided an explicit expression forthe vacuum modular Hamiltonian of any spatial region A whose boundary lies onthe lightcone of a point [118]. To describe this, consider the case where ∂A lies onthe past lightcone of a point p and let Aˆ be the region on the lightcone that forms thefuture boundary of the domain of dependence of A. For x ∈ Aˆ, define a functionf (x) that represents what fraction of the way x is along the lightlike geodesic fromp through x to ∂A (so that f (p) = 0 and f (x) = 1 for x ∈ ∂A). Now, define alightlike vector field on Aˆ byζµA(x) ≡ 2pi( f (x) − 1)(pµ − xµ) . (9.6)117Then the modular Hamiltonian can be expressed asHA =∫AˆζµATµνν . (9.7)In general, we cannot extend the vector field away from the surface Aˆ such that theexpression (9.7) remains valid when integrated over an arbitrary surface A′ with∂A′ = ∂A. In equation (9.38) we give an explicit expression for HA in a convenientcoordinate frame.Using this expression in (9.3), we can express the relative entropy for the regionA in a form that can be translated to a geometrical quantity using the HRRT formula.We would again like to understand the gravitational interpretation for this positivequantity.9.2.2 Gravity backgroundWe now focus on states in a holographic CFT dual to some asymptotically AdSspacetime with a good classical description. For any spatial subsystem A of theCFT, there is a corresponding region on the boundary of the dual spacetime (whichwe will also call A). The HRRT formula asserts that the CFT entanglement entropyfor the spatial subsystem A in a state |Ψ〉, at leading order in the 1/N expansion, isequal to 1/(4GN ) times the area of the minimal area extremal surface A˜ in the dualspacetime which is homologous to the region A on the boundary.For pure AdS, when the CFT region is a ball B, the spatial region Σ betweenB and B˜ forms a natural subsystem of the gravitational system, in that there existsa timelike Killing vector ξB defined on the domain of dependence DΣ of Σ andvanishing on B˜. At the boundary of AdS, this reduces to the vector ζB appearingin the modular Hamiltonian (9.4) for B. The vector ξB gives a notion of timeevolution which is confined toDΣ. From the CFT point of view, this time evolutioncorresponds to evolution by the modular Hamiltonian (9.4) within the domain ofdependence of B, which by a conformal transformation can bemapped to hyperbolicspace times time.For states which are small perturbations to the CFT vacuum state, it was shownin [117] that the relative entropy for a ball B at second order in perturbations to118the vacuum state corresponds to the perturbative bulk energy associated with thetimelike Killing vector ξB in DΣ (known as the canonical energy associated withthis vector).This result was extended to general states in [39]. While there are no Killingvectors for general asymptotically AdS geometries, it is always possible to define avector field ξB that behaves near the AdS boundary and near the extremal surfacein a similar way to the behavior of the Killing vector ξB in pure AdS. Specifically,we impose conditionsξa |B = ζaB, (9.8)∇[aξb] |B˜ = 2pinab, (9.9)ξ |B˜ = 0 , (9.10)where nab is the binormal to the codimension two extremal surface B˜. Given anysuch vector field, we can define a diffeomorphismg → g + Lξg . (9.11)This represents a symmetry of the gravitational theory, so we can define a corre-sponding conserved current and Noether charge. The resulting charge Hξ turnsout to be the same for any vector field satisfying the conditions (9.8) – (9.10). Itcan be interpreted as an energy associated to the vector field ξB or alternativelyas the Hamiltonian that generates the flow (9.11) in the phase space formulationof gravity. The main result of [39] is that the CFT relative entropy for a state |Ψ〉comparing the reduced density matrix ρB to its vacuum counterpart ρ(vac)B is equalto the difference of this gravitational energy between the spacetime Mψ dual to |Ψ〉and pure AdS,S(ρB | |ρ(vac)B ) = Hξ (Mψ) − Hξ (AdS) . (9.12)We will review the derivation of this identity in the next section when we generalizeit to our case.To writeHξ explicitly, we start with the Noether current (expressed as a d-form)Jξ = θ(Lξg) − ξ · L , (9.13)119where L is the Lagrangian density and θ is defined byδL(g) = dθ(δg) + E(g)δg . (9.14)Here, E(g) are the equations of motion obtained in the usual way by varying theaction. The Noether current is conserved off-shell for Killing vector fields andon-shell for any vector field ξ,dJξ = E(g) · Lξg. (9.15)Then, up to a boundary term, the energy Hξ is defined in the usual way as theintegral of the Noether charge over a spatial surface:Hξ =∫ΣJξ −∫∂Σξ · K . (9.16)Here, Σ is any spacelike surface bounded by the HRRT surface B˜ and by a spacelikesurface Σ∂M on the AdS boundary with the same domain of dependence as B. For aball-shaped region B, the quantity Hξ is independent of both the bulk surface Σ (asa consequence of diffeomorphism invariance) and also the spacelike surface Σ∂Mat the boundary of AdS (as a consequence of the fact that ζB defines an asymptoticsymmetry).The quantity K in the boundary term is defined so thatδ(ξ · K) = ξ · θ(δg) on ∂Σ . (9.17)As explained in [39], this ensures that the difference (9.12) does not depend on theregularization procedure used to calculated the energies and perform the subtraction.We can rewrite Hξ completely as a boundary term using the fact that on-shell,Jξ can be expressed as an exact form [39]Jξ = dQξ . (9.18)120Thus, for a background satisfying the gravitational equations, we haveHξ =∫∂ΣQξ −∫∂Σξ · K . (9.19)This shows that the definition of Hξ is independent of the details of the vector fieldξ in the interior of Σ. In our derivations below, it will be useful to have a differentialversion of this expression that gives the change in Hξ under on-shell variation ofthe metric. By combining (9.19) with (9.17), we obtainδHξ =∫∂Σ(δQξ − ξ · θ) (9.20)The interpretation of Hξ as a Hamiltonian for the phase space transformationassociated with (9.11) can be understood by recalling that the symplectic form onthis phase space is defined byΩ(δg1, δg2) =∫Σω(g, δg1, δg2) (9.21)where the d-form ω is defined in terms of θ asω(g, δ1g, δ2g) = δ1θ(g, δ2g) − δ2θ(g, δ1g) . (9.22)In terms of ω we have that for an arbitrary on-shell metric perturbationδHξ = Ω(δg,Lξg) =∫Σω(g, δg,Lξg). (9.23)This amounts to the usual relation dH = vH ·Ω between a Hamiltonian (in this caseHξ ) and its corresponding vector field (in this case Lξg) via the symplectic formΩ.9.3 Bulk interpretation of relative entropy for generalregions bounded on a lightconeConsider now a more general spacelike CFT subsystem A whose boundary lies onsome lightcone. In this case – unless the boundary is a sphere – there is no longer121a conformal Killing vector defined on the domain of dependence region DA andwe cannot write the boundary modular Hamiltonian as in (9.4) where the result isindependent of the surface Bˆ. Nevertheless, we have a similar expression (9.7) forthe modular Hamiltonian as a weighted integral of the CFT stress tensor over thelightcone region Aˆ (shown in figure 9.1). Thus, making use of the formula (9.3) forrelative entropy, together with the holographic entanglement entropy formula andthe holographic dictionary for the stress-energy tensor, we can translate the CFTrelative entropy to a gravitational quantity. In this section, we show that this canagain be interpreted as an energy difference,S(ρA| |ρvacA ) = Hξ (Mψ) − Hξ (AdS) (9.24)for an energy Hξ associated with a bulk spatial region Σ bounded by Aˆ and the bulkextremal surface A˜.To begin, we choose a bulk vector field ξ satisfyingξa |Aˆ = ζaA, (9.25)∇[aξb] |A˜ = 2pinab, (9.26)ξ |A˜ = 0. (9.27)The argument that the latter two conditions can be satisfied is the same as in [39],making use of the fact that we can define Gaussian null coordinates near the surfaceA˜. To enforce the first condition, we will make use of Fefferman-Graham (FG)coordinates for which the near-boundary metric takes the formds2 =1z2(dz2 + dxµdxµ + zdΓ(d)µν dxµdxν + O(zd+1)) (9.28)and choose a vector field expressed in these coordinates asξµ = ζµA+ zξµ1 + z2ξµ2 + . . .