Dressed States from Gauge Invariance

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Physics and Astronomy Faculty Publications Physics and Astronomy

6-7-2019

Dressed States from Gauge Invariance

Hayato Hirai Osaka University, Japan

Sotaro Sugishita University of Kentucky, [email protected]

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Repository Citation Hirai, Hayato and Sugishita, Sotaro, "Dressed States from Gauge Invariance" (2019). Physics and Astronomy Faculty Publications. 665. https://uknowledge.uky.edu/physastron_facpub/665

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Abstract The dressed state formalism enables us to define the infrared finite S-matrix for QED. In the formalism, asymptotic charged states are dressed by clouds of photons. The dressed asymptotic states are originally obtained by solving the dynamics of the asymptotic Hamiltonian in the far past or future region. However, there was an argument that the obtained dressed states are not gauge invariant. We resolve the problem by imposing a correct gauge invariant condition. We show that the dressed states can be obtained just by requiring the gauge invariance of asymptotic states. In other words, Gauss’s law naturally leads to proper asymptotic states for the infrared finite S-matrix. We also discuss the relation between the dressed state formalism and the asymptotic symmetry for QED.

Keywords Gauge Symmetry, Scattering Amplitudes

Disciplines Physics

Notes/Citation Information Published in Journal of High Energy Physics, v. 2019, issue 6, article 23, p. 1-22.

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This article is available at UKnowledge: https://uknowledge.uky.edu/physastron_facpub/665 JHEP06(2019)023 Springer -matrix for May 4, 2019 June 7, 2019 May 21, 2019 : S : : February 8, 2019 : Revised Published Accepted Received -matrix. We also discuss the S Published for SISSA by https://doi.org/10.1007/JHEP06(2019)023 [email protected] , a,b . 3 1901.09935 The Authors. and Sotaro Sugishita c Gauge Symmetry, Scattering Amplitudes a

The dressed state formalism enables us to define the infrared finite , [email protected] Toyonaka, Osaka, 560-0043, Japan Department of Physics andLexington, Astronomy, KY University 40506, of U.S.A. Kentucky, E-mail: Department of Physics, Osaka University, b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: correct gauge invariant condition. Werequiring show that the the gauge dressed invariance states ofleads can asymptotic be to states. obtained In proper just other by relation asymptotic words, between Gauss’s states the law for dressed naturally state the formalism infrared and finite the asymptotic symmetry for QED. Abstract: QED. In the formalism,dressed asymptotic asymptotic charged states states are originally are obtainedHamiltonian dressed by solving by in the clouds the dynamics of of farobtained the photons. past asymptotic dressed The or states future are region. not gauge However, there invariant. was We an resolve argument the that problem the by imposing a Hayato Hirai Dressed states from gauge invariance JHEP06(2019)023 ]. 8 – 3 ]). It 2 15 -matrix. – 9 S ]. Another 2 , -matrix [ 1 S 11 5 12 14 6 prescription 11 i 7 9 – 1 – 17 1 -matrix in QED and the dressed state formalism S 19 ]. Thus, the conservation laws should constrain the 18 17 6 , 16 1 Although the dressed state formalism was proposed many years ago, it has been re- 5.1 Softness of5.2 dresses Other future directions 2.3 Gauge invariant asymptotic states 3.1 Coulomb potential3.2 by point charges Coulomb in potential the from asymptotic dressed region states with 1.3 Our method and the differences from Faddeev and Kulish’s 2.1 Lagrangian and2.2 Hamiltonian in covariant Physical gauge states in the Schrödingerpicture 1.1 Introduction 1.2 Review of a problem of -matrix is a fundamental quantity of quantum field theories in Minkowski spacetime. cently reconsidered in the connectionhas with been the recognized that asymptotic QED symmetrygauge has transformations (see, [ an e.g., infinite [ numberOn of the symmetries other associated hand, with scatteringsum large amplitudes of vanish in IR the divergences conventional at approach all because the orders produces the exponential suppression. It was pointed famous example is quantumcause electrodynamics divergences of (QED). loop Virtual diagrams.cross-section photons of This with various problem processes small can including the energy beapproach was emission avoided also of by developed, real considering which soft enables the photonsIt us total [ to is treat called directly the the dressed IR finite state formalism, which will be reviewed in the next subsection. 1.1 Introduction S There is a systematicwe way then to often compute encounter it infrared perturbatively (IR) by divergences the for Feynman theories rules. with However, massless particles. A 1 Introduction and summary 4 Asymptotic symmetry in the5 dressed state formalism Conclusion and further discussion 3 Interpretation of the Faddeev-Kulish dresses 2 Necessity of dresses Contents 1 Introduction and summary JHEP06(2019)023 . . H 1.3 (1.8) (1.2) (1.3) (1.4) (1.5) (1.6) (1.7) (1.1) → ∞ t , 0 , i at 0 α , i ]. | 0 0 β ) | i 11 H α ) α ( | β ) ( out f in t of the Hamiltonian . Ω( ) dα g . β 0 dβ g t ∞ 0 E i − Z + t Z ) ( β ) | → s 0 t s f ) t t i − iH and − t f i − t t = lim ( α e ( 0 Ω( 0 E at which the Schrödingeroperators † ≡ iH out , ) iH i . s ) f − − 0 t t ], and will resolve the problem. In our α e of a free Hamiltonian e i in | 8 i β t, t ∞ Ω( β | β ( | | + 0 α E ) α h → →−∞ h β i f 0 t t ( out in – 2 – and ,U ) can be written as = lim = lim ) 0 = α t dβ g in 1.1 − →−∞ i E out t i ( i α,β Z β , , t | α S . 0 | iH ) -matrix in QED and the dressed state formalism ) i ) − s β β S →∞ ( e α | f ( t in ) t, t i ≡ ( t 0 out ) = lim 0 prescription for the dressed states. In addition, the relation U dβ g Ω( ) as follows: t, t i α,β , t dα g ) are the full and free time-evolution operators respectively: ( Z 0 g s S are eigenstates with energies ) t U s ( Z t →−∞ -matrix element ( t, t in i ) ( − U t i s S i 0 t t β ( | − ≡ = lim U f ) t iH in ( t i − with the same energies . We then present our results and the outline of the paper in subsection e ), the iH and β Ω( | 0 − ) and i 0 e 1.5 1.2 β ] that the vanishing of the amplitudes is consistent with the asymptotic sym- out | ∞ i lim t, t →−∞ i ( 11 + α t | U lim → f and t Motivated by these facts, we will investigate the dressed state formalism in this paper. 