JHEP03(2017)136 Springer March 13, 2017 March 27, 2017 January 3, 2017 : : : February 23, 2017 : Accepted Received Published Revised 10.1007/JHEP03(2017)136 d,e doi: , Published for SISSA by and G. Marmo d,e,f F. Lizzi b,c [email protected] [email protected] , , . 3 1612.05886 The Authors. c Gauge Symmetry, Space-Time Symmetries, Nonperturbative Effects

A.P. Balachandran, The meaning of local observables is poorly understood in gauge theories, not to , a [email protected] Departament de Estructura iUniversitat Constituents de de Barcelona, la Diagonal Mat`eria,Institut de 647,E-mail: Ci´enciesdel 08028 Cosmos, Barcelona, Catalonia, Spain [email protected] Taramani Chennai 600113, India Dipartimento di Fisica “E.Via Pancini” Cintia, Universit`adi Napoli 80126 Federico II, Napoli, Italy INFN — Sezione diVia Napoli, Cintia, 80126 Napoli, Italy Departamento Universidad de Te´orica, de F´ısica Zaragoza, C/Pedro Cerbuna 12, E-50009 Zaragoza,Physics Spain Department, Syracuse University, Physics Building Syracuse, NY 13244,Institute U.S.A. of Mathematical Sciences, C.I.T Campus, b c e d a f Open Access Article funded by SCOAP standard results and leaddoes to charge not conservation require and Lorentz low invariance. energy theorems. TheirKeywords: validity ArXiv ePrint: effects are captured allowing testoperators functions which generate do a not larger vanish at group.sectors, infinity. which The These differentiate quotient extended of different theselection infrared two sectors. sector, groups a generate The result superselection BMSbroken. long group known changes Ward for the identities its super- implied Lorentz by subgroup. the It gauge is invariance hence of spontaneously the S-matrix generalize the speak of quantum gravity. As(infrared) a transformations step in towards local afunctions, quantum better at physics. understanding first Our vanishing we at study observables infinity.can are asymptotic In be smeared this seen context by we as test show constraints, thatgauge which the generate equations transformations. of a motion group, This the is group of one space of and the time dependent main points of the paper. Infrared nontrivial Abstract: M. Asorey, Equations of motion asrules, constraints: Ward superselection identities JHEP03(2017)136 ] ) η 10 ( G 11 ], where the superscript 11 [ ], we introduce commutators ∞ 0 9 G 3 13 in canonical quantisation, and generate i E 9 of electromagnetic observables. The Maxwell – 1 – ) and charge conservation A ζ ( Q . Following the terminology of the canonical approach, 7 A 8 vanishing at infinity. They are the spacetime analogues of ]. 8 ) of µ – for the electric field A η 1 ( i E D i ∇ 14 3 1 17 vanishes at infinity and the subscript indicates that the group is connected η The focus in this paper is on the covariant formulation of Gauss law and infrared effects. 2.1 Smearing 2.2 Vacua and coherent states the Gauss law operator spacetime dependent gauge transformations.and They vanish are on first the domain class inwe call the the sense gauge of group Diracindicates they [ that generate on exponentiation as For these reasons we concentrateresults of first interest on to us. the free Followingfor the Maxwell work smeared of equations. Peierls fields, in That 1952 definingequations [ bring the are out algebra the formulated asdepending on quantum test constraints. functions They are defined by operators theory is well understood,for quantisation a on satisfactory spatial conceptual slicesspacetime basis and approach when its to one variants quantisation seem exploresergy is inadequate the therefore theorems limit suggested, have of and recentlyblack low attracted we hole energy will much information photons. follow attention loss it. in A [ connection Low with en- the problem of 1 Introduction In this paper, weconcepts approach of the local quantum study physics. ofstood Since asymptotic in the (infrared) meaning quantum of transformations gravity, local we using observables focus the is poorly instead under- on quantum electrodynamics. Although the 8 Conclusions 5 The superselection operator 6 Charge conservation and low energy7 theorems The BMS group 3 The superselection algebra 4 Gauge invariance and Ward identities Contents 1 Introduction 2 Equations of motion as the covariant gauss law JHEP03(2017)136 ] 20 low ] = 0 of the ]. The  ,S is gauge 11 ) ]. J ζ ) generate S ( ζ ) effectively, ( 13 Q , A Q ( 12 D ]] since because of ) = 1, [ 15 ˆ k ( ) is an invariant of the e ζ A ( , Chaps. 