Dynamics of entanglement in expanding quantum fields

Journal of High Energy Physics, Apr 2018

Abstract We develop a functional real-time approach to computing the entanglement between spatial regions for Gaussian states in quantum field theory. The entanglement entropy is characterized in terms of local correlation functions on space-like Cauchy hypersurfaces. The framework is applied to explore an expanding light cone geometry in the particular case of the Schwinger model for quantum electrodynamics in 1+1 space-time dimensions. We observe that the entanglement entropy becomes extensive in rapidity at early times and that the corresponding local reduced density matrix is a thermal density matrix for excitations around a coherent field with a time dependent temperature. Since the Schwinger model successfully describes many features of multiparticle production in e+e− collisions, our results provide an attractive explanation in this framework for the apparent thermal nature of multiparticle production even in the absence of significant final state scattering.

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Dynamics of entanglement in expanding quantum fields

HJE Dynamics of entanglement in expanding quantum elds Jurgen Berges 0 1 3 Stefan Floerchinger 0 1 3 Raju Venugopalan 0 1 2 0 Bldg. 510A, Upton, NY 11973 , U.S.A 1 Philosophenweg 16 , 69120 Heidelberg , Germany 2 Physics Department, Brookhaven National Laboratory 3 Institut fur Theoretische Physik, Universitat Heidelberg We develop a functional real-time approach to computing the entanglement between spatial regions for Gaussian states in quantum entropy is characterized in terms of local correlation functions on space-like Cauchy hypersurfaces. The framework is applied to explore an expanding light cone geometry in the particular case of the Schwinger model for quantum electrodynamics in 1+1 space-time dimensions. We observe that the entanglement entropy becomes extensive in rapidity at early times and that the corresponding local reduced density matrix is a thermal density matrix for excitations around a coherent eld with a time dependent temperature. Since the Schwinger model successfully describes many features of multiparticle production in e+e collisions, our results provide an attractive explanation in this framework for the apparent thermal nature of multiparticle production even in the absence of signi cant nal state scattering. ArXiv ePrint: 1712.09362 Conformal Field Theory; Field Theories in Lower Dimensions; Quark-Gluon - Plasma 2.6 Symplectic transformations, Williamson's theorem and entanglement entropy 12 Eigenvalue problem and boundary conditions Field- eld and conjugate momentum correlation functions Entanglement entropy 1 Introduction 2 Entropies and entanglement of Gaussian states Gaussian pure states Gaussian density matrices Projections and reduced density matrix Correlation functions Entropy and entanglement entropy 3 Entanglement entropy in Minkowski space 4 Entanglement entropy for expanding systems The Schwinger model General coordinates and background evolution Dynamics of perturbations Entanglement entropy of an expanding string 4.5 Local density matrix of an expanding string 5 Conclusions A Relative entropy B Symmetries, anomalies and bosonization B.1 Symmetries, conservation laws and anomalies B.2 Bosonization 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 4.1 4.2 4.3 4.4 1 Introduction for quantum computation and quantum communication devices [2]. Entanglement in quantum eld theory was investigated initially mainly with a view on black hole physics [3{5] and more recently in the context of holography [6, 7]. { 1 { For globally pure states, a good measure of the entanglement between a region A and its complement region B is the entanglement entropy. If one considers the reduced density matrix for A that follows from tracing over the Hilbert space associated with region B, A = TrB ; (1.1) (1.2) this reduced density matrix is of mixed state form as a result of the entanglement between A and B. This can be quanti ed by the entanglement entropy de ned as the von Neumann entropy associated with A, SA = Trf A ln Ag: (More general Renyi entanglement entropies will be discussed later in the text.) A detailed understanding of entanglement entropy exists for 1+1 dimensional conformal eld theory [8{12]. Technically, a replica trick in the Euclidean formulation of the theory leads to a partition function on an n-sheeted Riemann surface which can be evaluated. One can, as a result, compute the entanglement entropy not only of vacuum states but also of nite temperature states [10, 11]. The formalism has even been extended to discuss nonequilibrium dynamics [13{15]. Further, a replica method has also been developed to compute the relative entanglement entropy between two density matrices [16{18]. Novel methods have also been developed for free eld theories, in particular for massive (or massless) scalars and fermions [19{21]. These methods are based on the Euclidean formulation of these theories and also employ the replica trick. Analytical insights are supplemented by detailed numerical investigations using lattice techniques. In ref. [21], alternative real time methods are mentioned but not fully developed. In this work, we will be interested in the real-time dynamics of entanglement resulting from the rapid expansion of a system. We will concentrate on Gaussian states and on bosonic theories, a setup that will allow us to discuss the dynamics of an expanding string. The latter, as we shall soon discuss, will be a signi cant focus of this work. To follow the dynamics in real time, we will use a Schrodinger functional formulation for the corresponding density matrix. In order to determine the entanglement entropy of general Gaussian states (equilibrium or nonequilibrium, pure or mixed), it will be convenient to start with a somewhat abstract but fully general discussion of the corresponding mathematics. This discussion follows an approach that goes back to refs. [ 3, 4 ] and develops it further. (See also [22{25] for more recent work in this direction.) We will arrive at results that express the von Neumann and Renyi entanglement entropies (as well as relative entanglement entropies) directly in terms of traces over combinations of two-point correlation functions within the region A. Quantum eld theory when applied to the temporal evolution of systems, as for example in early universe cosmology or in the presence of time dependent background elds, reveals surprising features [26, 27]. For example, excitations can be described with different mode functions; these (and the accompanying creation and annihilation operators corresponding to inequivalent vacuum states) are related by nontrivial Bogoliubov transformations. In a static Minkowski space problem, one basis is preferred by having mode functions with positive frequencies; this is not the case in time dependent situations with { 2 { fewer symmetries. It is possible that an incoming vacuum state has nonvanishing particle number with respect to mode functions that are distinguished in having positive frequency states at asymptotically late times. In this case, the time dependence of the background eld or of expanding geometry lead e ectively to multiparticle production [26{29]. Entanglement plays an important role in a deeper understanding of such intriguing phenomena. E ective particle production happens typically in terms of entangled EinsteinPodolsky-Rosen pairs. When they separate in space, they contribute to entanglement between spatial regions. This is particularly relevant in the presence of horizons, for example close to a black hole or in the closely related Rindler wedge of spacetime, because causality dictates that observers there cannot recover the full information about a quantum state. As a result, Hawking radiation [30] or the closely related Unruh e ect [31] lead to a thermal spectrum of particles. While Hawking and Unruh radiation concern idealized static situations governed by an event horizon, similar horizon phenomena can occur in time dependent situations. For example, observations at a given space-time point are, by reasons of causality, only sensitive to the interior of their past light cone. If the quantum eld theoretic state is speci ed on some Cauchy surface in the past of an observer, only regions on this Cauchy surface inside the light cone are of relevance to her. This intersection of the light cone with the Cauchy surface constitutes a kind of particle horizon similar to the cosmological light horizon. The interior of this particle horizon typically lls a large volume; in this case, the entanglement entropy, if it scales with the area of its boundary, has a negligible e ect. The situation can be very di erent in an expanding situation and can therefore lead to interesting unanticipated consequences. Further, this phenomenon can help to explain certain puzzling experimental observations in high energy collision experiments that have thus far de ed explanation. A long standing puzzle in elementary electron-positron collisions is that experimental results for particle multiplicities are well described by a thermal model corresponding to Boltzmann weighted distributions with a certain temperature T [32{36]. This is surprising because the theoretical picture we have about these collisions makes thermalization by multiple collisions unlikely. A popular theoretical model for soft QCD processes has been developed in Lund [37, 38] and underlies, for example, the PYTHIA event generator [39, 40]. It is based on expanding QCD strings from which hadrons and resonances are produced by tunneling processes via the Schwinger mechanism. In the standard implementation of the Lund model, as noted in ref. [41], the thermal-like features seen in experimental data are hard to understand. One attempt to cure this problem by allowing for uctuations of the string tension, is discussed in [42]. In a quantum eld theoretic description, di erent regions in a QCD string are in fact