0 i -matrix elements of scatterings in quantum field theories are defined by inner products α Using eq. ( are defined. We can formally write the condition as Here, where we have introduced an arbitrary finite time with a smooth function The condition of in-states and out-states are then given by where (which is assumed to be| time-independent) such thatMore they can precisely, be we regarded should as consider eigenstates wave packets which are superpositions of eigenstates subsection 1.2 Review of aS problem of of in-states and out-states: In particular, we will revisit theis gauge a invariance in problem the formalism. onmethod, the We will gauge dressed argue that invariant states there condition are inWe obtained [ will just also from discuss thebetween the appropriate the gauge dressed invariant state condition. formalismorder and to the explain asymptotic our symmetry results will more be precisely, considered. we first In review the dressed state formalism in metry of QED. Initial andto final different states sectors used with inbetween respect the them should conventional to approach vanish the generally sincethat belong otherwise asymptotic we it symmetry. need breaks dressed the Therefore, states conservation in the law. order amplitude It to was obtain argued non-vanishing amplitudes [ out in [ JHEP06(2019)023 - – 3 S (1.9) (1.13) (1.10) (1.11) (1.12) cannot -matrix . Since ]. d ) can be S . E ) 21 s 1.9 t − i t ∼ ∞ ( , 0 ) t iH − t, ~x ( e -matrix ( ) µ cl i t S j ) − f ~x t ( ( , and define the interaction s µ -matrix. Fortunately, there iH V , . S ) into another time-evolution − xA . ), it is convenient to play in the + e 3 ) ) ) 0 s d ) s 0 1.6 0 t t 1.9 t H Z ( − -matrix as usually adopted in QFT ( t, t I f ( I S t − = in ( 0 ( V V 0 0 0 0 H iH dt U ) = s VU dt -matrix in QFTs. However, this e t t ∞ 1 ( t S -matrix ( s + Z − ]. This inclusive method actually works and as −∞ ) S i V s 2 Z , − →−∞ i i 1 – 3 – t, t as , t − ( 0 ], it was argued that the asymptotic dynamics of U →∞ 8 Fock f such that photons with energies less than t ≡ matrix. It is called the dressed state formalism [ H ] ) with d ) t t S ( E ( 18 ) = lim ) = T exp s I i as = T exp t t ) can be written as V V S Ω( Ω( + 1.6 † s 0 ) ]. It is argued that in any experiment for particle physics the f -matrix elements on the Fock space by the standard perturbation H t S 20 ) in ( , t Ω( usually have the Fock state representation, we finally obtain the ) = 19 t 0 ( i s as β →−∞ | i H -matrix in the Coulomb potential, we need such a modification [ , t which contains contributions of the long-range interaction in the asymptotic S and as →∞ f U 0 t i α = lim | S In Faddeev and Kulish’s paper [ Nevertheless, it is better that we have a well-defined One way to address this problem is giving up the The above is the standard treatment of the For computations of the Fock space basis ]. IR divergences in the conventional approach originate from the assumption that the the IR finite QED can be approximated bypicture: the following “interacting” Hamiltonian in the Schrödinger photons are massless particles,should i.e., take the account electromagnetism ofshould is the modify a interaction long-range the even interaction,operator in free we the time-evolution asymptotic operator region.region. In It fact, means even that for scatterings we in quantum mechanics (not QFTs), in order to obtain the measured cross-section is IR finite. is a way8 to define anasymptotic IR scattering finite states can be regarded as free particle states at textbooks such as [ detector has a minimumbe energy detected, and thereforeevents emitting the undetectable measured soft cross-section photons [ is the sum of cross-sections for all on the Fock space iswe try not to well-defined compute in the QEDtheory, we because encounter of the the IR infrared divergences. (IR) divergences. If where the symbolrepresented T as represents the Dyson the series time-ordered [ product. The operator in the interaction picture as Then the operator Ω( matrix operator on the Fock space interaction picture. We divide the Hamiltonian as Since JHEP06(2019)023 ) ). is s s 0 , we t, t t ( H 1.17 (1.22) (1.23) (1.18) (1.19) (1.20) (1.14) (1.15) (1.16) (1.17) as U , ) , s This current t , t i k p 1 · t ) for any E p ( t , i ]. The exponent ( as )] | − is time-dependent I 2 e as k ~ t U | ) ( ) V , s i as I k ~ t as ) ( = ~p H − ,V † µ ( ) , f t a 1 ρ ( t ) . ( , ω p ) − I ) 0 iH as )] . t t 2 V k ~ − [ ( k . t p · e ) 2 ( I E p ) ) as ) (1.21) s ω, 0 i I ~pt/E s dt as s t 1 V with e s t t ( t 0 ) , t ( t, t − R ,V s k ( = ( as f ~ as ] for the derivation) that 1 t dt ) R ( s t 0 ~x t 1 8 ( U ( H dt µ µ t s U t Z 0 3 t − a ( † k as R ) i δ I ) t )] commutes with as t dt U 2 1 t s , ( p 2 − t µ ) is a classical operator in the same sense ( V ) k t s − [ µ p Z as E ( ~p · e R ) into p 2 ( i V ) I ) as p 0 d p t 1 1.20 − dt →−∞ ) ( 1 ) 1.6 i t − E ,V s ~p I ) = as ω ) is a “classical” current operator given by t ) ) , t 0 ( p V s t 1 † Z – 4 – (2 3 0 k (2 t ( 3 d in ( ) T exp ) where 1 d 3 3 ( t, ~x ) I dt s s )] in ( s t, t as ) →∞ t ( d I 0 t 2 ( π as t dt − f s V R π µ t cl t 0 T exp t U V i 0 t, t ) t, t ( j , the asymptotic Hamiltonian (2 ( Z U ~p (2 − I µ as 0 cl t ≡ dt ( s e i Φ( 2 Z j t b Z ) = lim i U ≡ ) ) ) i ,V e ), one can find (see [ Z s s ) t ) ) ~p ≡ i ~p ( t ( 1 ( ( ) t † t, t t, t ) = as − ( s ρ b X I ( ( s 1.11 as I ) 0 Ω as as V p † t, t U V t, t U ) E ) = ) = ( f ~p Φ( p denotes that operators are in the Schrödingerpicture, and t as -integral, we have (2 ( 3 ( t ) = 3 t, ~x ρ s U d ) runs over all charged particles, and we omit the label for simplicity. s ) ( as π µ cl Ω t, t j (2 ( 1.14 as Z U e →−∞ i , t ) is obtained by replacing X ) are annihilation operators of photons [ t -matrix is then given by k ( ~ ) are creation operators of the charged particles (and antiparticles). ), and we represent it as →∞ ( ≡ S † as f µ d t ) a t t, ~x ( ( = lim We can proceed further by computing this asymptotic time-evolution operator ( The R µ cl The creation and annihilation operators have labels for a spinor basis, if the particle is a fermion. S j 1 (and † including the commutator [ as with where and by performing the where Furthermore, since the commutatorobtain [ Similar to the derivationis of given ( by where Ω where the sum in ( b is “classical” in theof sense the that explicit it time-dependence iseven of a in diagonal the operator Schrödingerpicture. on the usual Fock space. Because the usual free Hamiltonian for QED. where the superscript JHEP06(2019)023 . . In is just (1.26) (1.24) (1.25) as