Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Chaurette, Laurent
2018-08-09T19:23:20Z
2018
Doctor of Philosophy - PhD
University of British Columbia
Scattering amplitudes in massless gauge field theories have long been known to give rise to infrared divergent effects from the emission of very low energy gauge bosons. The traditional way of dealing with those divergences has been to abandon the idea of measuring amplitudes by only focusing on inclusive cross-sections constructed out of physically equivalent states. An alternative option, found to be consistent with the S-matrix framework, suggested to dress asymptotic states of charged particles by shockwaves of low energy bosons. In this formalism, the clouds of soft bosons, when tuned appropriately, cancel the usual infrared divergences occurring in the standard approach. Recently, the dressing approach has received renewed attention for its connection with newly discovered asymptotic symmetries of massless gauge theories and its potential role in the black hole information paradox.
We start by investigating quantum information properties of scattering theory while having only access to a subset of the outgoing state. We give an exact formula for the von Neuman entanglement entropy of an apparatus particle scattered off a set of system particles and show how to obtain late-time expectation values of apparatus observables.
We then specify to the case of quantum electrodynamics (QED) and gravity where the unobserved system particles are low energy photons and gravitons. Using the standard inclusive cross-section formalism, we demonstrate that those soft bosons decohere nearly all momentum superpositions of hard particles. Repeating a similar computation using the dressing formalism, we obtain an analogous result: In either framework, outgoing hard momentum states at late times are fully decohered from not having access to the soft bosons.
Finally, we make the connection between our results and the framework of asymptotic symmetries of QED and gravity. We give new evidence for the use of the dressed formalism by exhibiting an inconsistency in the scattering of wavepackets in the original inclusive cross-section framework.
https://circle.library.ubc.ca/rest/handle/2429/66714?expand=metadata
Infrared Quantum InformationbyLaurent ChauretteB. Sc., Universite´ de Montre´al, 2012M. Sc., University of British Columbia, 2014a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Physics)The University of British Columbia(Vancouver)August 2018c© Laurent Chaurette, 2018The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the dis-sertation entitled:Infrared Quantum Informationsubmitted by Laurent Chaurette in partial fulfillment of the require-ments forthe degree of Doctor of Philosophyin The Faculty of Graduate and Postdoctoral Studies (Phyics)Examining Committee:Gordon SemenoffSupervisorMoshe RozaliSupervisory Committee MemberFei ZhouSupervisory Committee MemberAriel ZhitnitskySupervisory Committee MemberiiAbstractScattering amplitudes in massless gauge field theories have long been knownto give rise to infrared divergent effects from the emission of very low energygauge bosons. The traditional way of dealing with those divergences hasbeen to abandon the idea of measuring amplitudes by only focusing oninclusive cross-sections constructed out of physically equivalent states. Analternative option, found to be consistent with the S-matrix framework,suggested to dress asymptotic states of charged particles by shockwavesof low energy bosons. In this formalism, the clouds of soft bosons, whentuned appropriately, cancel the usual infrared divergences occurring in thestandard approach. Recently, the dressing approach has received renewedattention for its connection with newly discovered asymptotic symmetries ofmassless gauge theories and its potential role in the black hole informationparadox.We start by investigating quantum information properties of scatteringtheory while having only access to a subset of the outgoing state. We givean exact formula for the von Neuman entanglement entropy of an apparatusparticle scattered off a set of system particles and show how to obtain late-time expectation values of apparatus observables.We then specify to the case of quantum electrodynamics (QED) andgravity where the unobserved system particles are low energy photons andgravitons. Using the standard inclusive cross-section formalism, we demon-strate that those soft bosons decohere nearly all momentum superpositionsof hard particles. Repeating a similar computation using the dressing for-malism, we obtain an analogous result: In either framework, outgoing hardmomentum states at late times are fully decohered from not having accessto the soft bosons.Finally, we make the connection between our results and the frameworkof asymptotic symmetries of QED and gravity. We give new evidence for theuse of the dressed formalism by exhibiting an inconsistency in the scatteringof wavepackets in the original inclusive cross-section framework.iiiLay SummaryField theories like quantum electrodynamics and perturbative gravity havelong been known to have issues arising from the emission of long wavelengthphotons and gravitons. The standard approach to curing those problemshas been to accept that such particles can not be observed by finite sizeddetectors and trace them out of any computation. However, a more recentproposal suggests that using states of charged matter dressed by incomingradiation in a very specific way can also cure the infrared problems of thetheory.In this thesis, we investigate quantum information properties of the longwavelength radiation after scattering. We evaluate relevant quantities suchas the entanglement entropy of the radiation and demonstrate that both ap-proaches predict complete decoherence of the charged particles at late times.We then demonstrate that the dressed formalism is the correct frameworkto perform scattering.ivPrefaceA version of chapter 2 has been uploaded to arxiv.org. Dan Carney, Lau-rent Chaurette & Gordon Semenoff, Scattering with partial information,arXiv:1606.0310. My main contributions were related to establishing thesetup in terms of density matrices and calculations of entanglement entropy.A version of chapter 3 has been published. Dan Carney, Laurent Chau-rette, Dominik Neuenfeld & Gordon Semenoff, Infrared quantum informa-tion, Phys.Rev.Lett. 119 (2017) no.18, 180502. My role was primarily re-lated to calculations of the decoherence exponent and the proof of its posi-tivity.A version of chapter 4 has been published. Dan Carney, Laurent Chau-rette, Dominik Neuenfeld & Gordon Semenoff, Dressed infrared quantuminformation, Phys.Rev. D97 (2018) no.2, 025007. My contribution wasmostly related to calculations on the damping factor D and its connectionto decoherenceA version of chapter 5 was submitted for publication and is currently un-dergoing peer-review. Dan Carney, Laurent Chaurette, Dominik Neuenfeld& Gordon, Semenoff, On the need for soft dressing, arXiv:1803.02370. Mycontributions were mostly related to decoherence calculations for entangledsuperpositions and wavepackets. I worked on the evaluation of the decoher-ence exponents and the proof of their positivity as well as the relation of thedecoherence in terms of conserved charges.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Infrared Catastrophe . . . . . . . . . . . . . . . . . . . . . . . 21.2 Inclusive Formalism . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Virtual Divergences . . . . . . . . . . . . . . . . . . . 61.2.2 Cancellation of Divergences . . . . . . . . . . . . . . . 81.3 Dressed Formalism . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 Second Order Cancellation of Divergences . . . . . . . 121.3.2 Dressed States as Eigenstates of the Asymptotic Hamil-tonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Asymptotic Symmetries . . . . . . . . . . . . . . . . . . . . . 171.4.1 Matching Conditions . . . . . . . . . . . . . . . . . . . 181.4.2 Conserved Charges . . . . . . . . . . . . . . . . . . . . 201.4.3 Vanishing of the S-matrix and Vacuum Transitions . . 231.5 Soft Hair and the Black Hole Information Paradox . . . . . . 251.5.1 Black Hole Information Paradox . . . . . . . . . . . . 251.5.2 Black Hole Soft Hair . . . . . . . . . . . . . . . . . . . 262 Scattering with Partial Information . . . . . . . . . . . . . . 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Scattering with Density Matrices . . . . . . . . . . . . . . . . 282.2.1 General Considerations . . . . . . . . . . . . . . . . . 282.2.2 Measuring the Apparatus State . . . . . . . . . . . . . 312.3 A-S Entanglement Entropy . . . . . . . . . . . . . . . . . . . 322.4 Examples with Two Scalar Fields . . . . . . . . . . . . . . . . 352.4.1 Entropy from 2→ 2 Scattering . . . . . . . . . . . . . 372.4.2 Verifying Spatial Superpositions . . . . . . . . . . . . 382.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43vi3 Infrared Quantum Information . . . . . . . . . . . . . . . . . 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Decoherence of the Hard Particles. . . . . . . . . . . . . . . . 463.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Entropy of the Soft Bosons . . . . . . . . . . . . . . . . . . . 513.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Dressed Infrared Quantum Information . . . . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 IR-safe S-matrix Formalism . . . . . . . . . . . . . . . . . . . 544.3 Soft Radiation and Decoherence . . . . . . . . . . . . . . . . . 564.4 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . 604.5 Black Hole Information . . . . . . . . . . . . . . . . . . . . . 604.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 On the Need for Soft Dressing . . . . . . . . . . . . . . . . . 625.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Scattering of Discrete Superpositions . . . . . . . . . . . . . . 655.2.1 Inclusive Formalism . . . . . . . . . . . . . . . . . . . 665.2.2 Dressed Formalism . . . . . . . . . . . . . . . . . . . . 685.3 Wavepackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3.1 Inclusive Formalism . . . . . . . . . . . . . . . . . . . 715.3.2 Dressed Wavepackets . . . . . . . . . . . . . . . . . . . 725.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4.1 Physical Interpretation . . . . . . . . . . . . . . . . . . 725.4.2 Allowed Dressings . . . . . . . . . . . . . . . . . . . . 735.4.3 Selection Sectors . . . . . . . . . . . . . . . . . . . . . 765.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.1 Scattering with Partial Information . . . . . . . . . . . . . . . 806.2 Decoherence from Infrared Photons and Gravitons . . . . . . 816.3 Wavepacket Scattering and the Need for Soft Dressing . . . . 826.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 83Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84A Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 89B Positivity of A, B Exponents . . . . . . . . . . . . . . . . . . 91viiC Dressed Soft Factorization . . . . . . . . . . . . . . . . . . . 94D Proof of Positivity of ∆A,∆B . . . . . . . . . . . . . . . . . . 95E The out-Density Matrix of Wavepacket Scattering . . . . . 97E.0.1 Contributions to the out-Density Matrix . . . . . . . . 97E.0.2 Taking the Cutoff λ→ 0 vs. Integration . . . . . . . . 100viiiList of FiguresFigure 1.1 The two diagrams contributing to first order bremsstrahlungemission for 1 → 1 potential scattering. The soft photoncan either be emitted a) before or b) after scattering offthe potential . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 The three diverging loop diagrams at order e2 for 1 → 1potential scattering. a) is the correction to the vertex dia-gram while b) and c) correspond to mass renormalizationfor the incoming and outgoing legs. . . . . . . . . . . . . . 7Figure 1.3 Three new diagrams for Bremsstrahlung up to first orderin e. The incoming photon can either a) not interact orb), c) by absorbed by either the incoming or outgoingelectron leg . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 1.4 In the dressing formalism, divergences from standard loopdiagrams are canceled by similar diagrams involving ex-changes between the soft clouds . . . . . . . . . . . . . . . 14Figure 1.5 Minkowski space Penrose diagram. I± represent past andfuture null infinity with their S2 boundaries identified byI±± . Each point (r, t) identifies two points on the diagram,one on the right and one on the left, which are related bythe antipodal mapping. The curved line represents themotion of massive particles while lightrays move alongstraight lines of constant u or v. . . . . . . . . . . . . . . 19Figure 2.1 A typical apparatus-system scattering process. Dottedlines denote the apparatus, solid lines the system. Timeruns from bottom to top. . . . . . . . . . . . . . . . . . . 33Figure 2.2 Verifying spatial superpositions of the system states |L〉,|R〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.3 Diagrams contributing to the lowest-order position-spacedistribution of the apparatus. . . . . . . . . . . . . . . . . 41ixFigure 5.1 (a) A plane wave goes through a single slit and emergesas a localized wavepacket. The scattering of the incomingwavepacket results in the production of Bremsstrahlung.(b) We can also define some Cauchy slice Σ and createthe state by an appropriate initial condition. (c) Evolv-ing this state backwards in time while forgetting aboutthe slit results in an incoming localized particle which isaccompanied by a radiation shockwave. . . . . . . . . . . 77xChapter 1IntroductionIt has long been known that field theories with massless gauge bosons areplagued with infrared divergences which effectively force the transition am-plitudes between any two states to be exactly zero. Historically, the methodof choice for dealing with such divergences was introduced by Bloch andNordsieck for quantum electrodynamics (QED) [1] and Weinberg for grav-ity [2] and has been to evaluate inclusive cross-sections between every phys-ically indistinguishable state. While this approach agrees with experimentson the transition probabilities between various states, it has the shortcomingof abandoning the S-matrix description of the theory as amplitudes are allzero. A second framework consistent with an S-matrix picture was proposedby Chung and Faddeev-Kulish [3, 4], suggesting that asymptotic states ofcharged particles could be dressed by a cloud of soft radiation. When thecloud is chosen to be a specific coherent state of soft photons and gravitons,the S-matrix elements between such states becomes non-singular. The twoapproaches were then mostly considered to be equivalent: choosing to evalu-ate probabilities could be done in either framework and would simply cometo a matter of choice.Seemingly unrelated recent findings [5–7] have demonstrated the exis-tence of an infinite number of broken symmetries in QED and gravity leadingto an infinitely degenerate vacuum for the theory. While scattering wouldallow transitioning from one vacuum to an other, the vanishing of the S-matrix can be seen as a statement about the conservation of the charges ofthose broken symmetries. A recent paper [8] showed there is in fact a strongconnection between the conservation of these charges and Faddeev-Kulishstates, leading to believe that dressed states could be the actual states foundin nature.Additionally, a new proposition [9] suggested dressed states could poten-tially hold the key to the resolution of the Black hole information paradox:If a black hole was formed from the collision of high energy dressed states,the long wavelength radiation contained in the dressing would not fall in theblack hole and could perhaps hold enough information to help distinguishbetween different states after the black hole evaporated.In this dissertation, we investigate quantum information properties of1the long wavelength radiation emitted from scattering using both the in-clusive cross section and dressed approaches. We investigate the late timedecoherence effects found in each formalism when radiation is left unob-served and argue that scattering requires the use of Faddeev-Kulish states.First, we wish to review some of the previous literature on the key conceptswe will be using in the thesis. In section 1.1, we review how the infraredcatastrophe comes about for QED and PG. Sections 1.2 and 1.3 give anoverview of the inclusive cross-section and dressing approaches respectively.Finally, section 1.4 reviews the program of broken asymptotic symmetriesof QED and gravity and how charge conservation is linked to the vanishingof the S-matrix.1.1 Infrared CatastropheWhen dealing with massless gauge theories like QED, one finds that theprobability of charged particles to emit photons diverges as the energy ofthe gauge bosons go to zero. Every scattering event is then dominated byoutgoing states which contain an infinite amount of soft photons renderingthe probability to emit only a finite number of photons to be zero. This isthe infrared catastrophe. In this section, we review how soft gauge bosonemission implies divergent amplitudes between any two Fock space states.We will start by reviewing the amplitudes of Feynman diagrams con-taining the emission of bremsstrahlung. Let us consider a scattering processα→ β where one of the outgoing legs emits a photon of momentum k. Thisadds a propagator to the diagram which has the effect of multiplying theamplitude by a factor of[ie−i (2pµ + kµ)(p+ k)2 +m2 − i](1.1)[ieγµ−i(/p+ /k) +m(p+ k)2 +m2 − i],for spin 0 and spin 1/2 particles respectively. Here, e, p and m are thecharge, momentum and mass of the particle emitting the photon. In thecase when the photon momentum k is nearly zero, both expressions havethe same behaviourepµp · k − i . (1.2)To obtain this result we used properties of gamma matrices, the fact thatp is on-shell (p2 = −m2) while keeping only the terms at lowest order in2qpipfk(a)qpipf k(b)Figure 1.1: The two diagrams contributing to first order bremsstrahlungemission for 1 → 1 potential scattering. The soft photon can either beemitted a) before or b) after scattering off the potentialthe photon momentum k. This expression is indeed universal in the sensethat it does not depend on the spin of the emitting particle. If instead ofan outgoing line, it was an incoming line that emitted the soft photon, thedenominator in equation (1.1) would instead behave as (p+ k)2 → (p− k)2.After taking the k → 0 limit, this effectively changes the sign of the pole toepµ−p · k − i . (1.3)Accounting for both possible first order emissions (fig 1.1), the amplitudegets modified byM(1)βα (k, l) = M(0)βα1√(2pi)32|k|∑n∈α,βηnenpn · ∗l (k)pn · k − iηn , (1.4)where the index n runs over all particles in α and β and ηn is defined as1(-1) if particle n is incoming(outgoing). The factor of 1√(2pi)32|k| came fromthe normalization of the emitted photon wavefunction while l(k) denotesits polarization. For gravity, the situation is completely analogous: the polestake the formM−1p ηn pµnpνnpn · k − iηn , (1.5)3where Mp is the Plank mass. The amplitudes are then modified accordinglyM(1)βα (k, l) = M(0)βα1√(2pi)32|k|∑n∈α,βM−1p ηnpµnpνn∗µν,l(k)pn · k − iηn , (1.6)following the same infrared behavior as for QED. We will therefore solelyfocus on photon emission from now on and simply give the results for gravityat the end.The probability to emit any low energy photon is given by the square ofthe amplitude summed over all possible outgoing photonsP(1)βα =∫d3k2∑l=1|Mµβα(k, l)|2 (1.7)= P (0)∫d3k(2pi)32k2∑l=1∑n,m∈α,βenemηnηmpµnpνm∗µ,l(k)ν,l(k)[pn · k − iηn] [pm · k − iηm]= P (0)I(α, β)∝ P (0)∫ Λλdkk2k3,which has a logarithmic divergence for small k. The probability of emittinga soft photon therefore seems to be infinitely larger than having no emission.We could then add a second soft photon and notice that the amplitude getsmultiplied by the square of the soft factor found in equation (1.4), that isevery emission is independent from each other and the amplitude of emittingN soft photons and M soft gravitons is simplyM(N)βα (k1, ..., kN , k′1, ..., k′M )→M (0)βαFαβ(k1, ..., kN )Gαβ(k′1, ..., k′M ) (1.8)with the functions F and G being the contributions from soft photons andsoft gravitons respectivelyFβα(k1, ..., kN ) =N∏j=12∑lj=1∑n∈α,βηnen√(2pi)3|kj |pµn∗µ,lj (kj)pn · kj − iηn (1.9)Gβα(k′1, ..., k′M ) =M∏j=12∑lj=1∑n∈α,βM−1p ηn√(2pi)3|k′j |pµnpνn∗µν,lj(k′j)pn · k′j − iηn. (1.10)4The probabilities are then found to follow a Poisson distributionP(N)βα = P(0)βαI(α, β)NN !, (1.11)where the factor of N ! in the denominator comes from all possible permu-tations of emission of N bosons. The expectation value of the number ofemitted soft bosons is infiniteN¯ =∞∑N=0NP(N)βα = I(α, β)→∞. (1.12)This shows that any scattering between charged particles in QED and grav-ity generate an infinite amount of soft radiation in average. It gets evenworst when we look at the normalization arising from1 =∑NP(N)βα (1.13)= P(0)βα∑NI(α, β)NN != P(0)βα eI(α,β),implying the probability that scattering process α→ β will emit no photonis P(0)βα = e−I(α,β) = 0. For any finite number of emitted photons N , theprobability remains exactly zero because of the negative divergence in theexponential.P(N)βα = e−I(α,β) I(α, β)NN != 0. (1.14)Therefore, every scattering between charged particles always emits an infi-nite number of low energy bosons and the transition probabilities betweenany two states in Fock space is zero. These dramatic results have beenknown as the infrared catastrophe and can actually be resolved. Two meth-ods of getting rid of those divergences have been found through the yearsand we will review each of them in the next sections.1.2 Inclusive FormalismIntroduced by Bloch-Nordsiek [1] for QED and extended to gravity by Wein-berg [2], the most widely used method of dealing with the divergences arisingfrom soft photon