entangled. If one considers a sub-region, for example the region A in gure 1, this interval can be described by a reduced density matrix corresponding to a trace over the complement region B as shown in eq. (1.1). Entanglement between di erent regions ensures that the reduced density matrix A is of mixed state form. One may now ask oneself whether this reduced density matrix could resemble locally a thermal state and whether this could explain the close-to-equilibrium distribution of hadron ratios as found experimentally. We { 3 { -------------------I------------------ I------------------- B A B FigFuirgeur1e.1R.Regeigoionnss iinn aann QQCCDDstsritnrgin.g. twhielyl esexppalroartee tinhisspqacuee,stthieoyncwonittrhibinuttehteoSecnhtawnignlegmerenmt obdetewleoefn1s+pa1t-idali mreeginosniso.nTahliQsiEs Dpa,rwtichuilcahrlyisrtehleevfraanmtienwtoherkpruesnednecrelyofinhgoritzhoensa, bfoorveexammepnleticolnoseedtophaebnlaocmkehnoloeloogriicnatlhLeuclnodselmyroedlaetle.d RReinmdlaerrkwaebdlgye, owfespancedtitmhea,tbaectavueserycaeuasrallyitytidmicetas,tetshtehadteonbssietryvemrsatthreixrefcoarnenxotcirteactoivoenr sthaer ofuulnldinfaorcmoahteiornenatbouetlda quantum state. As a result, Hawking radiation [29] or the closely related Unruh e↵ ect [30] lead to a thermal with a temperature describing spectrum that decays of particles. with proper time, T = ~=(2 While Hawking and Unruh radiation concern idealized static situations governed by an event horizon, seinmtialanrghleomrizeonntpahnendomtheneaecxapnaoncsciuornindytinmaemdiecpse.ndOenutrsirteusautilotnss.inFotrhiesxadmirpelce,tioobnseravraetisounsmamtaargiizveedn sinpaace-rteimceenptolientttaerre,[4b3y].reasons of causality, only sensitive to the interior of their past light cone. If the isfeeit rsecfasle.s[5w7i,th58th].e aBreeasidofesithsibgohunednaerryg,yhacsolalidneegrlipgihbylesiec↵ se,ctt.he Tdhyensaitmuaitcisonofcaenntbaenvgelreymdein↵terfeonrt iqnuaannteuxmpandeilndgtshiteuoarteiotinc amndodcealns itshearlesfooroef lreealdevtaonicneteirnesotitnhgeruncaonntticeixptaste.dOcnoenspeqrouemnicnese.ntFeuxrtahmer-, discussion of entanglement entropy in lattice gauge theory, tphlies ipshoenfocmouenrosne ccaonsmheollpogtoy e[x2p6l,ai2n7c]earntadinapnuoztzhlienrg ceoxnpecreirmnesnteaxlpoebrsiemrveantitosnswiinthhiuglhtreanceorgldy caotlolimsioinc eqxupaenritmuemntsgtahsaets hwavheertheuesnftaarndgefileemdeexnptlacnaantioanc.tually be investigated directly [59{62]. This paper is organized as follows. multiplicities are well described by a thermal model corresponding to In section 2, we discuss Gaussian A long standing puzzle in elementary electron-positron collisions is that experimental results for particle in quantum eld theory and their entanglement properties. with a certain temperature T [31–35]. This is surprising because the theoretical picture we have about these cdoellsicsiroinpstimonakoesf tphuerrmeaGliazautsiosinanbysmtautletispldeecsoclrliisbioends autnlikxeelyd. tAimpeopourlaornthseoomreeticaaplpmroodperliafotresCofatuQcChDy psurorcfeascses ihnasthbeenScdhevreoldopinegdeirn fLuunncdti[o36n,a3l7]reapnrdesuenndtearltieios,nf.or Wexeamupsle, athenoPtYaTtiHoInA wevitehnt agbensetrraatcotr Boltzmann We start in section density matrices weighted distributions 2.1 with a P the In section 2.2, we discuss general mixed Gaussian states in terms of a [i3n8d, ic39e]s. wIhtiicshbhaasesdthone aexdpvaanndtianggeQtChDat sittricnagns bfreomapwphliicehd htoadmroannsyancdonrcesroenteanpchesysairces psriotuduacteidonbsy. tunneling processes via the Schwinger mechanism. In the standard implementation of the Lund model, as variant of the Glaubernoted in ref. [40], the thermal-like features seen in experimental data are hard to understand. One attempt representation [63, of the density matrix. A key formula here is an expression for to cure this problem by allowing for fluctuations of the string tension, is discussed in [41]. In a quagnetunmerafielldGtahueosrseitainc ddesecnrispittiyonm,dai↵treirxent regions(i2n.1a4Q).CDI nstrsinugbasreectiniofnact entanwgeled. If one most in eq. 2.3, discuss ctohnesiddeerns saitsyubm-reagtiornix, fotrheaxtame mpleertgheesrewgiohnenA itnraficge.s1, othviesrinptearrvtasl coafn btheedecscornibegdubryaatiroenduscpedacdeenasirtey mpeartrfoixrmcoerrde.spoAndiknegytoraesturalcte hoveerrethise ctohmaptletmheentrreedguiocneBd ads esnhoswitnyinmeaqt.r(i1x.1)r.eEmnatainnsgleomf enGtabuestwsieaenn by explain certain this question characterized dfoi↵ remrenwt irtehgionneswencsouvraesritahnactethme arterdiucceesd tdheantsitcyanmabterixr e⇢lAatiesdoftomitxheed sotraitgeinfoarlmc.ovOanreiamnacye nmowatraisxk oneself whether this reduced density matrix could resemble locally a thermal state and whether this could projection simple representation of the von Neumann entanglement entropy emerges in eq. (2.59). In section 3 we illustrate the general formalism of section 2 by applying it to the derivation of the entanglement entropy of an interval of length L in 1 + 1 dimensional Minkowski space governed by a free scalar eld theory. We consider the corresponding eigenvalue problem in a discrete Fourier representation and discuss the nontrivial role of boundary terms in subsection 3.1. This is used in subsection 3.2 for the computation of eld- eld and conjugate momentum correlation functions, and in subsection 3.3 for the computation of the entanglement entropy. In section 4, we compute the entanglement entropy and observe that it generates a locally thermal state for relativistic particles in the rapidly expanding environment of a high energy electron-positron collision. In subsection 4.1, we investigate the Schwinger model and in subsections 4.2 and 4.3 the background eld and the dynamics of perturbations of the expanding string solution that capture essential aspects of the collision dynamics. The entanglement entropy of an expanding string is computed in subsection 4.4. We describe the excitations around an expanding coherent eld in terms of local density matrices that are of thermal form with a time dependent temperature T = ~=(2 ) in subsection 4.5 and provide a simple intuitive picture of how the apparent thermal character of these excitations arise. Our conclusions are stated in section 5. Appendix A generalizes our formal discussion of entanglement entropies of Gaussian density matrices to relative entanglement entropies and appendix B provides a brief review of the symmetries, quantum anomalies and bosonization of the Schwinger model for quantum electrodynamics in 1+1 dimensions. 2 2.1 Entropies and entanglement of Gaussian states Gaussian pure states We shall consider bosonic quantum eld operators m in the Schrodinger picture and employ a set of eigenstates mj i = mj i (2.1) with eigenvalues m. For a compact notation, the index m can label both continuous degrees of freedom such as position or momentum as well as discrete ones such as spin or internal quantum numbers. In particular, for complex elds, we will also take their complex conjugate to be part of the \Nambu eld" to also talk about the complex conjugate eld m.1 Nevertheless, it will often be convenient m although it is not independent of If the quantum system is in a pure state j i, then h j i represents its Schrodinger wave functional (to be understood as a functional of the eld ). We will be particularly interested in expanding systems, where at early times the expansion rate can be much larger than typical interaction or scattering rates of eld excitations. In this case, the quantum dynamics is typically well described in terms of Gaussian wave functionals, as will be discussed in more detail in section 4.1. The most general form of a Gaussian 1For instance, in the absence of any other components, we have 1 = and 2 = . { 5 { Using two sets of eigenstates of (2.1), whose eigenvalues we denote by + and pure-state density matrix can be written as ] = h +j ih j i = exp 1 2 12 y+h + + ( y+ + y ) + 2 1 y hy 2 1 y( + + + ( y+ i 2 ) : y ) + 2 i y( + ) The condition for a pure state density matrix Trf 2; g=Trf ; g 2 = 1 is satis ed as it should be. We will also work with the canonically conjugate momentum eld 2For an introductory discussion about most general Gaussian states, see for example [65] (section 2.4) for the case of real bosonic elds. (2.2) (2.3) (2.4) (2.5) , the (2.6) (2.7) In the second equation, we have used a condensed notation with denotes the collection of elds, and h, and are complex quantities that parametrize the state. Their physical meaning will be discussed below. Gaussianity refers to the maximum power of the eld appearing in the exponential (2.2) being quadratic. Correspondingly, we have The norm is given by a functional integral over , h j i = exp 1 yhy 2 i y 2 2 i y + 2 1 y + being unity. More generally, the scalar product between a state functional [ ] speci ed and another one [ ] speci ed by h, and is given by the functional 2 1 y(h + hy) + i y( 2 ) + i 2 ( )y + 2 1 y( + ) + 1 2 ( + )y : 1 2 Schrodinger wave functional representing a pure state for a complex bosonic eld theory can be written as2 i h j i = exp 2 mhmn n + 2 m m + 2 m m + 2 m m + 1 2 m m = exp i y + 2 i y + m = i m : { 6 { The representation (2.7) implies the canonical commutation relations [ m; m] = i mn; and correspondingly for the complex conjugate elds. make the transition to mixed states in terms of the density matrix, but restrict ourselves to situations where the density matrix remains of the Gaussian form.3 We can write such a mixed state density matrix in the form, ] is the pure state density matrix in (2.6) (dependent on the elds and ) and P [ ; ] is a quasi-probability distribution. When positive, P [ ; ] can be seen as the probability distribution for statistical noise in the parameter elds and . More generally, however, P [ ; ] need not be positive semi-de nite. (The density operator should of course be hermitian and positive semi-de nite.) Note that eq. (2.9) is closely related (although not identical) to the Glauber-P representation of a density matrix [63, 64]. In the following, for simplicity, we will take P [ ; ] to also be of the Gaussian form, " P [ ; ] = exp y jy; y 1 j ! # with a hermitian operator that we take to be of the form4 a i b ! i b ; a with a = y ; b = a y : b = y = i 2 (2.10) (2.11) (2.12) (2.13) For this to be a properly normalizable probability distribution, the eigenvalues of should be positive. One may also introduce the linearly transformed parameter elds, m = p ( m jm) + p m; m = jm) + p m; i 2 p ( m in terms of which the exponent in (2.10) becomes diagonal, P [ ; ] = exp y( a h b) 1 y( a + b) 1 + const : Substituting (2.10) in (2.9), one obtains straightforwardly that the mixed state density matrix [ +; ] is also Gaussian. In the limit where ! 