the → ∞ 1.2 t as FK H . ) H s ,t ) in the dressing i is known to be t to the subspace as t, ~x Φ( i ( , we will comment e µ cl FK FK ) j i FK t H H ( 5.1 H R e ) ) only in the dressing factors 0 t , ( I as we reviewed in subsec -matrix on 1.12 ) was modified by introducing a k. V ~ Fock , were required to satisfy S 0 as H ) is not adequate for dressed states. ) dt H f 1.22 FK s t i t Actually, the factors play a similar role t,t The asymptotic Hamiltonian Z 1.26 in ( 3 . i Φ( 4 i . µ ). e − i ∈ H c R ) ]. t ψ 3 FK ( | . We think that the gauge invariant condition − ) R 1.12 t H k = 0 for any µ ( e · is not needed for an appropriate gauge invariant – 5 – p p ) is not a good operator on the usual Fock space R i µ e ψ T exp c | ) →−∞ 1.12 ) to ) t k ~ Gupta-Bleuler condition was imposed on s ( ,t µ = lim f a t = 1. More concretely, the dressing operator was altered 1.22 µ free Φ( µ k i FK c in ( − µ H -matrix ( e are physical, and thus we have to restrict k ) k µ · S f p p t ( ) becomes FK R − H ), the dressing operator is given by usual one ( e ) was deleted by a requirement for an initial condition. Permitting 1.16 FK 1.26 -matrix differs from usual Dyson’s one ( , the energy of the excited photons by the dressing factor is soft in the limit 1.21 H S →−∞ ) satisfying i , the notion of particles for charged fields is still valid because k ~ , t 5.1 ( FK µ , we will present the appropriate condition. -matrix ( c ) in ( H ],if we impose the physical state condition. In subsection . Thus, if we formally introduce a dressed Hilbert space →∞ s is a diagonal operator on the Fock space. However, the standard interpretation of photons on S 3 2 t f Φ t ) ( i t e ( R = lim R the , it is well-defined on the dressed space e -matrix on ], ], to satisfy ( ] seems inappropriate. The S 2 Furthermore, we will show that the dressed Hilbert space can be obtained just by We will see that the artificial vector S 8 8 Even on In this paper, we do not take care of problems at ultraviolet regions. We assume that they can be We will see that we do not need to worry about this requirement in our approach. 8 and 4 2 3 Fock R factor the Fock space seems to besee lost in because subsection the dressingHence, factor excites the an particle infinite notion number for of photons. hard photons As we may will resolved be by valid. a standard renormalization procedure. to solve the dynamics offact, the although asymptotic the Hamiltonian Dyson’s H an approach to derive the dressing factor into the gauge invariant condition, asan dressed interaction. states The are obtained free by Gupta-BleulerIn taking condition section account ( of such requiring the gauge invariant condition. In our approach, it turns out that we do not need In [ null vector by shifting the coefficient condition. Our claim is that contributions of long-range interactions should be incorporated by imposing a gauge invariantin condition. [ However, the treatmentphysical for state the condition, gauge i.e., invariance physical states, IR finite [ on a subtlety of the proof of1.3 IR finiteness in [ Our methodNot and all the of the differences states from in Faddeev and Kulish’s e the as summing the contributions of soft photons, and the this, As a result, this In [ JHEP06(2019)023 ) as t ( H R (2.1) (2.2) (1.27) such that is the La- ) is just one Fock c, ). The operator H 1.25 matter prescription. µ . L free i ¯ 4 c ∂ µ i ∂ i ∈ H = ψ given by ( | , , FP FP FK L L = 0 H i + , ψ | 2 ) GF µ , k ~ -matrix in the usual perturbative ap- +). Here we have integrated out the L ( , A S µ µ free + + a ∂ , µ H k + as 1 2 , R − − e matter can also be integrated out since it is completely – 6 – L is a subspace of the Fock space = = c + as free GF H H L EM L ) acts takes the form , = µν ], the dressing factor for a charged particle with momentum 1.12 F 22 QED µν ), but not necessary. We will discuss the relation between the L s F , we will discuss the meaning of the original dressing operator 1 4 µ t, t c − Φ( -matrix ( i = S EM , as a support of this interpretation and also another justification that we do ) + t L 3 ( . We will reconfirm this fact especially taking care of the ~p R is a dressing factor, and ). As shown in [ as R 1.22 e can be We conclude that infrared divergences of the Besides, our method allows a variety of dresses, and In section corresponds to the potential Liénard-Wiechert for the uniformly moving charge with as ~p and the metric signatureNakanishi-Lautrap field. in this The ghost paper field decoupled is in ( QED. However,grangian we of keep charged matter it fields.scalars to In and obtain this fermions the paper, matter without BRST fields charge. derivative can self-interactions, be any and massive we complex do not write down the where picture to satisfy thea gauge systematic invariance. way, In we order use to the impose BRST formalism. the physical2.1 state condition in Lagrangian and HamiltonianIn in the BRST covariant gauge formalism with the Feynman gauge, the Lagrangian of QED is given by states at the same order, we can avoid IR divergences. 2 Necessity of dresses We show that states with charged particles must include photons even in the interaction ambiguity of dressing and the asymptotic symmetry of QEDproach in for subsec QED are causedthe by asymptotic the states usage of satisfying thelevel, the inappropriate they free asymptotic should Gupta-Bleuler states. not condition be Although may used be at used loop at levels. the tree If we instead use the correct gauge-invariant where the free Gupta-Bleuler condition areR satisfied ( momentum of them. We will seeon that which in Dyson’s our gauge invariant condition, the physical Hilbert space essentially just says thatfields if around there it is by a Gauss’s charged law. particle, The there fields should aroundnot exist the have electromagnetic to charge introduce indeed makein up eq. the ( dress. is a simpler approach to obtain the factor, and the interpretation is clear. The condition JHEP06(2019)023 . ) = s µ ) 2.1 † c A s ( (2.8) (2.9) (2.3) (2.4) (2.5) (2.6) (2.7) ¯ π (2.10) (2.11) , ) c . s ( ) π : ~y − V − = ) ~x † c . ( , s ( commutes with the 3 4 π iδ ij c. EM 0 = F H i∂ ij } ) as ) F = ~y 4 1 ( 2.1 ) ) c . c ( s + ( , ¯ π ) ) ¯ π 0 | , x , φ A ) ( | ) ~x ( φ i ¯ c, ( φ µ s 0 Π ghost V i ¯ c D i∂ ∂ { H ) . For example, if we have a charged − − x = + ( + ( , ¯ ¯ = φ φφ } i µ V, ) 2 j ) , A and the other interacting part matter c ~y − ) ( ) + ( m − c L ) 0 0 ) i matter 0 c x s ( − = Π H ( ¯ i c∂ H H i – 7 – φ φ µ , π ∂ , π i ∂ µ ) ) + = 0 A x ~x − F + ( ( ( ¯ H