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 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|