0, the functional P [ ] approaches 3Deviations from Gaussianity can be treated in this formalism perturbatively but shall not be discussed any further here. 4More general Gaussian forms for P [ ; ] are possible but will not be needed for our construction. 1 2 i Z 1 2 { 7 { a delta distribution functional and one recovers the pure state we started with. Performing the functional integral over and an overall shift of elds + ! + , h a b a + hy b ! + ! ! # ) : 1 2 which corresponds to the eld expectation value below. Interestingly, this is the most general Gaussian density matrix that satis es the her( ^y+ y)P; ( ^y y)P " 1 2 P ( + P ( ) ) ! i 2 + ( y+ y )P j + 2 i jyP ( + h a d(++) b d ( +) # ) : { 8 { gives (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) a + hy b d ( ) d(+ ) ! and in the following we will assume canonical normalization, trf g = 1, or divide by the appropriate power of the above expression. 2.3 Projections and reduced density matrix For the computation of the entanglement entropy and related quantities, we shall require reduced density matrices that result from performing partial traces over some of the degrees of freedom. Towards this end, one may introduce a projection operator P = P y whereby ~ m = ( mn Pmn) n+ = ( mn Pmn) n are the elds we want to trace out, and ^ m = Pmn n are those we wish to retain. For example, P might be a projector in position space which equals unity in some interval and is zero in the complement region. The reduced density matrix is formally given by Z R[ ^+; ^ ] = D ~ [P ^+ + (1 P ) ~; P ^ + (1 P ) ~] : (The projectors have been inserted here for clarity.) For the Gaussian density matrix (2.14), one can formally perform the functional integral over ~ and obtain yet again a Gaussian form for the reduced density matrix, We have used here the abbreviations d(++) =(h d(+ ) =(h a + The operator inversions used here are to be done within the subspace that is integrated out. Note that the reduced density matrix (2.20) is of the same general structure as the original density matrix (2.14); in particular, it still satis es the hermiticity property (2.15). However if one starts with the density matrix of a pure state where reduced density matrix (2.20) contains in general terms that mix a = + and b = 0, the (the terms d(+ ) and d( +) in (2.21)) and describes therefore a mixed state, as expected. Because the reduced density matrix is again Gaussian, it is completely determined by the eld expectation values and the two- eld correlation functions. As we will discuss, these can be computed directly on the domain of interest without further reference to projection A Gaussian density matrix can be completely characterized in terms of eld expectation values and correlation functions of two elds. More speci cally, for (2.9) one has for the eld m and the canonical conjugate momentum eld m = i = h mi = R D R D m [ ; ] For the connected correlation functions hABic = hABi hAihBi one nds h m nic = [(h + hy h m nic = [h 2 a) 1]mn; = [hy = [ a = [ a + (h (hy a + a b + (h b + (hy b)(h + hy 2 a) 1(h b)(h + hy 2 a) 1(hy a b)]mn a + b)]mn a + a b)(h + hy b)(h + hy 2 a) 1(hy 2 a) 1(h a + a b)]mn b)]mn; h m nic = i[(h + hy h n mic = h m nic = i[(h h n mic = i[(h + hy 2 a) 1(hy i[(hy a b)(h + hy a + b)(h + hy 2 a) 1]mn; a + b)]mn; 2 a) 1]mn : 2 a) 1(h a b)]mn; These are compatible with the canonical commutation relations, h m n h m n n mic = i mn; n mic = i mn : { 9 { = jm: (2.22) (2.23) (2.24) bmn [(h+hy 2 a) 1 b]mn = h m n + n mic h m kich k l + l kic [h They follow with some algebra from (2.23). 2.5 Entropy and entanglement entropy In this subsection we will determine the Renyi entropy for the general Gaussian density matrix (2.14) and extract from it the von Neumann entropy. The Renyi entropy is de ned by (we assume standard normalization trf g = 1), and the von Neumann entropy follows from this [8, 11] as a limit ic]lr1 h r n + n ric; For the purposes of the calculation that follows, it is clear5 that one can drop and j, as they do not enter trf N g. Using otherwise the general expression for in (2.14) leads after a straight-forward exercise in Gaussian integration to [20] Trf N g = exp Tr ln det (MN ) ; 1 2 which contains the N dimensional cyclic matrix (with operator valued entries) Note that the matrices h, hy, a and b are xed in terms of the connected correlation functions of , and the momenta , . For later convenience, we note the relations amn [(h+hy 2 a) 1 a]mn = X h m kich k nic i 4 k 1 X 4 k;l;r i 4 1 We have used here the abbreviations introduced in (2.25). We de ne further 1 N -dimensional unit matrix and Z N is the N -dimensional cyclic matrix (ZN )mn = (m+1)n. Here m; n are in the range 1; : : : ; N and the index m = N + 1 is to be identi ed with the N to be the MN = AN (a; b) ATN (aT ; bT ) ; AN (a; b) = 1 2 ! 1 N 1 2 ! ZN : 5This can be seen directly by writing out the expressions for trf N g in the functional integral formalism or more formally by noting that and j can be changed by unitary transformations. We have rederived here in general terms, and in the the functional integral representation, a result previously known in the operator formalism [66], see also [21] and references therein. From the result above, one can directly obtain an expression for the von Neumann entropy, 1 2(N ( 1) 0 0 + a + b2 b + + a + b2 b 1 2 !N 4 4 1 2 !N 1 2 ! ln ! ! ! 1 2 1 2 1 2 ln ln ln + a + b2 + b + a + b2 b + + a + b2 b 1 2 !) 1 2 1 2 1 2 !) !) !) : The determinant of the matrix AN (a; b) is found to be !N 4 detAN (a; b) = Combining terms leads to a compact expression for the Renyi entropy, S = Tr Note that this is positive semi-de nite because a b and a + b are positive semi-de nite. Note also, that a and b can be expressed in terms of correlation functions of elds and canonical momenta using (2.25). The above expression simpli es for b = 0 to S =Tr ( r 1 4 + a + 1 2 ! ln As a rst example and check of this formalism, consider a free real massive scalar eld in n-dimensional in nite Minkowski space. The correlation functions in this case can be written as S (~x S (~x ~y) = h (t; ~x) (t; ~y)i = ~y) = h (t; ~x) (t; ~y)i = Z Z p~ eip~(~x ~y) 1 p~ pp~2 + M 2 2 eip~(~x ~y)pp~2 + M 2 1 + n(p~) ; 2 + n(p~) ; (2.36) 1 2 !N because a = (h + hy) 1 a as given in (2.25) is diagonal in momentum space. One nds i=2 = 0 and therefore b = 0. Here one can evaluate the (full) entropy easily, Z p~ S = f(n(p~) + 1) ln (n(p~) + 1) n(p~) ln (n(p~))g ; (2.37) which is the standard result for free bosonic elds. As expected, the entropy vanishes for n(p~) ! 0. We note that similar considerations also hold for relative entropies involving more than one density matrix. This concept is discussed further later and the relative entropy for free scalar elds is discussed in more detail in appendix A. Symplectic transformations, Williamson's theorem and entanglement entropy The above expressions can be further simpli ed with the help of canonical transformations. One considers unitary changes of the eld basis, m ! Umn n ; m ! Umn n ; m ! m ! n(U y)nm ; n(U y)nm : They can be used to diagonalize hermitian operators such as h + hy as is the case in going from position to momentum space. These transformations have unitary representations as transformations of the Schrodinger functionals. This is clear as the scalar product (2.5) remains unchanged by unitary transformations of the eld basis due to D = D(U ). In addition to this, there is a larger class of transformations, which transform elds and momenta into each other. Consider the combined eld = ! ; = ! : Their canonical commutation relation de nes a symplectic metric, Indeed one can con rm that this relation de nes a Lie algebra. J A = (J A)y : The transformations such that formation. becomes [ m; n] = mn ; = y = 0 i1 i1! 0 : m ! Smn n; m ! n(Sy)nm ; S Sy = ; are compatible with the canonical commutation relations. This de nes a symplectic transWritten in terms of the Lie algebra, S = exp[i AJ A], the condition (2.43) (2.38) (2.39) (2.40) (2.41) (2.42) (2.43) (2.44) Notice that in the eld basis where m is real, there is no change in R for real symplectic transformations. Hence Smp = Smp, as expected. In a eld basis that di ers from this by a unitary transformation, the symplectic transformation has di erent form (and is not necessarily real). Speci cally, the symplectic transformation matrix in (2.42) and the generator J A transform under the unitary block diagonal transformations (2.38) as S ! U SU y; J A ! U J AU y: We need to show that the symplectic transformations (2.42) have unitary representations as transformations of the states of the eld theory. This is best done in the eld basis m are real elds and R = 1. The symplectic transformations are then real and The matrix R transforms also by the symplectic transformations (2.42), HJEP04(218)5 where one has ([67], see also [68]) Recall that and contain also the corresponding complex conjugate elds so that there are relations m = Rmn n ; n = Rn m1 m = m(Ry)mn : One may assume without loss of generality that there is a eld basis where all elds are real such that there Rmn = mn. Of course, the matrix R changes under the unitary, block diagonal transformations (2.38). More speci cally, one has (2.45) (2.46) (2.47) (2.48) (2.49) (2.50) R ! U R(U y) : R ! S R(S 1) : J A = (J A) = (J A)y = (J A)T : There is now a representation of the Lie algebra speci ed by (2.44) in terms of the operators XA = 1 2 m mn(J A)nk k ; acting on a Schrodinger functional. Indeed, one can con rm that they have the same commutation relations as the generators J A. Moreover, one has (XA)y = XA in the sense of the bilinear form (2.5) so that the symplectic transformations indeed have unitary representations. This is an important result because it allows one to use the symplectic transformations to simplify calculations, for example of the entropy. Because symplectic transformations have unitary representations, they do not change the entropy by construction. Finally, we note that (2.50) is invariant under the block diagonal unitary transformations (2.38) and can therefore be used in any eld basis. It is also clear that the corresponding unitary transformation maps Gaussian states to Gaussian states. In particular, the hermitian and positive covariance matrix corresponding to the symmetrized correlation function mn = 1 2 h m n + n mic = ! h m nic ; (2.51) ! S Sy = S S 1 ; which indeed satis es the properties of a similarity transformation. The eigenvalues of this combination, j , are directly related to the symplectic eigenvalues. It is therefore convenient to determine the eigenvalues of and to thereby relate observables such as the entanglement entropy to the Williamson form. We rst note that the Williamson form expression (2.53) of the symmetric correlation matrix (2.51) results in a very simple form for the quantities in (2.25): aij = 2 j 1 4 ij ; bij = 0 : Since the Heisenberg uncertainty relation tells us that aij has to be positive-de nite, this indicates that j 1=2. The symplectic transformations we have discussed and the resulting use of Williamson's theorem leads to a very convenient form for entropies. The entropy in (2.35) can be directly expressed in terms of the symplectic eigenvalues as = diag( 1; 2; : : : ; 1; 2; : : :); (2.53) HJEP04(218)5 This is not a similarity transformation because Sy 6= S 1. In other words, the eigenvalues of are not invariant under symplectic transformations. Williamson's theorem states (see [69] for a discussion) however that there must exist a symplectic transformation that brings to diagonal form, which is a key ingredient in our discussion of Gaussian states, transforms under symplectic transformations as with real and positive j > 0. These latter are the symplectic eigenvalues of the symmetrized covariance matrix. This is realized by considering the combination . One can show that it transforms as (2.52) (2.54) (2.55) (2.57) (2.58) 1 2 j D = + 1 2 1 ; relation gives us j simply as which has the eigenvalues !j+ = 1=2 + j and ! j = 1=2 1=2; therefore, !j+ 1 and ! j S = X n!j+ ln(!j+) + !j ln( !j )o ; Moreover, the symplectic eigenvalues m follow as pairs of conventional eigenvalues j of the combination . Following Sorkin ([24], see also [ 25 ]), a further simpli cation can be obtained by considering the matrix j . As noted, the uncertainty 0. One can then write (2.56) where the sum is over pairs of eigenvalues. More simply, m S = X !m ln (j!mj) = Tr D ln D2 In the last expression, the sum is now over all the eigenvalues of D. Each negative eigenvalue !m < 0 is paired with a positive one 1 !m. A pure state without entropy has !m 2 f0; 1g. Finally, we note that the matrix D in symbolic form can be expressed as, Dmn = Hence, the entropy associated with a Gaussian density matrix is fully determined from the set of connected correlation functions of (2.60) evaluated in the domain of interest. It is understood that the operator trace in (2.59) is also restricted to this domain. Thus far, we have concentrated on the Renyi and von Neumann entropies of a single Gaussian density matrix. It is also possible to determine relative entropies between two Gaussian density matrices and in a similar way and we discuss this in appendix A. (See ref. [70] for a general exposition on the concept of relative entropy.) Let us remark here that Williamson's theorem is not as useful for the determination of the relative entropy of two Gaussian density matrices as it is for the determination of the entropy of a single one. This is because it is not guaranteed that there is a basis in which the covariance matrices ( ) and ( ) (de ned in (2.51)) simultaneously assume their Williamson diagonal form. However, this should be the case when the matrices ( ) and ( ) (which transform under symplectic transformations as similarity transforms, see (2.54)) commute, i.e. [ ( ) ; ( ) ] = 0. One can then write the relative entropy as S( j ) = X !m() ln j!m( )j ln j!m()j ; m where the sum goes over all pairs of simultaneous eigenvalues (!m(); !m( )) of D( ) = 2 1 1 and D( ) = ( ) Entanglement entropy in Minkowski space In this section, we will illustrate our general formalism by applying it to the derivation of the entanglement entropy of an interval of length L in 1+1 dimensional Minkowski space governed by a free scalar eld theory. This is a well studied problem with alternative approaches and results are available from both numerical and analytical methods [21]. The conventional numerical method to deal with this problem is to discretize the entire theory on a spatial lattice. The interval L corresponds to a nite subset of lattice sites and they are entangled with the lattice sites in the complement region. Numerically, one can study the continuum theory in the limit of ner and ner lattices. The entanglement entropy itself is ultraviolet (UV) divergent and a regulator is provided by the lattice spacing. However universal quantities such as derivatives of the entanglement entropy with respect (2.61) ( ) + to the interval length L can be extracted as well. Alternatively, one may consider relative entropies that are nite in the continuum limit as discussed in appendix A. The approach we developed in section 2 has the advantage that it depends only on the correlation functions inside the interval considered. Information about entanglement with degrees of freedom outside this interval is taken into account by appropriate boundary conditions in a nontrivial way. While it is rather straightforward to treat a eld theory on a nite interval with periodic boundary conditions, which corresponds in fact to an isolated system, it is more involved to properly treat the theory on an interval that is not fully isolated but entangled with a complement region. To illustrate the subtle di erences properly is a major focus for the following discussion. Eigenvalue problem and boundary conditions We will consider an interval ( L=2; L=2) in Minkowski space with one spatial dimension. A free scalar eld will be governed by a Gaussian reduced density matrix on this interval. Moreover, the corresponding matrix entries, namely the functions h, hy, a and b introduced in section 2, will be such that the correlation functions have the same form as in in nite space; they are just restricted to the interval. The technical di culties arise now from the fact that products of these functions involve integrals over the interval ( L=2; L=2), only. For example, the quantity a = (h + hy 2 a) 1 a de ned in (2.25) becomes a(x; y) = dz S (x z) S (z Z L=2 L=2 = Z p;q ( pq2 + M 2 sin 12 (p 4pp2 + M 2 12 (p y) q)L q) 1 4 (x 2 4 y) ) (p q) eipx iqy: (3.1) Note that this is a nondiagonal matrix in momentum space for nite L. Only in the limit L ! 1 does one obtain sin( 12 (p q)L)=( 12 (p q)) ! (2 ) (p q) and a becomes diagonal in momentum space (and zero). The challenge is now to nd the eigenvalues of the matrix a(x; y) on the interval ( L=2; L=2). To solve the eigenvalue problem, we will use a discrete basis involving Fourier expansion on the interval ( L=2; L=2). In doing so, we will not assume periodic boundary conditions. We rst divide the relevant function (or eld) into a symmetric and an anti-symmetric part, '(x) = '(s)(x) + '(a)(x) ; '(s)(x) = '(x) + '( x) 2 ; '(a)(x) = '(x) '( x) 2 : (3.2) The symmetric part can be expanded into a Fourier series '(s)(x) = 'n = dx '(s)(x) e in x=L (n even) : (3.3) In a similar fashion, one can expand the anti-symmetric part '(a)(x) = 'n = dx '(a)(x) e in x=L (n odd): (3.4) 1 L 1 X nn=eve1n 1 L 1 X nn=od1d Z L=2 L=2 Z L=2 L=2 We can summarize this as 1 L 1 X n= 1 Z L=2 L=2 '(x) = 'n = dx '(x) 1 he in x=L + ( 1)nein x=Li : (3.5) For '(x) 2 R, one has 'n = ' n. Note that this type of Fourier expansion does not assume periodic boundary conditions for '(x). In the limit of large interval length L ! 1, eq. (3.5) becomes formally (with p = n =L and '^(q) = 'n), '(x) = 2 2 Z dp eipx'^(p) ; '^(p) = Z dx e ipx 1 h'(x) + ( 1) pL '( x) : i (3.6) It is useful to relate these expressions to the standard momentum space representation de ned as usual by 2 2 Z + One has for nite interval length L and for very large L formally '(x) = Z dq eipx'~(p); 2 '~(p) = dx e ipx'(x): 'n = Z dp 2 sin pL 2 n 2 p 1 n L 1 strongly alternating, term above is neglected. eld 2'^(p) ! '~(p) only if the second, 3.2 Field- eld and conjugate momentum correlation functions We shall now determine the matrix expressions for the correlation functions S (x y) = h (x) (y)i and S (x position space can be decomposed as y) = h (x) (y)i. We rst note that the correlation functions in S (x y) = (ss)(x; y) + (aa)(x; y); S (x y) = (ss)(x; y) + (aa)(x; y) ; (3.10) with (3.7) (3.8) (3.9) (ss)(x; y) =h (s)(x) (s)(y)i = (aa)(x; y) =h (a)(x) (a)(y)i = 1 2 1 2 1 2 1 2 [ (x) + ( x)] [ (y) + ( y)] ; [ (x) ( x)] [ (y) ( y)] ; and similar for the canonical momentum eld correlator. The cross terms like by parity symmetry. Further, y ! y while (ss)(x; y) is symmetric with respect to x ! (aa)(x; y) is anti-symmetric with respect to parity. (3.11) (sa) vanish x as well as For the combined operator in (3.1), it is natural to decompose a(x; y) = a(ss)(x; y) + a(aa)(x; y) with a(ss)(x; y) = a(aa)(x; y) = Z L=2 Z L=2 L=2 L=2 dz dz (ss)(x; z) (ss)(z; y) (aa)(x; z) (aa)(z; y) 1 8 1 8 (x (x 1 8 1 8 y) (x + y) ; y) + (x + y) : (3.12) (3.13) (3.14) (3.15) 1 L 1 L [ S ]m( n) = [ S ]m( n) = L L 1 Z dp 1 Z dp p p 2 2 sin sin 1 1 m L m L + + pL 2 pL 2 1 1 p + mL m 2 m 2 p + mL p p sin