s φD ¯ ) EM φ µ c c µ = ( µ { H Π i π D D µ ) Π c = − ( Π , and their conjugate momenta are anti-Hermitian 0 π ie 2 1 ) s = (¯ , c ~y H x µ = φ 3 = ¯ x − A L µ 3 d † µ j s d ~x are not related by the Hermitian conjugation. They are Grassmann-odd , the matter current is given by ¯ c Z ( ∂ c 3 , i Z φ which are defined from the Lagrangian ( − s δ such as µ c ) to represent that an operator is in the Schrödingerpicture like = = c e = µν ( = s and ¯ ¯ π ieA 0 † iη ) is different from the Hamiltonian obtained in a canonical way from Lagrangian ( c , s EM ) Π c − c ghost 2.9 H ( φ H )] = µ , π ~y ∂ µ is the free part of the Hamiltonian of matter fields. ( s ν = is the matter current derived from Π with charge , 6 φ given by ( µ by Π ) µ j φ matter ~x ¯ c ( D EM . The Lagrangian and the Hamiltonian are real under this Hermiticity. H 5 We now consider the Hamiltonian. We represent the conjugate momentum fields of s µ ) H The ghost fields c , c, s ( 5 6 A [ µ ¯ π by a total derivative term.BRST We charge have without eliminated a the boundary boundary term. term, and This then differenceHermitian this is operators not important except− in sec put the superscript The equal-time (anti)commutation relationsgiven for by the gauge fields and the ghost fields are and 2.2 Physical states inWe now the quantize Schrödingerpicture the system by imposing the canonical (anti)commutation relations. We with where A The Hamiltonian is then given by free part with the gauge field is given by where scalar explicit form of the Lagrangian. They are coupled to the gauge field so that the EoM of JHEP06(2019)023 ) as (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.12) (2.13) (2.14) (2.15) (2.23) 2.4 is given . ) 0 ~ phys k , . H s i − t ~x · k ~ k ~ ( , , i , 3 i i − δ i s ~x ~x 3 , · · ~x , ) · k k ~ ~ i i i k ~ iωt , i . π i ~x e i · ~x − − · i − ) k s s ~ ~x k s ~ i · s ~x i k )(2 . ~ · . 0 k ~ − ( i k ~ iωt iωt − j ω s i iωt † µ s µν s e e − e + ) ) s a c η (2 ) s iωt ) i µ k k iωt ~ ~ 0 k e ~ ( ( + e iωt k )] = 0 ~ ( ) − k † † iωt ) e † ~x c c k ~ is − k ¯ c ~ − ( ( − )) = ( e † Π + ¯ + ~x k † µ ~ , c · µs k − s } ~ )) ( s k ( a ~ ) ~x ~x c 7 † i i · · 0 ~x k ~ i 3 , j · φ − k k ( ~ ~ ~ k + δ ∂ s i i k ~ + i i s ( ~x 3 ωa · + + † ~x ) + ec k − s s · ~ i s ωa iωt π k ~ , c s i s BRST + − ) 0 iωt iωt ) + s + iωt e Q ] = k ~ k − − s ~ )(2 Π [ ) ) + ( − s ( ) e e iωt ¯ c ω k e ~ c k † 0 ) ) ~ s iωt ( ( ) ( – 8 – { − , φ a k k ~ ~ , ¯ µ 0 i π − k e ~ ( ( i ( ) e a a k ¯ c c i h ¯ c ) ( k , µ ~ h h h k ( ) k ~ k x 0 ) ) ( c ( s BRST − )] = (2 3 h 1 2 ~ k h 0 ] = 0 h µ ω ω d s Q ~ ) ) k 3 a [ 2 1 ( ) − k k (2 (2 k h Z ω ω † ν 3 3 3 3 3 3 π k ,H ) ~ ) ) ) d d k d k k (2 (2 − ( , a (2 ω 3 3 3 3 3 π π π 3 ) ) ) d d d δ = k k (2 ~ Z (2 (2 (2 π π 3 ( 3 3 ) s BRST ) µ d Z Z − Z (2 (2 π Q a π [ [ Z Z s BRST (2 )(2 i i ) = ) = ) = ) = Q ω − Z − ~x ~x ~x ~x ( ( ( ( (2 ) ) s s i c c ¯ c c . For example, the BRST operator acts on the charged scalar in ( s s ( ( ) = ) = ) = ¯ π π = µs ~x ~x ~x j ( ( ( } s i s 0 s µ ) 0 Π Π A ~ k ( † ¯ c ) leads to , ) k ~ ( 2.11 It is convenient to move to the momentum representation because the ghost fields are The obtained Hilbert space is too large, and the physical Hilbert space c This is always possible because this is just a change of canonical variables at a time { 7 with We also introduce the ladder operators of photons as eq. ( free. If we write matter current which is the BRSTwith transformation the for total the Hamiltonian and charged the scalar. matter This current BRST charge commutes by the BRST cohomology. In the Schrödingerpicture, the BRST operator is expressed as Note that, since the BRST charge acts on charged matter fields, it has a term containing the JHEP06(2019)023 , i . } ) i k (2.27) (2.26) (2.28) (2.30) (2.31) (2.24) (2.25) ~ } ( ) s k ~ 0 j ˜ t, s ( I 0 iωt j ˜ e = 0 are gener- should satisfy i 0 iωt i ψ e ) + | β k , ~ ) | ( ) k ~ t µ ) + ( ), ( a k ~ µ ( µ a . µ 1.5 k . µ -matrix is related to the is a physical state in the a 0 { k . k ~ I BRST S i ) µ in k β k ~ = 0 Q i | ( { ) † 0 ) β ) t -matrix is given by the usual | i c i k ~ t S ( β . ). Therefore, the physical state | . + † ) k Ω( ~ c i , we have } ( ~x ) s ) ). This fact holds in the interaction c ( = 0 + k ) = Ω( ~ k ~ s µ H s , 0 } j i = 0 for all i − ) s BRST t ) defined as ~x ( ). Since 1.26 · ( β i t, t k ~ s ~x | k Q ~ I ( i 0 ( ψ 0 ) − 0 | s j ) and ¯ ˜ 1.5 i − j ˜ 0 s t U i k i ~ t, j ( ( ) ( x e c I iωt k = 0. Thus, from eq. ( ~ →−∞ iωt 3 0 i ( − e t – 9 – d j ˜ s , as ( e in s BRST 0 i I BRST Z in j which is acted on the ˜ = lim ) (2.29) iωt Q i β s ) + s Q | ) − 0 ) + β in k ~ | i e i ( iωt , t k ~ ) = t, t β s ( e µ β ( | k ~ t | † µ a 0 ( ( lim ) + a →−∞ s BRST µ s µ U i U µ k ~ t j ˜ ) + k Q ( k h k ~ † µ ) is then written as ( { s BRST a ) µ ) = , cannot be the asymptotic physical states. µ Q k ~ t a s BRST i ( k µ 2.12 0 lim c →−∞ Q { | k Ω( i h ) t 1 h ) 0 = k ) ~ − ~p ( ) ( ω c s † = 0 becomes h b s BRST k (2 i ) 3 3 t, t Q ) ψ d ( ω | 0 π satisfies k (2 ) is the BRST operator in the interaction picture: U (2 3 3 t 0 commutes with the exact Hamiltonian ) ( d i ≡ Z π s BRST ) is the Fourier transformation of β ) | k t ~ − Q (2 ( ( ). The asymptotic state s I BRST Z s BRST = 0 Q j ˜ Q − 1.12 I BRST Since the ghost fields are decoupled, we can always restrict the ghost-sector of physical The BRST operator ( = Q s BRST Q It means that states satisfyingally the not free the Gupta-Bleuler physical condition asymptoticFock states. space, Thus, such the as charged 1-particle states in the standard Therefore, By restricting the ghost-sector to the ghost-vacuum, this condition becomes where Schrödingerpicture, it satisfies Since 2.3 Gauge invariant asymptoticWe states now move toone the ( interaction picturein-states in such the that Schrödingerpicture, the This is the physical statedifferent condition from in the the usual Schrödingerpicture. Gupta-Bleulerpicture Note condition as that ( we this will condition see is in next subsection. states to the ghost-vacuumcondition annihilated by where JHEP06(2019)023 ) ]: 17 ) by (2.36) 2.31 (2.37) (2.38) (2.32) (2.34) (2.35) (2.39) (2.40) (2.41) (2.33) ∼ ∞ t , i is a subspace of ) k ~ 8 ( free µ , a ) are dressed states. H µ ˜ k ) as ) k ~ k 2.31 ). One example is ~ )] = 0 . t , t, − ( ) ( ˜ ~p t, ˆ ) where R 2.32 G ( = 0 ( ] for details) , k ~ I . . ρ ) 0 t 0 ) ) 23 i t, k ~ . j k ˜ ~ p . k ~ ( · ) β ) E ~p I t, | ˆ ) by x. t, t, ~x i 0 k ( ~ iωt ρ ( ( i ), we can simplify it by recognizing ( t, ~x j ˜ I − | t − ) ( 0 0 0 µ cl cl e ( τ e . k i ~ cl j j ˜ j ˜ | a ) 2.38 iωt [ ˜ j , R p µ i e − t iωt = k free E ( ) ) such that e ) , t )+ p t k 0 H ~ cl ) ( (2 ( →∞ →∞ 3 ( ) k j ~ ˜ 3 k i ). It is convenient to use the hyperbolic co- τ ~ ˜ τ i ( R † µ d t R ) ) + ( e µ ) as a t, π ˜ k ~ R iωt a ( τ, ~x µ ( ) = lim I e e µ = ˜ 1.14 k ) = lim 2 (2 µ 0 , we can easily confirm that the Faddeev-Kulish – 10 – ) k ) ρ t 2.31 a j ˜ t t, ~x k ~ Z ( k ~ p ( µ · t, ~x ) + e E ~p R k t, ( iωt k ~ lim i ( →−∞ e I e iI 1 + free ( i ) I 0 j − t ≡ µ 0 − j k e ~ X p a j ˜ ) t, µ k ~ iωt ( = given by ( k lim iωt →∞ e t, ˆ t h ) = e )] = τ G ( 0 ) satisfies lim cl t →∞ h k ~ j = ˆ can be approximated in the asymptotic regions ( ( τ G 2 ˜ t, t I R ( 1 k ω p 0 , 1.22 · lim 0 →−∞ 2 cl ) E j p i j ˜ i t k ) ~ ( − are annihilated by ω µ e ) in ( a means that the current is that of the free theory. k (2 t µ free 3 3 ( ) k d ). [ H R π ) k ~ t free ( (2 − ˜ R ) to look at the asymptotic behaviors of massive charged fields as follows [ e Z ω, τ, ρ = ( ) = t µ ( ˜ ˜ k satisfying the free Gupta-Bleuler condition. Thus, the Hilbert space satisfying ( R Noting that the momentum representation of the classical current operator is given by Although one may use such a dressing operator We will show below that the states satisfying the condition ( Other components of the current also satisfy similar equations: 8 Fock Here, the subscript and a trivial equation dressing operator For later convenience, we represent the operator in ( Therefore, we can rewrite the condition ( ordinates ( Then we can straightforwardly obtain (see appendix C in [ where that the current operator the classical current operator is given by There are various choices of the dressing operator satisfying ( In fact, if there is an anti-Hermitian operator then states in H JHEP06(2019)023 . ) ) t ( free as (3.1) (3.2) 1.25 H (2.42) (2.43) (2.44) R ] that the in ( 24 FK ) is obtained H 1.22 , ), Φ is not relevant )) t , ( τ x · to the subspace ( R y ik e ) − Fock i x H ( ) and 4 + k ~ δ . prescription, and see that we ) is given by ) t, ω ). Although the question that -matrix is beyond the scope of ) ( τ k ~ S ( ˆ i + ( G µ dτ µ 2.38 0 2.38 a , k dy . µ 1 )( k free dτ ) free i = 0 is not relevant when we consider the t ( H ∞ t , if we restrict H + ) −∞ as µ i ) t i Z R c ω ( t e ( e as R − R that the existence of many choices is natural = e 0 e – 11 – ) ) = k 4 t ( ( x ) commutes with ( as 4 lim ) µ →−∞ k R lim i →−∞ 4 π e t i t ) d , j 1.23 (2 k ~ . The classical equation of motion for the gauge field in ) µ x t, ). Then we can define another asymptotic physical Hilbert p ( Z ( µ ˆ G j − ). Therefore, the Faddeev-Kulish dressed space 2.34 − = 0. The position at is the trajectory of the charged particle, which is supposed to = 0 can contribute to subleading orders, and it was shown in [ t t ) = t 2.38 in ( p x µ ) = ( p E 0 x cl ( j ) other than the Faddeev-Kulish dressing operator ( ˜ ret t µ By using the retarded Green’s function for the Klein-Gordon equation, = ( G A ] that the Faddeev-Kulish dressing factor for a charged particle with 9 as τ µ with R 22 m p corresponds to the classical potential Liénard-Wiechert around the particle. I 0 µ j ˜ p ) = τ ( µ y Besides of the phase operator, there are other choices of the dressing operator Since the phase operator Φ in ( However, the position at 9 asymptotic region. position is important for the subleading memory effect. where pass through the origin at 3.1 Coulomb potentialHere, by point we will charges in recallpoint the the particle asymptotic expression with region of momentum thethe Lorenz electromagnetic gauge potential is created given by by a charged momentum This fact supportssection, our statement we that willshould Gauss’s reconfirm use law this different require prescriptions fact for theexplicit taking initial computation dressing care and of factor. final scattering of states, amplitudes. the which In might this be useful for the this paper, we willfrom discuss the point in of subsection view of asymptotic symmetry. 3 Interpretation of theIt Faddeev-Kulish dresses is shown in [ space: which is a solutionwhat of types the of gauge dressing invariant cancel condition the ( IR divergences in the satisfying One example of by replacing for the gauge-invariance ( is gauge invariant without introducing a vector Thus, an asymptotic physical Hilbert space satisfying ( JHEP06(2019)023 . t (3.6) (3.7) (3.8) (3.3) (3.4) (3.5) k p · 0 E p t i p ~p . − E . e − ii defined by . ) and charge ) t ~x k ~ t µ p ( · ~p ( µ t p E † µ k ~ p p i p || a ~p − e E ) ~x ) t 0 i as − t ( · ~x match the classical k p ~ − µ + i t µ · p hh ( − p k ~ k 10 . i , and the second term e · − − e t i p e x · insertion. How to insert p ~p ) i E − ik 0 µ − t i − p − t k − − µ e → −∞ ~x k t p · · ) ( p prescription k t E p · p k ~ i · k ~ iω ( i e e ) is not normalized. i † µ p ) − + , a k e ~ − 3.5 ( i + t ) i µ 0 + 0 p | ~p t a t E x ) · − − µ p t ) up to the ~p ~p i − ( p ik E ( k ~x e † · − iω − ) · µ b ~x ) k − ~ p p k ~ 1.22 k t i ( e · ( · e ) as k ~ µ in + ) i ~p 0 p a i e ; R ω x 1 t e x – 12 – 2 ( i + ( p µ 0 ) ) ~p µ E p j k ω ω + dt ret µ µ ) · − 0 ii ≡ p t k k ~x (2 k (2 A x ) p −∞ 3 3 3 3 · t · ) ) Z d ( d k ~ − p i ) π π p 3 e || x ) ω k (2 (2 ) ( 3 π i k (2 ω d 3 3 Z ret Z (2 + ) µ k d (2 ) ) are given by G p 3 3 π k 0 ~p Z ) d ( · x ) = (2 3.1 π p ρ 4 µ p x ) d p E ( (2 p Z I µ ie e Z ) E Z A ii is given by the superposition of the Coulomb field created by each p ω ) e (2 − 3 t ). The generalization to multi-particle states is trivial, and the expec- 3 µ k (2 ( ) + ) ) + d ) 3 3 p x x x A ) π || d ( ( ( ) created by a charged point particle with momentum 3.5 ≡ − ) π (2 in µ in µ in µ x ) ( 3.4 (2 ) is the incoming free wave, which is specified at ~p A A A ) is an operator dressing the incoming single particle state. The gauge field in I µ Z ; x t ( ( x Z e A = = ( || in e in µ ) = ) t ret µ R x − A ( ) = ( A t p µ = ( hh More precisely, we should use a wave-packet since the state ( A in the dressing operator is determined by how the initial condition of gauge fields is in 10 . We represent this second term by R specified. Thus, therelativistic dressed states Coulomb stand fields for createdparticle the by state states themselves. ( of (anti-)electronstation We surrounded value have by considered of a single charged This operator matches thei dressing operator ( We can easily check that the following dressing operator satisfies the above condition, Then we demand that its expectationgauge value for field the ( above dressed state where the