7 and 11]. They D and is an abelian group. The 22 ) commutes with elements of 2 that acts on which do not vanish at spatial on ζ S ( -matrix, Ward identities follow. ∞ 0 ζ ). The group that S ∞ Q 0 G on ζ . G / ( e ζ ) does not vanish at infinity [ 0 / ) lead to other low energy theorems. ˆ 0 G k introduced by one of us (APB) and G ζ ˆ k G ( ∞ , k 0 e ζ 0 ) if G k / . We argue that it acts as automorphisms ( A 0 4 ( e ζ G 0 D M → – 2 – ) becomes k ζ . This is the familiar statement that ( S Q ), which are also constructed from the equations of ) = lim ζ ], they are described in [ ˆ k ( ( e ζ 21 Q separately. Further this action is non-trivial and changes ∞ 0 . consists of real functions 0 G ) and defines a superselection sector. G ∞ 0 ). The representation of . It is the quotient group A or G ( 0 / A 0 0 G ( D G ) need not vanish on G ). In that manner the BMS group action changes the superselection ζ ζ ( ( on ]. ⊆ D Q Q vanishes at infinity, 19 A ) – of A ζ . ( 16 ) generate gauge transformations and hence commute with all observables, ], but in the latter the sphere arises from blowing up spatial infinity. D η ( 0 14 is normal in G G acts as identity. But it is also the case that , but not on ∞ ∞ 0 0 ∞ 0 G G G / 0 Low energy theorems involving photon were already present in the work of Low [ The BMS group first arose as an asymptotic group in the analysis of asymptotically Incidentally, since charge conservation comes from the behaviour of QED as the pho- As is well-known, from the gauge invariance of the We then consider gauge invariance, Ward identities and low energy theorems. Since The operators The Lie algebra of Infrared effects are captured allowing test functions eigenvalues ) and G emerges from the infrared cutoff, blowing up the tip of the light cone in momentum space: ζ so that ( 2 sector. By definition, thenknown it for is its spontaneously Lorentz broken. subgroup, This but result, is as here mentioned alsoand above, extended Gell- is to Mann supertranslations. andproved Goldberger that [ the zero energy total cross section in Compton effect is exactly given by the energy theorem flat gravity. It was later understoodconformally as compactified the Minkowski group space whichon acts on the null infinities the We do not use Lorentzinfrared invariance effects, to the deduce Lorentz charge groupimplemented conservation is [ [cf. spontaneously [ broken in QED: it cannot beton unitarily frequency goes to zero, i.e. at large distances, it is appropriate to call it as a spacetime gauge transformations. Thatproof is of important: the gauge if invariance these of operators QED will do not notBut be exist, we complete. the can alsogives deduce charge low conservation, energy while theorems other therefrom. choices of Thus if Vaidya in [ Q they commute also withinvariant. the S-matrix But in contrast to the usual treatments, we have the operator realisation of S The group is isomorphic to the Sky group group since A representation of been previously considered by several authors in the canonicalinfinity. formalism That [ leads tomotion. operators When on exponentiation is called to the identity. The global analysis of gauge constraints in QED with test functions has JHEP03(2017)136 ], , ) 16 x (2.5) (2.1) (2.2) (2.3) )( . A 4 ( being any µν µ F η ) in contrast to , x µ )( η η ν ( . ∂ µν − = 0 , but it is necessary to ν theorems x f µ η A = 0 (on the mass shell): 4 µ f ,... d µ 3 µ ∂ , k ∂ Z 2 ,... µ ] to obtain subleading terms , 3 1 2 k ); ) = , 23 4 η 2 . ( R , x = 1 . In such a case by integrating by · ) = ( µ µν x ik n ∞ f ( f 0 − = 1 µ C e = for A ) ) ∈ µ x , n x ( ) , η µ µ (2.4) k )( f  ( ∞ η ) where µ ( x f ˜ η f < ( 4 µ µν ); d – 3 – ) = 0 ) ∂ µν f x k k ) ( f n ( ( Z µ i x ν ˜ ˜ ( f f ∂ ∂ A ν · · ) ) = x k k = ··· ( k x ∂ , which are real, smooth, vanishing at infinity and obey = 0, i.e. 0 1 ( µ n µ 4 ] and references therein). These µ i i µ µ → d f f ∂ lim e k f f 26 x f , ∂ µ – Z 4 ) of functions which satisfy the following constraints: ∂ ··· d k are gauge invariant because of the transversal character of the 24 1 C ( i ) = µ Z ∂ ˜ A f x 0 ( µ → lim ) = k A 0. Hence in particular f ) ( x ( A → = 0) solution of the equation µ k µ η x f µ 4 can be rewritten as ∂ d µ f Z ) = f smeared with test functions ( µ A are all finite as Let us introduce the space which obviously is gaugesupport. invariant. For the purposes Usuallythe of test functions this to paper functions vanish this at are requirementof infinity, but is taken the the too Fourier to useful drastic. transform requirement be for We is certainly small of in need momenta. reality compact on We the define, behaviour for test functions. Thisfunction can be made moretransverse explicit ( by pointingparts our we that have any transverse test A the Lorentz gauge condition Notice that the elements 2.1 Smearing The observables we should consider“smear” are them the with quantum connections appropriatethe test algebra of functions. electromagnetic observables We is will taken to do be generated this by following quantum connections Roepstorff [ of zero mass. 2 Equations of motionIn as this the section we covariant will gausspreted argue law as that constraints. the We equations restrict of ourselves motion to of free electrodynamics electromagnetism can until be section inter- in the photon frequency.being These results valid are in based the ontheorems Standard analyticity from Model. and are the and non-perturbative, scattering Goldstone