ξz = zξz1 + z2ξz2 + . . . . (9.29)We will now evaluate δHξ for this vector field starting from (9.20) and find thatit matches with a holographic expression for the change in relative entropy. First,122we evaluate the part at the AdS boundary. Explicit calculations in the FG gauge,which are done in appendix D.1, show thatδQξ − ξ · θ |z→0 = d16piGN ξaδΓ(d)abˆb |z→0 = d16piGN ζµAˆδΓ(d)µν µ , (9.30)where was defined in the previous section andˆa1...ak =√−g(d + 1 − k)!a1...akb1 · · ·bd+1−k dxb1 ∧ · · · ∧ dxbd+1−k . (9.31)Using the standard holographic relation between the asymptotic metric and the CFTstress tensor expectation value, we obtain∫Aˆ(δQξ − ξ · θ) = d16piGN∫AˆζµAˆδΓ(d)µν µ =∫AˆζµAˆδ〈Tµν〉µ = δ〈HAˆ〉. (9.32)Here, HAˆ is the boundary modular Hamiltonian for the region A, so this termrepresents the variation in the modular Hamiltonian term in the expression (9.3) forrelative entropy.Next, we look at the part of (9.20) coming from the other boundary of Σ, at theextremal surface. By condition (9.27) we have that ξ vanishes on A˜ and we are leftwith the integral over δQξ . Qξ can be brought into the form 116pi∇aξb ˆab [120] andby virtue of (9.26) we obtain the entanglement entropy using the HRRT conjecture,∫A˜δQξ =14GN∫A˜= δS. (9.33)Combining both contributions to (9.20), we have thatδHξ = δ〈HAˆ〉 − δS, (9.34)where the variation corresponds to an infinitesimal variation of the CFT state.Integrating this from the CFT vacuum state up to the state |ψ〉, we have thatS(ρA| |ρvacA ) = ∆〈HAˆ〉 − ∆S = ∆Hξ . (9.35)Thus, we have established that for a boundary region Awith ∂A on a lightcone, the123CFT relative entropy is interpreted in the dual gravity theory as an energy associatedwith the timelike vector field ξ.The energy Hξ is naturally associated with a certain spacetime region of thebulk, foliated by spatial surfaces bounded by the boundary lightcone region Aˆ andthe bulk extremal surface A˜. That such spatial surfaces exist is a consequence ofthe fact that the extremal surface A˜ always lies outside the causal wedge of theregion A (the intersection of the causal past and the causal future of the domain ofdependence of A) [121].9.4 Perturbative expansion of the holographic dual torelative entropyIn this section, we consider the expression for Hξ in the case where the CFT stateis a small perturbation of the vacuum state so that the density matrix can be writtenperturbatively as ρA = ρvacA + λρ1 + λ2ρ2 + . . . . In this case, the CFT state will bedual to a spacetime with metric gµν(λ) = g(0)µν + λg(1)µν + λ2g(2)µν + . . . .We recall that relative entropy vanishes up to second order in perturbations;making use of the expression (9.23), we will check that the gravitational expressionfor relative entropy also vanishes up to second order for general regions A boundedon a light cone. We then further make use of (9.23) to derive a gravitationalexpression dual to the first non-vanishing contribution to relative entropy, expressingit as a quadratic form in the first order metric perturbation.9.4.1 Light cone coordinates for AdSIt will be convenient to introduce coordinates for AdSd+1 tailored to the light coneon which the boundary of A lies. Starting from standard Poincaré coordinates withmetricds2 =1z2(dz2 − dt2 + d ®x2), (9.36)we assume that the point pwhose light cone contains ∂A is at ®x = z = 0 and t = ρ+0 ,where ρ+0 is an arbitrary constant. On the AdS boundary, we introduce polarcoordinates (t, ρ,Ω) = (t, ρ, φ1, . . . , φd−2) centered at ®x = 0 and define ρ± = t ± ρ.The surface ∂A is then described by ρ+ = ρ+0 and some function ρ− = Λ(φi).124With these coordinates, the vector field (9.6) defining the boundary modular flowtakes the formξ |Aˆ =2pi(ρ+0 − ρ−)(ρ− − Λ(φi))ρ+0 − Λ(φi)∂−. (9.37)and the modular Hamiltonian (9.7) may be written explicitly asHA = 4pi∬ ρ+0Λ(φi )dρ−dΩ(ρ+0 − ρ−2)d−1 [ρ− − Λ(φi)ρ+0 − Λ(φi)]T−−. (9.38)For the choiceΛ(φi) = −ρ+0 the region A is a ball of radius ρ+0 centered at the originon the t = 0 slice and the expression reduces to the usual expression for a modularHamiltonian of such a ball-shaped region.In the bulk, we similarly define polar coordinates (t,r, θ, φ1, . . . , φd−2) where(ρ, z) = r(cos θ, sin θ) and define r± ≡ t ± r so that the bulk light cone of the pointp is r+ = ρ+0 . We will see below that for pure AdS, the extremal surface A˜ lies onthis bulk light cone on a surface that we will parameterize as r− = Λ(θ, φi), whereΛ(θ = 0, φi) is the function that parameterized the surface ∂A.The AdSd+1 line element in these coordinates readsds2 =1sin2 θ(− 4dr+dr−(r+ − r−)2 + dθ2 + cos2 θgΩi jdφidφ j), (9.39)where gΩi j is the metric on the unit d − 2 sphere and only depends on φi.9.4.2 HRRT surface in pure AdSIn this section, we derive an analytic expression for the extremal surface A˜ inpure AdS whose boundary is the region ∂A on the lightcone of p. This will beuseful in giving more explicit expressions for the relative entropy at leading orderin perturbations.We choose static gauge, parameterizing the surface using the spacetime coor-dinates θ and φi and describing its profile in the other directions by ρ±(θ, φi). The125equations which determine its location areγab∂γab∂r±= − 1√γ∂a(8√γγabsin2 θ(r+ − r−)2 ∂br∓). (9.40)Let us make the ansatz that even away from the boundary the extremal surfacelives on the lightcone r+ = ρ+0 and r− = Λ(θ, φi). The induced metric of thiscodimension two surface isγab =1sin2 θ(δθaδθb + cos2 θgΩi jδiaδib), (9.41)where a, b ∈ {θ, φ1, ..., φd−2} and i, j ∈ {φ1, ..., φd−2}. This metric is independentof r±; we will see in section 9.5 that this is related to the Markov property of CFTsubregions with boundary on a lightcone.Since the inducedmetric is independent of r±, the left hand side of the equationsof motion (9.40) vanishes and we can see from the right hand side that the ansatzr+ = ρ+0 solves the equations. The remaining equation for f (θ, φi) ≡ ρ+0 − Λ(θ, φi)reads0 = ∂a(√γγabsin2 θ∂b1f (θ, φi)). (9.42)The solution which corresponds to ball-shaped entangling surfaces is well knownto be located at ρ2 + z2 = const. In order to obtain the solution for entanglingsurfaces of arbitrary shape (but still on a lightcone) we substitute the expression forthe induced metric and separate the equation for r− intocos3 θ tand−1 θ∂θ(1cos θ tand−1 θ∂θ1f (θ, φi))= − 1√gΩ∂i(√gΩ(gΩ)i j∂j 1f (θ, φi)).(9.43)Here, we followed our conventions and used indices i, j for the angular coordinatesφi. If we write 1f = h(θ)Φ(φi) we find that the left hand side can be solved if Φ(φi)is a spherical harmonic. In d − 2 dimensions, the eigenvalues of the Laplacianon Sd−2 are given by n(3 − d − n) for the n-th harmonic. Every level n has a126corresponding set of degenerate eigenfunctions Φln with l = 1, . . . , 2n+d−3n(n+d−4n−1)[122]. The left hand side readscos2 θh′′(θ) − cot θ(cos2 θ + (d − 2))h′(θ) + n(3 − d − n)h(θ) = 0. (9.44)This differential equation can be solved in terms of hypergeometric functions,h(θ) =c1 cos3−d−n θ 2F1(2 − d − n2,3 − d − n2;5 − d − 2n2; cos2 θ)+ c2 cosn θ 2F1(n − 12,n2;d − 1 + 2n2; cos2 θ).