interaction picture can be written as 3.2 Coulomb potentialLet’s from consider dressed a states dressed with state of a single incoming electron with momentum where is the potential Liénard-Wiechert created bye the particle with momentum the general solutions of ( JHEP06(2019)023 . out t = 0. R k p (3.9) · − given E p (3.12) (3.13) (3.10) (3.11) e i in − e R ) . k ~ ( ). Thus, the † µ t , ) for outgoing in ) a t p i ~p ( t R E ( i − − in out ~x − µ R R · = e p out k k ) ~ i · i ii † in ) − p t , t e R ( f p t − i || ( t ) 0 t k + p µ · S ( E p ) p k p i f · -matrix. Including the dressing t ]. , e hh ), and the dressing factor ( ) ) , guarantees that the asymptotic t S p 25 ( k ~ out out out ( out + R µ out ) R R a − ~p R nonzero. After computing IR finite physical t − − ; e − i e p ~p x e E ( = ) + µ ~p − ) = ( p ~x i k ) agree with the advanced potential for the is anti-Hermitian ( adv µ and b † out · · x | , t A k ( ~ R in in 0 p f i ]. This is another reason to think that such a I µ t e R 8 R = ( e – 13 – A ) i S ω || ≡ h out − µ ) ii t k (2 p k ) ( as in [ 3 3 t · ) p d ( µ π hh p p c || ) (2 prescription for the dressing operator ) out 11 x ) with i ( ω Z i ]) that the dressing factor is not a unitary operator on the Fock space I µ ) , t 8 k (2 ~p f A 3 3 ( t || ) d ρ ( ) π t ) prescription for initial and final states may be related to the prescription used S ( p (2 p i E is also anti-Hermitian ( to 0. is unitary. terms is opposite to that in the initial dressing operator hh p Z (2 3 in e 3 →−∞ i out in the normal ordering, the normalization factor has an IR divergence if we set d out i ) R − ,t π R − in e R (2 − -matrix acting on the Fock space takes the form (up to phase operators) →∞ e ) = f S ~p Z t This ; e x = lim ( 12 S ). ) = adv µ t ( Here, we also give a formal proof of the unitarity of The unitarity of the dressing factors, Similarly, we can fix the A 3.8 If we write This difference of the out 11 12 R Thus, it is oftenin said a (see, rigorous sense. e.g.,quantities, [ However, we it can does take not matter if weto keep define in-out and in-in propagators in nonstationary spacetime [ niteness. factors, the Hilbert space is positivethe definite. unitary The dressing asymptotic dressed factorsdressed states states by are thus states given have a by satisfyingcondition positive multiplying are the norm, positive free because definite Gupta-Bleuler states and satisfying any condition. the unitary free transformation The Gupta-Bleuler preserves the positive defi- Thus, the sign of by ( is thus unitary. The requirement can be satisfied by the following dressing operator and require that the expectationpoint value particle, of which is given by modification is unnatural. Notedressing also factor that states. We consider a dressed outgoing state by electromagnetic fields in thevelocities. asymptotic region This where result the is particles natural(gauge have since invariant) nearly condition our constant without dressed ignoring states thealso are interaction obtained would in by like the solving to asymptotic the comment regions.operator BRST that by We this introducing expectation a value vector changes if we modify the dressing particle. In other words, in the dressed state, the charged particles are properly dressed JHEP06(2019)023 and (4.1) (4.2) (3.14) (3.15) in R takes non- ) ) i i . t t ( ( ) s . in in R R , t ). Since i e e ) t ) ) ( i i ) 0 c , t , t 3.14 s ( U f f ) ] is BRST exact up to t t i iπ ( ( [ 0 0 , t , ] for related discussions). + s as f S S t ) ) Q 15 ( f f 0 s t t ) ) as – ∂ 0 ( ( i i U 9 t s j ) ( ¯ c , t s out out in f + − R R t , t R ( ( − − f e i x t e e ) S ∂ 3 i ( ) ) † f f d 0 is t t , t ( ( U f Π Z t , † out out ( = − 0 R R e S − 0 ) ) ) e i i 0 ∂ s BRST = 0. This charge ) t s i , t , t ( 0 Q I -matrix: f f , t – 14 – Π t t f S ( ( V t − † † 0 0 0 ( is a constant, the corresponding charge represents † 0 S S ) + dt ) ) f ) vanishes in the asymptotic regions, this charge does x S i i t i 3 t t ) x t ( ( i is ( d t Z ( in in i Z (Π R R † in i . − R − − ) satisfies e e e x ] = x ∂ ( 3 = = = 1 [ ], the asymptotic charges are physical charges, i.e., the asymp- d ) = s as i 23 Z ) is invariant under a class of gauge transformations which keep ) simply follows from the expression of eq. ( Q i , t − f , t 2.1 t f ( t ) = T exp ( S ] = i ) 0 i [ , t S -matrix is unitary. f s as , t t S f ( Q t 0 ( † S S denotes the usual (finite time) 0 ] for a recent review). The symmetries are given by “large” gauge transformations S intact. Neother’s charge for the transformations in the is Schrödingerpicture given by -matrix in the dressed state formalism. 26 are anti-Hermitian, we can show the unitarity of The Lagrangian ( µ S A out µ charges. As discussed in [ totic symmetry generated by thephysical charges symmetry. is not For a example,the redundancy if total of electric the Hilbert charge.with space There zero but is the total no electric reason charge. to restrict Similarly, we the should Hilbert not space restrict to the the subspace Hilbert space to the Therefore, if the gauge parameter not play any role on thevanishing physical values Hilbert in space. the However, asymptotic this regions. is not Such the nontrivial case if charges are called asymptotic where the gauge parameter the boundary term: We will show that theasymptotic Faddeev-Kulish dressed symmetry, states and carry investigate the thethe charges conservation associated law with of the the asymptotic charges for ∂ Gauge theories in 4-dimensional(see Minkowski [ space have ansuch infinite that the number gauge parameters of can symmetries they be nonvanishing preserve functions the in the asymptotic asymptotic behaviors regionsasymptotic but of symmetry fields. in We QED now and discuss dressed the states relation (see between the also [ Therefore, the 4 Asymptotic symmetry in the dressed state formalism The unitarity of R where JHEP06(2019)023 ). ) is ) is ) = t 3.8 (4.5) (4.6) (4.7) (4.8) (4.3) (4.4) ( 0 ∂ 3.4 in . ). This 2 is R ]. All of − ) r t , ( p 16 + Π . ~p O E ) , the canonical ) and i t x − ∂ 5 ( p ~x ~p iI s i ( ). The boundary E 0 · cl 13 0 s k ~ 0 − i in the interaction F (Π ~x A ) given by ( e i . ( ) given in eq. ( . ] · ~p = 0. On the other is I as t ≡ k ~ ; − ( i x ∂ ) ] can act nontrivially Q k x ~ 3 ) e )] , decays as in t , we can neglect the effect iI ( [ i d ) ~p + Π ) p R s R ~p c − ; Π Π s as s E i ( 0 i ret µ i is ) x j x t Q ∂ − − ( A A → ∞ p ~x π s 0 ~p ( + E 0 · ret 0 r + ( k ~ (Π − i is A i i∂ ~x i )] = − i ( Π s · ∂ e i + 0 ) at x ∂ ∂ k ~ 1 3 i I i ∂ s