emission is to calculate inclusive cross-sections. That is,5to account for the emission of soft photons but to trace them out at theend of the computation as these states are physically indistinguishable forany finite sized detector. In this context, the divergences coming from softemissions are exactly canceled by divergences coming from loop diagrams.However, the cancellation of divergences does not work at the level of theamplitudes but only for probabilities. The inclusive cross-section paradigmtherefore abandons the idea of a well-defined S-matrix for solely calculatingtransition rates between processes. Losing the S-matrix description of thetheory may sound disturbing, but the inclusive formalism has enjoyed widesuccess as it is in strong agreement with every experiment ever performed.1.2.1 Virtual DivergencesThere is a second type of infrared divergence occurring in the computationof Feynman diagrams for QED and gravity. Indeed, adding loop correctionsto diagrams also creates logarithmic divergences when the momentum in theloop approaches zero.Considering some arbitrary process α → β, let us add photon loopsbetween external legs and calculate the correction to the amplitude. Foreach loop added, the photon propagator provides a factor of∫ Λλd4k(2pi)4−iηµνk2 − i . (1.15)Here, the upper and lower bounds of the integral need to be carefully defined.We must cut the integral up to momentum Λ which is taken to be the upperbound on what we define to be soft photons. However, the lower cutoff λ isakin to a photon mass and needs to be taken to zero.Adding a loop also adds a propagator to the diagram for each particleconnected by that loop. The total contribution to the diagram at lowestorder in |k| is−i pn · pm(2pi)4enemηnηm∫ Λλd4k[k2 − i] [pn · k − iηn] [−pm · k − iηm] , (1.16)if the loop connects particles n and m. Let us define this quantity to beenemηnηmJnm. Considering the fact that the loop can connect any incomingand outgoing lines, all first order loop diagrams can be counted by summingthe result for n, m in α, β. Note that we did not need to include thecontribution from loops that are connected to internal lines as the propagator[(p± k)2 +m2 − i]−1 arising there would not be on-shell, leaving only finite6qpipfk(a)qpipfk(b)qpipfk(c)Figure 1.2: The three diverging loop diagrams at order e2 for 1→ 1 poten-tial scattering. a) is the correction to the vertex diagram while b) and c)correspond to mass renormalization for the incoming and outgoing legs.terms in |k|. The integral over d4k would not be singular, allowing us todrop those diagrams as we only interest ourselves in the divergent parts ofthe scattering.When adding multiple loops, each loop is independent from one anotherand the contributions simply multiply. For any number of internal loops N ,we findMN,λβα = MΛβα12NN ! ∑n,m∈α,βenemηnηmJnmN . (1.17)The upper indices λ(Λ) meaning that the amplitudes are computed solelywith loops of momentum above λ(Λ). Summing the contribution of virtualboson loops for any number of such loops exponentiatesMλβα = MΛβα exp12∑n,m∈α,βenemηnηmJnm . (1.18)The quantity Jnm can be evaluated by first performing the k0 integralby the method of residues. The resulting integral is simple and can be foundin [2]Jnm =2pi2βnmln[1 + βnm1− βnm]ln(Λλ)+ phase, (1.19)where βnm =(1− m2nm2m(pn·pm)2)1/2is the relative velocity between particles nand m, satisfying 0 ≤ βnm ≤ 1. Jnm also has a divergent imaginary part but7we will not bother writing it explicitly as it will drop out of the calculationsonce we square the amplitudes to get probabilities. Having an expressionfor Jnm we can finally compute the contribution of soft virtual loops to theamplitudesMλβα = MΛβα(λΛ)Aβα/2, (1.20)where the exponent Aβα is defined asAβα = − 18pi2∑n,m∈α,βenemηnηmβnmln[1 + βnm1− βnm]. (1.21)Once again, the story is the same for gravity where a similar integral overd4k needs to be performed. We then find that the amplitudes get a factorof(λΛ)Bβα/2 with the gravity exponent B defined asBβα =116pi2M2p∑n,m∈α,βmnmmηnηm1 + β2nmβnm√1− β2nmln[1 + βnm1− βnm]. (1.22)It is important to note that the exponents A and B are positive numbers forany scattering states α,β. Therefore, virtual boson loops always contributeto make the amplitudes vanish as we take the limit λ→ 0.Probabilities are computed by squaring the amplitudes yieldingΓλβα = ΓΛβα(λΛ)Aβα+Bβα. (1.23)1.2.2 Cancellation of DivergencesWe have now encountered two types of divergences occurring during scat-tering for massless gauge field theories: Bremsstrahlung of soft photons andvirtual loop diagrams. Let us now review how the two cancel each other atthe level of probabilities.Coming back to our expression for the soft factor Fβα, this time we willbe more careful and investigate the differential rate for the emission of Nsoft photons. That is the amplitude squared taken only over an element of8volume where all N photons are softdΓλβα(k1, ...,kN ) = ΓλβαN∏j=1d3kj(2pi)32|kj | (1.24)2∑l=1∑n,m∈α,βenemηnηmpµnpνm∗µ,l(kj)ν,l(kj)[pn · kj − iηn] [pm · kj − iηm] .The sum over polarizations can be performed and simply gives a factor of themetric ηµν . The integral over angles then happens to be exactly the same aswe encountered in equation (1.16) which allows us to write the differentialrate only in terms of frequencies emitteddΓλβα(ω1, ..., ωN ) = ΓλβαANβαdω1ω1...dωNωN, (1.25)where we recognize Aβα as defined in section (1.2.1) arising from the angularintegral. A nuance now comes from the integral over frequencies. Naively,we would expect to integrate each frequency from the lower bound λ, whichwill eventually need to be taken to zero, all the way to Λ which was ourdefinition of soft. However, the problem with this is that as the numberof emitted photons goes to infinity, so would the energy carried by thosephotons. Instead, we need to make sure there is only a finite amount ofenergy carried away by soft photons. Let us denote that energy ET and themaximum energy of each individual photon as E. These energies may seemsomewhat arbitrary but would in fact correspond to the energy resolutionof the experimenter’s detector.With this in mind, the total probability of emitting any number of softphotons can be written asΓλβα(E,ET ) = Γλβα∞∑N=1ANβαN !∫ωj<E∑j ωj<ETN∏j=1dωjωj. (1.26)Now we will spare the details of the calculation of this integral which canbe found in [2] to simply give the resultΓλβα(E,ET ) = Γλβα(Eλ)AβαF(Aβα, E,ET ), (1.27)with F being a complicated but smooth function that depends on the scat-9tering data and the detection cutoff. We finally have everything in hand tosee the cancellation of divergences. Virtual photons provide a factor(λΛ)Awhile emission of real soft photons yield(Eλ)A, the final probability beingΓβα(E,ET ) = ΓΛβα(EΛ)AβαF(Aβα, E,ET ). (1.28)As before, the calculation for gravity goes in the same way where the expo-nent Aβα gets replaced by Bβα defined above.In the inclusive cross-section formalism, we had to accept that the scat-tering amplitudes are ill defined with divergences coming from both real softphoton emission and virtual loops. However, the dependence on the photonregulator λ drops out of the calculation when evaluating probabilities. Themethod claims that while the QED S-matrix is zero everywhere, we can stillrecover the correct values for scattering probabilities which correspond towhat is observed in experiments.1.3 Dressed FormalismWhile the inclusive cross-section approach to dealing with infrared diver-gences has enjoyed wide success for its accurate agreement with experi-ments, the idea of abandoning the S-matrix description of QED and gravityis unsettling. A second program for curing the infrared behavior of QEDwas initiated by the works of Chung [3] and Faddeev-Kulish [4] where itwas shown that one can obtain a finite S-matrix between states dressed bylong-wavelength photons. In this section, we will review what it means forcharged particle states to be dressed and how the cancellation of divergencescomes about at the level of the S-matrix.Following the work of Chung, we will focus on scattering of a singleelectron off a potential. The generalization of the following procedure isexplained in details in [3] but the 1 → 1 case is sufficient to get the properunderstanding of this approach. Instead of working with standard free-fieldFock space electron states |p〉, each electron state gets promoted to a dressedstate, denoted ‖p〉〉. To define the dressing that accompanies the electron,let us define the soft factorFl(k,p) =p · el(k)p · k φ(k,p), (1.29)which is the bremsstrahlung emission pole found above multiplied by an10arbitrary function φ. This function captures the non-divergent behaviorof the emission and we do not care about what φ is except that it mustsmoothly approach 1 as |k| goes to zero. Then for every soft momentumλ < |k| < E, we construct a coherent state of transversely polarized photonsfrom the soft factorRp = e2∑l=1∫λ<|k|<Ed3k√2k[Fl(k,p)a†l (k)− F ∗l (k,p)al(k)], (1.30)with the dressing operator being defined asWp = exp(Rp). (1.31)The electron states we consider are simply given by this dressing operatoracting on free-field electron states‖p〉〉 = Wp |p〉 . (1.32)These states correspond to standard free field electrons accompanied bya shockwave of low energy photons. λ is once again the low energy cutofffor each photon and we will be interested in the λ→ 0 limit.We can simplify this expression of dressed states by using the Baker-Campbell-Hausdorff formula‖p〉〉 = exp{e2∑l=1∫λ<|k|<Ed3k√2k[Fl(k,p)a†l (k)− F ∗l (k,p)al(k)]}|p〉(1.33)= Np exp{e2∑l=1∫λ<|k|<Ed3k√2kFl(k,p)a†l (k)}× exp{−e2∑l=1∫λ<|k|<Ed3k√2kF ∗l (k,p)al(k)}|p〉= Np exp{e2∑l=1∫λ<|k|<Ed3k√2kFl(k,p)a†l (k)}|p〉 ,where Np is the normalization of the coherent state given byNp = exp{−e222∑l=1∫λ<|k|<Ed3k2k|Fl(k,p)|2}. (1.34)111.3.1 Second Order Cancellation of DivergencesThe dressing defined above was chosen precisely such that the standard S-matrix defined between such states shows no sign of infrared divergence. Wewill now review the argument leading to the cancellation to second order inthe electron charge e. For small values of the electric charge, we expand thedressed state into a first part containing only an electron and a second onecontaining an electron plus a single soft photon‖p〉〉 '(1− e222∑l=1∫d3k√2k|Fl(k,p)|2)(1 + e2∑l=1∫d3k2kFl(k,p)a†l (k))|p〉 .(1.35)Applying the S-matrix on such a state provides multiple types of interac-tions. First of all, we need to account for the vertex diagrams described infigure (1.2). As discussed before, these diagrams provide a divergence ofMp,p′∑n,m∈p,p′e2ηnηmJnm, (1.36)where Mp,p′ is the tree level vertex diagram between incoming electron pand outgoing electron p′. Note that up to order e2, the loop diagrams canonly be acting on the part of the dressed states that contain no photon.Their contribution therefore ends up being〈〈p′‖Sloop‖p〉〉 = Mp,p′(1− e222∑l=1∫d3k√2k|Fl(k,p′)|2)1 + ∑n,m∈p,p′e2ηnηmJnm×(1− e222∑l=1∫d3k√2k|Fl(k,p)|2)(1.37)= Mp,p′1 + e2 ∑n,m∈p,p′ηnηmJnm−e222∑l=1∫d3k√2k|Fl(k,p)|2 + |Fl(k,p′)|2).Turning to the bremsstrahlung diagrams, we notice an important differ-ence from the previous computation: This time the incoming state can alsocontain a photon. At zeroth order, the incoming photon does not interactand simply goes through unscattered (fig 1.3 a). At first order, the incoming12pipfqk(a)pipfqk(b)pipfqk(c)Figure 1.3: Three new diagrams for Bremsstrahlung up to first order in e.The incoming photon can either a) not interact or b), c) by absorbed byeither the incoming or outgoing electron legelectron can be absorbed by either the incoming or the outgoing electron leg(fig 1.3 b,c). We then need to account for three new interactions as well asthe two old ones from figure (1.1). Taking into account the normalizationof states containing one photon, diagram 1.3 a provides a factor of〈〈p′‖S1.3a‖p〉〉 = 〈p′|(e2∑l′=1∫d3k′2kF ∗l (k′,p′)al(k′))×(e2∑l=1∫d3k2kFl(k,p)a†l (k))|p〉 (1.38)= e2Mp,p′2∑l=1∫d3k2kF ∗l (k,p′)Fl(k,p).The bremsstrahlung diagrams (fig 1.1) have already been computed but thistime we account for the normalization of the dressed states〈〈p′‖S1.1‖p〉〉 = 〈p′|(e2∑l′=1∫d3k′2kF ∗l (k′,p′)al(k′))×(e2∑l=1∫d3k2k[Fl(k,p′)− Fl(k,p)]a†l (k))|p〉 (1.39)= e2Mp,p′2∑l=1∫d3k2kF ∗l (k,p′)[Fl(k,p′)− Fl(k,p)],13(a) (b) (c) (d)Figure 1.4: In the dressing formalism, divergences from standard loop di-agrams are canceled by similar diagrams involving exchanges between thesoft cloudswhile diagrams (fig 1.3 b,c) give similar results〈〈p′‖S1.3b,c‖p〉〉 = e2Mp,p′2∑l=1∫d3k2k[−F ∗l (k,p′) + F ∗l (k,p)]Fl(k,p).(1.40)Adding the contribution of all these terms, we find at order e2 that theS-matrix is〈〈p′‖S‖p〉〉 = e2Mp,p′(2∑l=1∫d3k2k[12|Fl(k,p)|2 + 12|Fl(k,p′)|2 − F ∗l (k,p′)Fl(k,p)]∑n,m∈p,p′ηnηmJnm . (1.41)Finally, we recall the form of the real part of Jnm after performing thek0 integral to beRe ∑n,m∈p,p′ηnηmJnm = 2(2pi)3∫d3k2k[p′µ2p′ · k −pµ2p · k]2. (1.42)Evaluating the sum over polarizations on the top line of equation (1.41)provides a factor of −ηµν and it now becomes clear that all infrared divergentterms cancel out of the S-matrix at second order.The full cancellation of IR divergences in the S-matrix between dressedstates can be shown order by order in a similar way, leaving the S-matrix14only with the non-divergent pieces.1.3.2 Dressed States as Eigenstates of the AsymptoticHamiltonianHaving found one set of states between which the S-matrix is non-singular,it is reasonable to ask why infrared divergences cancel for this precise choiceof dressing and if there exists other dressings that give non-zero amplitudes.The answer to this question was partially answered by investigating theasymptotic dynamics of charged particles [4]. The S-matrix is defined asthe scattering amplitudes between asymptotic states and, in Quantum FieldTheory, the LSZ procedure considers these asymptotic states to be free fieldFock space states. However, this is not correct for QED and gravity thatare long range forces with a 1/r potential. We can see this with from asimple non-relativistic quantum mechanical argument. We start by writingthe energy of the system asH = H0 + V =p22m+kr, (1.43)and notice that the potential dies off at infinity. A particle flying off totimelike infinity will, after a very long time, follow a trajectory of the formr(t) =pmt+ r0, (1.44)with r0 being an arbitrary vector representing the position of the particleat a given time. When time increases to infinity, r(t) → pt/m and thepotential becomeslimt→∞V (t) = Vas =mkpt. (1.45)Asymptotic states should therefore respect the Schrodinger equationwith the asymptotic HamiltonianiddtΨ(r, t) = [H0(t) + Vas(t)] Ψ(r, t). (1.46)The family of solutions to this differential equation takes the formΨ(r, t) = i∫d3p(2pi)3c(p) exp{−it p22m− imkplntt0}eipt, (1.47)with the function c being an arbitrary function of the momentum and t015the time at which initial conditions are given. This argument of course is aquantum mechanical treatment of the asymptotic dynamics exhibiting theproblem of expanding incoming and outgoing fields into free fields. Thefull field theory derivation needs to take into account that the interactionpotential is made out of free Dirac and gauge fieldsV = −e∫d3x : ψ¯(x)γµψ(x) : Aµ(x), (1.48)where ψ, ψ¯, Aµ are expanded in momentum creation and annihilation oper-ators in the usual wayψ(x, t) =∑±s∫d3p(2pi)3/2√mp0[b(p, s)u(p, s)e−ip·x + d†(p, s)v(p, s)eip·x]ψ¯(x, t) =∑±s∫d3p(2pi)3/2√mp0[b†(p, s)u¯(p, s)γ0eip·x + d(p, s)v¯(p, s)γ0e−ip·x]Aµ(x, t) =2∑l=1∫d3k√2ω (2pi)3/2[l,µ(k)al(k)e−ik·x + ∗l,µ(k)a†l (k)eik·x].(1.49)At large time, terms in the potential which maintain a non-suppressed ex-ponential in time will oscillate rapidly and cancel out, which allows us towrite the asymptotic potential asVas(t) = −e2∑l=1∫d3kd3p√2ω (2pi)3/2[b†(p)b(p)− d†(p)d(p)](1.50)pµp0[l,µ(k)a†l (−k) + ∗l,µ(k)al(k)]eip·kp0t.The asymptotic time evolution operator is given by the free time evolu-tion operator times the time ordered exponential of the asymptotic potentialUas(t) = e−iH0tT exp{− i ∫ dt′eiH0t′Vas(t′)e−iH0t′}. (1.51)The result was evaluated in [4] and givesUas(t) = exp{−iH0t} exp{R(t)} exp{iΦ(t)}, (1.52)16withR(t) = e2∑l=1∫d3kd3p√2ω (2pi)3/2[b†(p)b(p)− d†(p)d(p)](1.53)[Fl(k,p)a†(k)eip·kp0t − F ∗l (k,p)a(k)e−ip·kp0t]and Φ(t) is a phase which we will not care about. The S-matrix of a givenprocess then becomesS(t1, t2) = limt1→∞t2→−∞U †as(t1) exp{−iH(t1 − t2)}Uas(t2), (1.54)which differs from the standard QED S-matrix from the fact that Uas(t)now has an extra factor of exp{R(t)} exp{iΦ(t)}. However one can keep theDyson definition of the S-matrix if it is defined between states dressed byW = limt→±∞ exp{R(t)}. Applied on a free electron state of momentum p,we recover the dressed states defined in (1.32).When taking into account the long-ranged properties of the interactionpotential in QED, we notice that the S-matrix does not act between freeelectron states as the Dyson S-matrix but as this standard S-matrix dressedby W operators. The alternative which we adopt is to dress the in/outstates and keep the conventional definition of the S-matrix. Doing so, werecover the dressed states of Chung for which the S-matrix elements are freeof infrared divergences to all orders.1.4 Asymptotic SymmetriesIt has recently been discovered that abelian gauge theories possess an infi-nite number of symmetries. These symmetries are generated by large gaugetransformations and take a simple form at null infinity. Asymptotic sym-metries are spontaneously broken which results in an infinite degeneracyof the vacuum. At first glance, this topic seems unrelated to the study ofsoft theorems which are the primary focus of our attention in this thesis.However, it was found that the conservation of charges associated to thesesymmetries is directly related to the vanishing of the S-matrix in standardQED. By assuming unicity of the vacuum, conventional QED violates chargeconservation and forces the S-matrix to be null. However, when one takesinto account vacuum transitions, the S-matrix is restored. In those terms,the large gauge transformation program is a reformulation of Fadeev-Kulish17dressed states. We will now review the content of these large gauge trans-formations for QED and gravity and how charge conservation is linked tothe vanishing of the S-matrix. Finally, we will review the relation betweenFK states and conserved charges for QED.1.4.1 Matching ConditionsAs large gauge transformations take a simple form at null infinity, it issimpler to work in a coordinate system which exhibits this property. Ajudicious choice is to use advanced (v = t + r) and retarded (u = t − r)coordinates with metricds2 = −dv2 − 2dudr + 2r2γzz¯dzdz¯ (1.55)ds2 = −du2 − 2dudr + 2r2γzz¯dzdz¯,to parametrize points at negative and positive time t respectively. Thecoordinate r represents the radial distance while the location on the two-sphere is depicted via a stereographic projection unto the complex plane viacoordinates z, z¯. The metric on the complex plane is given byγzz¯ =2(1 + zz¯)2. (1.56)As we interest ourselves in the infrared behavior of QED, it is useful tolook at the expansion of fields at past/future null infinity denoted I−/I+(figure 1.5). In particular, we can examine the Lie´nard-Wiechert gauge fieldstrength generated by charged particles moving at constant velocityFrt(~x, t) =e24piN∑k=1Qkγk(r − txˆ · ~βk)|γ2k(t− rxˆ · ~βk)− t2 + r2|3/2, (1.57)where Qk is the charge of particle k in units of e, ~βk its velocity and γk is therelativistic factor γk =(1− β2k)−1/2. In advanced (retarded) coordinates,Frt(~x, t) = Frv(u)(~x, t). I± is located at null infinity and we can take the18I−+I−−I−I+−I+I++u=0v=0Figure 1.5: Minkowski space Penrose diagram. I± represent past and futurenull infinity with their S2 boundaries identified by I±± . Each point (r, t)identifies two points on the diagram, one on the right and one on the left,which are related by the antipodal mapping. The curved line representsthe motion of massive particles while lightrays move along straight lines ofconstant u or v.large r limit to find the expansion of the gauge field strength thereFrv∣∣I− =e24pir2N∑k=1Qkγ2k(1 + xˆ · ~βk)2 (1.58)Fru∣∣I+ =e24pir2N∑k=1Qkγ2k(1− xˆ · ~βk)2 .However, Lorentz invariance dictates that at the junction of I− and I+ thetwo fields must agree up to antipodal matching conditionFrv(xˆ)∣∣I−+ = Fru(−xˆ)∣∣I+− , (1.59)which is related to the fact that a boost towards the north pole in the pastcorresponds to a boost towards the south pole in the future. Instead ofworking with this antipodal matching, it is convenient to define coordinates19such that xˆ→ −xˆ in retarded coordinates so that we get the equalityFrv∣∣I−+ = Fru∣∣I+− . (1.60)1.4.2 Conserved ChargesThis matching condition for the gauge field strength is at the core of theargument leading to an infinity of conserved charges. Consider any function(t, r, z, z¯) respecting the same matching conditions(z, z¯)∣∣I−+ = (z, z¯)∣∣I+− , (1.61)then we can define chargesQ+ =1e2∫I+− ∗ F = 1e2∫I−+ ∗ F = Q− . (1.62)These charges being defined as integrals on a boundary, we can write themas full derivatives on I± and use Maxwell’s equations in form notation e2j =∗d ∗ F to rewrite them asQ+ =1e2∫I+d ∧ ∗F +∫I+ ∗ j + 1e2∫I++ ∗ F (1.63)Q− =1e2∫I−d ∧ ∗F +∫I− ∗ j + 1e2∫I−− ∗ F.Equation (1.62) and its rewritten form (1.63) exhibits the conservation ofan incoming charge Q− defined on past data I− into an outgoing chargeQ+ defined on future data I+. There exists an infinite number of functions which satisfy the matching conditions and therefore QED possesses aninfinite number of such conserved charges. While in theory any function obeying the matching conditions will provide a conserved quantity, it is moreinstructive to stick to a set of functions = (z, z¯) which only depend on thetwo sphere coordinates. One simple choice of basis for these functions couldbe spherical harmonics. Then, we can direcly notice using Gauss’ law thatthe conserved