beyond. amplitudes modes Later (see there [ ofthe appeared chiral above many symmetry work low in energy breaking photons, and were applied only approximate to as pion they treated the pions as particles Thompson formula. This work was generalised by Singh [ JHEP03(2017)136 . ∈ C (2.9) (2.6) (2.7) (2.8) (2.14) (2.10) (2.11) (2.13) in the µ defined f A indicates F s and the . From the mathe- † is ) g W k , . ( ) λ . µ y ] ] (2.12) ( a A x ) · and µ µ y g ik ∂ − ) f e x y † ( − · ) . But we choose to work with µ ik − k ( e W A x ) acts on the Fock space µ ( λ which is the the full Fock space. . − a f , ) ∂ ( . ) ) D k ) y + k A W . ( 0 ( x − x , † µ n k · ( µ x ) = . † µ a ( 2 ⊗ µ a · k ik ∗ 3 a A ) 3 ) ) ( − ik H k d s k e y f creation operators π − ( ( ) λµ 4 ), one notes that it depends on ) = 0 e µ µ n and (2 k d e F x f f )[ e ( =0 f , but the space contains also functions which do ( ( generate a Weyl algebra ∞ µ – 4 – ) k µ n M Z ) C ( a A ) D a k x k f ) = ( 4 = ( ( )[  with dµ k d k ( iA dµ ( F dµ e Z Z , dµ Z dµ Z ) = ) = )] for two such test functions Z )] = g y f ) = 0 g ( ( ( A − ) = has the mode expansion ( W ,A x ,A x ) , and therefore are not of compact support. We require µ ( ) ( λµ f µ f A ( F D ( satisfies the wave equation λ A A A ∂ [ and the usual invariant measure is → ∞ D | will play the role usually played by compact support functions. x k | C = are generated by the action of representation is smaller than that of 2 n k ⊗ F √ H is the causal Pauli-Jordan function = D 0 k We next consider the equations of motion. Classically they are The unitary operators The commutator [ Upon quantization, Compact support functions belong to ) which is better known in physics. Notice that the domain of the algebra f ( matical point of view,A it would beFock better space to work with The causal function where and its adjoint where the symmetrized states. To see the action of With standard commutation relations for as the Hilbert space completion of the multiphoton space of states with not vanish fast for For us the space JHEP03(2017)136 , ) µ ∂ 2.1 ) y (2.17) (2.18) (2.19) (2.20) (2.15) (2.16) ) ). The x − − x ( 2.13 y ( D , ) and transfer D − 4 ) first class vari- y R ( ) = 0 )( ) that fullfils ( ) = y λ ∞ 0 η y . η are ( ) for a test function )( C 2 λ 2 = 0 as in ( , − ∂ λµ η ∈ A 2.1 ( ( . F x D ) µ µ ρ ( y λ 4 = 1 . 4 η  F ∂ ) D in ( d R ρ a ( µ ) is the generator of a gauge A ∂ , µ ∂ ( ) η Z ) ∞ 0 η ( y ν µ A C , but G ( ∂ ∂ ) − µ ∈ ) or that − η λµ x − a,λ µ ( A F η i ∈ D ν ( λ )= D η λ ψ ). So ) ∂ | x µ ∂ µ x , η ∂ ( , or worse, will occur in commutators or − µ µ ⊆ D )( 2.13 ∂ 1 x η ) is not y A D ( 4 ) η ) ( ) = d − η A D ( η ( ) 2.15 ( λµ – 5 – . Z y a,µ λµ F ). µν A η )( λ F = 0 if AD F η λ  i ) = ( ) are first class constraints. That is the case, for 2.3 y ∂ η ). Here ψ η ) of 4 ( | ) may be noted: x ∂ λµ ( 4 ) 4 η A F ) = G R G η ( d ( . x d a λ ( ( ) 4 η G D x ( ∞ G d 0 Z y ∂ C 4 λµ Θ( Z d F µ , partial integration gives ∈ ) = λ µ η that appears in ( Z i∂ ∂ ,µ ( A 2 )] = µ G 2 := , η A η ( )] = ,µ 1 x ( η ,G µ ) 1 )] = 0. Indeed to get a sensible quantum operator, or even a sensible generator of canonical η ) generates gauge transformations. From there follows the second result, f ,A ( ( η ) η term vanishes after partial integration and use of ( η G ) being anti-symmetric. [ ( 2 ,A G . That is also the case. We show that in two steps. The first demonstrates  ) η G ( term vanishes after partial integration and use of [ η ( ρµ µ , and the smearing of the equation of motion, G we have for any pair which is zero by equationsconstraint on of the motion. domain Hence in quantum physics, we can set it as a because the box termtransformation. again vanishes by ( ables that [ The ∂ F Consistency demands that For classical fields Consistency also demands that The following properties of Towards this end, let us consider ∈ C c) a) b) µ The test function for and the Fourier transform condition ( derivatives to transformations. Otherwise distributionsPoisson like brackets. η We must smear the corresponding operator with test functions JHEP03(2017)136 ) i 2.9 ψ . | (2.21) (2.23) (2.24) (2.25) (2.26) (2.22) F i ) k . ( † µ F a ] ∗ ) , k 0 ( e η > · 0 k µ , k k − . It acts trivially in = 0 ]. ∗ 2 ∞ 0 ) has finite expectation values, G k 13 ( , , µ N , k x e ] η · 12 2 x · . ik k is a positive operator in , ) ik − ) , k e e . k ( † N ) ( ) µ ) + [ ) x ∞ k a is the number operator = 0 ( N k ( † such that ) ( µ ) < µ i µ F k k a i 2.22 ( ) in the Fock space representation. Use ( a ψ ( x η | ) is denoted by ψ µ )] 4 A + ) | η dµ ( a d 2 – 6 – k i ∈F k ( x · ( ( D ψ G e N Z η µ | ik Z | · a ) = − i ψ = µ e k k h ψ ) k ( | ) = µ ) k N k N ( k x ( ( µ µ µ a − e η ) )[ A ) k k ( ( ) trivially in x µ )( e η dµ η 2 ). This is crucial for the interpretation of the equations of motion ( 2.17 k η [ Z ( λµ h ) F G λ k ) = Because of the constraint ( ( is given by the states x . ( x ∂ dµ µ 4 N ∞ d . The group generated by A Z 2 < . with hindsight