sin 1 1 n L n L pL 2 + pL 2 + n 2 1 n 2 1 ; For a state with nonvanishing occupation number, one has to insert factors (1 + 2n(p~)) on the right hand side of these expressions. For the parity even-even (ss) (m; n even) and odd-odd (aa) (m; n odd) components The products of symmetric and anti-symmetric operators vanish under parity symmetry. With respect to the discrete indices m and n, one infers that amn is only nonzero when both indices are even or when both are odd, but that there can be no cross-terms. For the momentum space representation (2.36), using (3.8) (for n(p~) = 0) one nds that one has 1 L 1 L p 1 1 m L m L + + p + mL 1 1 p + mL p p 1 1 n L n L 1 1 , where the square root appears in the numerator. In computing these integrals, note rst that there are no poles on the real p-axis; we can therefore pull the contour slightly below the real axis. In addition, we can write [1 cos(pL)] = 1 eipL + 1 e ipL : The integral involving the rst bracket on the r. h. s. can be closed above the real paxis while the second can be closed below. The integral that is closed above picks up a 1 2 + 1 2 contribution from poles on the real p-axis for m2 = n2 as well as a contribution from the branch cut of the square root. The integral that is closed below has contributions from only the branch cut. The contribution from the integral over the poles is simply This result is in fact the ground state correlator one would have obtained by quantization of the scalar eld theory directly on the interval ( L=2; L=2) with periodic boundary conditions for parity-even modes and anti-periodic boundary conditions for parity-odd modes. It corresponds to a pure state without entanglement. Indeed, keeping only (3.16) together with the corresponding (poles only) approximation for the momentum-momentum correlation function, would lead to a vanishing entanglement entropy. We therefore see concretely from this example that the nontrivial contribution to the correlation function taking entanglement properly into account arises actually from the branch cut contribution to the integral in (3.14) and in its counterpart for . The contribution from the branch cuts to the eld- eld correlator, as determined by the right hand side of (3.14), is given by the integral ( 1) m2 n L Z 1 ML dy y 2 [y2 + (m )2][y2 + (n )2]py2 (M L)2 e y We shall rst discuss the result without the exponential in the last bracket. This should be a good approximation in particular for (M L) 1. For the opposite limit (M L) 1, we will add the contribution from the exponential term as well. Performing the integral gives for m2 6= n2, ( 1) m2 n L " 2 3(m2 jm j n2) pm2 2 + (M L)2 ln p(m )2 + (M L)2 pm2 2 + (M L)2 + jm j jm j ! jn j pn2 2 + (M L)2 ln p(n )2 + (M L)2 pn2 2 + (M L)2 + jn j jn j ! # ; L (M L)2 : (3.16) HJEP04(218)5 (3.18) (3.19) (3.20) while for m2 = n2 the result is ( 1) m2 n L " 4 2jmj((m )2 + (M L)2)3=2 2pm2 2(m2 2 + (M L)2) + (M L)2 ln Finally, for m = n = 0 one obtains pm2 2 + (M L)2 pm2 2 + (M L)2 + jm j jm j !# : Adding up these expressions gives the eld- eld correlation function in the massive case for M L coordinates that re ect the presence of a lightcone. The expressions we will derive in this section will be useful in describing the generation of entanglement entropy in both these cases. We will compute here the entanglement entropy and obtain its description in terms of a thermal density matrix in the bosonized massless Schwinger model in 1+1-dimensions. As noted previously, this is a popular model whose dynamics is observed to describe key features of particle production in elementary electron-positron collisions. After motivating the massless Schwinger model, we will discuss its bosonized version in terms of massive scalar elds. This will allow us to employ the machinery we developed in the previous sections to compute the time evolution of the entanglement entropy. We will also provide the corresponding results for fermionic elds towards the end of this section. The Schwinger model describes vector-like QED in 1 + 1 space-time dimensions. For a single massive fermion, the Lagrangian is L = ieA ) m 1 4 F F + im L R im R L F F : 1 4 ieA0 ieA0 + ieA1) R (4.1) The parameters of the model are the fermion mass m and the U(1) charge e. Note that the latter has dimensions of mass in two spacetime dimensions. The parameter e is related to the string tension, as one can see from the following consideration. The energy per unit length of the electric eld between charges e and e is given by E2=2. Moreover, from the Gauss law in 1+1 dimensions one has E2 = e2 such that the string tension is = e2=2. As is well known, the Schwinger model in two dimensions can be bosonized exactly [71]. (For completeness, we include a discussion of this feature of the model in appendix B.) The model can be reformulated as a scalar eld theory with Lagrangian (see eq. (B.20)) L = 1 2 me exp( ) 2 3=2 cos 2 p + : (4.2) Here is the Euler constant and is the vacuum angle. Note that QED in two dimensions has topologically non-trivial vacuum structure because the gauge group U(1) and the \boundary of Euclidean space at in nity" have the same topology S1. As is clear from the explicit bosonization procedure in appendix B, the scalar bosons are quadratic in the original eld and correspond to fermion-antifermion bound states . Hence the fermions themselves are not part of the physical spectrum | the theory displays (geometric) con nement. For the general case of a nonvanishing fermion mass m, (4.2) is still a nontrivial, interacting theory that cannot be solved easily. In particular, one expects a nontrivial renormalization of the e ective potential as well as propagator terms. Also additional bound states that are of quadratic or higher order in could arise. The situation simpli es substantially in the strong interaction limit e2 m2 where one may set m = 0. The massless Schwinger model becomes a free scalar eld theory after bosonization with a scalar boson of mass M = e=p . Entanglement in the Schwinger model, as well as in the 't Hooft model, has been investigated for static situations in ref. [72]. General coordinates and background evolution If one considers the free scalar eld theory with mass M in arbitrary curved coordinates, one has the action S = Z d2xpg 1 2 M 2 2 : (4.3) For the string that forms between a highly energetic quark-antiquark pair produced in electron-positron collisions, it is natural to consider a boost invariant expansion formulated in Bjorken coordinates with proper time = arctanh(x1=x0). The metric is g Christo el symbols are = , = = p(x0)2 (x1)2 and rapidity = diag( 1; 2) and pg = . The non-vanishing We will be interested in a situation where the external charges at the endpoints of a string generate a eld expectation value = h i, where the physics dictates that this background eld is boost invariant in the rapidity variable. The free- eld equation of motion in the Schwinger model for the \Bjorken expansion" of the -independent eld expectation value ( ) is then given by [73] 2 + 1 + M 2 = 0 : is satis ed. However here energy density and pressure are not related by a xed equation of state as in local thermal equilibrium but they are both determined dynamically by . The massless case, M = 0 is an exception which satis es the equation of state p = for pure radiation in 1+1 dimensions. More generally, depending on the initial conditions for , one can have initial conditions between p = in terms of J0(M ), one initially has p = with the result oscillating between this value and p = . The two independent solutions are ( ) J0(M ) and ( ) Y0(M ). While the former is regular for ! 0, the latter diverges. Both oscillate for late times be understood as multiple string breaking [74]. In the limit of vanishing mass M ! 0, the corresponding two independent solutions to the equation of motion (4.4) are ( ) const and ( ) ln( ). To ful ll the Gauss law, the electric eld E = e = must approach the p U(1) charge of the external quarks, E ! e for ! 0+. Therefore we obtain the solutions 1=M , which can The energy-momentum tensor has the form T = u u + p(u u + g ) with the \ uid velocity" u = (1; 0), energy density )2 + 12 M 2 2 and \pressure" p = ( ) = p J0(M ): + p = 0 ; (4.4) (4.5) (4.6) Note that the Bjorken metric g = diag( 1; 2) does not have a Killing vector eld pointing in direction; there is no solution = (f ( ); 0) of the equation L g However, there is a conformal Killing vector eld of this form which is a solution of L g g given by = (c ; 0) with = 2c with some constant c. These observations allow us to infer that no local thermal equilibrium state is conceivable which has Bjorken boost symmetry except for the case of a conformal eld theory. This insight will play a role in our discussion later on the emergence of a local thermal state and the dynamics generating entanglement entropy. Dynamics of perturbations . We begin by writing the eld as The uctuation eld has the equation of motion ( ; ) = ( ) + '( ; ): M 2 '( ; ) = 0 : = = 2 1 (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) In the following, we consider perturbations or uctuations around the background solution and the canonical equal-time commutation relations [ ( ; ); ( ; 0)] = i ( 0) imply and are implied by the commutator [a(k); ay(k0)] = 2 (k k0). g g 2 ( ; ) ; eld as We will now discuss the quantization of the eld '. Time dependent quantization problems of this type are best solved in terms of convenient mode functions. One writes the quantized '( ; ) = Z dk na(k)f ( ; jkj)eik + ay(k) f ( ; jkj)e ik o ; where a(k) and ay(k) are annihilation and creation operators. solutions to the di erential equation The mode functions depend only on the magnitude of the wave vector jkj and are 1 M 2 + f ( ; k) = 0: The inner product of these mode functions provides the normalization condition The canonical conjugate momentum eld is Note that (4.12) and (4.13) do not x the mode functions uniquely. The di erent possibile functions are related by the Bogoliubov transformations, f ( ; k) = (k)f ( ; k) + (k)f ( ; k); f ( ; k) = (k)f ( ; k) + (k)f ( ; k); f ( ; k) = (k)f ( ; k) (k)f ( ; k); f ( ; k) = (k)f ( ; k) (k)f ( ; k); (4.15) with j (k)j2 related by j (k)j2 = 1. The corresponding creation and annihilation operators are a(k) = a(k) = (k)a(k) + (k)ay(k); ay(k) = ay(k) = (k)a(k) + (k)ay(k); (k)a(k) + (k)ay(k): (4.16) p 2 k 2 p p2 sinh( k) The di erent sets of mode functions correspond to the vacuum states j i and j i respectively. These vacuum states contain no excitations in the sense that they are annihilated by the operators a(k)j i = 0 in the formed case and by a(k)j i = 0, in the latter case. Note however that the vacuum j i might contain excitations with respect to the operator a(k) and the vacuum j i with respect to a(k). However, the Bogoliubov transformations that connect the di erent choices are unitary transformations and therefore do not change entropy. Typically, the vacuum with respect to one set of mode functions corresponds to a squeezed state with respect to other sets of mode functions. In particular, (4.12) has the two independent solutions Jik(M ) and Yik(M ) or, equivalently, Hi(k2)(M ) and its complex conjugate H(1i)k(M ). The normalized mode functions corresponding to the latter choice are f ( ; k) = e k2 Hi(k2)(M ); f ( ; k) = e k2 H(1i)k(M ): (4.17) The set of mode functions f ( ; k) in (4.17) is distinguished by being a superposition of positive frequency modes with respect to time t of standard Minkowski spacetime [26]. This means that the standard Minkowski vacuum will be also a vacuum with respect to these mode functions in Bjorken coordinates. In the limit of vanishing mass M ! 0, the mode function (4.17) becomes f ( ; k) ! e ik ln( M=2) ie 2 (ik) p eik ln( M=2) p e k2 [cosh(k ) 2 (1 + ik) (M ! 0) : (4.18) One observes that it contains both positive and negative frequency contributions with respect to the logarithm ln( ) of Bjorken time. An alternative choice of mode functions is f ( ; k) = J ik(M ) ; f ( ; k) = Jik(M ) : (4.19) p 2 p p2 sinh( k) In the limit of vanishing mass M ! 0, the mode function (4.19) becomes f ( ; k) ! e ik ln( M=2) p (1 ik)p2 sinh k = e ik ln( ) i (k;M) 1 p2k (M ! 0) : (4.20) In this case, we see that it has only positive frequency contributions with respect to ln( ). The phase in the last equation is given by (k; M ) = k ln(M=2) + arg( (1 ik)): Note, in particular, that the factor multiplying k diverges in the formal limit M ! 0. The Bogoliubov coe cients that connect the mode functions (4.17) and (4.19) are The Gaussian density matrix or the vacuum state for this problem can be speci ed in terms of eld expectation values and the connected correlation functions, hay(k)a(k0)ic = n(k) 2 hay(k)ay(k0)ic = u (k) 2 (k k0) ; (k + k0) ; h (x1) (x2)ic = dz e iM cosh(z)(x01 x02) iM sinh(z)(x11 x12): (4.27) This agrees with (4.25) after substituting k0 = cosh(z), k1 = sinh(z), thereby conrming that the mode functions in (4.17) are indeed the ones corresponding to the standard Minkowski space vacuum. ( ; k), a complete set of connected correlation functions for the vacuum with n(k) = u(k) = u (k) = 0 in momentum space is given by For example, the vacuum state with n(k) = u(k) = u (k) = 0 results in the correlation function Z dk 2 h ( 1; 1) ( 2; 2)ic = f ( 1; k)f ( 2; k) eik( 1 2): It is instructive to compare (4.24) with the corresponding Minkowski space expression h (x1) (x2)ic = 1 Z 1 2 d2k 1 (2 )2 e ik0(x01 x02)+ik1(x11 x12)(2 ) (k2 + M 2): Employing the integral representation f ( ; k)eik = dz e iM cosh(z)x0 iM sinh(z)x1 ikz; where x0 = cosh( ) and x1 = sinh( ) are standard Minkowski coordinates, one obtains (k) = s e k (k) = s e : (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) (4.28) i p 2 Z 1 1 1 4 Re Z 1 1 h ( 1; k) ( 2; k0)ic = 2 h ( 1; k) ( 2; k0)ic = 2 h ( 1; k) ( 2; k0)ic = 2 h ( 1; k) ( 2; k0)ic = 2 (k (k (k (k k0) f ( 1; k)f ( 2; k); k0) f_( 1; k)f_ ( 2; k); k0) f ( 1; k)f_ ( 2; k); k0) f_( 1; k)f ( 2; k): where we have used the abbreviation f_( ; k) = @ f ( ; k). One can directly verify that with the above correlators, the matrix D in (2.60) has pairs of eigenvalues f0; 1g such that the entropy associated with the entire expanding string is zero. In fact, one does not need the precise form of f ( ; k) to show this; the normalization condition in (4.13) alone is su cient. Now with respect to the alternative set of mode functions, with only positive frequency solutions, the state with n(k) = u(k) = u (k) = 0 has the set of correlation functions, hay(k)ay(k0)ic = u (k) 2 (k k0) = j (k)j2 2 (k k0); (k + k0) = (k + k0) = (k) (k) 2 (k) (k) 2 (k + k0); In this alternative basis, the correlation functions do not look like those of an empty state but rather of one with occupation number n(k) = j (k)j2. From (4.22) one obtains, n(k) = j (k)j2 = 1 e2 k 1 : Recalling that the single particle energy in the expanding situation is E = pk2= 2 + M 2, for massless bosons, this distribution appearing in the \diagonal" elements of the correlation matrix (4.29) corresponds to a thermal spectrum with the time dependent temperature Such a thermal interpretation is not possible for a nonvanishing mass M , but the fact that the quasiparticles de ned by the mode functions f ( ; k) have a nonvanishing occupation number remains true. Of course, a thermal state would have vanishing entries for the other correlators appearing in (4.29). The \o -diagonal occupation function" is u(k) = u (k) = 1 One may use the relations in (4.29) to express the correlation functions in (4.28) in the alternative basis, h ( 1; k) ( 2; k0)ic = 2 (k (k (k (k +f ( 1; k)f ( 2; k)u(k) + f ( 1; k)f ( 2; k)u (k) ; k0) f ( 1; k)f ( 2; k)[1 + n(k)] + f ( 1; k)f ( 2; k)n(k) k0) nf_( 1; k)f_ ( 2; k)[1 + n(k)] + f_ ( 1; k)f_( 2; k)n(k) +f_( 1; k)f_( 2; k)u(k) + f_ ( 1; k)f_ ( 2; k)u (k)o ; f ( 1; k)f_( 2; k)u(k) + f ( 1; k)f_ ( 2; k)u (k)o ; +f_( 1; k)f ( 2; k)u(k) + f_ ( 1; k)f ( 2; k)u (k)o : k0) nf ( 1; k)f_ ( 2; k)[1 + n(k)] + f ( 1; k)f_( 2; k)n(k) k0) nf_( 1; k)f ( 2; k)[1 + n(k)] + f_ ( 1; k)f ( 2; k)n(k) (4.30) (4.31) (4.32) From these relations, one obtains the equal time correlation functions by setting 1 = 2 = . Moreover, in the limit M 1, one can use (4.20) for the mode functions f ( ; k). This gives the correlators h ( ; k) ( ; k0)ic = 2 h ( ; k) ( ; k0)ic = 2 h ( ; k) ( ; k0)ic = 2 h ( ; k) ( ; k0)ic = 2 (k (k (k (k k0) 1 k j j + n(k) + cos [2k ln( ) + 2 (k; M )] u(k) ; + n(k) + cos [2k ln( ) + 2 (k; M )] u(k) ; sin [2k ln( ) + 2 (k; M )] u(k) ; + sin [2k ln( ) + 2 (k; M )] u(k) : (4.34) ! 0. correlators at separation j ! 0. Note that the \o -diagonal occupation functions" u(k) that appear here are always multiplied with sine or cosine functions that are strongly oscillating with k in the limit M An interpretation of this term in position (rapidity) space is obtained by Fourier transforming the correlators above back to position (rapidity) space and examining the structure, for instance, of h ( ; ) ( ; 0)i. The strongly oscillating terms correspond then to pronounced structures in the rapidity di erence cos[2k ln(M =2)] 0 of the spatial 0 j 2j ln(M =2)j. This rapidity separation becomes large Entanglement entropy of an expanding string We shall now investigate the entanglement entropy of an interval with length in rapidity at some xed Bjorken time . Following the discussion in section 3 for the static Minkowski space case, one has to use a discrete Fourier expansion scheme for the nite interval ( =2). As discussed there, to ensure boundary conditions are not restricted, it is most convenient to split the elds into symmetric and anti-symmetric components. The calculation proceeds by determining correlation functions as in eq. (2.60) at xed Bjorken time and in the nite interval. As previously, one can use eq. (2.59) to determine the entanglement entropy. The correlation functions on the nite rapidity interval can be most conveniently obtained from the momentum space representations (4.28) or (4.33), together with the analog of the relation (3.8) expressing the discrete eld basis in terms of standard Fourier modes with an appropriate integral kernel. In our case here, this becomes (at xed time ) n = Z dk 2 sin k 2 n 2 k + n (k) : (4.35) When one calculates correlators in the discrete basis, such as h n mic, from (4.34) using the kernel (4.35), one observes that the terms proportional to the o -diagonal occupation number u(k) in (4.34) contain a term that oscillates very fast with k in the ! 0. This is because the combination 2k ln( ) + 2 (k; M ), with the phase (k; M ) in (4.21), is strongly dependent on k with in nite derivatives contributing in the h n ! 0. Hence these terms e ectively do not contribute to the correlators such as mi for a nite length interval One is then left with correlators that describe vacuum uctuations and the occupation numbers n(k) corresponding to a thermal distribution at very early times . This shows that the two limits M ! 1 do not commute. If one considers M , one nds the entanglement entropy of a thermal state with the time dependent temperature given by (4.31). This holds even if one then takes the limit ! 1. In contrast, if one considers an in nite interval ! 1 for nite M , one nds a pure state with vanishing entanglement entropy. This holds also if one considers subsequently the limit of vanishing mass M ! 0. Remarkably, one nds that at the very early times M ! 