charge associated to Y 00 is simply the total charge of the state.Q+Y 00= Q−Y 00is the statement that total charge is conserved during scattering.In the Y ml basis, the infinite number of conservation laws is an extension toelectric charge conservation stating that every incoming multipole momentsof the electric field are also antipodally conserved.20Massless charged particlesA simple situation arises when all charged particles are massless. Thencharged particles fly out to I+ and never reach I++ leaving the electric fieldto vanish at that point. This has the effect of canceling the last term on therhs of (1.63). We can thus express the charges as a sum of two termsQ+ =1e2∫I+dud2z (∂zFuz¯ + ∂z¯Fuz) +∫I+dud2zγzz¯ju (1.64)The point of dividing the charges in such a way can now become apparent.The second term on the rhs is constructed from the charged current andcorresponds to hard outgoing data. We will therefore call this term thehard charge denoted Q+H,. On the other hand, the first term on the rhsis generated by zero energy photons and thus called the soft charge Q+S,.While this statement may not seem obvious at first sight, one can expandthe integral of the gauge field strength on I+ in its Fourier componentsN+z =∫ ∞−∞duFuz = limω→0∫ ∞−∞duFuzeiωu, (1.65)where only the zero mode contributes. Q+S, is therefore the contribution tothe charge coming from creating and annihilating zero energy photons withpolarization ∂z and ∂z¯. On the other hand, the hard charge being made ofthe conserved current of the global U(1) symmetry ju, its action on a stateof N incoming particles of charge Qk is simplyQ−H, |in〉 =∫I−dud2zγzz¯ju |in〉 (1.66)=N∑k=1Qink (zk, z¯k) |in〉 . (1.67)This stems from the fact that ju acts on a momentum eigenstate of chargeQ asju(u′, ω, ω¯) |Q(u, z, z¯)〉 = Qγzz¯δ2(ω − z)δ(u′ − u) |Q(u, z, z¯)〉 . (1.68)Massive charged particlesThe case for massive particles is slightly more complicated. As massive par-ticles follow trajectories that start at I−− and end at I++ , they can never21reach the regions I− and I+. Tt is therefore only the third term in (1.63)which will contribute. While advanced and retarded coordinate v, u are con-venient to describe the geometry of Minkowski space near null infinity, theseare poor choices of coordinates at past/future timelike infinity. There, wewill instead consider the hyperbolic slicing consisting of surfaces of constantτ2 = t2 − r2 > 0, (1.69)denoted H3. We define a new coordinate ρ such thatρ =r√t2 − r2 =rτ, (1.70)with the metric taking the formds2 = −dτ2 + τ2(dρ21 + ρ2+ ρ2dΩ22). (1.71)This set of coordinates is well justified when analyzing the motion of aparticle moving with constant velocity at late times. Then the motion ofthe particle follows a trajectory of the formr =pp0t+ r0 (1.72)and at late times t→∞, we findρ =|p|m, τ =mp0t.Massive particles therefore approach I++ on trajectories of constant ρ. Toevaluate the action of the hard charge on massive particles, we need toextend the definition of (z, z¯) beyond I± and into the bulk of Minkowskispace. To do so, it is simpler to work in Lorenz gauge ∇µAµ = 0. Then, thegauge parameter follows the wave equation = 0. (1.73)We can solve this equation in the bulk of Minkowski space by using a Green’sfunction integration kernelH(ρ, xˆ) =∫d2qˆ G(ρ, xˆ; qˆ)(qˆ), (1.74)22where the kernel needs to satisfyG(ρ, xˆ; qˆ) = 0 (1.75)limr→∞G(ρ, xˆ; qˆ) = δ2(xˆ− qˆ).This Green’s function has solutionγ1/2ωω¯4pi(√1 + ρ2 − ρqˆ · xˆ)2 , (1.76)where qˆ points in the direction of ω, ω¯. We can then simply evaluate theaction of the massive hard charge using Gauss’ law. When applied on anoutgoing state of momentum p and charge Q∫I++ (ρ, xˆ) ∗ F |out〉 = Q( |p|m, pˆ) |out〉 , (1.77)the massive hard charge singles out the electric charge of the outgoing par-ticle times the function evaluated along lines of constant ρ = |p|m .1.4.3 Vanishing of the S-matrix and Vacuum TransitionsHaving derived an expression for the hard and soft charges, we are now readyto explain via the large gauge transformation formalism why conventionalQED finds a vanishing S-matrix. Assuming some incoming state |in〉 evolvesinto an out state |out〉, the amplitude for that process is given by the S-matrix element〈out|S|in〉 . (1.78)In those terms, the statement that charge is conserved can be written〈out|Q+ S|in〉 = 〈out|SQ− |in〉 . (1.79)The equality (1.79) is valid for any choice of the function and in par-ticular it holds for (ω, ω¯) = 1z−ω for which the soft charge integral simplifiessubstantially. In this special case, we can use the identity∂z¯1z − ω = 2piδ2(z − ω), (1.80)which follows from Stoke’s theorem and Cauchy’s integral theorem. Equa-23tion (1.79) then becomes4pi 〈out|N+z S − SN−z |in〉 =([ ∑k∈masslessQinkz − zk −∑k∈masslessQoutkz − zk](1.81)+[ ∑k∈massiveQink ( |pk|mk, pˆk)−∑k∈massiveQoutk ( |pk|mk, pˆk)])〈out|S|in〉,where we used the definition of Nz from (1.65). The lhs then representsthe difference of soft charges between the outgoing and incoming states. Onthe rhs, the index k lists all incoming (outgoing) particles with charge Qkexiting (entering) I− (I+) and I−− (I++ ) at position zk on the two-sphere atinfinity . N±z being comprised of zero-mode photon creation and annihilationoperators does not affect the hard content of the in and out states. We canthen consider incoming and outgoing states of the form |in〉 = |in;N inz 〉 suchthatN−z |in;N inz 〉 = N inz |in;N inz 〉 . (1.82)N inz is the zero-mode photon content of the incoming vacuum and corre-sponds to the infinite degeneracy of the vacuum state due to the brokenlarge gauge symmetries. The eigenvalue on the rhs of (1.81) is usually re-ferred to as 4piΩz and the conservation of charges becomes(Noutz −N inz) 〈out;Noutz |S|in;N inz 〉 = Ωz 〈out;Noutz |S|in;N inz 〉 . (1.83)For this equality to hold, two options are possible: Either the difference ofsoft charges equals the difference of hard charges(Noutz −N inz)= Ωz, orevery S-matrix element has to be zero〈out;Noutz |S|in;N inz 〉 = 0. (1.84)In standard QED, the vacuum is taken to be non-degenerate which resultsin the vanishing of the S-matrix generally attributed to infrared divergencesas explained in section 1.2. However, this problem can be avoided if oneworks with degenerate vacua containing soft photons. Then the outgoingvacuum is highly correlated with the hard data as the sum of soft and hardcharges need to remain conserved. This is reminiscent of the Faddeev-Kulishformalism where a pure state is formed out of hard data entangled with acoherent state of soft radiation. In fact, this connection was made morerigorous recently [8, 10] where it was argued that large gauge symmetries24are a reformulation of the FK dressing.1.5 Soft Hair and the Black Hole InformationParadoxAn analogous situation arises in gravity where the asymptotic symmetriesof asymptotically flat Minkowski space were found by Bondi, van der Burg,Metzner and Sachs (BMS) [11, 12]. In this case, the relevant symmetriesform a subgroup of the BMS group and are not large gauge transformations,but angle-dependent gauge-preserving diffeomorphisms called supertransla-tions. Just as in QED, the associated conserved charges can be divided intoa hard and a soft component, where the soft charge identifies the zero-modegraviton content of the vacuum. It was recently argued [9] that this degen-erate vacuum may play a crucial role in the black hole information paradox.In this section, we will review the long-standing argument leading to the in-formation paradox and how asymptotic symmetries of spacetime could helpresolve it.1.5.1 Black Hole Information ParadoxIn 1975, Hawking’s seminal paper [13] demonstrated the process from whichblack holes evaporate. While classical black holes may only absorb particles,quantum mechanics allow for the vacuum to create particle-antiparticle pairsin the vicinity of the black hole horizon. When a positive energy particleescapes the black hole region and leaves the negative energy one to fall in,the black hole loses a small amount of mass. Over long periods of time,this process leads to the evaporation of any black hole in Minkowski space.Furthermore, Hawking argued using the no-hair theorem that the outgoingradiation does not contain any information about the black hole formationprocess but follows the spectrum of a blackbody with temperatureTHawking =~8piGM. (1.85)The final state is therefore a completely mixed thermal state where all theinformation about the arrangement of particles that created the black holein the first place has been lost during the formation/evaporation process.This however contradicts unitarity. If one sends in a pure state made of welllocalized incoming particles which will form into a black hole, unitary timeevolution states that the global state of the system remains pure at all times.25Hawking’s proposal of a pure state time evolving into a mixed thermal statetherefore violates unitarity. This is the longstanding black hole informationparadox. If information is indeed lost during the formation/evaporationprocess, then we must accept the disturbing fact that unitarity is not afundamental restriction of nature. Alternatively, we need to come up withan explanation as to where all that information is stored.1.5.2 Black Hole Soft HairEven though the paradox is still unresolved, the asymptotic symmetries ofQED and gravity program may provide some insight towards a resolution.Recent work initiated by [9] claimed that the soft graviton vacuum degen-eracy plays an important role. While the hard matter content of the stateis lost inside the black hole, the soft gravitons of the degenerate vacuum,having wavelengths far larger than the black hole size, are very unlikely tobe lost inside the black hole. As we have previously seen, charge conser-vation implies a strong correlation between the soft vacuum and the harddata. Contrary to classical belief, black hole states would be characterizednot solely by their mass, charge and angular momentum, but also by aninfinite number of supertranslation charges. Supertranslations would thenplay the role of infinitely many soft hairs distinguishing the state of any dif-ferently formed black holes. An even more recent proposal [14] speculatedthat the graviton vacuum, being strongly entangled with the hard data con-tent, could be sufficient to purify the outgoing Hawking radiation. If thiswere true, such an explanation could be sufficient to resolve the paradoxas the incoming pure state would then unitarily evolve into an other purestate made of thermal radiation entangled with soft gravitons. The globaloutgoing state would take the form of a tensor product between thermal andsoft graviton states|ψ〉 =∑a|a〉Thermal |a〉soft , (1.86)but an observer with a finite sized detector would not observe the gravitonvacuum and only see the mixed thermal state predicted by HawkingTrsoft |ψ〉 〈ψ| = ρThermal. (1.87)26Chapter 2Scattering with Partial Infor-mationWe study relativistic scattering when one only has access to a subset of theparticles, using the language of quantum measurement theory. We give anexact, non-perturbative formula for the von Neumann entanglement entropyof an apparatus particle scattered off an arbitrary set of system particles,in either the elastic or inelastic regime, and show how to evaluate it per-turbatively. We give general formulas for the late-time expectation valuesof apparatus observables. Some simple example applications are included:in particular, a protocol to verify preparation of coherent superpositionsof spatially localized system states using position-space information in theoutgoing apparatus state, at lowest order in perturbation theory in a weakapparatus-system coupling.2.1 IntroductionThe purpose of this paper is to make contact between concepts from quan-tum information and relativistic scattering theory. In particular, we studyhow to use interacting fields as measurement devices.In standard formulations of measurement theory, one imagines perform-ing a measurement of a system S by coupling it to an apparatus A. We startthe apparatus in some register state |0〉A while the system is in an arbitrarysuperposition, and then entangle these in such a way that measurements onA can determine the initial state of S. Schematically, one writes things like|0〉A ⊗∑ici |i〉S →∑ici |i〉A ⊗ |i〉S , (2.1)with the arrow referring to time evolution under some total Hamiltonian(see eg. [15, 16]). This process necessarily generates entanglement betweenS and A. The goals of this paper are to study to what extent we canunderstand the scattering of system particles S by another particle A inthis language and to quantify how much entanglement is generated in suchscattering events.27To this purpose, we consider an arbitrary system of fields and append anapparatus field φA which we can scatter off the system, so we consider Hilbertspaces formed by tensor products of apparatus and system fields. The S-matrix generates entanglement between the factors. This approach differsfrom and complements other ways of dividing field-theoretic systems; onecan also consider, for example, divisions by spatial area [17, 18], momentumscale [19], or multiple non-interacting CFTs [20].We begin by reviewing and slightly extending the textbook treatment[21] of scattering theory to incorporate density matrices as initial condi-tions in section 2.2. We explain how to calculate expectation values ofoperators probing only the apparatus. In section 2.3, we present an exact,non-perturbative formula for the von Neumann entropy of the apparatus Aafter the scattering event, assuming only that the state at early and latetimes contains exactly one particle of φA.We then apply these results to the simplest possible example, in whichthe apparatus and system both consist of a single particle of some scalarfields φA,S , with A and S weakly coupled. In section 2.4.1 we give an explicitformula for the entropy generated when we scatter a product momentumstate |p〉A |q〉S , recovering and slightly correcting a result of [22, 23].In section 2.4.2, we consider a somewhat different problem. Supposethat we think we are preparing the system S in a superposition of two well-localized position states. We show how to do a measurement with A toverify that the superposition is really coherent, as opposed to (say) havingdecohered into a classical ensemble. We find that a good observable touse to determine the coherence of S is position-space interference fringesin the outgoing distribution for the apparatus particle A. These show upat lowest order in perturbation theory in the S-A coupling λ, whereas themomentum-space distribution of A is only sensitive at second order.2.2 Scattering with Density Matrices2.2.1 General ConsiderationsLet’s consider the general problem of scattering where we know the stateof the total system at very early times t → −∞, and we want to knowhow this evolves at very late times due to a scattering event. We want toconsider any density matrix for the full system as an initial condition. Thetreatment here is a straightforward generalization of Weinberg’s textbook[21], and our conventions throughout follow his. In particular, the metric28signature is −+ ++ and ~ = c = 1.Assume the total Hamiltonian can be writtenH = H0 + V, (2.2)and denote the energy eigenstates of the free Hamiltonian H0 asH0 |α〉 = Eα |α〉 . (2.3)Here the label α = p1σ1n1,p2σ2n2, . . . covers the momentum, spin, andparticle species of the free-particle states. We define in- and out-states asHeisenberg-picture states which have the energies Eα but are eigenstates ofthe full Hamiltonian,1H |α±〉 = Eα |α±〉 , (2.4)satisfying the condition that as t → ∓∞, for any reasonably smooth func-tions g±(α) of the particle labels,|ψ〉 =∫dα e−iEαtg±(α) |α±〉 →∫dα e−iEαtg±(α) |α〉 . (2.5)This condition says that at very early or late times, the in/out statesbehave like the free-particle states of the corresponding particle labels α.The notation is that + indicates an in-state while − denotes an out-state.Both the free and scattering states are taken to be Dirac delta-normalizable〈α|α′〉 = 〈α±|α′±〉 = δ(α− α′).If the system is in a wavepacket like (2.5), and we know the matrixelements 〈α|O|α′〉 of some observable in terms of free-particle states, we cancompute the expectation value of O at early or late times in the state |ψ〉as follows. In the Heisenberg picture we have O(t) = eiHtOe−iHt, so using(2.4) and (2.5), we have that as t→ ∓∞,〈ψ|O(t)|ψ〉 →∫dαdα′ ei(Eα−Eα′ )tg∗(α)g(α′) 〈α|O|α′〉 . (2.6)More generally, the system may be in a density matrix. This can be decom-posed into any complete basis, including the scattering states:ρ =∫dαdα′ρ±(α, α′) |α±〉 〈α±| . (2.7)1Notice that the conditions (2.3) and (2.4) mean that the “free” states and scatteringstates have the same energy spectrum. This means in particular that the masses appearingin the Hamiltonian are the physical (“renormalized” or “dressed”) masses of the particles.29Then the expectation value of O is given asymptotically by〈O(t)〉 = tr ρO(t)→∫dαdα′ ρ±(α, α′)ei(Eα−Eα′ )t 〈α|O|α′〉 (2.8)as t→ ∓∞.Since the states |α+〉 and |α−〉 separately form complete bases for positive-energy states of the system, we can express one base in terms of the other.The S-matrix is the unitary operator with elements given by the inner prod-uctSβα = 〈β−|α+〉 . (2.9)The in- and out-coefficients of the density matrix are thus related byρ−(β, β′) =∫dαdα′ SβαS∗β′α′ρ+(α, α′). (2.10)We will always consider Poincare´-invariant systems. We can therefore writethe S-matrix as an identity term plus a term with the total four-momentuminvariance factored out,Sβα = δ(β − α)− 2piiMβαδ4(pβ − pα). (2.11)In appendix A, we use the unitarity of the S-matrix,∫dβSβαS∗βα = δ(α− α′) (2.12)to derive the optical theorem, (A.4), which will play a role repeatedly in thecalculations that follow.Box NormalizationsIn computing various quantities it will be useful to work with discrete states.We can do this by putting the entire process into a large spacetime volumeof duration T and spatial volume V = L3. Periodic boundary conditions onV allow us to retain exact translation invariance. We define dimensionless,box-normalized states|α±〉box = N˜nα/2 |α±〉 , N˜ = (2pi)3V, (2.13)where nα is the number of particles in the state α. When working di-rectly with box-normed states, delta functions and S-matrix elements are30all dimensionless, integrals over states are replaced by sums, and the delta-functions are Kroneckers. We haveSboxβα = N˜(nα+nβ)/2Sβα, (2.14)by definition of the S-matrix. Delta functions are then regulated asδ3V (p− p′) = N˜−1δp,p′ , δT (E − E′) =12pi∫ T/2−T/2dt ei(E−E′)t. (2.15)Note in particular that this implies δT (0) = T/2pi. We then define a box-normalized transition amplitude:Sboxβα = δβα − 2piiM boxβα δpβpαδEβEα ⇐⇒ M boxβα = N˜ (nα+nβ−2)/2Mβα. (2.16)Note that M box has mass dimension one, since δT (E) has dimensions ofinverse mass.2.2.2 Measuring the Apparatus StateSuppose now that we divide the total system into an apparatus A and sys-tem S and only have direct access to A. Here we work out a formula forcomputing observables only of A, and for the von Neumann entropy of A.In what follows, we assume that A and S are distinguishable; a simpleway to achieve this is to just have A and S described by different fields.We will make this assumption in everything that follows. We will hereaftermake a slight abuse of the previous notation and label states with two indices(a, α) where a labels apparatus eigenstates and α labels system eigenstates.We can decompose the total Hilbert space as a product over free, in, or outstates:H = HA ⊗HS = H±A ⊗H±S . (2.17)The total S-matrix provides a unitary map between the in- and out-state de-compositions. In particular, a product in-state is a generally non-separablemixture of out-states:|aα〉+ =∫dbdβ Sbβ,aα |bβ〉− . (2.18)At early or late times, we want to compute the expectation value of anyobservable OA : HA → HA. Note that here OA is an operator on the freeapparatus Hilbert space factor in (2.17). Take O = OA⊗1S and apply (2.8).31By the asymptotic conditions on the scattering states, a simple calculationshows that at early or late times〈OA(t)〉 := 〈O(t)〉 →∫dada′dα ρ±(a, α; a′, α)ei(Ea−Ea′ )t 〈a|OA|a′〉 .(2.19)To derive this formula, we assumed that the free Hamiltonian has an additivespectrum H0 |aα〉 = (Ea +Eα) |aα〉. The result (2.19) holds for any densitymatrices; in particular, we do not need to assume that the total state factorsinto a product of a density matrix for A and a density matrix for S at eitherearly or late times.We would also like to define the entanglement entropy between apparatusand system. To do this, we again use the decomposition (2.17) to performpartial traces over the system. We can do this using either in- or out-states,ρ±A := trH±S ρ (2.20)from which we can in turn define the entanglement entropyS±A = − trH±A ρ±A ln ρ±A. (2.21)2.3 A-S Entanglement EntropyOur goal in this section is to calculate the entanglement entropy betweenthe system and apparatus at late times. Consider the system and apparatusboth prepared in definite momentum eigenstates at early times,|ψ〉 = |p+〉A |α+〉S . (2.22)Here as before α = q1n1σ1,q2n2σ2, . . . labels all the momenta, species, andspin of the system particles, while p is simply the initial momentum of theapparatus, which we take to be a scalar for notational simplicity. For theentirety of this section until the end, we will work in a spacetime box asdescribed above, but will refrain from writing “box” superscripts. At theend of the computation we will discuss the continuum limit.We assume that one and only one apparatus particle exists in both theinitial and final state. This can be arranged for example by assigning φAsome global charge, or by taking φA to have high mass and studying scat-tering events below its production threshold.Using the formalism from section 2.2, we can express the density matrix32pp qqpp qqppp q1q2q3q4p q1 q2↵↵1Figure 2.1: A typical apparatus-system scattering process. Dotted linesdenote the apparatus, solid lines the system. Time runs from bottom totop.in terms of out-states,ρ =∑pp′αα′SpαpαS∗p′α′pα |pα−〉 〈p′α′−| . (2.23)From here out we use underlines to denote outgoing variables. Expandingthe S-matrix with (2.16), one can see from this expression that ρ will havethe correct norm tr ρ = 1 if and only if the optical theorem (A.4) is satisfied(see appendix A). In particular, if one is working in perturbation theory, theoptical theorem mixes orders, so one needs to be careful about including thecorrect set of loop and tree diagrams at a given order.Now trace over the system, using out-states:ρ−A =∑pp′αSpαpαS∗p′αpα |p−〉 〈p′−| . (2.24)Decompose the S-matrix with (2.16). We get three types of terms: from thedelta-squared we get a term on the diagonal with momentum given by theinitial momentum p:ρ−A,1 = |p−〉 〈p−| . (2.25)33The cross-terms −iMρ+ iρM † give a contributionρ−A,2 = −2T Im [Mpαpα] |p−〉 〈p−| , (2.26)again to the density matrix element for the initial momentum p. This isthe forward scattering term that appears in the optical theorem. Finally,we need the terms from MρM †. One obtainsρ−A,3 = 2piT∑pα∣∣∣Mpαpα∣∣∣2 δp+pα,p+pαδ(EAp + ESα − EAp − ESα ) |p−〉 〈p−| .