aa r X i v : . [ h e p - t h ] S e p Memory and the Infrared
Cesar Gomez a,b , Raoul Letschka b a Arnold Sommerfeld Center for Theoretical PhysicsDepartment f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchenTheresienstr. 37, 80333 M¨unchen, Germany b Instituto de F´ısica Te´orica UAM-CSIC, C-XVIUniversidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Abstract
Memory effects in scattering processes are described in terms of theasymptotic retarded fields. These fields are completely determined bythe scattering data and the zero mode part is set by the soft photontheorem. The dressed asymptotic states defining an infrared finite S-matrix for charged particles can be defined as quantum coherent statesusing the corpuscular resolution of the asymptotic retarded fields. Im-posing that the net radiated energy in the scattering is zero leads tothe new set of conservation laws for the scattering S-matrix which areequivalent to the decoupling of the soft modes. The actual observabil-ity of the memory requires a non-vanishing radiated energy and couldbe described using the infrared part of the differential cross sectionthat only depends on the scattering data and the radiated energy.This is the IR safe cross section with any number of emitted pho-tons carrying total energy equal to the energy involved in the actualmemory detection. [email protected] [email protected] the cross section , sum over amplitudes for different numberof soft emitted photons. These two contributions, namely the one associatedwith virtual photons and the one coming from summing over different numberof final infrared soft photons lead to infrared divergent pieces that canceleach other in the final cross section . What remains is an infrared finitecross section. The concrete form of this cross section depends on an infraredscale ε . More precisely and using the notation of [14] the final differentialcross section factorizes into a pure infrared part that only depends on thescattering data but not on the details of the scattering and a non-infraredpart. This factorization of the cross section depends on the infrared scaleused to define the upper limit on the energies for real infrared, measurablephotons. The infrared part of the cross section, after infrared divergencesare cancelled, depends on the infrared scale ε in the general form d σ d ε ∼ e G ( ε ) with G ( ε ) ∼ ln (cid:0) Eε (cid:1) .The former solution of the infrared problem is not defining an infrared fi-nite S-matrix. The definition of such S-matrix was first addressed in [15]. Thekey ingredient used in [15] consisted in modifying the definition of asymptoticstates. These are defined using a coherent state dressing operator determinedby the asymptotic dynamics. Using these new asymptotic states for chargedparticles you can define an IR finite S-matrix for scattering processes withonly charged particles in the in and out states. In this context it can be easilyobserved that for the so-defined IR finite S-matrix soft photon modes are byconstruction decoupled. This FK-decoupling has been discussed in [16–19].Note that for this scattering S-matrix, with only charged particles in the2symptotic states, conservation of energy implies that no net amount of en-ergy is radiated. This condition leads to a set of conserved charges commutingwith S which are equivalent to the decoupling of soft modes. The physicalmeaning of these symmetries will be discussed in the next section.In spite of its beauty the S-matrix formalism of [15] is not directly ad-dressing the most physical discussion on how to deal with real processeswhere some energy is radiated in the form of real infrared photons.How are electromagnetic memory effects related with this issue? Thequick answer to this question is the following. In scattering processes amongmassive charged particles the charges and momentum of the in and out parti-cles determine the non-radiative part of the asymptotic retarded field. Thesein and out scattering data are enough to extract the zero mode part of the in-terpolating retarded field and consequently they account for the informationcontained in the soft photon theorems. In the IR finite S-matrix these zeroenergy modes are decoupled and moreover they don’t lead to any observable (in a finite amount of time) memory effect.The scattering data, although enough to derive the soft photon theo-rem, are not enough to fix the radiative component of the retarded field thatdepends on concrete information on how the scattering process is actuallytaking place, in particular (in the classical case) on the accelerations. Thisradiative part of the retarded field carries energy as well as radiative modeswith typical frequencies of the order of the inverse of the time scale on whichthe scattering process is taking place. The observability of the memory effectusing a physical detector depends crucially on this radiated energy. In QEDthis information is partially encoded in the infrared part of the differentialcross section, namely on the dependence on the infrared scale ε that we cantake as equal to the energy involved in the actual detection of the memoryeffect. In particular we shall associate to memory the infrared part of thecross section that only depends on the scattering data and where we con-sider an arbitrary number of emitted real infrared photons with total energy ε equal to the energy involved in the memory detection.In this note we shall reduce the discussion of memory to the electromag-netic case and only at the end we will make few comments on similaritiesand differences with the gravitational case.3 Classical memory