0, this entanglement entropy in a nite rapidity interval is equivalent to that of a 1+1 dimensional conformal eld theory at nite temperature when T = 1=(2 ). In such a 1+1 dimensional conformal eld theory at temperature T , the entanglement entropy of an interval of length L is given by [10, 11] (4.36) (4.37) S(T; L) = ln sinh( LT ) with central charge c = 1 in the present situation and where is a small length serving as ultraviolet regulator. We can use eq. (4.36) to determine the entanglement entropy of a rapidity interval one must set L = leads to the result at xed Bjorken time . Because the metric is ds2 = d 2 + 2d 2 . Using also the -dependent temperature T = 1=(2 ) in (4.36) S( ; ) = ln (2 sinh( =2)= ) + const: However, the result in eq. (4.37) can also be derived via an entirely di erent consideration as we have discussed in a recent letter [43]. In the following we brie y outline this derivation but refer to [43] for a more detailed discussion. We rst make use of the fact that the coherent eld does not contribute to the connected correlation functions in the covariance matrix in eq. (2.60) and can be dropped from the computation of the entanglement entropy. Furthermore, the entanglement entropy of an interval is unchanged by unitary evolution as long as the boundaries are kept xed. More speci c, the entanglement entropy of a rapidity interval at constant proper time corresponding to the dashed red line in gure 3 is actually equal to the entanglement entropy associated with the dotted red line at constant time t because the spatial boundaries agree and there is a unitary operator describing the transition between the states on both hyper surfaces. The determination of S( ; ) is therefore reduced to calculating the entanglement entropy of an interval with length z = L = 2 sinh ( =2) at a constant time t = cosh( =2) in the static Minkowski space vacuum. Using (3.30), this leads to (4.37) as well. The additive constant in eq. (4.37) is not universal but the derivatives of S with respect to and = (c=6) coth( =2). For a large rapidity interval 1, one has S = (c=6)[ + 2 ln( )] + const. This shows the existence of a time-independent piece of the entanglement entropy that is extensive in rapidity and a -independent piece that grows logarithmically with the proper time. c 3 c 1 T τ = const η = const region A region B z =2) at xed proper time (region A, dashed red line). The complement region B corresponds to ( 1; =2) and ( =2; 1) (dot-dashed orange line). The point p is the origin of the past light cone that delimits region A and q is the endpoint of the future light cone. For better orientation we also show lines of constant proper time and rapidity . At later times, for the nonconformal case of free massive scalars, the universal part of the entanglement entropy behaves as in the conformal case for M z 1 and decays 1 [21]. In this case, the derivative with respect to of the entanglement for M z entropy gives exponentially. S( ; ) = = cE (2M sinh( =2)) coth( =2)=2 ; (4.38) where cE (M z) = z@S=@ z is the entanglement entropy c-function for a massive scalar eld. Taking the conformal limit, one obtains, as anticipated, that cE (0) = c=3 = 1=3. For large values of the argument, this function has the form cE (x) ! xK1(2x)=4, which decays We can employ the general expression for cE (x) [21] for massive free bosons to compute for the massless Schwinger model. The result is displayed in gure 4 for the di erent values of M that are shown in the caption (dashed lines). For short times 1, one observes a signi cant entanglement over rapidity intervals = O(1). At intermediate values of M and approaches a plateau at 1=6 as a function of at early times. One also sees that it decays both for very large and for later times . The plateau is governed by the conformal limit M by the solid black line in the gure. ! 0 which is shown Since the entanglement entropy c-function in the conformal limit is identical for real massless scalar bosons and for massless Dirac fermions, this indicates that the conformal limit is consistently described in the Schwinger model with or without bosonization. One may also determine the entanglement entropy of free massive Dirac fermions using similar manipulations as described above and using the corresponding c-function given in ref. [21]. The result is also shown in gure 4 (dotted lines). The approach to the universal plateau at dS=d = 1=6 for M ! 0 is even faster for the free fermion case. dS/dΔη 25 Δη scalar elds (dashed curves) and free massive Dirac fermions (dotted curves). From left to right, the curves correspond to M = 1, M = 10 1, M = 10 2, M = 10 3, M = 10 4, and M = 10 5. At su ciently early time, a plateau forms corresponding to the conformal case (solid black line). A more intuitive (but also more heuristic) description of why the vacuum state j i in the conformal limit looks thermal in a nite rapidity interval is as follows. Although a pure state, the state describing the expanding string contains entangled pairs of quasiparticles with opposite momentum in the Bogoliubov basis (4.19) which consists of modes with positive frequencies with respect to ln( ) for M ! 0. This particular basis is special because only there can one interpret quasiparticles (in the classical quasiparticle limit) as moving in space on well de ned trajectories. For a rapidity interval , this implies that quasiparticles constantly come in via the left and right boundaries. They are entangled with other quasiparticles moving in the opposite direction but that is not seen locally. Because these quasiparticles have a thermal spectrum, local observables will e ectively look thermal. A related argument was employed previously to understand the time evolution of entanglement entropy after a quantum quench [13, 15]. 4.5 Local density matrix of an expanding string One can go even further and make stronger statements regarding the thermal character of entanglement entropy in a nite interval of the expanding string. Note that the correlators in (4.34), when projected to any interval of nite length with the kernel (4.35), are at M ! 0 exactly those of the 1+1-dimensional conformal eld theory in thermal equilibrium if the temperature is identi ed to be T = 1=(2 ). Gaussian density matrices are fully speci ed by one-point expectation values and two-point correlation functions. Because the correlation functions assume their thermal form on any nite interval, we can infer that in the limit ! 1 the density operator has the thermal form, where K is given by the local expression 1 Z = e K Z = Tr e K Z K = d T : (4.39) (4.40) Equations (4.39) and (4.41) specify the density matrix of a locally thermal state on a hypersurface where = u =T is the so-called inverse temperature vector given by the uid velocity u (pointing here in -direction) and the temperature is T = 1=(2 ). Moreover, T is the energy-momentum tensor of excitations '( ; ) around the coherent eld ( ) according to (4.9). Note also that = (1=T; 0) (in coordinates ; ) is actually a conformal Killing vector according to (4.8). Note that we have taken here the limit ! 1 within the conformal theory, i.e. after taking the limit M ! 0. In contrast, if ! 1 is taken for nite M , the density matrix describes a pure state even if ! 0 is taken afterwards. The limits M ! 1 do not commute, as has already been discussed in section 4.4. The result in eqs. (4.39) and (4.41) can also be understood as a limit of a more general result in conformal eld theory, as discussed already in ref. [43]. A conformal eld theory in the vacuum state in a region with a boundary formed by the intersection of two light cones (see gure 3), has a reduced density matrix of the form (4.39) on any hypersurface in that region with boundary on the intersection of the two light cones. The quantity K is then the so-called modular or entanglement Hamiltonian and it is a local expression given by [75, 76] (see also [77]) Z K = T : Here T is again the energy-momentum tensor of excitations and is a vector eld that can be written as 2 x) (x p) + (x p) (q p) (x where x is the space-time position on the hypersuface we consider, q is the end-point of the future light cone and p the starting point of the past light cone that form the boundary of the region. Note that (4.41) is again of the same form as a density matrix of a thermal state if one identi es = T1 u the vector of (inverse of) temperature T and uid velocity u . However, the vector vanishes on the boundary of the region enclosed by the two light cones corresponding formally to an in nite temperature. Consider now a situation where the two enclosing light cones intersect on a constant hypersurface in Bjorken coordinates with a rapidity di erence . If we concentrate on the point in the middle of this rapidity interval, the vector points in -direction and has length 2 T = 1=(2 the limit =2) 1)= sinh( =2). The associated temperature approaches precisely ! 1. More general one has in the limit ! 1. Note that ! 1 is crucial here because it leads to a rapidity-independent uid velocity and temperature. matrix we have The state we are considering here is vacuum-like but may have a nonvanishing coherent eld and therefore, a corresponding nonzero energy. Further, excitations that can be formed from local unitary (entropy preserving) transformations lead to a modi ed density 1 [76]. Using the notion of a relative entropy to as explained in appendix A, S( 1j ) = hKi = P ; (4.41) (4.42) (4.43) = R d precisely the characteristic of a thermal state. h T i is the four-momentum associated with the perturbation. This is Within a large but nite rapidity interval, the relative entropy of the state with a small perturbation compared to the coherent eld state of the expanding string has the same value as in a thermal state with temperature T = 1=(2 ). In the expanding geometry of interest, quantum uctuations of this kind are therefore as likely as in a thermal state. Such uctuations should therefore be observed with a distribution corresponding to that of a grand canonical ensemble. 