(2.27)We see that the reduced density matrix for A is diagonal in an arbitraryreference frame. This is due entirely to translation invariance and our as-sumption that we always have precisely one apparatus particle. Writing theapparatus state in matrix form, we haveρ−A =1 + I0 + F (p)F (p1)F (p2). . . , (2.28)where the piare all the outgoing apparatus momenta p 6= p. The coefficientsareI0 = −2T ImMpαpαF (p) = 2piT∑α∣∣∣Mpαpα∣∣∣2 δp+pα,p+pαδ(EAp + ESα − EAp − ESα ). (2.29)The coefficients F (p) could be called “conditional transition probabilities”.They are given by fixing an apparatus out-momentum p and then summingover the transition probabilities to all the possible system states consisentwith total momentum conservation. Note that F (p) = 0 for momenta vio-lating energy conservation, that is when EAp > EAp + ESα − ES0 .22In 2→ 2 scattering, we can write the return-amplitude term I0 + F (p) in a way thattreats the two particles more symmetrically: by the optical theorem (A.4), we haveI0 = −(2pi)2∑pq∣∣Mpqpq∣∣2 δp+q,p+qδEAp +ESq ,EAp +ESq (2.30)while by definition, F (p) = (2pi)2 |Mpqpq|2. So the shift in the initial-momentum density34The von Neumann entanglement entropy of the apparatus is given bySA = −(1 + I0 + F (p)) ln(1 + I0 + F (p))−∑p 6=pF (p) lnF (p). (2.32)The result (2.32) is exact and non-perturbative. It follows completely fromLorentz invariance and our assumption that precisely one A particle is inboth the initial and final state. It can be simplified by invoking perturbationtheory: we assume that the scattering amplitudes are significantly less thanunity. Then |I0 + F (p)| 1, so we can Taylor expand the first term in(2.32) and get a term linear in this expression. But the other terms stillhave logarithms, so we have an expression like small +∑small ln(small),and the log terms will dominate. So we are left withSA = −∑pF (p) lnF (p). (2.33)In a large box, it is immaterial if the sum on outgoing apparatus momentap includes p = p or not, since this term is individually of measure zero.2.4 Examples with Two Scalar FieldsWe will now consider some simple applications of the above theory, withboth system and apparatus described by scalar fields φA,S with a weakcoupling λ. Throughout, we will assume that the initial energies are belowthe threshold for on-shell pair-production, so that we can work entirely with2→ 2 matrix elements.In the first subsection, we study entropy generated during a 2 → 2scattering event. In the second subsection, we show how to verify thatthe system S has been prepared in a spatial superposition by scatteringwith A. More precisely, we show how to read out the coherence of such asuperposition using position-space information in A, at lowest order in λ.Let us fix our conventions. We take the apparatus and system to bematrix eigenvalue is∆0 = −(I0 + F (p)) = (2pi)2∑(p,q)6=(p,q)∣∣Mpqpq∣∣2 δp+q,p+qδEAp +ESq ,EAp +ESq . (2.31)35described by the actionS = −∫d4x12(∂µφS)2 +12(∂µφA)2 +12m2Sφ2S +12m2Aφ2A+λ4φ2Sφ2A +λA4!φ4A +λS4!φ4S + Lct.(2.34)In particular, the fields φS,A are considered to be distinguishable and renor-malized. The term Lct contains the counterterms; here we use the standardon-shell renormalization conditions that the on-shell propagators have unitresidue at the physical masses and the interactions are given exactly by theirphysical couplings at threshold. This way we can work with amputated di-agrams only, and the lowest order in perturbation theory is just tree level.We will take up loop corrections in a future publication. We assume thatthe self-couplings λA,S 1 and ignore them hereafter.The free single-particle states and operators are normalized as〈k′|k〉 =[ak, a†k′]= δ3(k− k′). (2.35)More generally, a free n-particle state of a given species is |k1 · · ·kn〉 =a†kn · · · a†k1|0〉, where |0〉 is the free vacuum. In what follows we use p todenote the 3-momentum of the apparatus and q that of the system. Therelevant S-matrix elements are thenSpqpq = δ3(p− p)δ3(q− q)− 2piiMpqpqδ4(p+ q − p− q) (2.36)with the amplitude given by, to lowest order in perturbation theory,iMpqpq =pp qq=iλ(2pi)3√16EApESqEApESq. (2.37)Here the single-particle energies areES,Ak =√m2S,A + k2. (2.38)362.4.1 Entropy from 2→ 2 ScatteringTo begin, we study the simplest possible process: scattering with the systemand apparatus both prepared in definite momentum eigenstates at earlytimes,|ψ〉 = |p+〉A |q+〉S . (2.39)This is precisely what we studied in section 2.3 and, as we did there, we willwork with box-normalized states until the end of the calculation.After the scattering event, the von Neumann entropy of the apparatusis given directly by our formula (2.33), viz.SA = −∑pF (p) lnF (p). (2.40)Again the sum runs over all outgoing apparatus momenta p, and the co-efficients F (p) are defined in (2.29). Because scattering in this theory isisotropic, it is straightforward to compute the apparatus density matrixeigenvalues explicitly. Move to the center-of-momentum frame p = −q.ThenF (p) = 2piT∑q∣∣∣Mpqpq∣∣∣2 δp+q,p+qδ(EAp + ESq − EAp − ESq )= 2piT |M(pcm)|2 δ(f(|p|)),(2.41)where we used isotropy of the interaction to write this asM(pcm) = Mp,−p;p,−p, pcm = |p| = |p|, f(|p|) = EA|p|+ES|p|−EA|p|−ES|p|.(2.42)The entropy of A at late times is thus given bySA = −2piT∑p|M(pcm)|2 δ(f(|p|)) ln[2piT |M(pcm)|2 δ(f(|p|))]. (2.43)At this stage, we can take the continuum limit. We replace the sum∑p →V/(2pi)3∫d3p, and do the integral in spherical coordinates. The delta-function outside the log enforces energy conservation, and so the delta insidethe log is replaced by δT (0) = T/2pi. We also have to insert the appropriatefactors of N˜ = (2pi)3/V to convert from the box-normalized amplitude to37the continuum-normalized one, see eq. (2.16). Finally, we obtainSA = −2(2pi)5 TVp2cm(EA + ES) |M(pcm)|2 ln[(2pi)6T 2V 2|M(pcm)|2], (2.44)where the energies are understood to be evaluated at pcm. This holds atany order of perturbation theory. If we wanted to work to lowest order inperturbation theory, we can use our matrix element (2.37) given above, inwhich case we have explicitly[22]SA = −TVλ216pipcm(EA + ES)(EAES)2ln[T 2V 2λ216(EAES)2]. (2.45)This formula bears some remarking. For one thing, recall that the totalcross-section for this theory at this order of perturbation theory is given byσ = λ2/16piEAES in the center-of-momentum frame. So we have that theentropy is proportional to this quantity, integrated over time and againstthe flux of incoming particles.3 We always have a large spatial volume Vin mind, so SA ≥ 0. The argument of the logarithm likewise cannot betoo small: if Tλ/16V EAES ≤ 1 then the entropy will be negative. Thisis essentially the statement that the Compton wavelengths of the particlesneed to be within the spacetime box. As we take the spatial volume V →∞with T fixed, SA goes to zero from above; this follows from the fact that theprobability of the waves to interact at all goes to zero. Finally, one mightworry about V fixed and T → ∞, in which case the entropy goes to −∞,but this corresponds to an infinite number of repeated interactions, whichwould also violate the basic assumption of the S-matrix setup that we aredescribing an isolated event.2.4.2 Verifying Spatial SuperpositionsLet’s consider now a rather different problem. Suppose we prepare thesystem and apparatus in a separable state, but the system state may ormay not be pure. We would like to know how this system informationwould show up in the outgoing apparatus state.For definiteness, we consider the following problem: suppose that someblack box machine in our lab prepares the system as either a classical en-semble or coherent superposition of two system states, each localized to adifferent point in real space. The question is: how do we verify the coherence3In this frame, the flux is Φ = u/V with the relative velocity u = pcm(EA+ES)/EAES .38of the superposition from a scattering experiment?We will see that it is sufficient to look at the position-space wavefunctionof the outgoing apparatus at order λ. The signature of the system superpo-sition is interference fringes in the apparatus state. They show up at orderλ because the position-space projector |x〉 〈x| is sensitive to off-diagonalmomentum-space apparatus density matrix elements, which are generatedat first order in the perturbation, as we now demonstrate explicitly.We begin by defining a pair of states |L〉 , |R〉 that describe the apparatusprepared in an incoming state of momentum p and the system centered atdifferent positions xL,R in real space.4 Define the usual Gaussian wavefunc-tiong(q) = NS exp{−q2/4σS} , NS = 1(2piσS)3/4(2.47)and take the system to be initialized at rest in a lab frame, so we define thestate as follows: let i ∈ {L,R} and put|i〉 = NA |p+〉A∫d3q fi(q) |q+〉S ,fi(q) = g(q) exp {iq · xi} , NA =√(2pi)3V.(2.48)See figure 2.2. These states are not orthogonal; their overlap is = 〈L|R〉 = exp{−σS |∆x0|2 /2}, ∆x0 = xL − xR. (2.49)We have in mind that the system states are localized in real space, so thatthe momentum spread σS is large. The two states are well-separated if 1; we assume this below for mathematical ease, but the results do notdepend qualitatively on this condition.5 We are assume that the scatteringis done in a sufficiently short time so that we can ignore the spreading ofthese wavepackets.Now consider an arbitrary density matrix in the space spanned by the4In this section we will use continuum-normalized states, regulating squares of Diracdeltas as [δ3(p− p′)]2 = V(2pi)3δ3(p− p′), [δ(E − E′)]2 = T2piδ(E − E′). (2.46)5When working with the following formulas, the non-orthogonality of |L〉 , |R〉 shouldbe kept in mind; in particular traces should be done with momentum eigenstates. A usefulrelation is tr |i〉 〈j| = 〈i|j〉 = for i 6= j and 1 for i = j.39|L i |Ri1/ 2 SppxLxL xFigure 2.2: Verifying spatial superpositions of the system states |L〉, |R〉.|L〉 , |R〉 states:ρ = Γij |i〉 〈j| , i, j ∈ {L,R} . (2.50)For example, we can form a convex family of density matrices, with coeffi-cientsΓij(α) =12(1 + )(1 + − α αα 1 + − α), 0 ≤ α ≤ 1. (2.51)These linearly interpolate between the classical ensemble proportional to|L〉 〈L|+|R〉 〈R| at α = 0 and the perfect coherent superposition proportionalto (|L〉+ |R〉)(〈L|+〈R|) at α = 1. These all have unit trace, while the puritytr ρ2(α) = [1+(α+−α)2]/2 vanishes when = α = 0 and goes up to unityif either = 1 or α = 1. We will refer to α as the coherence parameter. Notein particular that the off-diagonal element ΓLR is linear in α. The reduceddensity matrix for the apparatus expressed with out-states isρ−A = N2A∑ij∫d3pd3p′d3qd3qd3q′Γijfi(q)f∗j (q′)SpqpqS∗p′qpq′ |p−〉 〈p′−| .(2.52)Let’s study some outgoing apparatus observables. Consider first theoutgoing momentum distribution P (p) of the apparatus, so that we takeOA = |p〉 〈p| and use (2.19); the expectation value can be read off fromthe diagonal elements of (2.52). We can work these out a bit more explic-40pp qqpp qqFigure 2.3: Diagrams contributing to the lowest-order position-space distri-bution of the apparatus.itly. The identity-squared term from decomposing the S-matrix with (2.11)contributes to P (p) as P0(p) = δ3(p− p). The interaction terms givePint(p) = (2pi)3 TV∫d3q |g(q)|2 (1 + α cos 2q ·∆x0){− 2ImMpqpqδ3(p− p)+∣∣∣Mp,q−k;p,q∣∣∣2 δ (EAp + ESq−k − EAp − ESq)}(2.53)where here k = p − p is the momentum transfer, and we took 1 towrite the result in a simple way. We see that the overall probability isproportional to T/V , as expected. Both terms receive a contribution fromthe coherence α of the initial superposition. In our specific theory (2.37),both of these contributions are of order λ2, with the forward-scattering termin (2.53) coming in only at one-loop order. So to measure α by doing suchan observation, we would have to be sensitive at order λ2.However, it is possible to see signatures of the coherence α at first or-der in λ if we instead look at position-space observables. Consider theposition-space probability distribution for the apparatus at late times afterthe scattering, P (x). This can be obtained by again applying (2.19) butnow using the observable OA = |x〉 〈x|, the single-particle position projec-tor. The delta-squared terms from the S-matrix result in P0(x, t) = V−1by direct computation. Next we need both the cross terms Mρ− ρM † andthe amplitude-square MρM † term; the latter will start at O(λ2), so let usconsider the former. A straightforward calculation using hermiticity of Γij41givesP1(x) =4piV∫d3qd3qδ(EAp+q−q + ESq − EAp − ESq )g(q)g∗(q)× ImMp+q−q,q;p,q∑ijΓij exp{−iφij(q,q)} , (2.54)where the subscript 1 means we are thinking of this in first-order perturba-tion theory, and the phases areφij(q,q) = −ESq t+ (x− xi) · q+ ESq t− (x− xj) · q. (2.55)Consider measuring the location of the outgoing A particle when t and |x−xi| are of the same order and large. Then the integral may be approximatedby its stationary phase value, which here is given whenq = qi = mSγivi∆̂xi, q = qj = mSγjvj∆̂xj , (2.56)where∆xi = x− xi, ∆̂xi = ∆xi|∆xi| , vi =|∆xi|t, γi =1√1− v2i. (2.57)Note that vi = vi(x, t) and likewise γi = γi(x, t) depend on the point ofobservation x and the time t; we suppress this dependence in the formulasthat follow. At these values for the momenta, we have thatESqi = mSγi, φij = −mSt[γ−1i − γ−1j]= −φji. (2.58)In particular, we see that the LL and RR terms have zero phase, and thusgive real contributions in (2.54) since our amplitude (2.37) is real at lowestorder, so they do not contribute to the outgoing position distribution. Theinterference terms LR and RL do contribute, however, and we getP1(x, t) = A(x, t) sin (φLR(x, t)) (2.59)where at this point we have finally used the reality of our amplitude (2.37).The position-space amplitude isA = α2(2pi)4Vγ5/2L γ5/2R m3St−3g(qL)g(qR) [δLRMLR − δRLMRL] (2.60)42where we defined for brevityMij = Mp0+qi−qj ,qj ;p0,qi , δij = δ(EAp+qi−qj + ESqi − EAp − ESqj ). (2.61)The delta-functions localize the distribution to the stationary-phase wave-fronts, and are an artifact of the way we did the integrals. In reality, theyshould be smoothed out.The key physics is in the sine term in (2.59), and the fact that A islinear in both the coupling λ and coherence parameter α. The amplitudeA is a rather complicated function of x, t, but the point is clear enough: ifwe arrange an array of particle detectors in a sphere around the origin, itwill pick up the interference pattern given by the sine term in (2.59). Theheights of the interference fringes, in turn, are set by the coherence α: inparticular, if the system is initialized in a classical ensemble, α = 0 andthere are no fringes.Physically, these are interferences between the process where no scatter-ing occurs and the process where the apparatus scatters off one or the othersystem locations, see figure 2.3. Mathematically, this is in the Mρ1− 1ρM †terms in the action of the S-matrix on the density matrix. This is why theinterference appears at order λ and not λ2. This should be contrasted withmomentum-space observables, which are only sensitive to the interferenceat λ2: the position-space observable is sensitive to off-diagonal momentum-space density matrix elements, which are generated at lowest order in per-turbation theory.2.5 ConclusionsWe have studied some prototypical examples of an apparatus particle scat-tering off a collection of system particles, applying the language of quantummeasurement theory to a field-theoretic problem. Our general density matrixformalism allows for the computation of arbitrary apparatus observables atearly and late times, and we showed how to compute the apparatus-systementanglement entropy generated during scattering.Our scenario contrasts standard formulations of measurement theory insome significant ways. For one thing, our system and apparatus are rela-tivistic and have continuous spectra. For another, we do not imagine that wecan precisely engineer some interaction Hamiltonian; here we are just stuckwith whatever our effective field theory happens to give us. Nonetheless wehave found that it is straightforward to use standard measurement-theorytechniques.43A potential application is detection of system properties at lower ordersof perturbation theory than usually considered in scattering. For example,one often hears that λφ4 scattering is only sensitive to λ2 as opposed to λ,because the cross-section scales like λ2. On the contrary, one can clearly doan interference measurement as described above to measure the coupling atorder λ.More theoretically, these kinds of calculations may help shed some lighton certain aspects of black hole physics. In particular, a recent proposal isthat the black hole information is radiated out to null infinity by soft bosonicmodes.[24] This information should thus be quantified by precisely the kindof von Neumann entropy we have considered here. Implications of the softboson theorems for the entropy calculations presented above will appear ina future article.44Chapter 3Infrared Quantum InformationWe discuss information-theoretic properties of low-energy photons and gravi-tons in the S-matrix. Given an incoming n-particle momentum eigenstate,we demonstrate that unobserved soft photons decohere nearly all outgo-ing momentum superpositions of charged particles, while the universalityof gravity implies that soft gravitons decohere nearly all outgoing momen-tum superpositions of all the hard particles. Using this decoherence, wecompute the entanglement entropy of the soft bosons and show that it isinfrared-finite when the leading divergences are re-summed a` la Bloch andNordsieck.3.1 IntroductionThe massless nature of photons and gravitons leads to an infrared catastro-phe, in which the S-matrix becomes ill-defined due to divergences comingfrom low-energy virtual bosons. The usual solution to this problem, origi-nally given by Bloch and Nordsieck in electrodynamics [1] and extended togravity by Weinberg [2], is to argue that an infinite number of low-energybosons are radiated away during a scattering event; this leads to divergenceswhich cancels the divergences from the virtual states, and physical predic-tions in terms of infrared-finite inclusive transition probabilities.In this letter, we study quantum information-theoretic aspects of thisproposal. Since each photon and graviton has two polarization states andthree momentum degrees of freedom, one might suspect that the low-energyradiation produced during scattering could carry a huge amount of informa-tion. Here we demonstrate that, according to the methodology of [1, 2, 25], ifthe initial state is an incoming n-particle momentum eigenstate, the “soft”bosonic divergences can lead to complete decoherence of the momentumstate of the outgoing “hard” particles. This decoherence is avoided only forsuperpositions of pairs of outgoing states for which an infinite set of angle-dependent currents match, see eq. (3.11). In simple examples like QED, thiswill be enough to get complete decoherence of all momentum superpositions.In less simple cases, one is still left with an extremely sparse density matrixdominated by its diagonal elements.45Having traced the radiation in this fashion, we obtain an infrared-finite,mixed reduced density matrix for the hard particles. In the simple caseswhen we get a completely diagonal matrix, we compute the entanglemententropy carried by the soft gauge bosons. The answer is finite and scaleslike the logarithm of the energy resolution E of a hypothetical soft bosondetector.While the tracing out of the soft radiation can be viewed as a physicalstatement about the energy resolution of a real detector, in this formalism,the trace is also forced on us by mathematical consistency: it is the onlyway to get well-defined transition probabilities from the infrared-divergentS-matrix. There is an alternative approach to the infrared catastrophe, inwhich one constructs an IR-finite S-matrix of transition amplitudes between“dressed” matter states.[3, 4, 26, 27] In such an approach, there are no diver-gences and so one is not forced to trace over any soft radiation. Whether thetwo formalisms lead to the same physical picture is an interesting question,and we leave a detailed comparison to future work.Recently, the infrared structure of gauge theories has become a topicof much interest due to the proposal that soft radiation may encode infor-mation about the history of formation of a black hole.