For a given classical scattering where some initial charges q j with velocities v j lead to a final state with charges q i and velocities v i the electromagneticmemory is determined by the retarded field created by the currents j µ definedby these scattering data. In four dimensions the retarded electromagneticfield at some observation point O = ( x , t ) is given by A µ ( x , t ) = Z j µ (cid:0) x ′ , t − rc (cid:1) r d x ′ (1)with r = ( x − x ′ ) . For small velocities we can Taylor expand the currentand define the retarded field as a series in 1 /c . The field tensor F µν generatedby the moving charges can be expanded in powers of 1 /r . It contains a piecethat goes like 1 /r that only depends on the velocities of the sources and apiece that goes like 1 /r that accounts for the radiation emitted during thescattering process.For an idealised point-like scattering taking place at the origin the radia-tive part of the retarded field has support on the u = 0 null hypersurface t = rc . This simply reflects the fact that only at the origin the moving par-ticles entering into the scattering are accelerated. At large distances x ≫ x ′ the retarded field is given by A µ ( x ) = X out θ ( u ) r q i v µi − v i · ˆ x + X in θ ( − u ) r q j v µj − v j · ˆ x (2)where u is determined by the equation u = t − r , ˆ x is the norm vector of x and from now on c = 1.The field tensor is then given by F µν = X out q i κ [ µ v ν ] i κ α v αi " r δ ( u ) κ α v αi + 1 r v iβ v βi ( κ α v αi ) θ ( u ) + X in q j κ [ µ v ν ] j κ α v αj " − r δ ( u ) κ α v αj + 1 r v jβ v βj (cid:0) κ α v αj (cid:1) θ ( − u ) (3)where κ µ = (1 , ˆ x ) and v µi = (1 , v i ) (4)4nd indices are raised and lowered in cartesian coordinates by the metric η µν = diag(1 , − , − , − /r part dependson the concrete classical modelling of the scattering, in this simple case in theform of an instantaneous change of the velocities taking place at the origin.The non-radiative part that goes as 1 /r depends only on the in and outscattering data.The classical memory effect associated with a given scattering processwhere we use as data the in and out four-momenta of the scattered particlesis given by the non-radiative fields F in and F out . However the actual detectionof the memory is determined by the interaction of some charged detector withthe interpolating radiative field . This effect on the memory detector is non-vanishing and observable due to the fact that the interpolating radiative fieldcarries non-vanishing energy ε . For future convenience it would be important to work out the spectral de-composition of the asymptotic retarded fields defined by the in and out setof free moving charged particles. The Fourier modes of the retarded field aregiven by A µ ( t, k ) = X out πq i | k | (cid:16) k · v i (cid:17) p µi e − i k · pi Ei t p αi k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t> + X in πq j | k | (cid:16) k · v j (cid:17) p µj e − i k · pj Ej t p αj k α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t< (5)where p µi is the 4-momentum of the i th particle and k µ = ( | k | , k ).The important thing to be noticed is that the Fourier components ofthe retarded field created by a moving charge with constant velocity v i arewaves with wave vector k but frequency ω i = k · v i . These Fourier modesare obviously not real photons with the exception of the soft k = 0 mode.Once we move into quantum field theory these modes will define the quantumconstituents of the coherent state dressing of free moving charged particles. Classically we can associate with a given scattering process among chargedparticles the non-radiative retarded fields defined by the in and out scat-5ering data i.e. by the charges, masses and velocities of the incoming andoutgoing particles. Let us generically denote A in and A out these retardedfields. Associated with these data we can formally define a transformation T : A in → A out . This transformation is not a gauge transformation since A in and A out , although satisfying the conditionlim r →∞ r ( F µν ( A in ) − F µν ( A out )) = 0 , (6)have, at order 1 /r , different values of the corresponding stress tensor.Let us now fix the asymptotic kinematical data for the incoming andoutgoing charges in such a way that energy and momentum are conservedi.e. P in E j = P out E i . In this case conservation of energy will imply thatthe only possible radiated mode is a zero energy zero mode. In classicalelectrodynamics this constraint is not easy to impose. Indeed if we fix thescattering data and we use those kinematical data to derive the classicalradiated field we will only achieve total energy conservation if in addition wetake into account the backreaction of the radiated field i.e. the Abraham-Lorentz forces on the outgoing scattering data. As it is well known thisproblem cannot be fully solved in classical electrodynamics.We can however formally impose the conservation of energy on the scat-tering data for the charged particles which is effectively equivalent to settingthe net amount of radiated energy to be equal to zero. To understand thephysical meaning of this zero radiated energy constraint it can be illustra-tive to recall the attempt of Wheeler and Feynman (WF) [20, 21] to definein classical electrodynamics the radiative reaction on sources in the contextof the absorber theory. Indeed if we think that all the radiation emitted isabsorbed leading to zero radiated energy we get that asymptotically we canimpose the WF condition: F µν ret = F µν adv (7)for the radiative part of the total advanced and retarded fields. Generically,although in Maxwell theory we have the advanced and retarded solution,only the retarded part of the radiative field is actually considered as physical.Thus the former condition makes sense if we have a formal absorber and nonet radiation carrying non-zero energy is left unabsorbed. Note that the condition (7) allows us to define the WF field associated with a movingcharge as 1 / F ret + F adv ).