(9.45)To fix the constants in (9.45) it helps to use intuition from the solutions in the casewhere the boundary of a subregion is located on a null-plane instead of a lightcone(see appendix B). In that case it is clear that effects from perturbations away froma constant entangling surface on the extremal surface die off as z → ∞. Under atransformation which maps the Rindler result to a ball-shaped region, the distantpart of the extremal surface corresponds to θ = pi/2. Consequently, we requirethat hn(pi/2) → 0 for n ≥ 1 and hn(pi/2) = 1 for n = 0. At the same time, forθ → 0 we need that hn(θ) is constant and different from zero. These constraintsare easily solved with c1 = 0, c2 = 1. Introducing a normalization factor to ensurethat hn(0) = 1, we are left withhn(θ) = cosn θΓ( d+n2 )Γ( d−1+n2 )Γ( d−12 + n)Γ( d2 )2F1(n − 12,n2;d − 1 + 2n2; cos2 θ). (9.46)In conclusion this shows that extremal surfaces in the bulk are located at r+ = ρ+0and r− = Λ(θ, φi) withΛ(θ, φi) = ρ+0 −1C0 +∑∞n=1∑l Cn,lhn(θ)Φln(φi). (9.47)Here, n runs over spherical harmonics in d−2 dimensions and l over their respective127degeneracy. They intersect the boundary atΛ(φi) = ρ+0 −1C0 +∑∞n=1∑l Cn,lΦln(φi). (9.48)Thus, the constants Cn,l are determined in terms of the function parameterizing theboundary surface by performing the spherical harmonic expansion1ρ+0 − Λ(φi)= C0 +∞∑n=1∑lCn,lΦln(φi) . (9.49)As a simple example, one choice of surface involving only the n = 1 harmonicsfor the AdS4 case takes the formρ+(φ) = ρ+0 , ρ−(φ) = ρ+0 −2ρ+0√1 − β21 + β cos φ, (9.50)and correspond to ball-shaped regions in a reference frame boosted relative to theoriginal one by velocity β in the x-direction.9.4.3 The bulk vector fieldOur next step is to provide an explicit expression for the vector field on the extremalsurface which obeys equations (9.25) – (9.27), such that the quantity Hξ is dual torelative entropy.Using (9.37), the explicit form of equation (9.25) isξ |Aˆ =2pi(ρ+0 − ρ−)(ρ− − Λ(0, φi))ρ+0 − Λ(0, φi)∂−. (9.51)Equation (9.26) requires knowledge of the unit binormalnµν = nµ2 nν1 − nµ1 nν2, (9.52)but thanks to the knowledge about the expression for the extremal surface whichwe found in the preceding section it is possible to calculate it explicitly. Here,n1,2 denote two orthogonal normal vectors to the RT surface. The calculation is128delegated to appendix D.3. The non-zero components of the unit binormal readn+− = g+−, na− = −∂aΛ(θ, φi), (9.53)where a again runs over coordinates (θ, φi). One possible choice of a vector fieldsatisfying the boundary conditions given by equations (9.26) is:24ξ =2pi(ρ+0 − r+)(r+ − Λ(θ, φi))ρ+0 − Λ(θ, φi)∂+ +2pi(ρ+0 − r−)(r− − Λ(θ, φi))ρ+0 − Λ(θ, φi)∂−+4pi(ρ+0 − r+)sin2 θ∂a(1ρ+0 − Λ(θ, φi))∂a .(9.54)Here, the ∂− and ∂+ components are chosen to match with the expression for theKilling vector ξ in the case when Λ is constant. On the light cone, only the∂− component (along the lightcone) is nonzero, and this has the same qualitativebehavior as the vector ζ on the boundary lightcone. It is immediately clear thatconditions (9.25) and (9.27) are satisfied. It is also straightforward to verify thecondition involving the unit binormal using the fact that for a torsion free connectionwe have ∇µξν − ∇νξµ = ∂µξν − ∂νξµ.Calculating the Lie derivative of the metric with respect to this vector field giveszero on the light cone r+ = ρ+0 but not away from the light cone. This is in contrastto the case of a ball-shaped region, where the Lie derivative vanished everywhereinside the entanglement wedge.9.4.4 Perturbative formulae for ∆HξTo write an explicit perturbative expression for ∆Hξ , we begin with the on-shellresultδHξ =∫Σω(g(λ), ddλg,Lξg(λ)). (9.55)24Upon expanding the sums in equation (9.54) it looks like the φi components of the vector fielddiverge as θ → pi2 and for d > 3 as φi → 0, pi due to the metric on the Sd−2 sphere. However,these divergences can be shown to be mere coordinate singularities: From equation (9.46) we seethat ∂iΛ ∼ cos θ. Hence the φi components of the vector field go only as cos−1 θ. This happens asa consequence of the coordinate singularity at θ = pi/2 in polar coordinates which can be removedby going into Poincaré coordinates (t, z, ®x). Similar arguments also hold for singularities due to theSd−2 metric. Coordinate independent quantities like the norm of the spatial part of the vector fieldremain finite as can be seen from inspecting the metric.129Here, the symplectic d-form ω is explicitly given byω(g(λ), ddλg,Lξg(λ))=116piGNˆµPµναβσρ(Lξgνα∇β ddλgσρ −ddλgνα∇βLξgσρ),(9.56)wherePµναβσρ = gµσgνρgαβ − 12gµβgνσgρα − 12gµνgαβgσρ − 12gναgµσgβρ +12gναgµβgσρ.(9.57)Since both Pµναβσρ and ˆµ depend on the metric they will have a series expansionsin λ when we express the metric as a series. Also in this case we will use sub-or superscripts in parenthesis to indicate the order of the term in λ. Here andin the following we will use ∇µ to denote covariant derivatives with respect togµν(0) = g(0)µν .It will be convenient for us to choose a gauge for the metric perturbations suchthat the extremal surface stays at the same coordinate location for any variation ofthe metric. It was shown in [120] that this is always possible. In this case, we haveat first order∆H(1)ξ =∫Σω(g(λ), ddλg,Lξg(λ)). (9.58)Wewill see in the next section that this vanishes, in accordwith the general vanishingof relative entropy at first order (also known as the first law of entanglement).At second order, we haved2dλ2S(g(λ)| |g0)|λ=0 =∫Σddλω(g(λ), ddλg,Lξg(λ))λ=0. (9.59)We will calculate this more explicitly in section (9.4.4).Vanishing of the first order expressionIn this section, we demonstrate that our gravitational expression for the relativeentropy vanishes for first order perturbations as required. Expanding the first order130expression (9.58) for ω yields∫Σω(g(λ), ddλg,Lξg(λ))λ=0= − 132piGN∫Σ(0)+(g+−(0))2g(1)−−gab(0) ∂+Lξg(0)ab,(9.60)where repeated lower case letters a, b imply summation over angular coordinates(θ, φi). Using the definition of the Lie derivativegab(0) ∂+Lξg(0)ab= 2gab(0) ∂+∇aξb, (9.61)and the fact that since gab is independent of r± all Christoffel symbols of the formΓ±abvanish at leading order, the problem reduces to a problem of only the angularcoordinates. We obtaingab(0) ∂+Lξg(0)ab= 2∇a∂+ξa = 2√γ∂a(√γ∂+ξa) . (9.62)Substituting the general form of ξa from equation (9.54) and using that g(0)ab= γ(0)abwe end up withγab(0) ∂+Lξγ(0)ab= −8pi 1√γ∂a(√γγab(0)sin2 θ∂b1ρ+0 − Λ). (9.63)g(0)µν and g(1)µν are the bulk metric and its perturbation and γ(0)ab, γ(1)abare the inducedmetric and the induced metric perturbation, respectively. This expression is pro-portional to the equation for an extremal surface, equation (9.42), and thereforevanishes.If we drop the assumption that the Einstein equations are satisfied, one canshow that the first law of entanglement entropy implies that the Einstein equationshold at first order around pure AdS. This was done in [119] where only ball-shapedCFT subregions were considered. Utilizing more general subregions bounded by alightcone does not yield new (in-)equalities at first order.131Relative entropy at second orderWe will now provide a more explicit expression for the leading perturbative con-tribution to relative entropy, which appears at second order in the perturbations.Starting from (9.59) and using our explicit expression for ω, we obtain four poten-tially contributing terms,d2dλ2S(g(λ)| |g0)|λ=0 = 116piGN∫Σ(1)+ P+ναβσρ(0)(Lξg(0)να∇βg(1)σρ − g(1)να∇βLξg(0)σρ)+116piGN∫Σ (0)P+ναβσρ(1)(Lξg(0)να∇βg(1)σρ − g(1)να∇βLξg(0)σρ)+116piGN∫Σ (0)P+ναβσρ(0)(Lξg(0)να∇βg(2)σρ − g(2)να∇βLξg(0)σρ)+116piGN∫Σ (0)P+ναβσρ(0)(Lξg(1)να∇βg(1)σρ − g(1)να∇βLξg(1)σρ).