[24, 28, 29] We hopethat our work can make this discussion more quantitatively grounded; wecomment on black holes at the end of this letter. More generally, it is ofinterest to understand the information-theoretic nature of the infrared sec-tor of quantum field theories, and our paper is intended to make some firststeps in this direction.3.2 Decoherence of the Hard Particles.Fix a single-particle energy resolution E. We define soft bosons as thosewith energy less than E, and hard particles as anything else. Consider anincoming state |α〉in consisting of hard particles, charged or otherwise, ofdefinite momenta.1 The S-matrix evolves this into a coherent superpositionof states with hard particles β and soft bosons b = γ, h (photons γ andgravitons h),|α〉in =∑βbSβb,α |βb〉out . (3.1)1Our field theory conventions follow [21]. Labels like α, β, b mean a list of free-particlequantum numbers, e.g. |α〉in = |p1σ1, . . .〉in listing momenta and spin of the incomingparticles.46Hereafter we drop the subscript on kets, which will always be out-states.Tracing out the bosons |b〉, the reduced density matrix for the outgoinghard particles isρ =∑ββ′bSβb,αS∗β′b,α |β〉 〈β′| . (3.2)Using the usual soft factorization theorems [2, 25, 30], we can write theamplitudes in terms of the amplitudes for α→ β multiplied by soft factors,one for each boson:Sβb,α = Sβ,αFβ,α(γ)Gβ,α(h), (3.3)where the soft factors F,G areFβ,α(γ) =∑n∈α,β∑±∏i∈γenηn(2pi)3/2|ki|1/2pµn∗µ,±(ki)pn · ki − iηnGβ,α(h) =∑n∈α,β∑±∏i∈hM−1p ηn(2pi)3/2|ki|1/2pµnpνn∗µν,±(ki)pn · ki − iηn .(3.4)Here the index n runs over all the incoming and outgoing hard particles, iruns over the outgoing soft bosons; ηn = −1 for an incoming and +1 for anoutgoing hard particle. The en are electric charges and Mp = (8piGN )−1/2 isthe Planck mass, and the ’s are polarization vectors or tensors for outgoingsoft photons and gravitons, respectively. By an argument identical to theone employed by Weinberg [2], and assuming we can neglect the total lostenergy ET compared to the energy of the hard particles, we can use thisfactorization to perform the sum over soft bosons in (3.2), 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).(3.5)Here λ E is an infrared regulator used to cut off momentum integralswhich we will send to zero later; one can think of λ as a mass for the photon47and graviton. The exponents areA˜ββ′,α = −∑n∈α,βn′∈α,β′enen′ηnηn′8pi2β−1nn′ ln[1 + βnn′1− βnn′]B˜ββ′,α =∑n∈α,βn′∈α,β′mnmn′ηnηn′16pi2M2p1 + β2nn′βnn′√1− β2nn′ln[1 + βnn′1− βnn′],(3.6)and f is a complicated function which can be found in [21]; for E/ET = O(1)and for small A, f may be approximated as f(1, A) ≈ 1−pi2A2/12+O(A4).In these formulas, βnn′ is the relative velocity between particles n and n′,βnn′ =√1− m2nm2n′(pn · pn′)2 ,For future use, we note that 0 ≤ β ≤ 1, and both of the dimensionlessfunctions of β appearing in (3.6) run over [2,∞) as β runs from 0 to 1. Wehave βnm = 0 if and only if pn = pm.The divergences as λ → 0 in (3.5) come from summing over an infinitenumber of radiated, on-shell bosons. There are also infrared divergencesinherent to the transition amplitude Sβ,α itself coming from virtual bosons.Again following Weinberg, we can add these divergences up, and we havethatSβ,α = SΛβ,α(λΛ)Aβ,α/2(λΛ)Bβ,α/2, (3.7)where now SΛβ,α means the amplitude computed using only virtual bosonsof energy above Λ, andAβ,α = −∑n,m∈α,βenemηnηm8pi2β−1nm ln[1 + βnm1− βnm]Bβ,α =∑n,m∈α,βmnmmηnηm16pi2M2p1 + β2nmβnm√1− β2nmln[1 + βnm1− βnm].(3.8)An infrared-divergent “Coulomb” phase is suppressed in (3.7). We will seeshortly that this phase cancels out of all the relevant density matrix elements.Putting the above results together, we find that the reduced density48matrix coefficient 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).(3.9)The question is how this behaves in the limit that the infrared regulatorλ→ 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.(3.10)In the appendix, we prove that both of these exponents are positive-definite,∆Aββ′,α ≥ 0 and ∆Bββ′,α ≥ 0. The density matrix components (3.9) whichsurvive as the regulator λ→ 0 are those for which ∆A = ∆B = 0; all otherdensity matrix elements will vanish.To give necessary and sufficient conditions for ∆A = ∆B = 0, we definetwo currents for each spatial velocity vector v. We assume for simplicity thatonly massive particles carry electric charge. For massive particles, there areelectromagnetic and gravitational currents defined asjEMv =∑ieiai†pi(v)aipi(v)jGRv =∑iEi(v)ai†pi(v)aipi(v).(3.11)Here i labels particle species, ei their charges and mi their masses; the kine-matic quantities pi(v) = miv/√1− v2 and Ei(v) = mi/√1− v2 are themomentum and energy of species i when it has velocity v. For lightlike par-ticles we have to separately define the gravitational current, since a velocityand species does not uniquely determine a momentum:jGR,m=0v =∑i∫ ∞0dω ωai†ωvaiωv. (3.12)Momentum eigenstates of any number of particles are obviously eigenstatesof these currents and we denote their eigenvalues jv |α〉 = jv(α) |α〉.The photonic exponent ∆Aββ′,α is zero if and only if the charged currents49in β are the same as those in β′; the gravitational exponent ∆Bββ′,α is zeroif and only if all the hard gravitational currents in β are the same as thosein β′. This is demonstrated in detail in the appendix. For any such pair ofoutgoing states |β〉 , |β′〉, (3.9) becomes independent of the IR regulator λand is thus finite as λ→ 0,ρββ′ = SΛ∗β′αSΛβαFβα (E,ET ,Λ) , (3.13)whereFβα = f(EET, Aβα)f(EET, Bβα)(EΛ)Aβα+Bβα. (3.14)This is the case in particular for diagonal density matrix elements β = β′,for which we obtain the standard transition probabilitiesρββ =∣∣SΛβα∣∣2Fβα (E,ET ,Λ) . (3.15)On the other hand, if there is even a single v for which one of the currents(3.11) or (3.12) does not have the same eigenvalue in |β〉 and |β′〉, then thedensity matrix coefficient decays as λ∆A+∆B → 0 as the regulator λ → 0.We see that the unobserved soft bosons have almost completely decoheredthe momentum state of the hard particles. Only a very sparse subset ofsuperpositions in which all the jv(β) = jv(β′) survive.3.3 ExamplesTo get a feel for the results presented in the previous section, we considera few examples. First, consider any scattering with a single incoming andoutgoing charged particle, like potential or single Compton scattering. Letthe incoming momentum be α = p and the outgoing momenta of the twobranches β = q, β′ = q′. We have either directly from the definition (3.10)or the theorem (B.1) that∆Aqq′,p = − e28pi2[2− γqq′], (3.16)where γqq′ = β−1qq′ ln[(1 + βqq′)/(1− βqq′)]. This ∆A is easily seen to equalzero if and only if q = q′. Thus other than the spin degree of freedom, theresulting density matrix for the charge is exactly diagonal in momentumspace.To see an example where the current-matching condition is non-trivially50fulfilled, consider a theory with two charged particle species of charge e ande/2 and the same mass. Then we can get an outgoing superposition of astate β = (e, q) and one with two half-charges β′ = (e/2, q′1)+(e/2, q′2). Thedifferential exponent for such a superposition is∆Aββ′,p = − e28pi2[3 +12γq1q2 − γqq1 − γqq2], (3.17)which is zero if q = q1 = q2. In other words, the currents (3.11) cannotdistinguish between a full charge of momentum q and two half-charges ofthe same momentum.3.4 Entropy of the Soft BosonsWe have seen that the reduced density matrix for the outgoing hard particlesis very nearly diagonal in the momentum basis. In a simple example like atheory with various scalar fields φi of different, non-zero masses mi, the softgraviton emission causes complete decoherence into a diagonal momentum-space reduced density matrix for the hard particles. More generally, we mayhave a sparse set of superpositions, and in any case spin and other internaldegrees of freedom are unaffected by the soft emission.In a simple example with a purely diagonal reduced density matrix, itis straightforward to compute the entanglement entropy of the soft emittedbosons. The total hard + soft system is in a bipartite pure state, with thepartition being between the hard particles and soft bosons, so the entan-glement entropy of the bosons is the same as that of the hard particles.Following the calculation in [22, 31, 32], we can simply write down the en-tropy:S =∑β∣∣SΛβα∣∣2Fβα ln [∣∣SΛβα∣∣2Fβα] . (3.18)This sum is infrared-finite; again, F is given in (3.14), and the superscriptΛ means the naive S-matrix computed with virtual bosons only of energiesgreater than Λ. Given the explicit form of F , we see that the entropy scaleslike the log of the infrared detector resolution E.3.5 DiscussionAccording to the solution of the infrared catastrophe advocated in [1, 2, 25],an infinite number of very low-energy photons and gravitons are produced51during scattering events. We have shown that if taken seriously, consideringthis radiation as lost to the environment completely decoheres almost anymomentum state of the outgoing hard particles. The basic idea is simple:the radiation is essentially classical, so any two scattering events are easy todistinguish by their radiation.The physical content of this result is somewhat unclear. A conservativeview is that the methodology of [1, 2, 25] is ill-suited to finding outgoingdensity matrices. As remarked earlier, in this formalism, one must trace theradiation to get well-defined transition probabilities. An alternative wouldbe to use the infrared-finite S-matrix program [3, 4, 26, 27], in which notrace over radiation is needed at all. But then we need to understand wherethe physical low-energy radiation is within that formalism–since after all, aphoton that is lost to the environment certainly does decohere the system.The decoherence found here is for the momentum states of the particles:at lowest order in their momenta, soft bosons do not lead to decoherence ofspin degrees of freedom. However, the sub-leading soft theorems [33–35] doinvolve the spin of the hard particles, so going to the next order in the softparticles would be interesting. We would also like to understand to whatextent our answers depend on the infinite-time approximation used in theS-matrix approach.To end, we comment on potential applications to the black hole informa-tion paradox. The idea advocated in [24, 28] is that correlations between thehard and soft particles mean that information about the black hole state canbe encoded into soft radiation. In [29, 36, 37], the dressed-state formalismand soft factorization has been used to argue that the soft particles simplyfactor out of the S-matrix and thus contain no such information. In theapproach used here, it is manifest that the outgoing hard state and outgo-ing soft state are highly correlated, leading to the decoherence of the hardstate. The outgoing density matrix for the hard particles, while not com-pletely thermal, has been mixed in momentum as much as possible whileretaining consistency with standard QED/perturbative gravity predictions.It is tempting to conjecture that this generalizes to all asymptotically mea-surable quantum numbers.At high center-of-mass energies√s, black holes should have productioncross-sections given by their geometric areas σprod ∼ pir2h(√s).[38] Using thisin (3.18), one obtains a hard-soft entanglement entropy scaling like the blackhole area times logarithmic soft factors. In this sense one might view thesoft radiation as containing a significant fraction of the black hole entropy.52Chapter 4Dressed Infrared Quantum In-formationWe study information-theoretic aspects of the infrared sector of quantumelectrodynamics, using the dressed-state approach pioneered by Chung, Kib-ble, Faddeev-Kulish and others. In this formalism QED has an IR-finiteS-matrix describing the scattering of electrons dressed by coherent statesof photons. We show that measurements sensitive only to the outgoingelectronic degrees of freedom will experience decoherence in the electronmomentum basis due to unobservable photons in the dressing. We makesome comments on possible refinements of the dressed-state formalism, andhow these considerations relate to the black hole information paradox.4.1 IntroductionThere are two common methods for dealing with infrared divergences inquantum electrodynamics. One is to form inclusive transition probabilities,tracing over arbitrary low-energy photon emission states.[1, 2, 25] However,one may wish to retain an S-matrix description instead of working directlywith probabilities. To this end, a long literature initiated by Chung, Kib-ble, and Faddeev-Kulish has advanced a program in QED where one formsan infrared-finite S-matrix between states of charges “dressed” by long-wavelength photon modes.[3, 4, 8, 26, 39–41] The extension to perturbativegravity in flat spacetime has been initiated in [27].In the inclusive probability formalism, one is forced to trace out softphotons to get finite answers. In previous work, we showed that this leadsto an almost completely decohered density matrix for the outgoing stateafter a scattering event.[42] This paper analyses the situation in dressedstate formalisms, in which no trace over IR photons is needed to obtain afinite outgoing state. However, consider the measurement of an observablesensitive only to electronic and high-energy photonic degrees of freedom. Weshow that for such observables, there will be a loss of coherence identicalto that obtained in the inclusive probability method. Quantum informationis lost to the low-energy bremsstrahlung photons created in the scattering53process.The primary goal of this paper is to give concrete calculations exhibitingthe dressed formalism and how it leads to decoherence. To this end, wework with the formulas from the papers of Chung and Faddeev-Kulish. Theresult of this calculation should carry over identically to any of the exist-ing refinements of Chung’s formalism. In section 4.4, we make a numberof remarks on possible refinements to the basic dressing formalism, give anexpanded physical interpretation of our results, and relate our work to liter-ature in mathematical physics on QED superselection rules. In section 4.5we make remarks on how this work fits into the recent literature on the blackhole information paradox; in brief, we believe that our results are consistentwith the recent proposal of Strominger [43], but not the original proposal ofHawking, Perry and Strominger.[24, 28]4.2 IR-safe S-matrix FormalismFollowing Chung, we study an electron with incoming momentum p scatter-ing off a weak external potential. This 1→ 1 process is simple and sufficientto understand the basic point; at the end of the next section, we show howto generalize our results to n-particle scattering. The electron spin will beunimportant for us and we supress it in what follows. The standard free-field Fock state |p〉 for the electron is promoted to a dressed state ‖p〉〉 asfollows. For a given photon momentum k we define the soft factorF`(k,p) =p · e`(k)p · k φ(k,p). (4.1)Here ` = 1, 2 labels the photon polarization states, and φ(k,p) is any func-tion that smoothly goes to φ→ 1 as |k| → 0. We introduce an IR regulator(“photon mass”) λ and an upper infrared cutoff E > λ, which can be thoughtof as the energy resolution of a single-photon detector in our experiment.LetRp = e2∑`=1∫λ<|k|<Ed3k√2k[F`(k,p)a†`(k)− F ∗` (k,p)a`(k)](4.2)54and define the single-electron dressing operatorWp = exp {Rp}= Np exp{e2∑`=1∫d3k√2kF`(p,k)a†`(k)}× exp{−e2∑`=1∫d3k√2kF ∗` (p,k)a`(k)},(4.3)where in the second line, we have put this coherent-state displacement op-erator into its normal-ordered form, with normalization factor 1Np = exp{−e222∑`=1∫d3k2k|F`(p,k)|2}. (4.4)Here and in the following all momentum-space integrals are evaluated in theshell λ < |k| < E. The dressed single-electron state ‖p〉〉 is then defined as‖p〉〉 = Wp |p〉 . (4.5)This consists of the electron and a coherent state of on-shell, transversely-polarized photons.Consider now an incoming dressed electron scattering into a superpo-sition of outgoing dressed electron states. The outgoing state is, to lowestorder in perturbation theory in the electric charge,|ψ〉 =∫d3qSqp‖q〉〉. (4.6)At higher orders there will be additional photons in the outgoing state; asexplained in the next section, these will not affect the infrared behavior stud-ied here, so we ignore them for now. Here the S-matrix is just the standardFeynman-Dyson time evolution operator, evaluated between dressed states.That is,Sqp = 〈〈q|S|p〉〉, (4.7)1This factor diverges, so these states have zero norm. In this sense, the dressed-state formalism simply re-organizes the calculations such that the divergences are in thedefinitions of the states ‖p〉〉 instead of the S-matrix elements. We view this is a majordifficulty with these formalisms, and understanding this better would be very useful. Seeeg. [44] for some ideas in this direction.55with S = T exp(−i ∫∞−∞ V (t)dt) as usual.[21] As calculated by Chung, thedressed 1→ 1 elements of this matrix are independent of the IR regulator λand thus infrared-finite as we send λ→ 0. We can write the matrix elementSqp =(EΛ)ASΛqp (4.8)whereA = − e28pi2β−1 ln[1 + β1− β], β =√1− m4(p · q)2 . (4.9)The undressed S-matrix element on the right side means the amplitudecomputed by Feynman diagrams with photon loops evaluated only withphoton energies above Λ and evaluated between undressed electron states,that is, with no external soft photons. By definition, this quantity is infrared-finite; moreover, the dependence on the scale Λ cancels between the prefactorand SΛ.4.3 Soft Radiation and DecoherenceThe state (4.6) is a coherent superposition of states, each containing a bareelectron and its corresponding photonic dressing. The presence of hardphotons in the outgoing state will not change our conclusions below, so forsimplicity we ignore them. In particular, the density matrix formed fromthis state has off-diagonal elements of the formS∗q′pSqp‖q〉〉〈〈q′‖. (4.10)These states have highly non-trivial photon content. However, if one isdoing a measurement involving only the electron degree of freedom, thenthese photons are unobserved, and we can make predictions with the re-duced density matrix of the electron, obtained by tracing the photons out.The resulting electron density matrix has coefficients damped by a factorinvolving the overlap of the photon states, namelyρelectron =∫d3qd3q′S∗q′pSqpDqq′‖q〉〉〈〈q′‖ (4.11)where the dampening factor is given by the photon-vacuum expectationvalueDqq′ = 〈0|W †q′Wq|0〉 . (4.12)56Straightforward computation gives this factor asDqq′ = exp{−e222∑`=1∫d3k2k∣∣F`(q)− F ∗` (q′)∣∣2}= exp{−e2∫d3k2k(q − q′)2(q · k)(q′ · k)}.(4.13)In this integrand, since q and q′ are two timelike vectors with the sametemporal component, we have that the numerator is positive definite andthe denominator is positive. It is therefore manifest that we have D = 1 ifq = q′ and D = 0 otherwise, since the integral over d3k diverges in its lowerlimit. Thus, tracing the photons leads to an electron density matrix that iscompletely diagonalized in momentum space.It is noteworthy that the factor (4.13) depends only on properties of theoutgoing superposition; it has no dependence on the initial state. This mayseem surprising since we are tracing over outgoing radiation, the productionof which depends on both the initial and final electron state. The pointis that the damping factor measures the distinguishability of the radiationfields given the processes p → q and p → q′. The radiation field fora scattering process consists of two pieces added together: a term Aµ ∼pµ/p · k peaked in the direction of the incoming electron and a term Aµ ∼qµ/q · k peaked in the direction of the outgoing electron. The outgoingradiation fields with outgoing electrons q, q′ are then only distinguishableby the second terms here, since both radiation fields will have the same polein the incoming direction.The damping factor (4.13) is precisely what was found in [42], reducedto the problem of 1 → 1 scattering. The mechanism is the same: physical,low-energy photon bremsstrahlung is emitted in the scattering. These pho-tons are highly correlated with the electron state and thus, if one does notobserve them jointly with the electron, one will measure a highly-decoheredelectron density matrix. The only difference is bookkeeping: in the dressedformalism, the bremsstrahlung photons are folded into the dressed electronstates (the incoming/outgoing parts of the bremsstrahlung in the incom-ing/outgoing dressing, respectively). However, referring to “an electron” asa state including these soft photons is just an abuse of semantics. In anactual measurement of the electron momentum, one does not measure thesesoft photons.The dressed states are not energy eigenstates, and in fact contain statesof arbitrarily high total energy. This should be contrasted with the inclusive-57probability treatment used by Weinberg, which has a cutoff on both thesingle-photon energy E and the total outgoing energy contained by all thephotons ET ≥ E in the outgoing state [2]. This additional parameter,however, appears only in the ratio ET /E in Weinberg’s probability formulas,and one finds that the dependence on ET vanishes as ET → ∞. This canbe understood because what is important is the very low-energy behavior ofthe photons, so moving an upper cutoff has limited impact.We note that (4.6) does not include effects from the bremsstrahlung ofadditional soft photons beyond those in the dressing. There is no kinematicreason to exclude such photons, so the outgoing state should properly bewritten as|ψ〉 =∞∑n=0∑{`}∫d3qd3nkSq{k`};p‖q〉〉. (4.14)Here {k`} = {k1`1, . . . ,kn`n} is a list of n photon momenta and polariza-tions. By the dressed version of the soft photon factorization theorem (seeappendix), we have thatSqk`;p = Sqp × eO(|k|0) , (4.15)or in other words lim|k|→0 |k|Sqk`;p = 0. Thus, when we take a trace overn-photon dressed states in (4.14), we obtain a sum of additional decoherencefactors of the formDnmqq′ = 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〉 .