6n scattering language we can think of the advanced field as associatedwith some incoming radiation and the retarded field as the outgoing radi-ation, so if we consider a scattering with in state defined by a set of only charged particles (and zero radiation) the former condition (7) only makessense for the zero mode part that does not carry any energy.The equality between retarded and advanced fields (7) leads to a set ofconservation laws where the classical charges can be defined by the convolu-tion of (7) with arbitrary test functions [22]. The so called soft charges canbe defined as those determined by the zero mode part of the retarded andadvanced fields.In a scattering process among charged particles where we use as scat-tering data a set of in and out momenta for the charged matter satisfyingconservation of energy i.e. with no net radiation, we can impose the con-dition (7) and these charges will act as symmetries of the S-matrix. Sincethere is no radiation the only relevant piece is the zero mode soft part. In thiscase any memory effect defined as the difference between the non-radiativepart of the retarded fields created by the incoming and outgoing particlesis physically unobservable . This unobservability becomes equivalent, in theS-matrix language, to the decoupling of the radiative zero mode.In summary the “new symmetries” of the QED S-matrix [12, 23] are aconsequence of imposing what we can call the WF condition or in morephysical terms, the absence of any loss of energy in the form of radiatedinfrared photons. This condition is naturally implemented in any S-matrixformulation where in and out states are sets of charged particles. However inorder to have observable memory effects a certain amount of energy should beradiated and in that case we need to work with the differential cross section.Once some energy is actually radiated we cannot impose (7) since this energyis only contained in the retarded part of the field.In reality the probability that in a physical scattering we have zero netradiated energy is indeed zero, so these symmetries of the IR finite S-matrixonly account for the soft theorem part. We can think of the symmetriesfor zero energy radiation processes as being spontaneously broken with the k = 0 soft mode as a Goldstone boson. However we would like to stressthat whenever we have a real amount of energy radiated with no incomingradiation, which is actually always the case, the condition (7) can only beimposed for the zero mode part which is what, as we shall discuss in moment,you actually do in the definition of the IR finite S-matrix.7 Memory and infrared QED
As was already pointed out in [15] a prescription to define an IR finite S-matrix was partially developed. The key ingredient in this construction wasto use the asymptotic dynamics in order to define new asymptotic states bydressing standard Fock matter states | i i with the coherent state of photonssourced by the asymptotic current J µas . We can represent this dressing as | i i → e R ( J µas ) | i i (8)We can now easily identify the operator R . Using the spectral decompositionof the retarded field (5) created by the asymptotic free moving charges we candefine the quantum resolution of this field using as quantum constituents,quanta with momentum k and frequency ω i = k · v i . (9)Denoting the creation and annihilation operators for these quanta b k and b † k the corresponding coherent state will be defined by the operatore R d k P πqi | k | ( k · vi ) pµi e − i k · pi Ei tpαi kα b † k µ (10)acting on the vacuum defined by b k | i = 0. If we want to use the creationand annihilation operators a k and a † k of the Fock space of free photons withdispersion relation ω = | k | we need to transform b modes into a photons. Thisleads to the FK expression derived from the asymptotic dynamics, namelye R d3 kk pµpαkα e i (cid:18) pαkαp (cid:19) t a † k µ . (11)By construction on these coherent states the expectation value of the fieldoperator ˆ A is given by the classical retarded field. Note that these coherentstates contain an infinite number of k = 0 photons. If in the scatteringprocess we impose zero energy radiated then the total number of modes inthe in and out states will be conserved. For other examples of the same technique see [24] and [25].