we have here just verified that the Fock space of physical states k Z − i √ ψ = = =0 | ) to write i , we find that = N ψ | | 0 2.2 ) ψ k One characteristic operator of Fock space We can now verify ( Usually deviations from classical equations of motion can appear in quantum field h η ( is in the kernel of i ∈ F G ψ The domain of This domain is ai.e. subset of the larger set of states where where Remark: F as constraints. where and with the constraint which we can assume by| the gauge invariance of (2.1) and (2.16). Thus, for any Fock state symmetry, the preservation of canonical Gaussthe law absence constraint and of general anomalies covariance for imply all Maxwell equations. and ( see that the role ofThe the smearing role functions, and of of smearinganalysed their in infrared test behaviour, the functions is canonical fundamental. formalism for by a several global authors [ analysistheory of due gauge to constraints anomalies. has been In the present case since there is no anomaly in the gauge This establishes the connection between the equations of motion and the constraints. We JHEP03(2017)136 = 0 φ (2.30) (2.31) (2.32) (2.33) (2.27) (2.28) (2.29) it follows 2 , replacing , but by the x k φ ] and we shall √ µ 31 ∂ = – . 0 = ] k 27 x µ · , f ik 18 − . e ∗ . diverges: it has an infinite ) (2.34) ∞ ) , k k i ( , = ∞ g ∞ 0 | µ i e k f g ( . | < ) = g ) − ) ) ∗ . ∞ k k = 0, and since ) x x k ∗ ( , · ( ) k 0 µ < ik µ e , − N k k . As is known [ f e , e ) f ) ( 0 ( i µ y ∗ ) µ k g k N 0 ( ) e k ( | ( f ∗ k and vanishes at infinity in ) i ) k ( g f, f D is compactly supported implies that ). In the latter case in the state f ( 0 µ ) k ( dµ e µ k k = f , y φ  ( e ) ( ), we see that 0 f iA = ( N ) on the vacuum, generates a state which does i = 0, e ≥ going to 0, so that the divergence is at worst ) φ 0. This is a consequence of the transversal and k µ e k )[ ∈ C g 0 f i ( ( µ – 7 – k µ ( | k k 0 2.6 f ( g ) k f ≥ ( = | y f Z k f ) dµ µ k 4 iA | ) i k ( dµ  N ∂ d dµ g ( f in ( | µ h | ≥ ) = 0 f a 0 µ Z | f, f Z Z k k dµ ) for → ( is a transverse, but not vanishing at infinity, function i i f A Z  0 h µ ≤ h 0 − − e >ε g f 0 /k = lim 0 ) := → k  = = i Z (1 g i 0 | = lim = 0, which since O f, f f ( → | N lim φ ε | E ) = 0 or g x  ) where ) is h ( µ . Indeed, since k µ f f , A 0 | 2.27 k f ( h built by applying exp( g in ( i g ) | = 0. g 0 unless ( µ and iA . It is easy to show that ( f > e ) with 4 µ properties of e f ∈ C R by ∗ ( µ k i ∈ F e f f ) ∞ The state Also f f | ( C iA can remain finite. not belong to the domainlater of see the in number section operator 5,operator infrared in dressing Fock space. does not leave invariant the domain of the number while its energy We assume that logarithmic. Then the expectationnumber of value of infrared photons, Hence We now removee the requirement that of Thus from the mode expansion of and, thus, i.e. small that transversality condition Let us now consider the coherent state defined by with 2.2 Vacua and coherent states JHEP03(2017)136 , f 0 ) if G ζ (3.3) (3.4) (3.5) (3.1) (3.2) ( and ) near G k η ) can be ( . 6= ˜ ζ σ ) 2.4 ζ ( Q is normal in ). ∞ 0 ). Hence the group G 2.5 A ( controls the behaviour )( D ) acts without divergent µ which need not vanish at ζ and the index ζ 2.4 ( µ 2 . ζ Q S . ) µ . and ) ζ , ) as before since both 2 x (2) µ f ), we can say that ). But in general ( ) by . ζ ( η µ η η } ≤ . Its subgroup ( A 0. The conclusion is that in view ( and ) vanishes can be labelled by the 0 − A 2.5 0 ). σ η )] = 0 { ) G G µ G ( f , x A η → (1) ( ), ( ( G in ζ )( ( ) = k ) = D ζ ) and µ 6= 0 ,A with ( 2.4 η ∈ C G ) A η η acts as identity on ( ) ( ( ζ ) as λµ σ k ( Q | b k (2) D G ∞ ( 0 F η ( ) e k ) = Q ζ e : ζ λ G behaviour of | e ζ · . This defines a new class of operators, analogs C – 8 – ) where . (2) − (2) x C k f ζ 0 ζ ∞ x ∂ 0 ( ( 0 4 → (1) )] = [ G − A d → Q e k ζ / η −−−→ lim k ( 0 ) − (1) Z G ζ k ) ,G . Further ( ( ): the latter are superselected. ) 0 e ζ ζ ζ (1) G belongs to G ) = ( The large · ( ζ ζ ( µ Q k ( Q η [ ) need not vanish on Q 0. We now explain this point. Q are characterised by functions on ζ = ’s differing by an ( ζ → ∞ 0 µ Q ζ G 0 ) commute with both / k ζ 0 ( G ) if Q η ) as ( k . This conclusion is reached by requiring that G | ( has elements with the properties ( µ k | e ζ ) generates on exponentiation the group is not of compact support the behaviour of its Fourier transform / , and, thus, ∞ 0 ζ , letting us do the needed partial integrations: k C ( ζ G C Q = ’s, which we will indicate with a different symbol, to stress the difference: ) and b k G ) elements of k ( Since Since Since We now comment briefly on the asymptotics of The The operators µ 3.3 cannot have compact support as shown by the derivation of ( e η is not in µ we can identify allof such ( ζ if where terms on the infrared dressed charged states. When the origin is different, itreplaced may by diverge at the origin and we shall later see that ( commuting with all elements of classifying superselection sectors is Remarks on asymptotics. of So the irreducibleirreducible representations representations of of They reduce to ζ belong to Suppose next that we replaceinfinity, the and test therefore functions does notof belong the to 3 The superselection algebra JHEP03(2017)136 2 . S with (4.1) (4.3) (3.6) (3.7) i ) 2 e ζ k , have the 1 − e ζ ( 2 ∗ ζ 2 e ]) and refer- ζ · ] which uses a 1, the integral k 27 and 1 , 32 µ ζ 1 k 17 ζ σ > to  ) 1 η k ( ∗ 1 e ζ , and outline the approach · I k S µ k . )] = 0 if both .  