(4.16)Evaluating the inner product using (4.3), one findsDnmqq′ ∼[2∑`=1∫d3k√2kRe(F`(q)− F`(q′))]n+m, (4.17)which is infrared-finite. Summing these contributions, which exponentiate,will not change the conclusion that (4.13) leads to vanishing off-diagonalelectron density matrix elements.Finally, we explain the generalization to n-electron states. We will findthat the same decoherence is found in the dressed formalism as in the inclu-sive formalism.[42] Following Faddeev-Kulish [4], we write the multi-particle58dressing operator by replacing (4.3) withRp → e2∑l=1∫d3p(2pi)3∫d3k√2k[Fl(k,p)a†l (k)− F ∗l (k,p)a†l (k)]ρ(p),(4.18)where we have introduced an operator which counts charged particles withmomentum p.ρ(p) =∑s(b†p,sbp,s − d†p,sdp,s), (4.19)and the b and d are electron and positron operators, respectively.2 As inthe one-particle case, additional photons do not affect the IR behaviour ofscattering amplitudes. Hence, we will ignore them and only consider thecase where the out-state is a linear superposition of dressed electron states.In that case we have to replace the outgoing momentum by list of momenta,q → β = {q1,q2, . . .} and similarly q′ → β′ = {q′1,q′2, . . .}. This results toa replacement in (4.13) ofF`(q)→∑n∈βF`(qn)F ∗` (q′)→∑m∈β′F ∗` (q′m).(4.20)Using the explicit form of F in the limit k → 0, the damping factor (4.13)then then becomesDββ′ = exp−e2 ∫ d3k2k∑m,n∈β,β′ηmηnpm · qn(qm · k)(qn · k) . (4.21)In this equation the labels m,n both run over the full set β∪β′, and ηn = +1if n ∈ β while ηn = −1 if n ∈ β′. This is precisely the quantity ∆Aββ′,αdefined in [42], so we see that the results of that paper carry over to thedressed formalisms used here.2Note that in the multi-particle case there is an infinite phase factor which needs tobe included in the definition of the S-matrix. Since this phase factor does not affect ourdiscussion, we ignore it in the following.594.4 Physical InterpretationDressed-state formalisms are engineered to provide infrared-finite transitionamplitudes, as opposed to inclusive probabilities constructed in the tradi-tional approach studied in [42]. In the dressed formalism, the outgoing state(4.6) is a coherent superposition of states ‖p〉〉 consisting of electrons plusdressing photons. However, if one does a measurement of an observable sen-sitive only to the electron state, the measurement will exhibit decoherencebecause the unobserved dressing photons are highly correlated to the elec-tron state. We have given a concrete calculation showing that the dampingfactor (4.21) is identical in either the dressed or undressed formalism.The physical relevance of this calculation rests on the idea that the ba-sic observable is a simple electron operator in Fock space. What would bemuch better would be to use a dressed LSZ reduction formula to understandthe asymptotic limits of electron correlation functions. [41, 44] Neverthe-less, the basic physical picture seems clear: in a scattering experiment, onedoes not measure an electron plus a finely-tuned shockwave of outgoingbremsstrahlung photons, just the electron on its own. This is responsiblefor well-measured phenomena like radiation damping.QED has a complicated asymptotic Hilbert space structure which is stillsomewhat poorly understood. For example, although Faddeev-Kulish tryto define a single, separable Hilbert space Has [4, 44] other authors haveargued that one needs an uncountable set of separable Hilbert spaces.[26, 41]Formally, this is related to the fact that the dressing operator (4.3) does notconverge on the usual Fock space. A related idea is that one can arguethat QED has an infinite set of superselection rules based on the asymptoticchargesQ(Ω) = limr→∞ r2Er(r,Ω) (4.22)defined by the radial electric field at infinity.[45, 46] We believe that thecalculations presented here and in [42] demonstrate the physical mechanismfor enforcing such a superselection rule. The charges (4.22), the currentsdefined in our previous work [42], and the large-U(1) charges defined in[7, 47] are presumably closely related, and working out the precise relationsis an interesting line of inquiry.4.5 Black Hole InformationThe recent resurgence of interest in infrared issues in QED and gravity wassparked largely by a proposal of Hawking, Perry and Strominger suggesting60that information apparently “lost” in the process of black hole formation andevolution could be encoded in soft radiation.[24, 28] The original proposalwas that there are symmetries which relate “hard” scattering (like the blackhole process) to soft scattering and thus led to constraints on the S-matrix.As emphasized by a number of authors, this is not true in the dressed stateapproach.[29, 36, 37, 48] As we review in the appendix, soft modes decouplefrom the dressed hard scattering event at lowest order, in the sense thatlimω→0[aω, Sdressed] = 0. Dropping a soft boson into the black hole willnot yield any information about the black hole formation and evaporationprocess.However, a more recent proposal due to Strominger is to simply positthat outgoing soft radiation purifies the outgoing Hawking radiation.[43]That is, the state after the black hole has evaporated is of the form |ψ〉 =∑a |a〉Hawking |a〉soft, such that the Hawking radiation is described by athermal density matrix, i.e. ρHawking = trsoft |ψ〉 〈ψ| ≈ ρthermal. We believethat both the results presented here and those in our previous work areconsistent with this proposal. In either the inclusive or dressed formalism,the final state of any scattering process contains soft radiation which ishighly correlated with the hard particles because the radiation is createddue to accelerations in the hard process. The open issue is to explain whythe hard density matrix coefficients behave thermally, which likely relies ondetails of the black hole S-matrix.4.6 ConclusionsWhen charged particles scatter, they experience acceleration, causing themto radiate low-energy photons. If one waits an infinitely long time (as man-dated by an S-matrix description), these photons cause severe decoherenceof the charged particle momentum state. This was first seen in [42] in thestandard formulation of QED involving IR-finite inclusive cross section, andhere we have shown the same conclusion holds in IR-safe, dressed formalismsof QED; they should carry over in a simple way to perturbative quantumgravity. These results constitute a sharp and robust connection between theinfrared catastrophe and quantum information theory, and should provideguidance in problems related to the infrared structure of gauge theories.61Chapter 5On the Need for Soft DressingIn order to deal with IR divergences arising in QED or perturbative quan-tum gravity scattering processes, one can either calculate inclusive quantitiesor use dressed asymptotic states. We consider incoming superpositions ofmomentum eigenstates and show that in calculations of cross-sections thesetwo approaches yield different answers: in the inclusive formalism no in-terference occurs for incoming finite superpositions and wavepackets do notscatter at all, while the dressed formalism yields the expected interferenceterms. This suggests that rather than Fock space states, one should useFaddeev-Kulish-type dressed states to correctly describe physical processesinvolving incoming superpositions. We interpret this in terms of selectionrules due to large U(1) gauge symmetries and BMS supertranslations.5.1 IntroductionQuantum electrodynamics and perturbative quantum gravity are effectivequantum field theories which describe the two long-ranged forces seen innature. They also both suffer from infrared divergences coming from vir-tual boson loops in Feynman diagrams in the perturbative computation ofthe S-matrix. These divergences exponentiate when resummed and set theamplitude for any process between a finite number of interacting particlesto zero. This is known as the infrared catastrophe.One proposed resolution of the infrared catastrophe is to consider onlyinclusive quantities, for example soft-inclusive transition probabilities in thecontext of scattering theory, which are defined by summing over the pro-duction of any number of soft photons and gravitons. In the case of elec-trodynamics, this resolution dates back to Bloch and Nordsieck [1, 25] and,in perturbative quantum gravity, it was developed by Weinberg [2]. Thecontributions from emitted soft bosons cancel the IR divergences from thevirtual loops. An upshot of this solution of the infrared problem is the factthat, in QED, any non-trivial scattering process involving charged particlesinevitably produces a cloud of an infinite number of arbitrarily soft photons.In the case of quantum gravity, soft gravitons are produced, and, since allparticles carry gravitational charge, IR divergences arise in any scattering62process. The use of inclusive probabilities is justified by the assumptionthat the softest photons and gravitons must escape detection. These bosonscarry very little energy and have a negligible effect on the kinematics of theprocess. However, it was recently shown that they carry a lot of informationin the sense that their quantum states are highly entangled with those ofthe charged particles. The loss of the soft particles results in decoherenceof the final state of the hard particles, where the momentum eigenstates forelectrically or gravitationally charged particles are the pointer basis [42, 49].See refs. [50–52] for related work.The infrared catastrophe can be traced back to the long-ranged nature ofthe interactions which is in conflict with the assumption of asymptotic decou-pling needed to formulate scattering theory [53]. An approach to the infraredproblem, alternative to using inclusive probabilities, is to use dressed stateswhich are defined by including the aforementioned clouds of soft photons andgravitons with the asymptotic states [3, 4, 26, 27, 41, 44, 54–56]. Faddeevand Kulish argued that such an approach diagonalizes the correct asymp-totic Hamiltonian and therefore yields the asymptotic decoupling which isnecessary for a satisfactory formulation of scattering theory. The detailedstructure of the coherent states can be adjusted so as to cancel the infrareddivergences in the S-matrix, providing an IR-finite S-matrix and scatteringprobabilities. However, the out-going states still contain particles accompa-nied by soft photon and graviton clouds. One can ask the same question:given these infrared safe states, what is the nature of the state of the out-going hard particles? The answer is that precisely the same decoherence isfound to occur in either the inclusive or dressed approaches [57], i.e. thereis still a lot of information in the entanglement between the hard particlesand the radiation.1Both the dressed and inclusive formalisms are designed to give the samepredictions for the probability of scattering from an incoming set of momentap1, . . . ,pn into an outgoing set of momenta p′1, . . . ,p′m. The measurementof observables which only depend on the hard particles should be predictablefrom the reduced density matrix obtained by tracing over soft bosons, whichare invisible to a finite size detector. Given an incoming momentum eigen-state the two formalisms agree. Thus, one might naively think for calculatingcross-sections it does not matter which formalism one chooses. We show inthis paper that this is not the case: the two approaches differ in their treat-ment of incoming superpositions. Consider a simple superposition of two1Note, there are also other proposals for how to define an IR finite density matrix [58],which we will not discuss here.63momentum eigenstates for a single charged particle|ψ〉 = 1√2(|p〉+ |q〉), (5.1)scattering off of a classical potential. We expect the out-state to be describedby a density matrix of the formρ =12S (|p〉 〈p|+ |p〉 〈q|+ |q〉 〈p|+ |q〉 〈q|)S†. (5.2)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〉 arecorrectly dressed states, this expectation is indeed correct. In the inclusiveformalism, however, where |p〉, |q〉 are Fock space momentum eigenstates,there is no interference between the different momenta as opposed to thediagonal terms of (5.2). We find that the diagonal entries of the densitymatrix which encode the cross-sections are of the formσψ→out ∝ 〈out| ρincl |out〉 = 12〈out| S (|p〉 〈p|+ |q〉 〈q|)S† |out〉 . (5.3)In other words, the cross-section behaves as if we had started with a classicalensemble of states with momenta p and q. The entire scattering history isdecohered by the loss of the soft radiation. This appears to contrast starklywith any realistic experiment.Moreover, as we will show, repeating the analysis for wavepackets, e.g.|ψ〉 = ∫ dpf(p) |p〉, leads to the nonsensical conclusion that a wave-packetis not observed to scatter at all. However, in the dressed state formalism ofFaddeev-Kulish the interference appears as in equation (5.2). This stronglysuggests that scattering theory in quantum electrodynamics and perturba-tive quantum gravity should really not be formulated in terms of standardFock states of charged particles. Formulating the theories using dressedstates seems to be a good alternative.Dressed states also arise naturally in the recent discussions of asymptoticgauge symmetries [7, 28, 47, 59–61], which imply the existence of selectionsectors[8, 10, 36, 62]. See also [63, 64] for work on soft charges and dressingin holography. Our findings have a nice interpretation in the language ofthis program: only superpositions of states within the same selection sectorcan interfere. This explains the failure of the undressed approach. In theinclusive formalism, essentially any pair of momentum eigenstates live in dif-ferent charge sectors. In contrast, the Faddeev-Kulish formalism is designed64so that all of the dressed states live within the same charge sector.Our results can also be viewed in the context of the black hole informa-tion problem [13, 65]. In particular, Hawking, Perry, and Strominger [9] andStrominger [14] have recently suggested that black hole information may beencoded in soft radiation. In black hole thought experiments, one typicallyimagines preparing an initial state of wavepackets organized to scatter withhigh probability to form an intermediate black hole. Our results suggestthen that one needs to use dressed initial states to study this problem. Seealso [29, 37] for some remarks on the use of dressed or inclusive formalismsfor studying black hole information.The rest of the paper is organized as follows. We start by presentingthe calculations showing that the dressed and undressed formalism disagreein section 5.2 for discrete superpositions and in section 5.3 for wavepackets.The discussion and interpretation of the results takes place in section 5.4.There, we will argue why our findings imply that dressed states are bettersuited to describe scattering than the inclusive Fock-space formalism. Wewill give a new very short argument for the known result of [10] that thedressing operators and the S-matrix weakly commute and argue for a moregeneral form of dressing beyond Faddeev-Kulish. We will then interpret ourresults in terms of asymptotic symmetries and selection sectors before con-cluding in section 5.5. The appendix contains proofs of certain statementsin sections 5.2 and 5.3.5.2 Scattering of Discrete SuperpositionsIn this and the next section we generalize the results of [42] to the case ofincoming superpositions of momentum eigenstates. We begin in this sectionby studying discrete superpositions |ψ〉 = |α1〉 + · · · + |αN 〉 of states withvarious momenta α = p1,p2, . . .. We will see that the dressed and inclusiveformalisms give vastly different predictions for the probability distributionof the outgoing momenta: dressed states will exhibit interference betweenthe αi whereas undressed states do not.655.2.1 Inclusive FormalismConsider scattering of an incoming superposition of charged momentumeigenstates|in〉 =N∑ifi |αi〉 , (5.4)with∑i |fi|2 = 1. The outgoing density matrix vanishes due to IR diver-gences in virtual photon loops. However, we can obtain a finite result if wetrace over outgoing radiation [1, 2, 25, 42]. The resulting reduced densitymatrix of the hard particles takes the formρ =∑bN∑i,j∫∫dβ dβ′fif∗j Sβb,αiS∗β′b,αj |β〉 〈β′| , (5.5)where β and β′ are lists of the momenta of hard particles in the outgoingstate, and the sum over b denotes the trace over soft bosons. We will beinterested in the effect of 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 diver-gences by introducing an IR cutoff (e.g. a soft boson mass λ). Followingthe standard soft photon resummation techniques [2], one finds that the to-tal effect of these divergences yields reduced density matrix elements of theformρββ′ =N∑i,jfif∗j SΛβαiSΛ∗β′αjλ∆Aββ′,αiαj+∆Bββ′,αiαjFββ′,αiαj (E,ET ,Λ). (5.6)Here we have introduced “UV” cutoffs Λ, E on the virtual and real softboson energies, so SΛ are S-matrix elements with the soft boson loops cutoff below Λ and we only trace over outgoing bosons with individual energiesup to E and total energy ET . The explicit form of the Sudakov rescalingfunction F can be found in [42]. What concerns us here is the behavior ofthis expression in the limit where we remove the IR regulator λ→ 0, which66is controlled by the exponents∆Aββ′,αα′ = −12∑n,n′∈α,α¯′,β,β¯′enen′ηnηn′8pi2β−1nn′ ln[1 + βnn′1− βnn′],∆Bββ′,αα′ = −12∑n,n′∈α,α¯′,β,β¯′mnmn′ηnηn′16pi2M2pβ−1nn′1 + β2nn′√1− β2nn′ln[1 + βnn′1− βnn′].(5.7)The factor ηn is defined as +1 (−1) if particle n is incoming (outgoing). Thequantities βnn′ =√1− m2nm2n′(pn·pn′ )2 are the relative velocities between pairs ofparticles and a bar interchanges incoming states for outgoing and vice versa.The expressions for ∆A and ∆B come from contributions of soft photonsand gravitons, respectively. The question now is which terms survive.The special case of no superposition, αi = αj = α, was discussed in [42].There it was shown that ∆Aββ′,αα ≥ 0 and ∆Bββ′,αα ≥ 0, so that in thelimit λ → 0, all of the terms in the sum except those with ∆A = ∆B = 0will vanish. The equality holds if and only if the out states β and β′ containparticles such that the amount of electrical charge and mass carried withany choice of velocity agrees for β and β′. This can be phrased in terms ofan infinite set of operators which measure charges flowing along a velocityv. These are defined asjˆemv =∑ieia†i,pi(v)ai,pi(v),jˆgrv =∑iEi(v)a†i,pi(v)ai,pi(v),jˆgr,0v =∑i∫dω ωa†i,vωai,vω,(5.8)for charged particles, massive particles and hard massless particles, respec-tively. The sum runs over all particle species. Clearly, momentum eigen-states are also eigenstates of these operators. Using these operators, theequality of currents can be expressed asjˆv |β〉 ∼ jˆv |β′〉 , (5.9)where the tilde means that the eigenvalues of the states are the same onboth sides for all velocities. In appendix D, we show that the more general67exponents ∆Aββ′,αα′ and ∆Bββ′,αα′ are positive. Similarly to the argumentin [57], one can show that ∆A and ∆B are non-zero if and only ifjˆv |αi〉+ jˆv |β′〉 ∼ jˆv |αj〉+ jˆv |β〉 , (5.10)that is if the list of hard currents in states |α〉 and |β′〉 is the same as thelist of hard currents in states |α′〉 and |β〉. An easy way to understandthe form of equation (5.10) is by looking at equation (5.7). There, thebar over α′ (which corresponds to αj) indicates that it should be treatedas an outgoing particle, i.e. similarly to β. On the other hand β¯′ shouldbe treated similarly to α. Hence, we obtain equation (5.10) from (5.9) byreplacing αi → αi + β′ and αj → αj + β. On the other hand it is clear thatin the case of |αi〉 = |αj〉 = |α〉 equation (5.10) reduces to equation (5.9).Armed with these results, we can calculate the cross-sections given anincoming superposition. These are proportional to the diagonal elementsβ = β′ of the density matrix; for simplicity we ignore forward scatteringterms. The diagonal terms of the density matrix (5.6) are proportionalto λ∆A+∆B. This factor reduces to unity if jˆv |αi〉 ∼ jˆv |αj〉 for all of thecurrents (5.8) and is zero otherwise. For a generic superposition, this impliesthat only terms with i = j contribute and we findσin→β ∝ ρββ =N∑i,jfif∗j Fββ,αiαjSΛβαiSΛ∗βαjδαiαj =N∑i|fi|2|SΛβ,αi |2Fββ,αiαi .