A − d ( − ¯ c − i should approach functions of angu- is R r i ) e [Π ( , 0 ~p − ωp x ∂ ). As mentioned in footnote O ; ∂ i i 3 k + Π s x ∂ d − · ( = 2.9 + H is i ωp ] = 0 p Z s k ret i A k 0 0 x = s 0 p − ∂ · 3 − up to the BRST exact term: A Π i d E i 0 p (Π s k i ∂ s BRST ) s Z can p ∂ – 15 – H )[ ) = ω i i H x ∂ E ~p ,Q 3 ∂ − ( k (2 in the interaction picture takes the following form ) iI d , s as 3 3 ρ ). Since R ω ) d x ) , I as ] requires that Q 0 (Π 3 p π [ k (2 i s as ∂ [ Q d 3 3 E (2 is Q ) s BRST d s as p Z ) is given by x ∂ π (2 t 3 i Q (Π Q 3 Z 3 ( i d d ) (2 − ]. Therefore, asymptotic charges in acts on the initial dressing operator ie π ) as [ x ∂ Z [ Z R 3 = I as (2 d ) 4.4 s as − ] = R ~p s ) = Q i Q ( Z = 0. Hence, one can say that the commutator of Π ~p and ρ ; ] = ,H ) x I i } − p = 0 because the integrand is not singular at [ ( s as ) )] = E I as i t Q )] ret 0 ( [ p ∂ t Q (2 s 3 ( A in 3 c i has extra boundary terms: d (¯ ) in ∂ i π ,R ) s − can ,R x ∂ (2 ) 3 ) H d ~p t, ~x R Z ), the asymptotic charge ; ( i t, ~x I i x ( ie − ( I i 4.2 , [Π = ret i [Π This fact naturally leads us to classify the asymptotic states by The finiteness condition of This is the reason why we adopted the Hamiltonian ( A s BRST 0 13 ∂ Q Hamiltonian terms affect the{ commutator ( of boundary termscondition if is the probably radial satisfied component for physical of scattering the states electric in a field reasonable operator, setup. ˆ where we have also set given by the “classicalfield operator” as which represents the classical electric Liénard-Wiechert hand, the classical electriccomputed field as for the classical configuration where we have set As in ( up to the BRST exact part: The commutator of Π Therefore, the spectrum of the physical Hilbert space ispicture. infinitely We degenerated. now see how and they commute with the Hamiltonian on the physical Hilbert space. lar coordinates around theasymptotic null charges corresponding infinities. tothe It the asymptotic means number charges commute of that with functions we the on have BRST two-sphere charge: an [ infinite number of subspace annihilated by JHEP06(2019)023 ]. out is a 23 ˆ Ψ (4.9) (4.15) (4.12) (4.13) (4.14) (4.10) . If i 0 . | ) ) x ~p . , ( , ( ) and a similar ) ) ] † 0 cl ) represents that 0 ∞ j out eb [ − 4.10 =+ t = 4.10 out ( + Λ such that they contain i Λ ) = | 0 out | p | + H as R ) ] are given by momentum- . Q out out − ] ~p in ( [ ( e Λ | Λ R † h 0 of charged particles and h ~pt/E cl , . From ( cl as e b j out † Q − out ] (4.11) ˆ Ψ h + [1] = 0 , d ] = | and ~x † 0 cl cl as ( i [ i ) carry the asymptotic charges for b → = i j 3 t ∂ + out Q δ in in i → −∞ + | I, as + 0 Λ ) cl Λ I, as i ) h 1.22 | ~p ) acts on Q F are arbitrary (c-number) functionals of i ( [ − Q | , t ∂ ρ = | I, as i x ) 4.5 | 0 3 cl out out p Q d 0 F as out Λ E [ ) is given by the classical current as h h out S Z | → ∞ t p h . The asymptotic symmetry implies x (2 ( ] in ( 3 3 in 3 − f d – 16 – in out [ ) , d , and Λ R t 0 b, d , i e i π Λ R I as Z S i h in in ] for details). Therefore, eq. ( (2 Q in + in | ≡ Λ I, as ] = | 23 ) Λ Z ] and | ] in and Q in e 0 [ ( R † in iI i [ | − cl as ˆ Ψ , e in Π in ] ) ], and Λ (see [ Q i + Λ Λ ∂ out [ | [ h − H which is an arbitrary function on two-sphere that determines )] = −∞ I as I as t ) with limits = Λ Q ( 0 = Q Q t i [ ( . in in ; that is, we consider states ]. Let us see the implication at the quantum level. To make our in = ( 3.14 3 , we have Λ R ,R | i 23 e ) ] out →∞ ∞ t in [ R | = t, ~x − − = i − ( e I, as t I, as iI in Q | Q Π asymptotically approaches. We then prepare the following dressed states by i ] = lim ∂ [ [ = 1, it gives just a total electric charge as is an arbitrary product of creation operators is given by ( , the eigenvalues agree with the classical leading hard charges computed in [ I, as † in 0 ˆ S Q Ψ At the classical level, the conservation of asymptotic charges leads to the electromag- Therefore, the asymptotic charge to which → ∞ 0 where computation for where is any product of annihilation operators where exciting charged particles on we have seen in section netic memory effect [ discussion simple, we supposetotic that charges the at radiation sectoronly is transverse photons given and by satisfy eigenstates of asymp- dependent functionals of the asymptotic behaviors of charged Fock particles with thethe dressing classical operator free ( chargedis particles natural with because their electric the Liénard-Wiechert fields. dressing corresponds This to result creating the potential Liénard-Wiechert as t The leading hard chargesmoving are charged the particles contributions and to theirconstant the Coulomb-like asymptotic electric charges fields. fromnontrivial uniformly For large example, gauge if parameter, we the eigenvalues take of a lim The integral is in fact the asymptotic charge operator on the Fock space of charged particles. In the limit Similarly, the commutator of JHEP06(2019)023 . ). 0 out ˆ Ψ | 4.12 (4.16) (4.17) (4.18) , there 0 h − H Q ) consist of ) should be and 6= . i + H 0 + H 4.12 | 1.22 Q Q † in ) holds for any ˆ Ψ out ˆ Ψ 4.18 | 0 ) in eq. ( h . t ( = R = 0 i cl as so that ( in Q | i , 0 , is memorized in the change of the in are spontaneously broken in the standard S out →∞ lim in | t Λ Λ -matrix in the dressed state formalism. + Λ out S − h − H out ) i → | ˆ Ψ Q out in in | Λ 0 = Λ , the dressing operators are not uniquely fixed h | prescription by giving the interpretation as the − ]. It also means a quantum analog of the classical – 17 – i out , 2.3 − H Roughly speaking, eigenstates in ( 11 i Q 0 | + Λ 14 − . in † + H I as ˆ Ψ Q out Q with the H − ) Q t ( + Λ R = + H e i Q 0 | ( . in † − H ˆ Ψ -matrix elements can take non-zero values only when the asymptotic Q ) makes a photon cloud only when there are charged particles. Thus, we represent the hard charge eigenvalues for the states S − cl as ] it was discussed that the dressing operator + H 8 + H Q 1.22 Q ]. Q 16 ) becomes lim and ) in ( →−∞ t t ( − H 4.14 R Q We close this section with further discussion and comments on future directions. We here comment on the possibility of other dressing operators. The standard Fock The asymptotic symmetries for general gauge parameters 14 carry the chargesconservation associated law of with the the asymptotic charges asymptotic for the symmetry, and have investigated the Fock vacuum [ original paper [ modified, we have found thatunmodified such dressing a factor modification is notCoulomb fields needed. around We charges. have also Ingauge-invariant