(9.64)The first and third terms vanish because of our first order results of section 9.4.4.The last term is reminiscent of the standard canonical energy associated with theinterior of the entanglement wedge, except that ξ is no longer a Killing vector. Thenon-zero contributions take the formδ(2)Hξ =∫Σω(g(λ), ddλg,Lξddλg)λ=0− 116piGN∫Σ(0)+(g+−(0))2 [g(1)−cgca(0)gdb(0) g(1)−d − g(1)−−gab(1)]∂+Lξg(0)ab.(9.65)Here, a, b, c, d run over angular coordinates, µ, ν run over all coordinates. Note thatalthough we are calculating relative entropy at second order, the expression onlydepends on first order metric perturbations. Due to the fact that ξ is no longer aKilling vector field, we appear to have a contribution in addition to the first termwhich appears for the case of ball-shaped regions.However, we have not yet imposed the Hollands-Wald gauge condition on thefirst order metric perturbations, for which the coordinate location of the extremalsurface is the same as in the case of pure AdS. We have additional gauge freedomon top of this, and it may be that for a suitable gauge choice, the final term in theexpression above can be eliminated. We have checked that this is the case for a132planar black hole in AdS4. We discuss this, as well as the procedure of choosingthe Hollands-Wald gauge condition in more detail in appendix D.4.9.5 Holographic proof of the Markov property of thevacuum stateIn [118] it was pointed out that the vacuum states of subregions of a CFT boundedby curves ρ− = ΛA and ρ− = ΛB on the lightcone ρ+ = ρ+0 saturate strongsubadditivity, i.e.,SA + SB − SA∩B − SA∪B = 0. (9.66)This is also known as the Markov property. Moreover, even for CFTs deformed byrelevant perturbations, the reduced density matrices for regions A and B describeMarkov states if A and B have their boundary on a null-plane. In its most generalform the proof used that the modular Hamiltonians for such regions obeyHA + HB − HA∩B − HA∪B = 0, (9.67)which can be proved using methods of algebraic QFT. In this section we willgive a holographic proof of the Markov property which uses the Ryu-Takayanagiproposal for entanglement entropy. We will start with the proof for a subregion of adeformed CFT with boundary on a null-plane and after that also show the propertyfor subregions of CFTs with boundary on a lightcone.9.5.1 The Markov property for states on the null-planeThe vacuum state of a deformed CFT is dual to a geometry of the formds2 = f (z)dz2 + g(z)(−2dx+dx− + dxµ⊥dx⊥µ). (9.68)An undeformed CFT corresponds to the special case f (z) = g(z) = 1z2. The entan-glement entropy of a subregion A can then be calculated using the RT prescription,following the same steps as in section D.2. We assume that the boundary ∂A isdescribed by x− = const and x+ = x+(®x⊥). To describe the corresponding extremal133surface we go to static gauge, where z and x⊥ are our coordinates and x±(z, x⊥) isthe embedding. The ansatz x− = const and x+ = x+(®x⊥, z) simplifies the equationto0 = ∂a(√γγab∂bx+g+−). (9.69)The relevant solution to this equation in the case of pure AdS is discussed inappendix D.2 and is given byx+(z, xi⊥) =2 2−d2 kd/2Γ(d/2)∫dd−2kak i zd/2Kd/2(zk)eiki xi . (9.70)Here, Kd/2 is the modified Bessel function of the second kind and the coefficientsak i are given in terms of the entangling surface x+(0, xi⊥) asak =∫dd−2x⊥(2pi)d−2 e−ik ·x⊥ x+(0, xi⊥). (9.71)More generally, the induced metric on the extremal surface in the bulk isds2 = f (z)dz2 + g(z)(dxµ⊥dx⊥µ) (9.72)and independent of the embedding x+(®x⊥, z). Thus, it is clear that the areas ofall extremal surfaces ending on x− = const are the same, potentially up to termswhich depend on how the area of the extremal surface is regularized as we approachthe boundary. The standard prescription given by cutting off z at some distance away from the boundary gives a universal cutoff term for all such extremal surfacesand therefore the entanglement entropies for all regions with boundary on x− areidentical and strong subadditivity is saturated. Our argument is an explicit versionof very similar arguments which have been used to show the saturation of theQuantum Null Energy condition [123].2525We thank Adam Levine for pointing this out to us.1349.5.2 The Markov property for states on the lightconeIf we consider an arbitrary region on the lightcone we expect the Markov propertyto hold for undeformed CFTs, since the lightcone is conformally equivalent tothe null-plane. The solution for an extremal surface in pure AdS ending on alightcone at the boundary was already discussed in section 9.4.2. Consider thecase where we have two different entangling surfaces given by ρ− = ΛA(φi) andρ− = ΛB(φi). We have seen before that the metric on the extremal surface is in factr− independent. However, again the dependence on the entangling surface can enterthrough regularization of the integral and would show up in the cutoff-dependentterm.In the coordinates of our choice θ, φi the divergent term in the area comes fromthe integral over θ. Following the standard way of regulating the surface integral weintroduce a cutoff z = , which translates into cutting off the integral at θ = r ≈ ρ .From this is follows that if we choose the canonical way of regulating the entropy,the θ integral runs from 2(ρ+0 −Λ) ≡ θ− to pi/2.The entropy which is proportional to the area term can now be calculated usingthe explicit form of the induced metric, equation (9.41), and is given by∫ √γ =∫dΩ∫ pi/2θ−dθcosd−2 θsind−1 θ. (9.73)The only way the shape of the entangling surface appears is through the cutoff, i.e.,the surface area can be expanded asA =0∑α=d−2cn(ρ+0 − Λ(φi)2)α, (9.74)where the coefficients cn are the same for all entangling surfaces. In the light ofequation (9.73) saturation of strong subadditivity for two regions on a lightcone135defined by ΛA and ΛB is guaranteed if∫dΩ((ρ+0 − ΛA(φi))α + (ρ+0 − ΛB(φi))α−max(ρ+0 − ΛA(φi), ρ+0 − ΛB(φi))α −min(ρ+0 − ΛA(φi), ρ+0 − ΛB(φi))α)= 0,(9.75)which is trivially pointwise true. This again shows that strong subadditivity issaturated, or in other words, reduced density matrices for regions on the lightconedescribe Markovian states. For more details on the form of the coefficients cn inthe expansion, see [124].The authors of [118] also speculated about the possibility of introducing a cutoffto regulate the area of extremal surfaces such that the area of the extremal surfaces ofsubregions on the lightcone are all exactly equal. The previous discussion explicitlyshows that choosing to introduce a cutoff θ = instead of z = realizes such aregularization procedure in which all entanglement entropies for regions on thelightcone are in fact the same.9.6 DiscussionThe results of this chapter imply that for any classical asymptotically AdS spacetimearising in a consistent theory of quantum gravity, the energy ∆Hξ must be positiveand must not decrease as we increase the size of region A. It would be interestingto understand if it is possible to prove this result directly in general relativity,by requiring that the matter stress-energy tensor satisfy some standard energycondition.It may be useful to point out that there is a differential quantity whose positivityimplies all the other positivity and monotonicity results considered here. If weconsider a deformation of the region A by an infinitesimal amount v(Ω), where vis some vector field on ∂A pointing along the lightcone away from p, the change inrelative entropy to first order must take the formδS(ρA| |ρvacA ) = ∫δΩv(Ω)SA(Ω) (9.76)136The monotonicity property implies that the quantity SA(Ω) must be positive for allA and all Ω.26 It would be interesting to make use of our results to come up with amore explicit expression for the gravitational analogue of the quantity SA(Ω). Oneapproach to providing a GR proof of the subsystem energy theorems would be toprove positivity of this.The Markov property discussed in section 9.5 suggests that it should be in-teresting to consider (for general states) the gravitational dual of the combinationS(A) + S(B) − S(A ∪ B) − S(A ∩ B) of entanglement entropies for regions A andB on a lightcone. Since strong subadditivity is saturated for the vacuum state,this gravitational quantity will vanish for pure AdS, but must be positive for anynearby physical asymptotically AdS spacetime according to strong subadditivity.Thus, strong subadditivity for these regions on a light cone will lead to a constrainton gravitational physics that appears even when considering small perturbationsaway from AdS. For two-dimensional CFTs, this quantity was already consideredpreviously in [34, 37]; the analysis there suggests that this gravitational constrainttakes the form of a spatially integrated null-energy condition. See [114] for someadditional discussion of gravitational constraints from strong subadditivity.26A special case of this positivity result was utilized in the proof of the averaged null energycondition in [125].137Chapter 10Conclusions10.1 Infrared quantum informationPart I of this thesis is concerned with the definition of information theoretic quanti-ties for scattering in four dimensions in the presence of long range forces mediatedby photons and gravitons. The presence of long range forces results in infrareddivergences in the calculation of scattering amplitudes which need to be dealt withby choosing one of two approaches. In the first, we only ask questions which canalso be operationally answered, and restrict our attention to inclusive observables.The construction of inclusive quantities involves summing over all possible stateswhich yield outcomes compatible with our measurements. We have seen in chapter4 that this treatment results in an essentially complete decoherence of the outgoingdensity matrix. The condition under which an off-diagonal density matrix elementin the momentum basis does not decohere can be phrased in terms of a condi-tion between an infinite number of current operators. The decoherence makes itparticularly easy to calculate the entanglement entropy between the hard and softmodes. However, this procedure makes quantum electrodynamics and perturbativequantum gravity inherently non-unitary.Alternatively we can use so-called dressed formalisms which add a finely tunedset of soft bosons to scattering states. These formalisms do not require a sum overoutgoing soft bosons and the S-matrix is formally unitary. Furthermore, they allowone to ask questions about amplitudes and other “unphysical” quantities. Also, in138this case we can calculate the entanglement entropy between soft and hard modes.In chapter 5 we found agreement with the calculation in the inclusive formalism.Chapter 6 discussed an important difference between the two formalisms. Inthe previous chapters, the calculations were done using incoming and outgoingmomentum eigenstates. If we replace momentum eigenstates by wavepackets, thepredictions of the inclusive and dressed formalism disagree; the reduced outgoingdensity matrix in the inclusive formalism becomes trivial. This behavior can betraced back to the fact that all components of the wavepackets after scattering areorthogonal. In the dressed formalism, however, everything works as expected. Thissuggests that the use of dressed states is not simply an alternative to the inclusiveformalism, but – at least in four dimensions – in fact required if one wants to treatquestions beyond scattering of momentum eigenstates.In chapter 7 we tackled two issues. First, the total decoherence found by tracingout soft modes only depends on the fact that S-matrix scattering assumes a limit inwhich incoming and outgoing states have had an infinite amount of time to interact,so that bosons of infinitely long wavelength can be produced. We thus tried tounderstand late-but-finite time behavior of decoherence. Second, all dressed stateproposals have the issue that they either do not come with a well-defined Hilbertspace, their Hilbert space is non-separable, or that their Hilbert space is not arepresentation of the canonical commutation relations of the soft-boson canonicalcommutation relations, but instead a set of vectors coming from different, unitarilyinequivalent representations. We solve both problems for quantum electrodynamicsby showing that, if charged asymptotic states are equipped with the correct electricfield and additional radiative dressing, they form states in a single representation ofthe CCR.Repeating the decoherence calculationwith states in such a representation,it was possible to extract the time-dependence of decoherence at late time.The above results can be used to calculate time dependence of quantum infor-mation theoretic quantities such as relative entropy between different photon states.A logical next step would be the extension of the Hilbert space construction in chap-ter 7 to the case of perturbative quantum gravity. As we have seen in the course ofthis thesis, many of our results are related to or can naturally be interpreted in thecontext of asymptotic symmetries related to Weinberg’s soft theorems. It would beinteresting to investigate the relation between our results and symmetries related139to subleading soft theorems. A better understanding of dressed states along thoselines will be a crucial contribution to understanding the Hilbert space structure offlat space holography.10.2 Quantum information and holographyIn part II of this thesis we turned to more established applications of quantuminformation theory in the context of theAdS/CFT correspondence. It was previouslyshown that relative entropy between the density matrices of the vacuum and someother holographic CFT state, reduced on a ball-shaped region, is dual to a measureof energy of the associated entanglement wedges. In chapter 9 we showed thatthis statement can be generalized to deformed ball-shaped regions which can beexpressed as a cone cut. This measure of energy inherits properties from relativeentropy, like monotonicity under inclusion of subregions and positivity. Moreover,we gave an explicit form for the bulk extremal surface as a function of the CFTentangling surface which bounds the deformed ball-shaped region. 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Wang, “Modular Hamiltoniansfor deformed half-spaces and the averaged null energy condition,” Journalof High Energy Physics 2016 no. 9, (Sep, 2016) 38, arXiv:1605.08072.151Appendix AInfrared quantum informationHere, we show that the exponents ∆A,∆B controlling the infrared divergencesare always positive or zero, and give necessary and sufficient conditions for theseexponents to vanish.The first step is to notice that the expressions for the differential exponents (4.8)between the processes α → β and α → β′ are the same as the exponents (4.6) forthe divergences in the process β→ β′, that is∆Aββ′,α = Aβ′,β/2,∆Bββ′,α = Bβ′,β/2.