(5.11)As we see, no interference terms between incoming states are present. In-stead, the total cross-section is calculated as if the incoming states were partof a classical ensemble with probabilities |fi|2. The reason is that in the in-clusive approach the information about the interference is carried away byunobservable soft radiation. To define the scattering cross-section, however,we need to trace out the soft radiation and we obtain the above prediction,which is at odds with the naive expectation, equation (5.2).5.2.2 Dressed FormalismThe calculation above was done using the usual, undressed Fock states ofhard charges, which required to calculate inclusive cross-sections. The al-ternative approach we will now turn to is to consider transitions betweendressed states. For concreteness, we will follow the dressing approach of68Chung and Faddeev-Kulish2, which contains charged particles accompaniedby a cloud of real bosons which radiate out to lightlike infinity [3, 4, 27].For a given set of momenta α = p1,p2, . . ., we write the dressed state as3‖α〉〉 ≡Wα |α〉 . (5.12)The operator Wα equips the state |α〉 with a cloud of photons/gravitons.For QED, Wα is the unitary operator (with a finite IR cutoff λ)Wα ≡ exp{e2∑l=1∫ Eλd3k√2k(Fl(k, α)al†k − F ∗l (k, α)alk)}, (5.13)where al†k creates a photon in the polarization state l and the soft factorFl(k, α) =∑p∈αl · pk · p φ(k,p) (5.14)depends on the polarization vectors l and some smooth, real functionφ(k,p) which goes to 1 as |k| → 0. Letting W act on Fock space statesfor λ = 0 gives states with vanishing normalization, hence in the strictλ→ 0 limit W is no good operator on Fock space. Thus, as before, we willdo calculations with finite λ and only at the end we will take λ→ 0.4The Faddeev-Kulish construction was adapted to perturbative quantumgravity in [27]. In this case the dressing has the same form as equation(5.13), the only difference being that a (a†) is now a graviton annihilation(creation) operator and the functions F depend on the polarization tensorµν [27],F grl (k, α) =∑p∈αpµµνl pνk · p φ(k,p). (5.15)S-matrix elements taken between dressed statesSβα ≡ 〈〈β‖S‖α〉〉 = 〈β|W †βSWα |α〉 (5.16)2Recently, a generalization of Faddeev-Kulish states was suggested [8]. We will extendour discussion to those states in section 5.4.3The double bracket notation is due to [29]. The previous paper of the authors [57]used |α˜〉 to denote dressed states. The authors regret this life decision.4Note that as argued in [4], a proper definition of W in the limit λ → 0 should bepossible on a von Neumann space.69are independent of λ and thus finite as λ → 0. The Sudakov factor F iscontained in the dressed S-matrix elements.5Consider now an incoming state consisting of a discrete superposition ofsuch dressed states,‖in〉〉 =∑ifi‖αi〉〉. (5.17)The outgoing density matrix is thenρ =∑i,j∫∫dβdβ′fif∗j SβαiS∗β′αj‖β〉〉〈〈β′‖. (5.18)This density matrix is formally unitary, however, every experiment shouldbe able to ignore soft radiation. Following [42], we treat the soft modes asunobservable and trace them out. This yields the reduced density matrixfor the outgoing hard particles,ρhardββ′ =∑i,jfif∗j SβαiS∗β′αj 〈0|W †βWβ′ |0〉 . (5.19)The last term is the photon vacuum expectation value of the out-state dress-ing operators. This factor reduces to one or zero as shown in [42]; one ifjˆ(β) ∼ jˆ(β′) and zero otherwise. This is responsible for the decay of mostoff-diagonal elements in (5.19). However, if we are interested in the cross-section for a particular outgoing state β, this is again given by a diagonaldensity matrix element,σin→β ∝ ρββ =∑i,jfif∗j SβαiS∗βαj. (5.20)In stark contrast to the result obtained in the inclusive formalism, equation(5.11), this cross-section exhibits the usual interference between the variousincoming states, c.f. equation (5.2). The reason for this is that in the dressedformalism, the outgoing radiation is described by the dressing which onlydepends on the out-state and not on the in-state. We will discuss this inmore detail in section 5.4. This establishes that the inclusive and dressedformalism are not equivalent but yield different predictions for cross-sections5The actual definition of the S-matrix should also contain a term to cancel the infiniteCoulomb phase factor. Since this is immaterial to the current discussion we neglect thissubtlety.70of finite superpositions.5.3 WavepacketsWe will now proceed to look at scattering of wavepackets and find that theresult is even more disturbing. After tracing out infrared radiation in theundressed formalism, no indication of scattering is left in the hard system.On the contrary, once again we will see that with dressed states, one getsthe expected scattering out-state.5.3.1 Inclusive FormalismWe consider incoming wavepackets of the form|in〉 =∫dαf(α) |α〉 , (5.21)normalized such that∫dα|f(α)|2 = 1. The full analysis of the precedingsection still applies, provided we replace∑αi→ ∫ dα, αi → α, fi → f(α)and similarly for aj → α′. The only notable exception is the calculation ofsingle matrix elements as in equation (5.11), which now readsρββ =∫∫dαdα′f(α)f∗(α′)SΛβ,αSΛ∗β,α′δαα′ . (5.22)Note that here, by the same argument as before, the λ-dependent factor isturned into a Kronecker delta, which now reduces the integrand to a measurezero subset on the domain of integration. The only term that survives theintegration is the initial state, which is acted on with the usual Dirac deltaδ(α − β), i.e. the “1” term in S = 1 − 2piiM. The detailed argument canbe found in appendix E. Thus we conclude thatρoutββ′ = f(β)f∗(β′) = ρinββ′ . (5.23)The hard particles show no sign of a scattering event.715.3.2 Dressed WavepacketsThe dressed formalism has perfectly reasonable scattering behavior. Con-sider wavepackets built from dressed states‖in〉〉 =∫dα f(α)‖α〉〉, (5.24)with ‖α〉〉 a dressed state in the same notation as in equation (5.12). TheS-matrix applied on dressed states is infrared-finite and the outgoing densitymatrix can be expressed asρ =∫∫dβdβ′∫∫dαdα′f(α)f∗(α′)SβαS∗β′α′‖β〉〉〈〈β′‖. (5.25)Tracing over soft modes, we findρββ′ =∫∫dαdα′f(α)f∗(α′)SβαS∗β′α′ 〈W †βWβ′〉 . (5.26)Again the expectation value is taken in the photon vacuum. The crucialpoint here is that this factor is independent of the initial states α. Uponsending the IR cutoff λ to zero, the expectation value for W †W takes onlythe values 1 or 0, leading to decoherence in the outgoing state, but thecross-sections still exhibit all the usual interference between components ofthe incoming wavefunction,ρββ =∫∫dαdα′f(α)f∗(α′)SβαS∗βα′ , (5.27)unlike in the inclusive formalism.5.4 ImplicationsIn this section we will discuss the implications of our results and general-ize and re-interpret our findings in particular in view of asymptotic gaugesymmetries in QED and perturbative quantum gravity.5.4.1 Physical InterpretationThe reason for the different predictions of the inclusive and dressed formal-ism is the IR radiation produced in the scattering process. The key idea isthat accelerated charges produce radiation fields made from soft bosons. In72the far infrared, the radiation spectrum has poles as the photon frequencyk0 → 0 of the form pi/pi · k, where pi are the hard momenta. These polesreflect the fact that the radiation states are essentially classical and arecompletely distinguishable for different sets of asymptotic currents jˆv.In the inclusive formalism, we imagine incoming states with no radiation,and so the outgoing radiation state has poles from both the incoming hardparticles α and the outgoing hard particles β. In the dressed formalism,the incoming part of the radiation is instead folded into the dressed state‖α〉〉, which in the Faddeev-Kulish approach is designed precisely so thatthe outgoing radiation field only includes the poles from the outgoing hardparticles. Thus if we scatter undressed Fock space states, a measurementof the radiation field at late times would completely determine the entiredynamical history of the process α → β, leading to the classical answer(5.11). If we instead scatter dressed states, the outgoing radiation has in-complete information about the incoming charged state, which is why thevarious incoming states still interfere in (5.20). Given that this type of in-terference is observed all the time in nature, this seems to strongly suggestthat the dressed formalism is correct for any problem involving incomingsuperpositions of momenta.Based on the result of section 5.2, one might argue that equation (5.11)perhaps is the correct answer and one would have to test experimentallywhether or not interference terms appear if we give a scattering processenough time so that the decoherence becomes sizable. After all, the in-clusive and dressed approach to calculating cross-sections are at least inprinciple distinguishable, although maybe not in practice due to very longdecoherence times. However, we have demonstrated in section 5.3 that theinclusive formalism predicts an even more problematic result for continu-ous superpositions, namely that no scattering is observed at all. We thuspropose that using the dressed formalism is the most conservative and phys-ically sensible solution to the problem of vanishing interference presented inthis paper.5.4.2 Allowed DressingsDressing operators weakly commute with the S-matrixIt was conjectured in [8] and proven in [10] that the far IR part of thedressing weakly commutes with the S-matrix to leading order in the energyof the bosons contained in the dressing. In particular, this means that the73amplitudes〈β|W †βSWα |α〉 ∼ 〈β|W †βWαS |α〉 ∼ 〈β|SW †βWα |α〉 (5.28)are all IR finite, while they might differ by a finite amount. A short proofof this in QED, complementary to [10], can be given as follows (the gravi-tational case follows analogously). Recall that Weinberg’s soft theorem forQED states that to lowest order in the soft photon momentum q of outgoingsoft photons〈l1al1q1 . . . lNalNqNS〉 ∼N∏i=1 M∑jηjejli · pjqi · pj 〈S〉 . (5.29)A similar argument holds for incoming photons. For incoming photons withmomentum q we find that〈S∗l1al1†q1 . . . ∗lNalN †qN 〉 ∼N∏i=1− M∑jηjej∗li · pjqi · pj 〈S〉 . (5.30)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 thatthe momentum in the denominator of the propagator changes (p−q)2+m2 →(p+q)+m2 and vice versa. For small momentum, the denominator becomes−2pq → 2pq. From this it directly follows that for general dressings atleading order in the IR divergences,〈SW 〉 = 〈Se∫d3k(Fl(k)al†k−F ∗l (k)alk)〉 ∼ N 〈Se∫d3kFl(k)al†k 〉∼ N 〈e−∫d3kF ∗l (k)alkS〉∼ 〈e∫d3k(Fl(k)al†k−F ∗l (k)alk)S〉 = 〈WS〉 .(5.31)In the first and third step we have split the exponential using the Baker-Campbell-Hausdorff formula (N is the normalization which is finite for finiteλ) and in the second equality we have used Weinberg’s soft theorem foroutgoing and incoming particles.74Dressings cannot be arbitrarily moved between in- and out-statesThis opens up the question about the most general structure of a consistentFaddeev-Kulish-like dressing. For example, one could ask whether one canconsistently define S-matrix elements with the dressing only acting on theout-state. To answer this question, we assume that the dressing of the out-state has the same IR structure as equation (5.13), but is more general inthat it may also include the momenta of (some) particles of the in-state, i.e.Wβ → WβWα˜ or any other momenta which might not even appear in theprocess, WβWα˜ → WβWα˜Wζ . The IR structure of the in-dressing is thenfixed by the requirement that the S-matrix element is finite. In addition tothe requirement of IR-finiteness we ask that the so defined S-matrix elementsgive rise to the correct rules for superposition and the correct scattering forwavepackets, even after tracing out soft radiation.Applying the logic of the previous sections and [57], one finds that tracingover the soft bosons yields for a diagonal matrix element ρββρhardββ =∑i,jfif∗j SβαiS∗β′α′j〈0|W †α˜′Wα˜|0〉 (5.32)andρhardββ =∫∫dαdα′f(α)f∗(α′)SβαS∗β′α′ 〈0|W †α˜′Wα˜|0〉 (5.33)for finite and continuous superpositions, respectively. Here, we have usedthat〈W †α˜′W †β′WβWα˜〉∣∣∣β=β′= 〈W †α˜′Wα˜〉 . (5.34)The expectation value is taken in the soft boson Fock space. The expressionin the case of α˜ = α and α˜′ = α′ was already encountered in sections 5.2and 5.3 in the context of inclusive calculations, where it was responsible forthe unphysical form of the cross-sections. By the same logic it follows thateven in the case where α˜ is a proper subset of α, we will obtain a Kroneckerdelta which sets α˜ = α˜′ and we again do not obtain the expected form of thecross-section. Instead, particles from the subset α˜ will cease to interfere. Wethus conclude that the dressing of the out-states must be independent of thein-states and it is not consistent to build superposition of states which aredressed differently. This means that building superpositions from hard andcharged Fock space states is not meaningful. In particular, we cannot useundressed states to span the in-state space by simply moving all dressings75to the out-state.Generalized Faddeev-Kulish statesHowever, it would be consistent to define dressed states by acting with aconstant dressing operator Wζ for fixed ζ on states ‖α〉〉,‖α〉〉ζ ≡W †ζWα |α〉 . (5.35)Physically this corresponds to defining all asymptotic states on a fixed, co-herent soft boson background, defined by some momenta ζ. This state doesnot affect the physics since soft modes decouple from Faddeev-Kulish am-plitudes [29] and thus this additional cloud of soft photons will just passthrough the scattering process. The difference between the Faddeev-Kulishdressed state ‖α〉〉 and the generalized states of the form ‖α〉〉ζ is that thestate ‖ζ〉〉ζ = W †ζWζ |ζ〉 = |ζ〉 does not contain additional photons. This alsoexplains why QED calculations using momentum eigenstates without anyadditional dressing give the correct cross-sections once we trace over softradiation. Such a calculation can be interpreted as happening in a set ofdressed states defined by‖α〉〉in = W †inWα |α〉 , (5.36)such that the in-state ‖in〉〉in does not contain photons and looks like astandard Fock-space state.Localized particles are accompanied by radiationWe also conclude from the previous sections that there are no charged, nor-malizable states which do not contain radiation. The reason is that withineach selection sector there is only one non-normalizable state which does notcontain radiation. Thus building a superposition to obtain a normalizablestate will necessarily include dressed states which by definition contain softbosons. A nice argument which makes this behavior plausible was given byGervais and Zwanziger [45], see figure 5.1.5.4.3 Selection SectorsEverything said so far has a nice interpretation in terms of the charges Q±εof large gauge transformations (LGT) for QED and supertranslation forperturbative quantum gravity. For a review see [61]. Large gauge transfor-76(a)Σ(b) (c)Figure 5.1: (a) A plane wave goes through a single slit and emerges as alocalized wavepacket. The scattering of the incoming wavepacket results inthe production of Bremsstrahlung. (b) We can also define some Cauchy sliceΣ and create the state by an appropriate initial condition. (c) Evolving thisstate backwards in time while forgetting about the slit results in an incominglocalized particle which is accompanied by a radiation shockwave.mations in QED are gauge transformations which do not die off at infinity.They are generated by an angle-dependent function ε(φ, θ). Similarly, su-pertranslations in perturbative quantum gravity are diffeomorphisms whichdo not vanish at infinity. They are constrained by certain falloff conditions.The transformations are generated by an infinite family of charges Q±ε atfuture and past lightlike infinity, parametrized by a functions ε(φ, θ) on thecelestial sphere. The charges split into a hard and a soft partQ±ε = Q±H,ε +Q±S,ε. (5.37)The soft charge generates the transformation on zero frequency photons orgravitons and leaves undressed particles invariant, while the hard charge gen-erates LGT or supertranslations of charged particles, i.e. electrons in QEDand all particles in perturbative quantum gravity. The action on particlescan be found in [7, 36, 47, 62].The charges Q±ε are conserved during time evolution (and in particularin any scattering process) and thus give rise to selection sectors of QEDand gravity. These selection sectors give a different perspective on the IRcatastrophe: Fock states of different momenta are differently charged underQ±ε and thus cannot scatter into each other. For dressed states, the sit-uation is different: It was shown in [8, 10, 36] that for QED and gravity,Faddeev-Kulish dressed states ‖α〉〉 are eigenstates of Q±ε with an eigenvalue77independent of α.It turns out that also our generalized version of Faddeev-Kulish states‖α〉〉ζ , equation (5.35), are eigenstates of the generators Q±ε with eigenvalueswhich depend on ζ. To see this note that [36][Q±ε ,W †ζ ] = [Q±S,ε,W †ζ ] ∝∫S2dqˆζ2ζ · qˆ ε(φ, θ), (5.38)and similarly for gravity [10]. Thus the generalized Faddeev-Kulish statesspan a space of states which splits into selection sectors parametrized byζ. The statement that we can build physically reasonable superpositionsusing generalized Faddeev-Kulish states translates into the statement thatsuperpositions can be taken within a selection sector of the LGT and super-translation charges Q±ε .In the context of these charges, zero energy eigenstates of Q±S,ε are ofteninterpreted as an infinite set of vacua. Note that the name vacuum might bemisleading as states in a single selection sector are in fact built on differentvacua. Our results also raise doubt on whether physical observables existwhich can take a state from one selection sector into another. If they did wecould use them to create a superpositions of states from different sectors.But as we have seen above, in this case interference would not happen, whichis in conflict with basic postulates of quantum mechanics.5.5 ConclusionsCalculating cross-sections in standard QED and perturbative quantum grav-ity forces us to deal with IR divergences. Tracing out unobservable softmodes seems to be a physically well-motivated approach which has success-fully been employed for plane-wave scattering. However, as we have shownthis approach fails in more generic examples. For finite superpositions itdoes not reproduce interference terms which are expected; for wavepacketsit predicts that no scattering is observed. We have demonstrated in thispaper that dressed states a` la Faddeev-Kulish (and certain generalizations)resolve this issue, although it is not clear if the inclusive and dressed for-malism are the only possible resolutions. Importantly, we have shown thatpredictions of different resolutions can disagree, making them distinguish-able.Superpositions must be taken within a set of states with most of thestates dressed by soft bosons. The corresponding dressing operators areonly well-defined on Fock space if we use an IR-regulator which we only78remove at the end of the day. In the strict λ→ 0 limit, the states are not inFock space but rather in the much larger von Neumann space which allowsfor any photon content, including uncountable sets of photons [41, 54]. Thissuggests an interesting picture which seems worth investigating. The Hilbertspace of QED is non-separable but has separable subspaces which are stableunder action of the S-matrix and form selection sectors. These subspaces arenot the usual Fock spaces but look like the state spaces defined by Faddeevand Kulish [4], in which almost all charged states are accompanied by softradiation. It would be an interesting task to make these statements moreprecise.Our results may have implications for the black hole information lossproblem. Virtually all discussions of information loss in the black hole con-text rely on the possibility of localizing particles – from throwing a particleinto a black hole to keeping information localized. We argued above thatnormalizable (and in particular localized) states are necessarily accompaniedby soft radiation. It is well known that the absorption cross-section of radi-ation with frequency ω vanishes as ω → 0 and therefore it seems plausiblethat, whenever a localized particle is thrown into a black hole, the soft partof its state which is strongly correlated with the hard part remains outsidethe black hole. If this is true a black hole geometry is always in a mixedstate which is purified by radiation outside the horizon.79Chapter 6ConclusionIn this dissertation we investigated quantum information properties of rel-ativistic scattering theory with an emphasis on the infrared behavior ofmassless gauge field theories. To do so, we used a density matrix approachmixed with the S-matrix machinery of quantum field theory to uncover longtime properties of interacting QFTs.6.1 Scattering with Partial InformationIn chapter 2, we analyzed interacting quantum field theories where an ob-server only has partial access to the full state of the system. We consideredthe case where a set of apparatus particles interact with an unobserved col-lection of system particles and used the more general framework of densitymatrices to express the global state. Incoming states were time evolved intooutgoing ones using the S-matrix formalism well known to field theorists.Outgoing states described in such a way were however still comprised ofapparatus and system particles. At asymptotic times, we expect the fullHilbert space to decompose into a tensor product between the apparatusHilbert space and the system Hilbert space allowing us to trace out the col-lection of unobserved particles. This procedure left us with a density matrixdescribing the precise state an observer could measure at late times. Wecould then compute various quantities like the late time expectation valueof apparatus observable but more notably we were able to produce an exactformula for the von Neumann entanglement entropy between the apparatusand the system states. The entanglement entropy in this case quantifies theamount of information of the total state that has been lost by not being ableto observe the system particles.For the simple example of a pair of interacting massive scalar fields withcoupling of the form λφ2Aφ2S , we could give a perturbative expression for theentanglement entropy which scaled like the total cross-section of the processintegrated over time against the flux of incoming particles. We also usedthis example to suggest a way to directly measure the parameter λ while theliterature usually suggests the theory is only sensible to λ2. Starting with thesystem scalar field in a spatial superposition of states, One should observe80interference fringes at specific locations on a spherical detector surroundingthe scattering. We demonstrated that the amplitudes of these interferencefringes depend on λ at lowest order in perturbation theory as the positionspace observable depends on the off-diagonal elements of the reduced densitymatrix.6.2 Decoherence from Infrared Photons andGravitonsIn chapters 3, we specialize to the case of QED and gravity where scatteringof charged particles radiates away an infinite number of low energy gaugebosons. Using again our density matrix based approach to scattering theorywe were able to extend Weinberg’s soft theorem to off-diagonal elements ofthe density matrix. Applying the S-matrix on both sides of the incomingdensity matrix we could obtain a late time density matrix as usual. Startingby tracing out the outgoing radiation in the standard way, we obtained diver-gences in the photon mass of the form λ−A˜ββ′,α , with an explicit formula forA˜. We then had to account for the divergences arising from soft boson loopdiagrams in the usual Weinberg fashion providing factors of λAβ,α/2+Aβ′,α/2.The