Θ . 2 ’s do not form a Lie algebra. Such -matrix in QED: µ ζ µ + ( ∂ S Q J  µ µ . Otherwise, or if ) U(1)) ,Q k k Θ (4.2) ) , ) after changing x J 1 µ 2 − x A 4 ζ ∂ ( is abelian or not. The answer depends on 4 S ( d 2 d e is constant. We can treat pairs ζ 2.29 + Q ∞ 0 · Z e µ Z ζ G (and not just as at constant time, as in the i k i / A µ 0 – 9 – is invariant under the gauge transformation is odd in k converges in the infrared because the measure is G = 0, [ I exp → e ζ exp k = Maps( S k I  µ · · ) T A ∞ k 0 k k TS ( = G 1 is thus isomorphic to the gauge group of maps from e / spacetime I ζ = Then 0 · S I = 1 of the above integral is logarithmic. There may be a ∞ G 0 . As 1 k S 3 G σ µ ) / )]. We find from ( k π 0 ) between the initial and final states are also gauge invariant. 2 ζ G (2 ( h 4.3 ) θ/ k ,Q ( ) 1 be the interaction representation sin ζ ) in powers of the coupling constant and taking its matrix elements, dµ | ( I ) in the whole k S | Q Z 4.3 ], which is supported by a considerable literature (cf. [ the group 4.2 16 σ )] = 2 dθdφ | ζ in the same manner. In the divergent cases ( k 1. The integral is also zero if | σ d 1 the coefficient integral of is a conserved current. ,Q ) µ 1 σ < J ζ ) = This treatment is not satisfactory for our purposes. The first point is that the operator The more serious problem is that the initial and final states are considered to have Expanding ( A natural question to ask is whether For fixed ( σ < We are now going beyond free QED to a slightly more general case. k = 0 gauge) is not shown. 1 ( Q 0 [ to be the case.of Below Roepstorff we [ focus onences just therein. this Specially infrared interesting partcovariant is formalism of the in approach momentum of space Gervais-Zwanziger [ quite close to the one developed in this paper. implementing ( A a finite number of photons. But because of infrared effects, it is known that this is not so that The matrix elements of ( one gets Ward identities order by order. Here 4 Gauge invariance andA Ward identities concise statement of gaugethe invariance following: and Ward let identities as formulated in textbooks is diverges. The divergence for regularisation to get adifferent finite answer evendomain if problems may not spoil physics. For dµ same to U(1). the commutator [ JHEP03(2017)136 I ), H 4.6 (4.8) (4.9) (4.4) (4.5) (4.6) (4.7) (4.11) (4.10) and no 2 ) in ( 4.7 . , 0 i k , i p, e | γ i p, e 0 | | γ i , . 1. Substituting (  , , 0 ) | )  i x )) τ (   τ µ ) ( p p, e m 0 0 | ζ A γ x ) τp/m i ( − ( − x I 0 ( | x m/k µ x µ H τ (  A ]:  0 4 , . Thus in the current µ 4 I i δ µ m µ δ p x J 14 the state of a charged particle of momentum dx p H p 3 m µ ˙ µ i 0 d ζ p, e m p 0 | ) = −∞ γ dx dτ Z dτ p, e i τ – 10 – Z | ]. dτ ( 0 i for the infrared model is that of a charged particle | 0 0 µ Z −∞ Consider the initial state with one particle −∞  µ Z ζ 34 state is e Z , Z J e i 33 ie in  exp ) = exp ) =  0 ) = T k x k ( ( exp ( µ = I µ J T H in J i = exp due to vanishingly small photon frequency γ in ; The current I i ) in an elegant from γ H ; p, e | and constant momentum 4.10 p, e | m state is then easily calculated [ )] is a multiple of identity. . One has to take into account the fact that charged particles radiate. If 0 0 e in x ( , mass is the photon Fock vacuum and I e γ i ,H 0 ) | 0 x The exponential is just the Wilson line integral along the particle trajectory. A gen- We can write ( ( A simple modification of what follows also covers the case of several charged particles. The results 2 I and charge H eralization of the samethe approximation quark for confinement non-abelian mechanism gauge [ theories gives some hintsdepend on only on the total momentum and total charge of the multiparticle system. Here the time[ ordering is not needed in the right hand side, since the commutator The infrared we get the current and the interaction Hamiltonian of classical electromagnetism, we set Thus the change ofneglected particle in the momentum approximation due of to interest the of large back reaction to photon emission is The infrared model. of charge so that this radiation hasmate accumulated the for effect an of infinite this period of time. We want to approxi- photons, where p is the interaction Hamiltonian, the Infrared dressing of initial state. JHEP03(2017)136 + i (5.1) E i (4.12) (4.17) (4.13) (4.14) (4.15) i ∂ . p, e | ). Thus the γ i ∈ C η 0 ( | ζ / ) G ), as in ( is meant invari- η ( x ( iG 3 e  e δ ) η ( = , iG 0 . i J −  Θ(0) e ) ) transforms only the Dirac- . ie . But for a charge particle at x ζ p, e  µ | − ( ) e γ A Θ( . x Q i 6= 0 ( µ = 2 a separating value, the most gauge invariance 0 in = ) | µ ) defines its superselection sectors. ∂ i ) ) ˆ ζ k σ γ η A  ( ( ( x ) (4.16) ) ; ) By e ζ ( x Q x x iG µ ( ( e 3 p, e µ | ) to get its contibution to xJ )Θ( 3 x e δ xJ = ) x Θ(0) , d 3 ( σ 0 ˆ k ie = 0 so that d 0 ) and charge conservation in ( k − i Z e ζ ζ ) = 0 e γ 0 ( – 11 – Z xJ x x 0 ; 3 , 0 = Q d dx dx → (0 p, e in 0 0 0 | i 0 0. That implies that the test function k =0 ) −−−→ −∞ J 0 −∞ γ η Z x ) ( Z i ; → Z k i − iG ( i 0 e − e ζ  p, e k | − ·  )  η k to vanish at ( exp ) as ζ ) iG exp η k e exp ( ( ) comes from gauge transforming ), while the response under e ζ iG . η · e show different behaviours, with (  k G 4.17 σ = → ∞ in 0 is fully gauge invariant. i x γ in ; i ) under γ = 0. This is to be smeared with Θ(0 0 as ; is ) p, e | 4.8 ) |·i in η → 4.8 ( p, e i | )) γ iG Before discussing charge conservation and Ward identities let us discussLet the us behaviour isolate the angular part and consider the infrared limit Thus since the charged particle is at spatial origin at time 0, the gauge transform of x ; e ( 3 p, e Different values of interesting case. cies. We can thusWilson choose line. of the nonvanishing eδ state 5 The superselection operator The superselection rules are associated with very large distances and very low frequen- | The first factor inorigin, ( the Gauss law has the additional term proportional to since ( and Θ which becomes after integration by parts The factor within the brackets is ance of ( Therefore we have to show that Now Gauge properties of infrared dressed state. JHEP03(2017)136 + ), to ζ (0) V ( 2 = 0 σ (5.5) (5.6) (5.7) (5.2) (5.3) (5.4) S Q 0 2) and k = 2 the ). Again ≤ e ζ σ i ( σ x · . Q ik in i − in γ e i ) ; γ k ; , (with − is not in the domain p, e σ 0 since it approaches with the tip | ( p, e e ζ | in  0 limit. Thus, the one- + · i →   V γ k 0 ) ) ; → µ k k x k 0 ) − ) must be odd. k p, e ( x Θ( | ˆ + k e ζ µ ( will have a constant limit x · − e ∂ · ζ ) i + y k ik ) is finite. Instead for x ( e ( V ζ ) − , ( µ D p, e k | ) ) ˆ k ( Q γ on y k e ζ xJ Ω i ( 3 in 3 · )( d 0 ) e ζ σ i | d 0 λ Θ: · π k γ ζ µ ; µ = k λ . In this case dk ). We can calculate (2 Z ) is a state with non-trivial response to ∂ k 0  ∂ i ζ k 0 h ( ( k ) p, e ) | y k ) Q 0 lies on a light cone dx 4 4.11 k ) as p, e – 12 – ( ζ | ( d 0 ( the point to a sphere. This procedure is common γ ) = ∞ = 2 the functions > i dµ 2.7 ) diverges. The gauge transformation is therefore iQ Z dµ k Z 0 σ e ( 0 | Z µ ) ie k 5.3 Z ζ ) need not be a constant as state ( is odd in ∂ ( dµ ie − k e ζ − − ( . This means these dressed states have values different ) expressing iQ in   e for ζ e σ · = blown up 0 k , we have to examine only the from ( k 2.20 )] = σ 0 µ x k ( = exp = exp k µ . In the case → ∞ in 2 i 0 ,A S γ ) k = 2 the integral exists and is well defined, but the would be function ; = 0 of the light cone is regarded as just a point, and we denote by ζ ( σ 0 Q k p, e ) on | ) ) diverges, unless ˆ k ζ ( ( 2 poses no problems, as in this case e ζ 5.7 2. For iQ Θ = [ e µ ), noting that i∂ σ < 0. But in our case σ < ). 4.12 0. If the tip ). This direction-dependent limit can be accommodated by attaching a sphere → vanishes fast as ζ ( We can conclude that the superselection sectors areSome labelled remarks by are in order. The photon momentum Likewise the infrared dressed Expressing the invariant measure ( k and not a point. We have 0 e ζ , → k Q + 0 (0 e in mathematics. k the light cone withas the tip, any smoothζ function V of the functions removed. The infrared features we have encountered are all concerned with the limit As particle dressed states havethe non-zero case values for theexponent superselection of operators ( we get as in ( given by a closed, butthis not limiting exact case, form. but this It is would be beyond interesting the to scopewhich study of depends the this on cohomology paper. of thefrom value zero of for the superselection operators exists if Θ, obtained by removing we can see that the integral ( JHEP03(2017)136 e ψ , on i e ) (6.3) (6.2) (5.8) (6.1) ζ ( iQ  e i ) -dependence µ e ζ τp/m . 1 requires more − i y γ ( ≥ · ; ik σ e  p, e ∗ ∗ | ) ) y k  , the vertex involved will ( ( ) 0 e λ ζ p , which for electron field ζ ·  µ λ ) ) is such that k to A ∂ ) , µ τp/m 2 charged particles of charge p i ) 6.2 + σ . J k k ) ) + Θ( + N ) is superselected. Hence its value · τp/m 1 k − µ ζ ( ( ( k σ p ∂ ( ( µ e e · ζ ζ i µ · · τp/m e A p m p iQ − k k µ y ) shifts . m µ µ p − ( := · dτ ζ )  2 ik ik ik i dτ k 0 iσ − → ∗ ( . Thus for . But −∞ − electrons and photons emitted with varying e ) e e i ζ p 0 η 1 Z ) + ) ) k e · ], our treatment does not invoke the fact that −∞ ( k k iσ y is – 13 – N ) are the sum of over all photon number of the Z e ie k ζ ( ( ( e )). The Sky group for QED is abelian being the 26 =1 · µ † µ λ N i − in i ie ζ a a i k Λ. Its presence in (  λ 3.7 k − 4.11 P γ − µ ∂ · ; − ) (with  Θ. This gives a shift of photon creation and annihila- → → ∂ Sky; h k = p ) µ · exp ) ) k 1 both groups are isomorphic. But ∈ q k p, e k k | p · ( i × 2.15 m ψ∂ ( ( e ζ  p µ † µ µ iσ · 0 on a a = exp e y σ < k → 4 ) ψγ  h i ζ in d is independent of ( ) lim ) i ) ] where the Sky group was introduced, spatial infinity was blown e γ k k k iQ to ( ( ; to U(1) with natural multiplication ( state are the same, so that charge is conserved. 