(A.1)To see this, note from the definitions (4.4), (4.6), and (4.8) that there are terms ineach of Aβ,α, Aβ′,α, and A˜ββ′,α coming from contractions between pairs of incominglegs, pairs of an incoming and outgoing leg, and pairs of outgoing legs. One caneasily check that the in/in and in/out terms cancel pairwise between the A and A˜terms in ∆A. The remainder is the terms involving contractions between pairs ofoutgoing legs:∆Aββ′,α =12∑p,p′∈βγpp′ +12∑p,p′∈β′γpp′ −∑p∈β,p′∈βγpp′ (A.2)where we defined γpp′ = epep′β−1pp′ log[(1+ βpp′)/(1− βpp′)]. We have used the factthat every ηp that would have been in (A.2) is a −1 since every line being summed is152an outgoing particle. But then we have a relative minus sign and factor of 2 betweenthe first two terms and the third; this is precisely the same factor that would havecome from the relative ηin = −1 and ηout = +1 terms in exponent for the processβ→ β′, namelyAβ′,β =∑p,p′∈βγpp′ +∑p,p′∈β′γpp′ − 2∑p∈β,p′∈β′γpp′ . (A.3)This proves (A.1) for ∆A; an identical combinatorial argument shows that thegravitational exponent obeys the analogous relation, ∆Bββ′,α = Bβ′,β/2.Now we prove that for the process α → β + (soft) the exponent Aβ,α is alwaysgreater or equal to zero with equality if and only if the in and outgoing currentsagree; we can then take α = β′ to get the results quoted in the text. Referring toWeinberg’s derivation [56], we can write Aβ,α asAβ,α =12(2pi)3∫S2dqˆ tµ(qˆ)tµ(qˆ). (A.4)Here,tµ(qˆ) ≡∑nenηnpµnpn · q = c(q)qµ + ci(q)(qi⊥)µ . (A.5)In this equation, we have defined a lightlike vector qµ = (1, qˆ) and qi⊥, i = 1,2are two unit normalized, mutually orthogonal, purely spatial vectors perpendicularto qµ. The sum on n ∈ α, β runs over in- and out-going particles. By chargeconservation, t · q = 0, which justifies the decomposition in the second equality in(A.5). With this decomposition we may writeAβ,α =12(2pi)3∫S2dqˆ(c21(q) + c22(q)) ≥ 0, (A.6)which immediately proves the statement that Aβ,α ≥ 0.Now it remains to be shown that equality holds if and only if all of the in-and out-going currents match. From the previous paragraph we know that Aβ,αvanishes if and only if both ci(q) = 0 for all q, that is if and only if t · qi⊥ = 0.Assume that Aβ,α = 0, so that q⊥ · t(q) = 0. Now suppose also that jv0(α) , jv0(β)153for some v0, where these are the eigenvalues of jv |α〉 = jv(α) |α〉 and similarlyfor β. We derive a contradiction. For any finite set of velocities, the functionsfv(qˆ) = (v · q⊥)/(1 − v · qˆ) are linearly independent. Therefore the terms in0 = t · q⊥ =∑nenηnvn · q⊥vn · q (A.7)must cancel separately for each velocity in the list of vn. Consider in particular theterm for v0. For this to vanish, the sum of the coefficients must vanish, i.e.,0 =∑n |vn=v0enηn =[jv0(α) − jv0(β)], (A.8)the relative minus coming from the η factors. But this contradicts our assumptionthat jv0(α) , jv0(β). This completes the proof for A.The proof for gravitons goes similarly. Again referring to Weinberg we writeB asBβ,α =G4pi2∫S2dqˆtµνDµνρσtρσ . (A.9)Here, Dµνρσ = ηµνηρσ − ηµρηνσ − ηµσηνρ is the numerator of the graviton propa-gator, andtµν =∑nηnpµnpνnpn · q = cq(µqν) + ciq(µqν)⊥,i + ci jq(µ⊥,iqν)⊥, j . (A.10)This symmetric tensor obeys tµνqν = 0 by energy-momentum conservation, whichjustifies the decomposition in the second equality. Using this we havetµνDµνρσtρσ = 2cijcji −(cii)2= (λ1 − λ2)2 (A.11)with λ1,2 the two eigenvalues of the matrix ci j . Plugging this into (A.9) weimmediately see that B ≥ 0. The condition for vanishing of Bβ,β′ is that theeigenvalues are equal λ1 = λ2, which means that ci j is proportional to the identity154matrix. Hence, if B vanishes we have that0 = tµνq⊥,1µ q⊥,2ν =∑nηnEn(vn · q1⊥)(vn · q2⊥)vn · q . (A.12)As before, any finite set of functions gv(q) = (v · q1⊥)(v · q2⊥)/(v · q) are linearlyindependent functions of q, and so by direct analogy with the previous proof, B = 0if and only if jgrv (α) = jgrv (β) for every v.155Appendix BDressed soft factorizationpp′kpp′kpp′kpp′kFigure B.1: Diagrams contributing to the dressed scattering with additionalbremsstrahlung. The first two diagrams correspond to photons comingfrom the dressing, while the latter two diagrams correspond to the usualFeynman diagrams where the photon is emitted from the electron lines.The soft photon theorem looks somewhat different in dressed QED. In standard,undressed QED, the theorem says that the amplitude for a process p → q accom-panied by emission of an additional soft photon of momentum k and polarization `has amplitudeSqk`,p = e[q · ∗` (k)q · k −p · ∗` (k)p · k]Sq,p. (B.1)156This is singular in the k → 0 limit. On the other hand, in the dressed formalism ofQED, the statement is thatS˜qk`,p = e f (k)S˜q,p, (B.2)where f (k) ∼ O(|k|0), so that the right-hand side is finite as k → 0. We can seethis by straightforward computation. In computing equation (B.2), there will befour Feynman diagrams at lowest order in the charge (see figure B.1). We will getthe usual pair of Feynman diagrams coming from contractions of the interactionHamiltonian with the external photon state, leading to the poles, equation (B.1).Moreover we will get a pair of terms coming from contractions of the interactionHamiltonian with dressing operators. These contribute a factor[f ∗` (k,p) − f ∗` (k,q)] → [q · ∗` (k)q · k −p · ∗` (k)p · k]+ O(|k|0), (B.3)times −e, where the limit as k → 0 follows from the definition (3.50). This extracontribution precisely cancels the poles in (B.1), leaving only the order O(|k|0)term.157Appendix COn the need for soft dressingC.1 Proof of positivity of ∆A,∆BThe exponent that is responsible for the decoherence of the system is defined as∆Aββ′,αα′ =12Aβ,α +12Aβ′,α′ − A˜ββ′,αα′ . (C.1)The factor in the first two terms, Aβ,α, is defined as in [56]Aβ,α =12(2pi)3∫S2dqˆ(∑n∈βenηnpµnpn · qˆ)gµν(∑m∈αemηmpµmpm · qˆ). (C.2)Performing the integral over qˆ yieldsAβ,α = −∑n,n′∈α,βenen′ηnηn′8pi2βnn′ log[1 + βnn′1 − βnn′]. (C.3)Similarly A˜ββ′,αα′ can be written asA˜ββ′,αα = −∑n∈α,βn′∈α′β′enen′ηnηn′8pi2βnn′ log[1 + βnn′1 − βnn′]. (C.4)158We rearrange the terms such that ∆A can be written as∆Aββ′,αα′ = −12∑n,n′∈α,α¯′,β,β¯′enen′ηnηn′8pi2β−1nn′ log[1 + βnn′1 − βnn′], (C.5)where a bar means incoming particles are taken to be outgoing and vice versa (orequivalently, ηα¯′ = −ηα′). From equation (C.5), it is clear that incoming particlesare found within the set {α, β′} while the outgoing particles are part of {α′, β}. Letus rename those sets σ and σ′ respectively. ∆A now takes the form∆Aββ′,αα′ = −12∑n,n′∈σ,σ′enen′ηnηn′8pi2β−1nn′ log[1 + βnn′1 − βnn′]=12Aσσ′ ≥ 0, (C.6)as was proven in [1]. This shows that ∆Aββ′,αα′ ≥ 0. The same proof goes throughfor ∆Bββ′,αα′.C.2 The out-density matrix of wavepacket scatteringIn this part of the appendix we flesh out the argument in section 6.3, namely thatafter tracing out soft radiation, the only contribution to the out-density matrix iscoming from the identity term in the S-matrix. We will focus on the case of QED.C.2.1 Contributions to the out-density matrixFirst, let us decompose the IR regulated S-matrix into its trivial part and the M-matrix element. For simplicity we ignore partially disconnected terms, where onlya subset of particles interact. Then,SΛα,β = δ(α − β) − 2piiMΛαβδ(4)(pµα − pµβ), (C.7)where the first term is the trivial LSZ constribution to forward scattering. This trivialpart does not involve any divergent loops and therefore exhibits no Λ-dependence.However, the factorization of the S-matrix into a cutoff dependent term times somepower of λ/Λ remains valid since all exponents of the form Aα,β vanish identicallyfor forward scattering. This decomposition of the S-matrix gives rise to threedifferent terms for the outgoing density matrix, containing different powers ofM.159“No scattering”-termThe case where both S-matrices contribute the delta function term results – unsur-prisingly – in the well-defined outgoing density matrixρ(I)ββ′ =∫dαdα′ f (α) f (α′)∗δ(α − β)δ(α′ − β′)δαα′ = f (β) f ∗(β′). (C.8)Contribution from forward scatteringWe would now expect to find an additional contribution to the density matrixreflecting the non-trivial scattering processes, coming from the cross-terms−2pii(δ(α − β)MΛα′βδ(4)(pµα′ − pµβ) − δ(α′ − β)M†Λαβδ(4)(pµα − pµβ)). (C.9)For simplicity, let us focus solely on the case in which S∗ contributes the deltafunction and S contributes the connected partρ(II)ββ′ = −2pii f ∗(β′)∫dα f (α)MΛβαδ(4)(pµα − pµβ)λ∆Aα,βG(E,ET ,Λ)β,α + . . . ,(C.10)where the ellipsis denotes the contribution coming from the omitted term of (C.9).The exponent of λ only vanishes if the currents in α and β agree. We will show inappendix C.2.2 that we can take the limit λ→ 0 before doing the integrals. Takingthis limit, λ∆Aα,β gets replaced byδαβ =1, if charged particles in α and β have the same velocities0, otherwise,(C.11)which is zero almost everywhere. If the integrand was regular, we could concludethat the integrand is a zero measure subset and integrates to zero and thusρ(II)ββ′ = 0. (C.12)160However, the integrand is not well-behaved. Singular behavior can come from thedelta function or the matrix element, so let’s consider the two possibilities.The singular nature of the Dirac delta does not affect our conclusion: for nincoming particles, the measure dα runs over 3n momentum variables while thedelta function constrains 4 of them, leaving us with 3n− 4 independent ones. If wemanaged to find a configuration for which ∆Aβ,α = 0, any infinitesimal variationof the momenta in α along a direction that conserves energy and momentum wouldmodify the eigenvalue of the current operator jv(α) − jv(β) and make ∆Aβα non-zero. Therefore, the integrandwould still be a zero-measure subset for the remainingintegrals.What could still happen is thatMΛβα is so singular that it gives a contribution.For this to happen it would need to have contributions in the form of Dirac deltafunctions. However, also this does not happen, for example for Compton scatteringwhich scatters into a continuum of states. Additional IR divergences also do notappear as guaranteed by the Kinoshita-Lee-Nauenberg theorem. We will not givea general proof since for our purposes it is problematic enough to know that noscattering is observed for some physical process.The scattering termIt is evident that a similar argument goes through for theM2 term. One findsρ(III)ββ′ = −4pi2∫dαdα′ f (α) f ∗(α′)MΛβαMΛ∗α′β′λ∆Aαα′ ,ββ′ (C.13)× F(E,ET ,Λ)ββ′,αα′δ(4)(pµα − pµβ)δ(4)(pµα′ − pµβ′). (C.14)The analysis boils down the the question whether the term∫dαdα′λ∆Aαα′ ,ββ′δ(4)(pµα − pµβ)δ(4)(pµα′ − pµβ′). (C.15)vanishes. As soon as there is at least one particle with charge, we need to obeythe condition that the charged particles in α and β′ agree with those in β and α′for the exponent of λ to vanish. Infinitesimal variations of α and α′ that preservethe eigenvalue of the current operator jv(α) − jv(α′) form a zero-measure subset of161the 6n − 8 directions that preserve momentum and energy, forcing us to concludethat the integration runs over a zero measure subset and the only contribution to thereduced density matrix comes from the trivial part of the scattering process. Thismeans thatρout, red.ββ′ = f (β) f ∗(β′) = ρinββ′, (C.16)or in other words it predicts that a measurement will not detect scattering forwavepackets. This is clearly in contradiction with reality and suggests that thestandard formulation of QED and perturbative quantum gravity which relies on theexistence of wavepackets is invalid.C.2.2 Taking the cutoff λ→ 0 vs. integrationOne might be concerned that the limit λ→ 0 and the integrals do not commute. Inthis part of the appendix, we will check the claim made in the preceding subsection,i.e., we will show that one can explicitly check that the integration and taking the IRregulator λ to zero commute. We assume in the following that we talk about QEDwith electrons and muons in the non-relativistic limit, which again is good enoughas it is sufficient to show that we can find a limit in which no sign of scatteringexists in the outgoing hard state. The wave packets are chosen to factorize for everyparticle and to be Gaussians in velocity centered around v = 0,f (v) =(2piκ)3/4exp(−v2κ). (C.17)In order to stay in the non-relativistic limit, κ must be sufficiently small. They arenormalized such that ∫d3v | f (v)|2 = 1. (C.18)In the exponent of λ we set α′ = β′ for simplicity, i.e., we consider the case offorward scattering. In the non-relativistic limit, we can expand the exponent of λ162into∆Aα,β =e224pi2∑n,m∈α,β(vα − vβ)2. (C.19)Thus, λ∆A has the formλ∆A ∝ exp(−12γ∑n,m∈α,β(vα − vβ)2), (C.20)where taking the cutoff λ to zero corresponds to γ ∝ − log(λ) → ∞. The state αconsists of a muon with well defined momentum and one electron with momentummv, where v is centered around 0. The state β consists of the same muon (weassume it was not really deflected) and one electron with momentum mv′. Toobtain the contribution to forward scattering, we have to perform the integral∝∫d3v(2piκ)3/4exp(−v2κ)exp(−γ(v − v′)2)· (other terms). (C.21)Here, we assumed that the other terms which include the matrix element in theregime of interest is finite and approximately independent of v. The integral yields(2piκ(1 + γκ)2)3/4exp(− γv′21 + γκ). (C.22)Taking the limit γ → ∞, it is clear that this expression vanishes. If we want toconsider an outgoing wave packet we have to integrate this over f (v′ − vout). Theresult is proportional to (2piκ(1 + 2γκ)2)3/4exp(− γv2out1 + 2γκ)(C.23)and still vanishes if we remove the cutoff, γ →∞.163Appendix DCone Relative EntropiesD.1 Equivalence of Hξ on the boundary and the modularHamiltonianIn this appendix we will show that Hξ reduces to the modular Hamiltonian onthe boundary, even in the case of a deformed entangling surface. We take theinfinitesimal difference between pure AdS and another spacetime that satisfies thelinearized Einstein’s equations around pureAdS, i.e., wewant to calculate δQξ−ξ ·θon a constant z slice near the boundary. We can find in the appendix of [39] thatδQξ−ξ ·θ = 116piGN ˆab[δgac∇cξb − 12δgcc∇aξb + ξc∇bδgac − ξb∇cδgca + ξb∇aδgcc].(D.1)The next step is to expand the sum over a and b. As we approach the boundary weconsider volume elements on on constant z slices and thus the term involving thevolume element ˆµν vanishes. In Fefferman-Graham gauge (δgzc = 0) we findδQξ − ξ · θ = 116piGN ˆµz[12δgνν∇zξµ − ξµ∇zδgνν − ξc∇µδgzc + ξµ∇cδgcz]+116piGNˆµz[δgµν∇νξz − 12δgνν∇µξz + ξν∇zδgµν − ξz∇νδgνµ + ξz∇µδgνν].(D.2)164Now all we need to do is find the leading order behaviour near z = 0. To this effectwe assume that the vector fields have a asymptotic expansion near the conformalboundary given in equation (9.29).We also take δgab = zd−2Γ(d)ab + zd−1Γ(d+1)ab+ .... The leading order terms ofequation (D.2) ared16piGNηµλˆµzΓ(d)λν ξνzd+1 + ... = O(1), (D.3)where we use the fact that for a CFT traceless stress-energy tensor implies thatηνρg(d)νρ = 0 and ˆµz = O(z−(d+1)). Finally, employing the relation between themetric perturbation in FG coordinates and the stress-energy tensor,∆〈Tµν〉 = d16piGN Γ(d)µνz=0(D.4)and the definition of given in section 9.2 we arrive atδQξ − ξ · θ = ρ〈Tρσ〉ξσ + O(z). (D.5)D.2 The HRRT surface ending on the null-planeIn order to derive the HRRT surface which ends on a curve located on a boundarynull-plane, we split the coordinates into x± = t ± x (here x is the spatial directionparallel to the null-plane), boundary directions xi⊥ orthogonal to the null-plane, andthe bulk coordinate z. The metric on the Poincaré patch in these coordinates isds2 =1z2(dz2 − 2dx+dx− + dxi⊥dx⊥i). (D.6)We choose static gauge for the coordinates on our extremal surface, such that