addition, justified dressed we the states. have shown We have the also possibility shown of that other the types Faddeev-Kulish of dressed states 5 Conclusion and further discussion In this paper, we havefrom shown the that gauge-invariant condition the without Faddeev-Kulish solving dressed the states asymptotic can dynamics. be While obtained in just the need other dressing operators thanAs Faddeev-Kulish’s in we order have to already prepare arguedfrom eigenstates in the ( subsection gauge invariant condition.symmetry, We and think leave that it this for varietyof a is future the related work asymptotic to to the charges. classify asymptotic gauge invariant dressed states in terms hard charges vacuum is not theclouds eigenstate of of soft photons without chargederator particles. However, the Faddeev-Kulish dressing op- between the out states and thememory in states effect. [ In a scatteringshould event, if be the a hard charges change areConversely, in a not conserved the change in radiation the sector radiation sector, Λ It means that the charges conserved, respectively as Thus, ( Here, JHEP06(2019)023 , - t S k p · E p ) or ]. If i (5.2) (5.3) (5.4) (5.5) (5.1) 3 e . This 3.8 , ) . ω ]. In fact, ) ( 3 δ ω ) affects the , ( is an on-shell i → ∞ i δ -matrix in our ) ) µ t i k ~ S k ~ p 1.22 ) ( ( † µ k ~ † µ . ( a ) a i † µ ), the operator takes t a ( − in ) + ) R k ~ k ~ 1.22 ) + ( e ( µ k ~ µ ( a a ) µ h h 0 = 0 because a t k ( h k ω µ µ I · . · p k p ) p µ V p · ) might be needed to make the α 0 p 3 ) ( p ) k dt ω f 3 3 π t ) probably suffers from an infinitely i ) 1.23 k d iπδ t k (2 3 (2 π 3 3 Z ∓ -inserted dressing operator eq. ( d 5.5 ) d i (2 Z π -matrix element and then take the limits , the phase operator cannot be determined = ) − Z S ~p (2 ( ) i ~p ρ 2.3 0) can be ignored in the limit ( ) soft iαt ). Then, owing to the oscillating factor | ρ p e Z – 18 – ~p ) α ) | E p T exp ~p p ω > ) ( (2 E 3 > 0 for the dressing operator ( f 3 ρ t p d ) ( (2 ) p 3 lim ∼ 3 p π →∞ d ) can be zero only when t ) E out E ˆ k k 0 (2 π R · p (2 → − 3 ~p (2 Z lim 3 e d ) e − Z π p e 0. Roughly, this is the dressing operator used in [ (2 E X ( →−∞ i ∼ Z . In addition, since eq. ( X 2 ω iπ , t e lim k − 2 − iπ = →∞ X f t k ) = 1 at ) = ∼ · → −∞ t t ( ( p i ∼ t in soft t out R k p R · R E p i and e ). We shall use the following identity as a distribution: . lim →∞ →−∞ lim t t 0 0 3.12 ]. Since only the soft momentum region is relevant for the proof of the IR finiteness, Nevertheless, it may be dangerous to use the above asymptotic limit directly. The However, one may worry that the hard momentum contribution in ( 3 → → → ∞ lim lim -matrix on the Fock space is given by → ∞ f t oscillating phase factor, the phasematrix operator well-defined. such As as we ( saidfrom in subsection the gauge invariance.dressed We would state like formalism to including report the a determination computation of of the the phase operator in future. S Thus, we should first compute the finite time Therefore, we can say thatt only soft photons constitute the dresses in the asymptotic limit From the identity, we have contributions from nonzero momentastatement ( can be madeeq. more ( rigorous by using this simplification may be justified. physical observables. We canFirst make note a that rough argument thatmomentum this of is not a the massive case particle as ( follows. because the behavior of thein dressing [ operator at the non-soft momentum region was not specified The infrared finiteness ofwe the extract dressed soft state momentum region formalismthe is form based on Chung’s analysis [ 5.1 Softness of dresses JHEP06(2019)023 , 27 ], we (5.6) ) are 36 5.6 , and path- 35 ]. However, µν 32 F is unnecessary. µν c , which is an analog of ]. It was found recently µν ). As in QED, we should c 40 ]). A similarity between – 31 1.26 38 – . 29 ]. ) x ( 34 φ ) ξ ( µ A µ dξ x Γ R ]). However, a tensor ie – 19 – − ] has additional terms depending on the choice 13 , -matrices directly. We think that the dressed state e ]. It might be possible to construct dressed states ] was extended to more general theories [ 32 S 12 2 ≡ 45 , – 1 41 ; Γ) x ), which permits any use, distribution and reproduction in ( ] for gravitational theories in AdS, operators like ( φ 33 ] (see also [ 37 CC-BY 4.0 This article is distributed under the terms of the Creative Commons ). Furthermore, we should also investigate the relation to the asymptotic ], was introduced by imposing a free “gauge invariant” condition which is a 8 1.22 in [ ], the behavior of Γ near the asymptotic region was not specified. Thus, it is µ c 28 , It is important to extend our analysis to other theories. Although the cancellation of IR Mandelstam developed a manifestly gauge-independent formalism of gauge theories [ 27 ]. In the formalism, the dynamical variables of QED are the field strength tion of Science (JSPS) Fellows No.16J01004. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. in a massless scalar theory from the gauge-invariant condition for the dual two-formAcknowledgments field. The work of SS is supported in part by the Grant-in-Aid for Japan Society for the Promo- gravitational counterpart of the free Gupta-Bleulerimpose condition an ( appropriate physical condition, andAsymptotic we symmetries expect for that scalar the theories tensor arethat also the studied asymptotic in symmetry [ oftwo-form a field massless dual scalar to is related the to scalar the [ gauge symmetry of the divergences in the inclusive method [ do not know how to defineformalism the is IR surely finite useful for thisgravity problem. was developed The in dressed state [ formalism fora the vector perturbative asymptotic boundary is important inIn determining the [ transformation law ofinteresting the to symmetry. understand more preciselydressed the state relations formalism among Mandelstam’s and formalism, the the asymptotic symmetry [ the dressing operatorof constructed the in path [ Γ.Kulish’s one Thus, ( the dressingsymmetry. operator As seems explained not intransformed to [ be under related the directly asymptotic to symmetry, Faddeev- and the behavior of the path Γ near the Such fields attached withconstruction Wilson in lines the areMandelstam’s also AdS/CFT formalism considered correspondence and in the (see the dressed e.g. context state of [ formalism the was bulk discussed in re- [ We would like to comment on other future directions. 28 dependent charged fields such as 5.2 Other future directions JHEP06(2019)023 , ]. 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