14 e ] µ 2 dµ dµ dµ p, e S 35 | out ) Z Z ψA Z ζ µ ( ie ie dτ iQ − − 0 ψγ e i −∞   Z ie state and − . We have [  = exp = exp in in i γ We now remark on going beyond the tree approximation, which isThe also gauge necessary transformation to ( Unlike traditional treatments like [ In an earlier work [ ; exp is not a true vector, nor any reference to Lorentz invariance. The diagrams summed µ p, e momenta, when an electron line changes momentum from tion operators Thus if we consider tree diagrams with get amplitudes which canfactors measure out. is the shift of A in the infrared dressingdiagrams operator Weinberg considers. in It ( isfixed the charge tree as approximation photon to momenta the go Feynman to diagrams zero. with The integral multiplying gets replaced by the totalin charge the Thus the eigenvalue of 6 Charge conservation andCharge low energy conservation theorems follows| directly from the infrared part in the action of up to a sphere tomomentum define space. this For group. fixed Herediscussion instead, (See we the get discussion thegroup of dual of eq. blow-up maps of ( from the origin in JHEP03(2017)136 . 4 M (7.8) (7.3) (7.4) (7.5) (7.6) (7.7) (7.1) (7.2) . µ ξ ) and metric µ 5 ξ , ξ 4 := . , ξ  ξ µ ) } ξ ◦ will be non-trivial.  x · ξ } ) γ . Such a calculation is beyond ζ . x ( ◦ . Consider the six dimensional : . ) = ( , ξ ) = 0 4 2 6= 0 6 ··· iQ − 5 ξ V 5 γ . ξ , e = 0  ξ ) 0 M , ξ 5 (1 ξ k − µ , + 4 ξ 2 − , 2 1 ξ of V, λ ◦ 1 2 4 γ 4 µ k , ξ − ξ 4 ξ )  ∈ 6= 0 4 x 1 2 2) does not act on Minkowski space )( − ξ ξ − M · , 5 , , . . . , ξ state to 2 i ξ . We explain its action to the extent we )( 1 1 x  ]; ξ 5 +  , ξ ξ ξ γ + , ξ in ) is superselected. That should give identities ) λξ 3 0 ). Since varying electron momenta will occur, J =1 ζ 4 2 γ 4 ◦ 0 i , γ ξ X + ( ξ ξ p (1 + – 14 – ( µ 4 ξ γ Q γ = − − 2 1 ξ ξ ] = [ − , ( − 5 2 0 ξ 4 p ξ x, 6= 0 is the Minkowski space: = µ − ( 1 +  ξ , ξ e µ ζ γ ]:[ µ µ µ  , ξ · x ξ ξ ξ ξ ) = 1 2 [ ) µ : ] = ξ 0  { , | ξ ξ p ξ ξ ξ ξ γ = ( with 2 ( = { ◦  − ξ γ , coordinates ( ξ = p 6 VP VP )( ] = R 6= 0 : [ V 0 ξ of 0 p [ 1 + . γ µ  is − ξ = 0 and 2 2 γ p , ]. ( is 5 4 , the projective space associated with · ξ = 38 ξ M ]. It acts on null infinity γ 4 + interior VP ξ 4 37 ξ , . with topology 36 VP 2 , = 0, 6= 0 4 ∈ γ γ M ξ will not factor out and the response of Setting Let If The four-dimensional conformal group SO(4 But this eigenvalue cannot change as ζ is a four-dimensional space. we see that the But if where on L.H.S., Then Let us consider VP We can write this metric as The null cone in It acts only onspace the Dirac-Weyl compactification 7 The BMS group The Bondi-Metzner-Sachs group was introducedradiation during the [ study of classicalrequire. gravitational See also [ the for scattering amplitudes involving photon momenta the scope of this paper. carry the factor JHEP03(2017)136 . 2 S (7.9) (7.15) (7.16) (7.17) (7.18) (7.19) (7.10) (7.11) (7.12) (7.13) (7.14) super- with ↑ + L , 2 . For convenience S J , ) . ) N 2 . . ◦ 0, we can write 4 Λ .

1 : Λ , . R > α 4 , Λ 0 ) 1] , 0 = 1 ∈ /ξ 2 . ≷ on 0 − N N ) µ α , ξ N 0 ∗ 1 a ◦ ↑ + 1 · N , + N is a real function on 2 L = 0 as its tip.It has the topology of . 4 ◦ (Λ N S J 4 + Λ . R α . α /ξ ξ ) 1 (Λ ; × ↑ + µ . With = 0; , + α N → ξ α ) as conformal transformation. + N R , 0, 0 2 · 0 N 2 ∈ L N J S = N 0 ≷ a – 15 – S ◦ ) = N = 0. The BMS group acts on ] = [ : ) = ( Λ ( 0  ξ 2 + N µ α ,N on ( ξ N Λ J 0 0 Λ), where Away from ) = ( , α a ↑ + 2 N,N ∗ N α, N  L α Λ , 6= 0 : [ 0 )( ) = = ( = 4 1 ξ N ) is N . We can regard Λ  ( N , N J α 1 J . We focus on , Λ)( α 0  ( α, has just angular momenta 0 and 1, N J ( α and distinguish µ N be the usual pull-back action of Λ) on ( ] spans a light cone for each sign of ξ α = ) shows that the BMS group is the semi-direct product of α, ∗ 4 . The composition law for the latter is addition of functions so that it is Λ denotes the action of α /ξ 7.16 µ 6= 0, [ → N ξ 4 ◦ α ξ In quantum theory, it is important to realize the BMS group as operators on the Then ( Let Let us call this space as gives the Poincar´egroup. quantum Hilbert space. Inincluding our infrared case, effects. the latter carries a non-trivial representation of QED The subalgebra where translations abelian. We find The action of ( where Λ The BMS group consists of a pair ( and Λ is a Lorentz transformation: Thus, with coordinates The BMS group acts on It is obtained by blowing uplet the us origin set Thus if JHEP03(2017)136 ) ζ ( sep- Q (7.22) (7.23) ), but ∞ 0 . There G 2.4 + or J . With that and changes 4 0 G ∞ M 0 as in ( G ) is fixed. Hence / 0 ∞ 0 ) (7.21) 3.3 G 0 G ζ 0 ) which give the gauge to all of Λ) 5.3 + 0 (7.20) , but not on α, J ( 0 ∞ 0 − which however coincide on the G → . k / −−−→ ] ζ µ 0 ˆ x e ζ ) · G ˆ k k Λ) ( r e ζ 0 α, k · for all Lorentz gauge transformations Λ. − (( k ) 0 r , r 0 G Θ + − µ u Λ) ( ∂ 0 0 . Substituting ) = k x 0 α, [ . ζ ( i . But the BMS group acts only on G 4 – 16 – ) also acts on the asymptotic values of the gauge 0 =  + by supertranslations or boosts is generically non- Θ = − e M 0 u µ Λ) J ) ∂ 7.16 = Θ k α, ( µ x e Λ) from boundary to bulk act in the same manner ζ · ∂ Λ) (( · ik Q α, α, k  e − Θ to ) Λ) µ ζ ∂ BMS group is spontaneously broken α, ( BMS acts on the superselection algebra Λ) α, ) and hence defines ζ (( , and vanishes on quantum states. ( Q ∞ 0 Q G are two such actions of this group on 0 ). It is convenient to do so, and the results are unaffected. ) is a rotationally invariant measure. Λ) k Θ to a new function ( ( 2.5 α, µ . ∂ dµ ∞ 0 G Λ), ( / 0 The transformation of Hence, the BMS group is spontaneously broken to its rotation subgroup. The BMS group transforms the values of the exponentials and hence the asymptotic The asymptotic BMS symmetry ( All possible extensions of ( Thus The action of the BMS group on the leading asymptotic terms ( But the BMS group acts only on We now suggest that the BMS group does act on We can approach the problem by extending the action of G α, trivial and changes itthe at superselection sector. infinity. The Thus exceptionbecause this is the action rotation is subgroup non-trivial which acts on trivially values of we get fields and their gaugetransformation transformations. of In particular, consider ( is a generator of on Thus, arately. The reasondropping is ( as follows. Let usif characterise ( elements of asymptotic terms, then in hand, we cantransformations. transform test functions and perhaps find operatorsare to many implement ways these to extend its action to JHEP03(2017)136 116 (2014) 152 Phys. Rev. Lett. groups such as BMS , 07 ]. arXiv:1506.02906 , SPIRE JHEP , IN ]. ][ symmetry SPIRE IN ][ Soft hair on black holes Gravitational black hole hair from event horizon ]. – 17 – arXiv:1601.03725 New symmetries of QED [ SPIRE IN ][ arXiv:1505.05346 Asymptotic symmetries of QED and Weinberg’s soft photon [ ), which permits any use, distribution and reproduction in ]. (2016) 088 of spacetime. However, because gauge invariance under local 06 still hold which permits us to generalize the standard results to SPIRE ∞ 0 IN (2015) 115 G ][ JHEP CC-BY 4.0 , 07 On BMS invariance of gravitational scattering arXiv:1601.00921 This article is distributed under the terms of the Creative Commons boundaries [ JHEP ]. , SPIRE arXiv:1312.2229 IN theorem (2016) 231301 supertranslations [ [ A crucial observation is that the proof of such results does not requires Lorentz The same analysis affects the role of other asymptotic S.W. Hawking, M.J. Perry and A. Strominger, A. Averin, G. Dvali, C. Gomez and D. L¨ust, A. Strominger, D. Kapec, M. Pate and A. Strominger, M. Campiglia and A. Laddha, [4] [5] [1] [2] [3] Attribution License ( any medium, provided the original author(s) and source are credited. References by CUR GeneralitatMDM-2014-0369 de of Catalunya ICCUB (Unidad under delike projects Excelencia to ‘ FPA2013-46570 acknowledge Maria the and deExcellence” support Maeztu’). 2014 Programme provided G. SGR 2016-2017. by Marmo the 104, would Banco de Santander-UCIIIMOpen “Chairs Access. of This article is based upon work(European from COST Cooperation Action in MP1405 QSPACE, Science supportedsupported by by and COST the Technology). Spanish MINECO/FEDER M. grant2015-E24/2. FPA2015-65745-P Asorey and F.L. DGA-FSE work is grant supported has by been INFN, I.S.’s partially GEOSYMQFT and received partial support deduce to charge conservation and low energy theorems. invariance. Acknowledgments tions because they change the charged superselection sector. which act on the gauge transformation are preserved by theof covariant Gauss the law S-matrix constraints, under Ward identities We have shown that equationsThis of interpretation motion allows us can to beformulation define considered of a as covariant quantization, version constraints we of inQED. analysed Gauss field In law. the theory. particular Using physics we thespontaneous effects have Peierls’ symmetry of shown breaking the that of the infrared some infrared behaviour space-time dressed of symmetries one-particle like states Lorentz induce transforma- a 8 Conclusions JHEP03(2017)136 , . A 95 96 Eur. 19 , (1967) 730 19 Phys. Rev. Lett. Phys. Rev. Phys. Rev. , Lecture Notes (1986) 331 , , , Photopion ]. B 174 SPIRE IN [ Proc. Roy. Soc. London Commun. Math. Phys. , ]. , Phys. Rev. Lett. (1966) 616. , ]. , Addison-Wesley, U.S. (1966). , Springer, Germany (2005). Phys. 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