8e can consider a more complicated coherent state of photons describingthe whole radiative part of the retarded field and to think of this coherentstate as a sort of domain wall interpolating between the asymptotic in andout retarded fields. The soft photon theorem accounts for the zero mode partof this domain wall . The radiated energy acting on the potential memorydetector is roughly what we can interpret as the mass of this photonic domainwall.The IR finite S-matrix is defined bylim t →∞ e R ( − t ) † S e R ( t ) (12)where e R ( t ) = e R dp d3 kk pµpαkα e i (cid:18) pαkαp (cid:19) t a † k µ − h.c ! ρ ( p ) for ρ ( p ) = P i δ ( p − p i ). ThisS-matrix satisfies the decoupling of soft modes [17–19] lim k → [ S, a k ] = 0 . (13)The so-defined S-matrix is IR finite due to the fact that the former dressingfactor cancels the infrared divergences (after resummation) coming from thevirtual photon self energies.Note that in this S-matrix we are imposing the zero energy radiation con-dition (7) and consequently the S-matrix commutes with the charges definedby convoluting (7) with arbitrary test functions. These Ward identities aresimply reflecting the kinematical constraints we are imposing on the scat-tering states, namely vanishing net energy in the form of radiation for instates without real photons and are fully equivalent to the decoupling of softphotons. Note also that, in this case, the so-called hard charges [12, 19, 23]are absorbed in the dressing.It is important to stress that the decoupling of soft modes should notbe confused with the absence of observable memory effects. Indeed as al-ready stressed observable memory requires a certain amount of energy in theretarded field to be radiated in the form of infrared emitted photons andtherefore does not satisfy the S-matrix matching condition for the chargedkinematical data. For a more rigorous proof see [26]. .2 QED measure of memory Given a scattering process in QED we can associate, as a way to characterizethe memory, the differential cross section d σ d ε for ε the radiated energy in theform of infrared photons. The dependence of the cross section on ε is wellknown in QED [14]. We shall be interested only in the infrared part of thecross section i.e. in the part that only depends on scattering data.d σ d ε (cid:12)(cid:12)(cid:12)(cid:12) IR ∼ A e ln Eε (14)with A being a finite coefficient depending only on the scattering data. Thisinfrared part of the cross section corresponds to having arbitrary number ofemitted infrared photons with total energy equal or less than ε , i.e. it is thecross section σ (2 → sof t ( E soft ≤ ε )) and is an IR safe quantity.The important message of these cross sections is the dependence on theenergy radiated. This is important for understanding the real nature of thememory. In fact we could think of nullifying the memory by pushing ε → ε is telling us that such a formal limit cannot be taken or equivalently thatthe actual probability to scatter without radiating is zero. The interpolatingradiative field measured by the memory detector contains energetic modesin addition to the Goldstone zero mode piece. The actual interaction of thedetector with these modes is what makes the memory effect, in scatteringprocesses, actually observable in a finite amount of time. To summarise we have observed that although in an IR finite S-matrix ofthe type discussed in [15] the Goldstone part of the radiative mode is decou-pled this does not nullify electromagnetic memory effects. These effects areclassically due to the non-vanishing energy carried by the radiative part ofthe retarded field created by the scattering process and show up quantummechanically already in the infrared dependence of the differential cross sec-tion on the amount of energy radiated in the form of real infrared quanta. Inthis sense memory is intimately connected with the infrared scale appearing10n the standard computation of differential cross sections in QED. Pushingformally this IR scale to zero and working with the IR finite S-matrix sat-isfying the condition (7) nullifies memory as an observable effect. Note alsothat memory accounts for the radiative backreaction on the outgoing chargedparticles of the radiated energy.Regarding gravity the discussion of classical gravitational memory is for-mally identical to the electromagnetic case. Again in this case observable memory effects are associated with the non-zero mode part of the interpo-lating radiative field. In quantum field theory language we can extract thedependence of the differential cross section on the infrared scale as we doin the electromagnetic case. This problem has been recently discussed forgraviton scattering in [27]. This construction can be used to define IR safequantities to be associated with gravitational memory.A very attractive idea discussed in [28–30] suggests a connection betweenthe gravitational memory and the information paradox. The basic point ofthe idea is to extend the gravitational memory associated with radiation go-ing through the null infinity to the analog problem for the black hole horizonitself. This extension is by no means straightforward for a simple reason.While in the case of the null infinity the radiated energy (involved in thememory) interpolates between two Minkowski asymptotic metrics (relatedby a supertranslation) in the case of the horizon any gravitational mem-ory interpolates between two black hole metrics with different mass . Thezero mode part of this interpolating metric can be formally used to define soft hair . However this zero mode part is effectively decoupled and unob-servable [17, 19]. The conservation laws associated with the gravitationalgeneralization of (7) can only deal, even in the presence of horizons, withthe zero energy part of the involved radiative modes and consequently doesnot lead to any observable effect. The observable gravitational memory, asit is the case with the electromagnetic memory, is a radiative backreactioneffect that can be only worked out quantum mechanically, in other wordsthe zero mode part which corresponds to no radiative backreaction (and canbe described in purely classical or semiclassical terms) is ineffective to solvethe information paradox. The quantum backreaction, as well as the relevantquantum hair, is O (1 /N ) for N the black hole entropy [31, 32].However, for a scattering process where some initial state leads to the From a purely historical point of view an interesting discussion on WF theory andgravity can be found in [20] → N approach [33] to black hole formation. We hope to come back to thisissue in a future publication. Acknowledgements
We would like to thank Gia Dvali for insightful comments related to theobservability of memory and Kepa Sousa and Mischa Panchenko for manydiscussions and comments.The work of C.G. was supported in part by Humboldt Foundation andby Grants: FPA 2009-07908 and ERC Advanced Grant 339169 ”Selfcomple-tion”. The work of R.L. was supported by the ERC Advanced Grant 339169”Selfcompletion”.
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