x± =x±(z, xi⊥). The entangling surface on the boundary is then given by x± = x±(0, xi⊥).The equations which determine the embeddings x±(z, xi⊥) are given byγab∂γab∂x±= − 1√γ∂a(2√γγabg+−∂bx∓), (D.7)165where the induced metric is denoted by γab. Having the extremal surface endingon a boundary null-plane means that either x+ or x− are constant. Without loss ofgenerality, we choose x− = x−0 = const. This reduces the two equations (D.7) to asingle equation for x+(z, xi⊥). Making the ansatz x+(z, xi⊥) = hk(z)gk(xi⊥) we canseparate the equation intozd−1∂z(z1−d∂zhk(z)) = −∆⊥gk(xi⊥). (D.8)The general solutions for the functions hk(z) and gk(xi⊥) are given bygk(x⊥) = ak i eiki xi⊥, (D.9)hk(z) = ck zd/2Id/2(zk) + dk zd/2Kd/2(zk), (D.10)where k = |k i | and xi⊥k i denotes the Euclidean inner product between the vectorsk i and xi. Iν and Kν denote the modified Bessel functions of first and second kind,respectively. We also define h0 = limz→0 hk(z). It turns out that we do not wantthe full solution for hk . Intuitively, it is clear that the effect of deformations of theentangling surface on the boundary should die off as z → ∞. At the same timewe also require that the shape of the extremal surface is uniquely determined byboundary conditions. The asymptotic behavior of hk as z →∞ and z → 0 islimz→∞ hk(z) = ck√12pikekz + dk√pi2ke−kz, (D.11)limz→0hk(z) = dk2 d−22 Γ(d2)k−d/2. (D.12)We can only fulfill above requirements if we set ck = 0. Hence any extremal surfaceending on the null-plane x− = x−0 is given byx+(z, xi⊥) =2 2−d2 kd/2Γ(d/2)∫dd−2ka®k zd/2Kd/2(zk)eik i xi . (D.13)The normalization is chosen such thatlimz→0x+(z, xi⊥) =∫dd−2kak i eiki xi⊥ (D.14)166determines ak in terms of the entangling surface x+(0, x⊥).D.3 Calculation of the binormalThe binormal nµν is defined asnµν = nµ2 nν1 − nν2nµ1 , (D.15)where n1 and n2 are orthogonal ±1 normalized normal vectors to the extremalsurface. To calculate them start by calculating the d − 1 tangent vectors to thesurface which will be labeled by n as tn = tµn ∂µ, n ∈ {1,2, ..., d − 1}. They satisfytµn ∂µ(r+ − ρ+0 ) = 0 and tµn ∂µ(r− −Λ(θ, φi)) = 0. A convenient set of tangent vectorsis given byt1 =√gθθ ((∂θΛ)∂− + ∂θ) , (D.16)t2 =√gφ1φ1((∂φ1Λ)∂− + ∂φ1), (D.17)t3 =√gφ2φ2((∂φ2Λ)∂− + ∂φ2), (D.18)t4 = . . . (D.19)and so on for all φi. It is easy to see that these vectors form an orthonormal basis onthe Ryu-Takayanagi surface. Requiring that n1 and n2 are orthogonal to all tangentvectors, gµνnµ1,2tνa = 0. This requirement is fulfilled by choosingn+1,2 = g+−, na1,2 = −∂aΛ, (D.20)where a stands again for all angular components. The condition that n1 and n2be orthogonal and normalized to +1 and −1, respectively, is obeyed provided wechoosen−1 =12(1 − ∂aΛ∂aΛ) , n−2 = −12(1 + ∂aΛ∂aΛ) . (D.21)167One can check that the only non-zero components of the binormal are given by:n+− = g+−, na− = −∂aΛ. (D.22)D.4 Hollands-Wald gauge conditionIn this appendix, we argue that for the example of a planar black hole in AdS4,considered as a perturbation of pure AdS, we can choose a gauge where g(1)−a |Σ =0 = g(1)−− |Σ which at the same time is compatible with Hollands-Wald gauge. In thiscase, the final term in our second order expression (9.65) for the relative entropyvanishes.Hollands-Wald gauge is determined by requiring that the extremal surface in thedeformed spacetime sits at the same coordinate location than the extremal surfacein the undeformed spacetime. In particular this means thatr− = Λ(θ, φ), r+ = ρ+0 . (D.23)The requirement that also after a perturbation of the metric the extremal surface A˜sits at its old coordinate location translates into0 = ∂−(γab(0) γ(1)ab)− ∂c(√γ(0)γca(0)∂axµg(1)−µ)A˜, (D.24)0 = − 12√γ(0)γab(0) ∂ar−g(0)+−∂b(γcd(0)γ(1)cd) + ∂c(√γ(0)∂dr−g(0)+−γca(0)γ(1)abγbd(0) )− ∂b(√γ(0)γab(0) ∂ar−g(1)+−) − ∂b(√γ(0)γab(0) g(1)+a) +12√γ(0)∂+(γab(0) γ(1)ab)A˜.(D.25)As a warm-up consider a ball-shaped entangling surface with a correspondingextremal surface at r+ = ρ+0 ,r− = −ρ+0 placed in a planar black hole background,ds2 =1z2(−(1 − µzd)dt2 + dz2(1 − µzd) + dx2), (D.26)at leading order in µ. The equations for the extremal surface now become at first168order0 =12√γ(0)∂±(γab(0) γ(1)ab)− ∂a(√γ(0)γab(0) g(1)±b)A˜. (D.27)We can use the symmetry of the perturbation under time translations and regularityat the boundary to find a vector field v that generates a diffeomorphism g → Lvgwhich locates the extremal surface in the perturbed geometry at the same coordinatelocation as the extremal surface in the unperturbed geometry.v+ = − µ64 sin θ(1 + sin2 θ)(r+ − r−)2, (D.28)v− =µ64sin θ(1 + sin2 θ)(r+ − r−)2, (D.29)vθ =µ64(r+ − r−)3 cos3 θ, (D.30)vφ = 0. (D.31)This diffeomorphism brings the metric perturbation into the formδds2 =µ8(r+ − r−)1 + sin2 θsin θdy+dy− +µ32(r+ − r−)3 cos θ cot θdθ2− µ32(r+ − r−)3 cos3 θ cot θdφ2.(D.32)The only non-vanishing components of the metric in the new coordinates areg+−,gθθ and gφφ. In particular, we have that g(1)−a = 0 = g(1)−−. The main benefit ofthese coordinates is that equation (D.27) holds automatically. Hence at least fora ball-shaped entangling surface we are in Hollands-Wald gauge and the extremalsurface is located at r± = ±ρ+0 . It can be seen from the metric that lines of constantr± are lightlike and therefore we know that the new entangling surface still is onthe bulk lightcone of a point p at the boundary.From this we can conclude that the entanglement wedge associated to any regionbounded by a lightcone does not contain any point outside the causal wedge. Aswe have seen this is true for ball-shaped regions. A deformation of the entanglingsurface cannot change this, since the boundary domain of dependence is smallerthan that of some ball-shaped region. At the same time, the extremal surface cannotlie within the causal domain of dependence and therefore we must conclude that169the extremal surface also lies on the lightcone.This means that the transformations (D.28) – (D.31) bring the HRRT surfaceto its correction r+ location. The only additional adjustment we need to make tothe coordinate system is to reparameterize r− around the extremal surface, e.g. byrescaling the r− coordinate in an angle-dependent way.To find a solution to the general Hollands-Wald gauge condition, equation(D.25), we alter the plus-component of the vector field, v+ → v+ + v˜+(θ, φ), aroundthe extremal surface such that it shifts the extremal surface into its new correctlocation on the lightcone. This vector field can be chosen such that at the extremalsurface A˜ it remains constant along r− and r+ and thus depends only on θ and φ.It should be clear that such a solution exists, since at the boundary the correctionv˜+(θ, φ) vanishes and is smooth everywhere else. More formally, in this caseequation (D.25) reduces toµ16∂θ(cot θ∂θ(ρ+0 − Λ(θ, φ))2) +µ16tan θ∂2φ(R − Λ(θ, φ))2A˜= ∂θ(cos θ(∂θ v˜+(θ, φ) + 2 cot θv˜+(θ, φ))) + 1cos θ ∂2φ v˜+(θ, φ)A˜.(D.33)For small deformations of the ball shaped entangling surface we can write Λ(θ, φ)as a series expansion in the deformations. At first order, this gives us a linearPDE which can be solved. Higher orders become inherently non-linear and thusthis equation is in general very hard to solve. An interesting observation one canmake for small n = 1 deformations of the entangling surface is that the linear ordercorrection is zero.170
Thesis/Dissertation
2019-09
10.14288/1.0380253
eng
Physics
Vancouver : University of British Columbia Library
University of British Columbia
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Graduate
Aspects of quantum information in quantum field theory and quantum gravity
Text
http://hdl.handle.net/2429/71167