limit when the infrared regulator λ is taken to zero told us that the finalinfrared divergences in the density matrix would depend on the exponent∆Aββ′,α =Aβ,α2+Aβ′,α2− A˜ββ′,α. (6.1)We were able to prove the positivity of the exponent, meaning that the λ→ 0limit would leave every term in the density matrix to be either 0 or finite.Then, we showed that the exponent ∆Aββ′,α is exactly 0 if and only if thesum of the charged currents in states β, β′ are identical. This has the effect ofsending nearly all off-diagonal terms in the reduced outgoing density matrixto zero. In the case of QED, we are left with some coherence in the statefrom the inclusion of uncharged matter. However, in gravity all particlesare charged leaving us with a completely decohered density matrix. Wetherefore arrive to the conclusion that after waiting for an infinite amountof time after scattering, long wavelength photons and gravitons are sufficientto provide nearly complete decoherence of the outgoing states: Instead ofleaving us with an entangled density matrix as one could have expected, atlate times we observe a classical ensemble of states with probabilities givenby Weinberg’s inclusive cross-sections approach.In chapter 4, we addressed the same question from the Faddeev-Kulish81dressing perspective. This implies considering scattering between states ofcharged particles accompanied by clouds of finely tuned soft bosons. There,the S-matrix elements are finite due to the cancellation of divergence arisingfrom cloud interactions. This means the S-matrix is well defined for anyvalues of the infrared regulator λ even as it is taken to zero. However, wecould still make the argument that an observer with a finite sized detectorwould not have access to those soft photon clouds. Tracing out the outgoingsoft radiation, we found that the terms in the reduced outgoing densitymatrix depended on the dampening factorDβ,β′ = 〈0|W †β′Wβ|0〉 , (6.2)made from the expectation value of dressing operators. A detailed analysisof the dampening factor allowed us to conclude the same result as in theWeinberg formalism: The dampening factor turned out to be exactly 0 forevery pair of states β, β′ that did not contain the same charged currentcontent. In the case where the charged currents agreed, the dampeningfactor was simply 1. Our result indicated that whichever formalism youchose to use, late-time decoherence would inevitably happen.6.3 Wavepacket Scattering and the Need for SoftDressingIn chapter 5, we gave an original argument advocating for the use of thedressing approach. While the recent literature has provided some evidencethat the dressing approach is intertwined with the topic of asymptotic sym-metries of massless gauge field theories, no specific examples were foundwhere the two approaches would give different results. We investigatedthe general case of scattering of an entangled superposition of particlesand found disturbing results. In the inclusive formalism, we demonstratedthat after scattering no trace of interference between incoming states werepresent. Scattering of an entangled superposition would therefore be thesame as scattering a classicle ensemble of particles which certainly goesagainst known observational evidence. From the dressing perspective how-ever, we find a more natural answer. As the outgoing dressing contains noinformation about the incoming dressing, the reduced outgoing density ma-trix does not destroy interference between incoming superpositions. Whiletracing out the outgoing radiation leads to decoherence, it still maintainsthe initial entanglement of the system.Moving on to wavepacket scattering, we showed that the continuous82superposition case is even more dramatic in the inclusive approach: There,no scattering can happen. The decoherence condition forces the reducedoutgoing density matrix to an integral where the integrand is a subset ofmeasure zero of the domain of integration. The only portion of the S-matrixwhich remains is the identity factor δ(α−β), leaving the outgoing state to beprecisely the same as the outgoing state. As we certainly live in a universewhere wavepacket scattering is possible, the inclusive formalism must berejected. However, wavepacket scattering from the dressed picture still yieldsthe expected results where outgoing states exhibit the usual interferencebetween components of the incoming wavefunction.Advocating for the use of the Faddeev-Kulish dressing, we used the newdevelopments from asymptotic symmetries of QED and gravity to interpretour results. Decoherence was explained in terms of conserved charges. Per-forming the trace over soft photons forced the soft charge to agree for thekets and bras of the reduced density matrix elements. Total charge conser-vation then required that the hard charges must also agree, leaving us witha decohered state. However, allowing for different outgoing soft vacuums re-stored the coherence of the hard data. We could also explain in the languageof soft charges why the inclusive formalism gave the correct results for non-entangled incoming states while it failed for entangled ones. In the inclusiveformalism, one makes a very specific choice of dressing where the incomingvacuum contains no soft photons. While this was possible to make sucha choice for non-entangled state, entangled states cannot accomodate thisrestriction. There, every entangled parts are found within different selectionsectors and cannot interact together.6.4 Closing RemarksThe primary focus of this thesis was to investigate quantum informationproperties of scattering theory with a focus on infrared divergent effectsfound in massless gauge field theories. We were able to extract meaningfulquantities related to real observations where soft particles are lost to theenvironment. For QED and gravity, we postulated the existence of a latetime loss of coherence effect on hard data driven by soft photons and gravi-tons. We gave new evidence in favor of the use of dressed charged statesand expressed our results in terms of the new language of asymptotic sym-metries of QED and gravity. 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Afshar, D. Grumiller, and M. M. Sheikh-Jabbari, Near horizon softhair as microstates of three dimensional black holes, Phys. Rev. D 96(2017), no. 8 84032.[64] R. K. Mishra and R. Sundrum, Asymptotic Symmetries, Holographyand Topological Hair, JHEP 01 (2018) 14.[65] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, Black holes:Complementarity or firewalls?, JHEP 2 (2013) 1–19.88Appendix AOptical TheoremHere we repeat Weinberg’s proof of the optical theorem, for completeness,because the same techniques appear repeatedly in the above. In particular,we explain how unitarity of the density matrices used in scattering is directlyrelated to the optical theorem.Our scattering states are supposed to be continuum-normalized〈α′±|α±〉 = δ(α− α′), (A.1)where the right hand side as usual means a product of Dirac deltas onthe spatial momenta. Now, this equation needs to be consistent with theunitarity of the S-matrix, i.e. we should haveδ(α− α′) = 〈α′+|α+〉 =∫dβdβ′SβαS∗β′α′ 〈β′−|β−〉 =∫dβSβαS∗βα′ . (A.2)Writing the usual decomposition of S as in (2.11) and doing some of theintegrals, we see that we need2pii[Mα′αδ4(pβ − pα)−M∗αα′δ4(pβ − pα′)]= (2pi)2∫dβMβαM∗βα′δ4(pβ − pα)δ4(pβ − pα′).(A.3)Specialize to the case α = α′. We obtain the optical theoremImMαα = −pi∫dβ |Mβα|2 δ4(pβ − pα). (A.4)Consider scattering an initial state |α+〉box, now in a finite spacetime boxas described in the main text. Then our density matrix should have unittrace. Writing this after applying the S-matrix and doing the trace usingout-states |β−〉box, we have1 = tr ρ =∑β∣∣∣Sboxβα ∣∣∣2 . (A.5)If we expand the S-matrix as in (2.16), then the delta-squared term on the89right hand side will give exactly the 1 on the left-hand side in our tracenorm condition here. So then the remaining three terms will have to cancelamongst themselves, which is exactly the case when (A.4) holds.Note that in perturbation theory in some weak coupling λ, the opticaltheorem mixes orders of λ. For our purposes above, for example to getthe entanglement entropy to O(λ2), we need to ensure that we normalizethe density matrix to tr ρ = 1 + O(λ3). But then we need the scatteringmatrix elements appearing in (A.4) to cancel on the two sides of the equationup to O(λ2). In other words, to explicitly check the normalization of thedensity matrix in perturbation theory at this order, we need to include thelowest-order loop diagram for forward scattering in computing the scatteringamplitudes.90Appendix BPositivity of A, B ExponentsAppendix. Here, we show that the exponents ∆A,∆B controlling theinfrared divergences are always positive or zero, and give necessary andsufficient conditions for these exponents to vanish.The first step is to notice that the expressions for the differential expo-nents (3.10) between the processes α → β and α → β′ are the same as theexponents (3.8) for the divergences in the process β → β′, that is∆Aββ′,α = Aβ′,β/2,∆Bββ′,α = Bβ′,β/2.(B.1)To see this, note from the definitions (3.6),(3.8), and (3.10) that there areterms in each of Aβ,α, Aβ′,α, and A˜ββ′,α coming from contractions betweenpairs of incoming legs, pairs of an incoming and outgoing leg, and pairs ofoutgoing legs. One can easily check that the in/in and in/out terms cancelpairwise between the A and A˜ terms in ∆A. The remainder is the termsinvolving contractions between pairs of outgoing legs:∆Aββ′,α =12∑p,p′∈βγpp′ +12∑p,p′∈β′γpp′ −∑p∈β,p′∈βγpp′ (B.2)where we defined γpp′ = epep′β−1pp′ ln[(1 + βpp′)/(1− βpp′)]. We have used thefact that every ηp that would have been in (B.2) is a −1 since every linebeing summed is an outgoing particle, cf. (3.3). But then we have a relativeminus sign and factor of 2 between the first two terms and the third; this isprecisely the same factor that would have come from the relative ηin = −1and ηout = +1 terms in exponent for the process β → β′, namelyAβ′,β =∑p,p′∈βγpp′ +∑p,p′∈β′γpp′ − 2∑p∈β,p′∈β′γpp′ . (B.3)This proves (B.1) for ∆A; an identical combinatorial argument shows thatthe gravitational exponent obeys the analogous relation, ∆Bββ′,α = Bβ′,β/2.Now we prove that for the process α → β + (soft) the exponent Aβαis always greater or equal to zero with equality if and only if the in and91outgoing currents agree; we can then take α = β′ to get the results quotedin the text. Referring to Weinberg’s derivation [2], we can write Aβα asAβα =12(2pi)3∫S2dqˆ tµ(qˆ)tµ(qˆ). (B.4)Here,tµ(qˆ) ≡∑nenηnpµnpn · q = c(q)qµ + ci(q)(qi⊥)µ. (B.5)In this equation, we have defined a lightlike vector qµ = (1, qˆ) and qi⊥,i = 1, 2 are two unit normalized, mutually orthogonal, purely spatial vectorsperpendicular to qµ. The sum on n ∈ α, β runs over in- and out-goingparticles. By charge conservation, t·q = 0, which justifies the decompositionin the second equality in (B.5). With this decomposition we may writeAβα =12(2pi)3∫S2dqˆ(c21(q) + c22(q)) ≥ 0, (B.6)which immediately proves the statement that Aβα ≥ 0.Now it remains to be shown that equality holds if and only if all of thein- and out-going currents match. From the previous paragraph we knowthat Aβα vanishes if and only if both ci(q) = 0 for all q, that is if and onlyif t · qi⊥ = 0. Assume that Aβα = 0, so that q⊥ · t(q) = 0. Now supposealso that jv0(α) 6= jv0(β) for some v0, where these are the eigenvalues ofjv |α〉 = jv(α) |α〉 and similarly for β. We derive a contradiction. For anyfinite set of velocities, the functions fv(qˆ) = (v · q⊥)/(1− v · qˆ) are linearlyindependent. Therefore the terms in0 = t · q⊥ =∑nenηnvn · q⊥vn · q (B.7)must cancel separately for each velocity in the list of vn. Consider in par-ticular the term for v0. For this to vanish, the sum of the coefficients mustvanish, i.e.0 =∑n|vn=v0enηn = [jv0(α)− jv0(β)] , (B.8)the relative minus coming from the η factors. But this contradicts ourassumption that jv0(α) 6= jv0(β). This completes the proof for A.The proof for gravitons goes similarly. Again referring to Weinberg we92write B asBβα =G4pi2∫S2dqˆtµνDµνρσtρσ. (B.9)Here, Dµνρσ = ηµνηρσ − ηµρηνσ − ηµσηνρ is the numerator of the gravitonpropagator, andtµν =∑nηnpµnpνnpn · q = cq(µqν) + ciq(µqν)⊥,i + cijq(µ⊥,iqν)⊥,j . (B.10)This symmetric tensor obeys tµνqν = 0 by energy-momentum conservation,which justifies the decomposition in the second equality. Using this we havetµνDµνρσtρσ = 2cijcji −(cii)2= (λ1 − λ2)2 (B.11)with λ1,2 the two eigenvalues of the matrix cij . Plugging this into (B.9) weimmediately see that B ≥ 0. The condition for vanishing of Bββ′ is that theeigenvalues are equal λ1 = λ2, which means that cij is proportional to theidentity matrix. Hence, if B vanishes we have that0 = tµνq⊥,1µ q⊥,2ν =∑nηnEn(vn · q1⊥)(vn · q2⊥)vn · q . (B.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 = 0 if and only if jgravv (α) = jgravv (β) for every v.93Appendix CDressed Soft FactorizationThe soft photon theorem looks somewhat different in dressed QED. In stan-dard, undressed QED, the theorem says that the amplitude for a processp→ q accompanied by emission of an additional soft photon of momentumk and polarization ` has amplitudeSqk`,p = e[q · e∗` (k)q · k −p · e∗` (k)p · k]Sq,p. (C.1)This is singular in the k → 0 limit. On the other hand, in the dressedformalism of QED, the statement is thatS˜qk`,p = ef(k)S˜q,p, (C.2)where f(k) ∼ O(|k|0), so that the right-hand side is finite as k → 0. We cansee this by straightforward computation. In computing the matrix element(C.2), there will be four Feynman diagrams at lowest order in the charge.We will get the usual pair of Feynman diagrams coming from contractions ofthe interaction Hamiltonian with the external photon state, leading to thepoles (C.1). Moreover we will get a pair of terms coming from contractionsof the interaction Hamiltonian with dressing operators. These contribute afactor[F ∗` (k,p)− F ∗` (k,q)]→[q · e∗` (k)q · k −p · e∗` (k)p · k]+O(|k|0), (C.3)times −e, where the limit as k → 0 follows from the definition (4.1). Thisextra contribution precisely cancels the poles in (C.1), leaving only the orderO(|k|0) term.94Appendix DProof of Positivity of ∆A,∆BThe exponent that is responsible for the decoherence of the system is definedas∆Aββ′,αα′ =12Aβ,α +12Aβ′,α′ − A˜ββ′,αα′ . (D.1)The factor in the first two terms, Aβ,α, is defined as in [2]Aβ,α =12(2pi)3∫S2dqˆ∑n∈βenηnpµnpn · qˆ gµν (∑m∈αemηmpµmpm · qˆ). (D.2)Performing the integral over qˆ yieldsAβ,α = −∑n,n′∈α,βenen′ηnηn′8pi2βnn′ ln[1 + βnn′1− βnn′]. (D.3)Similarly A˜ββ′,αα′ can be written asA˜ββ′,αα = −∑n∈α,βn′∈α′β′enen′ηnηn′8pi2βnn′ ln[1 + βnn′1− βnn′]. (D.4)We rearrange the terms such that ∆A can be written as∆Aββ′,αα′ = −12∑n,n′∈α,α¯′,β,β¯′enen′ηnηn′8pi2β−1nn′ ln[1 + βnn′1− βnn′], (D.5)where a bar means incoming particles are taken to be outgoing and vice versa(or equivalently, ηα¯′ = −ηα′). From equation (D.5), it is clear that incomingparticles are found within the set {α, β′} while the outgoing particles arepart of {α′, β}. Let us rename those sets σ and σ′ respectively. ∆A nowtakes the form∆Aββ′,αα′ = −12∑n,n′∈σ,σ′enen′ηnηn′8pi2β−1nn′ ln[1 + βnn′1− βnn′]=12Aσσ′ ≥ 0, (D.6)95as was proven in [42]. This shows that ∆Aββ′,αα′ ≥ 0. The same proof goesthrough for ∆Bββ′,αα′ .96Appendix EThe out-Density Matrix of WavepacketScatteringIn this part of the appendix we flesh out the argument in section 5.3, namelythat after tracing out soft radiation, the only contribution to the out-densitymatrix is coming from the identity term in the S-matrix. We will focus onthe case of QED.E.0.1 Contributions to the out-Density MatrixFirst, let us decompose the IR regulated S-matrix into its trivial part and theM-matrix element. For simplicity we ignore partially disconnected terms,where only a subset of particles interact. Then,SΛαβ = δ(α− β)− 2piiMΛαβδ(4)(pµα − pµβ), (E.1)where the first term is the trivial LSZ constribution to forward scattering.This trivial part does not involve any divergent loops and therefore exhibitsno Λ-dependence. However, the factorization of the S-matrix into a cutoffdependent term times some power of λ/Λ remains valid since all exponents ofthe form Aα,β vanish identically for forward scattering. This decompositionof the S-matrix gives rise to three different terms for the outgoing densitymatrix, containing different powers of M.“No scattering”-termThe case where both S-matrices contribute the delta function term results– unsurprisingly – 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′). (E.2)97Contribution from forward scatteringWe would now expect to find an additional contribution to the density ma-trix reflecting 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)). (E.3)For simplicity, let us focus solely on the case in which S∗ contributes thedelta function and S contributes the connected partρ(II)ββ′ = −2piif∗(β′)∫dαf(α)MΛβαδ(4)(pµα − pµβ)λ∆Aα,βF(E,ET ,Λ)β,α + . . . ,(E.4)where the ellipsis denotes the contribution coming from the omitted term of(E.3). The exponent of λ only vanishes if the currents in α and β agree. Wewill show in appendix E.0.2 that we can take the limit λ → 0 before doingthe integrals. Taking this limit, λ∆Aα,β gets replaced byδαβ ={1, if charged particles in α and β have the same velocities0, otherwise,(E.5)which is zero almost everywhere. If the integrand was regular, we couldconclude that the integrand is a zero measure subset and integrates to zeroand thusρ(II)ββ′ = 0. (E.6)However, the integrand is not well-behaved. Singular behavior can comefrom the delta function or the matrix element, so let’s consider the twopossibilities.The singular nature of the Dirac delta does not affect our conclusion:for n incoming particles, the measure dα runs over 3n momentum variableswhile the delta function constrains 4 of them, leaving us with 3n−4 indepen-dent ones. If we managed to find a configuration for which ∆Aβα = 0, anyinfinitesimal variation of the momenta in α along a direction that conservesenergy and momentum would modify the eigenvalue of the current operatorjˆv(α)− jˆv(β) and make ∆Aβα non-zero. Therefore, the integrand would stillbe a zero-measure subset for the remaining integrals.98What could still happen is thatMΛβα is so singular that it gives a contri-bution. For this to happen it would need to have contributions in the formof Dirac delta functions. However, also this does not happen, for examplefor Compton scattering which scatters into a continuum of states. Addi-tional IR divergences also do not appear as guaranteed by the Kinoshita-Lee-Nauenberg theorem. We will not give a general proof since for ourpurposes it is problematic enough to know that no scattering is observed forsome physical process.The scattering termIt is evident that a similar argument goes through for the M2 term. Onefindsρ(III)ββ′ = −4pi2∫dαdα′f(α)f∗(α′)MΛβαMΛ∗α′β′λ∆Aαα′,ββ′ (E.7)×F(E,ET ,Λ)ββ′,αα′δ(4)(pµα − pµβ)δ(4)(pµα′ − pµβ′). (E.8)The analysis boils down the the question whether the term∫dαdα′λ∆Aαα′,ββ′ δ(4)(pµα − pµβ)δ(4)(pµα′ − pµβ′). (E.9)vanishes. As soon as there is at least one particle with charge, we need toobey the condition that the charged particles in α and β′ agree with thosein β and α′ for the exponent of λ to vanish. Infinitesimal variations of α andα′ that preserve the eigenvalue of the current operator jˆv(α)− jˆv(α′) form azero-measure subset of the 6n − 8 directions that preserve momentum andenergy, forcing us to conclude that the integration runs over a zero measuresubset and the only contribution to the reduced density matrix comes fromthe trivial part of the scattering process. This means thatρout,red.ββ′ = f(β)f∗(β′) = ρinββ′ , (E.10)or in other words it predicts that a measurement will not detect scatteringfor wavepackets. This is clearly in contradiction with reality and suggeststhat the standard formulation of QED and perturbative quantum gravitywhich relies on the existence of wavepackets is invalid.99E.0.2 Taking the Cutoff λ→ 0 vs. IntegrationOne might be concerned that the limit λ → 0 and the integrals do notcommute. In this part of the appendix, we will check the claim made in thepreceding subsection, i.e. we will show that one can explicitly check thatthe integration and taking the IR regulator λ to zero commute. We assumein the following that we talk about QED with electrons and muons in thenon-relativistic limit, which again is good enough as it is sufficient to showthat we can find a limit in which no sign of scattering exists in the outgoinghard state. The wave packets are chosen to factorize for every particle andto be Gaussians in velocity centered around v = 0,f(v) =(2piκ)3/4exp(−v2κ). (E.11)In order to stay in the non-relativistic limit, κ must be sufficiently small.They are normalized such that∫d3v|f(v)|2 = 1. (E.12)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 exponentof λ into∆Aαβ =e224pi2∑n,m∈α,β(vα − vβ)2. (E.13)Thus, λ∆A has the formλ∆A ∝ exp−12γ∑n,m∈α,β(vα − vβ)2 , (E.14)where taking the cutoff λ to zero corresponds to γ ∝ − log(λ) → ∞. Thestate α consists of a muon with well defined momentum and one electronwith momentum mv, where v is centered around 0. The state β consistsof the same muon (we assume it was not really deflected) and one electronwith momentum mv′. To obtain the contribution to forward scattering, we100have to perform the integral∝∫d3v(2piκ)3/4exp(−v2κ)exp(−γ(v − v′)2) · (other terms). (E.15)Here, we assumed that the other terms which include the matrix elementin the regime of interest is finite and approximately independent of v. Theintegral yields (2piκ(1 + γκ)2)3/4exp(− γv′21 + γκ). (E.16)Taking the limit γ → ∞, it is clear that this expression vanishes. If wewant to consider an outgoing wave packet we have to integrate this overf(v′ − vout). The result is proportional to(2piκ(1 + 2γκ)2)3/4exp(− γv2out1 + 2γκ)(E.17)and still vanishes if we remove the cutoff, γ →∞.101
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