5 Conclusions We developed in this paper a powerful formalism exploiting Gaussian density matrices to examine di erent entanglement entropies that arise in a wide variety of equilibrium and nonequilibrium problems in quantum eld theory. The most important results of this exercise are expressions for the entanglement entropy in terms of two-point correlation functions that can be directly evaluated within the nite spacetime interval of interest. In particular, we revisited the computation of the entanglement entropy of an interval of length L in static, two-dimensional Minkowski space and discussed how it can be computed within our formalism. We nd that it can be fully determined by correlation functions within the interval. Our results clearly reveal the importance of boundary conditions in the emergence of the entanglement entropy in nite spacetime intervals. We then applied the insights gain from our general treatment to study entanglement in the expanding string formed between a highly energetic quark-antiquark pair after an electron-positron collision. In order to do so, we employed the Schwinger model of quantum electrodynamics, which is the basis of much of the phenomenology describing multiparticle production in these collisions. We exploit the fact that the Schwinger model can be bosonized and is (in the limit of vanishing fermion mass m) equivalent to a free bosonic theory. The expanding string is best described in Bjorken coordinates of proper time and rapidity and corresponds then to a coherent eld solution of the Klein-Gordon equation in an expanding geometry. The entanglement properties of this state are xed in terms of correlation functions for excitations around the coherent eld. We discuss them in terms of appropriate mode functions. We observe that di erent sets of mode functions are possible and that they are related by Bogoliubov transformations. We nd that the state corresponding to the standard Minkowski space vacuum appears at early time as an occupied squeezed state with mode functions that have positive frequency with respect to the logarithm of Bjorken time ln( ). Moreover, as we showed very explicitly in sections 4.3{4.5 and had discussed previously in [43], this state appears in any T = ~=(2 nite rapidity interval as a thermal state governed by a -dependent temperature ). This is a rather remarkable nding, since it implies that excitations around the coherent eld solution of the expanding sting are in fact thermal at very early times! Although our model for a QCD string is not fully realistic, we believe that the above described mechanism provides a compelling candidate for a deeper understanding of the approximately thermal distributions of hadron ratios found in many collider experiments even in contexts where scattering e ects contributing to thermalization of particles in the nal state are likely small. A crucial question beyond the framework investigated here is what happens when interactions are taken into account. For the bosonized massless Schwinger model, they arise from a nonvanishing fermion mass m, leading to a Sine-Gordon type interaction term as in eq. (4.2). At very early Bjorken time , the interaction term me is expected to be irrelevant but could start to play a role at intermediate times and | for large enough m | even before the boson mass term e2= . We expect that further correlations, and therefore entanglement, can be built up by interactions between excitations and we plan to investigate their consequences for the entanglement dynamics in future work. While we have concentrated here on dynamics in 1+1 dimensions, similar entanglement dynamics may be at play in a Bjorken-type expanding geometry with additional transverse dimensions. If this is the case, quantum entanglement may be an important feature of early time dynamics in heavy-ion collisions. We plan to explore this exciting possibility in future work. Similar considerations also apply to nonrelativistic condensed matter systems such as ultra-cold quantum gases whose \horizon" is set by the speed of sound. Entanglement in such many-body systems can be e ciently explored with table-top experiments enabling direct confrontation of theoretical predictions with experimental measurements. A Relative entropy Extending the discussion of section 2.5, we now consider two density matrices and . They may be reduced density matrices originating from the trace over some part of the Hilbert space. We will be interested in the quantum relative entropy or Kullback-Leibler divergence of with respect to , de ned by S( j ) = Trf (ln ln ) : The relative entropy measures in an information theoretic sense to which extend one can distinguish the distribution from the distribution , see [70] for a review. In particular, for S( j ) = 0, the two distributions cannot be distinguished and agree, (up to a set of measure zero). For any other choice 6 = , the relative entropy is positive, S( j ) > 0. An interesting special case is when is the thermal density matrix = 1 Z e H (A.1) (A.2) F (A.3) with = 1=T and Hamiltonian H. The partition function can be written as Z = e with free energy F = E T S, dF = SdT pdV . The relative entropy becomes S( j ) =Tr ln (hHi F ) = S + S + (hHi hHi ) ; where h i and h i denote expectation values with respect to the density matrices and , respectively, while S and S denote the corresponding von Neumann entropies. Possible UV divergent contributions to the entanglement entropy are independent of the state and cancel between the rst and second term in the second line of (A.3). (This is actually a general statement independent of the speci c choice for made here.) Moreover, also possibly UV divergent contributions to the expectation values of energy, e. g. from the zero-point uctuations of various modes cancel on the right hand side of (A.3). Note that for equal energy, hHi = hHi , the relative entropy (A.3) equals the the di erence of entropies. One may also consider a situation where can be obtained from by a unitary operation. In that case S = S and S( j ) = (hHi hHi ), which is reminiscent of the de nition of temperature starting from the microcanonical ensemble, dS = dE. These considerations can be extended away from the case where is thermal in terms of the modular Hamiltonian K = ln , S( j ) = hKi hKi Similar as for the von Neumann entropy we will perform the calculation in the replica formalism and obtain the relative entropy in (A.1) as a limit of the Renyi relative entropies [16] (the latter de ned in [18]) HJEP04(218)5 SN ( j ) = 1 1 N Trf N 1 Trf N g = 1 1 N ln Trf N 1 SN ( ): With these de nitions one has [16] S( j ) = lim SN ( j ): N!1 When both and are Gaussian density matrices, they are fully characterized by their respective expectation values h mi = Trf h mi = Trf mg = (m); mg = (m); h mi = Trf h mi = Trf mg = jm(); mg = jm( ); as well as connected two- eld correlation functions, speci ed for ( ) and a b ( ) and similar for in terms of h( ), h( )y, ( ) and a b in terms of h( ), h( )y, ( ) (see discussion in section 2.4). For simplicity we will assume in the following that the eld expectation values with respect to and agree, i.e. ( ) = m ( ) and jm() = jm( ). We make no such assumption m about the connected correlation functions, however, and in fact keep them fully general. (A.4) (A.5) (A.6) (A.7) Using similar manipulations as in section 2.5 we obtain 1 N 1 ln Trf N 1g = 1 2(N 1) Trln r14 + a( ) + b( )2 + b( ) + 1 r14 + a( ) + b( )2 + b( ) + 1 N 1 2 2 a( ) = h( ) + h( )y 2 (a ) 1 (a ); b( ) = h( ) + h( )y 2 (a ) 1 (b ); a( ) = h( ) + h( )y 2 (a ) 1 (a ); b( ) = h( ) + h( )y 2 (a ) 1 (b ): By subtracting (2.33) from (A.8) one obtains directly an expression for the relative Renyi entropy. In particular, we obtain for the quantum relative entropy by taking the limit N ! 1, the expression ln r14 + a( ) + b( )2 + b( ) + 1 2 (A.8) (A.9) r14 + a( ) + b( )2 + b( ) 1 2 r14 + a( ) + b( )2 b( ) 1 2 (A.10) ln ln r14 + a( ) + b( )2 + b( ) + 1 2 r14 + a( ) + b( )2 + b( ) 1 2 r14 + a( ) + b( )2 + b( ) 1 2 ln ln ln S( j ) = Tr + a( ) + + a( ) + + a( ) + r 1 4 r 1 4 + a( ) 1 2 1 2 r 1 4 r 1 4 1 2 1 2 r 1 4 r 1 4 + a( ) + a( ) 1 2 1 2 B Symmetries, anomalies and bosonization In this appendix we recall how the Schwinger model corresponding to vector-like QED in two dimensions can be bosonized [71]. One proceeds via a discussion of gauge symmetries and associated quantum anomalies. B.1 Consider the Lagrangian J = i J5 = i L = ieA 5) : The vector gauge symmetry is as usual ! e i ; e i ; A B ! B ; 1 or for left and right handed fermions L ! e i L; R ! e i R; L ! L e i ; R ! Re i : One can de ne the regularization such that the vector gauge symmetry above is not anomalous. The associated Noether vector current is in our notation In contrast, the axial gauge transformation or for left and right handed fermions is anomalous. The axial vector current ! ei 5 ; ei 5; A ! A ; B L; R ! e i R; L ! L e i ; R ! Rei ; is not conserved. There is an associated anomaly (see e. g. ref. [78], eq. (5.41)) 1 2 eF 1 (B.1) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) The Schwinger functional de ned by eiW = Z D D i R d2xL changes by an in nitesimal axial transformation according to W = Z Z d x 2 n F (x) + o Bosonization Z eiW [J] = D D D DA eiS+i Rx J A (B.9) (B.10) (B.11) (B.12) (B.15) (B.16) where S = Rx L is essentially the microscopic action of the mass-less Schwinger model and we have added an irrelevant constant in the form of an integral over a free bosonic eld . Decompose the gauge eld as A 1 1 with two functions and . The Lagrangian is 1 2 L = 5) F F : (B.13) The idea is now to perform now a vector gauge transformation with (which is simple to do) and an axial vector gauge transformation with . One needs to do this in in nitesimal steps taking into account the presence of the axial vector potential B Use now A + p1 Z n e 2 F (B.14) The fermionic functional integral becomes now a free one which contributes only an irrelevant factor to the partition function. Taking the anomaly into account leads therefore to eiW [J] = 4 which implies eF e A to obtain eiW [J] = Z 4 A +J A o This constitutes the bosonized form of the Schwinger model. In last step one can integrate out the gauge eld by performing the Gaussian integral or, equivalently, solving the corresponding eld equation. 1 4 (B.17) (B.18) (B.19) In two dimensions, the gauge eld A is not dynamical. One may chose the axial gauge A1 = 0 and the eld strength is F10 = F01 = E1 = @1A0. (There is no magnetic eld in two dimensions.) The equation of motion is in the bosonized theory and one can formally solve this = e 1 p where appears as an integration constant and is the vacuum angle. Using this in the partition function gives (for J = 0) eiW = Z 1 e2 2+ 8e22 2o 2 This is now the partition function of a massive scalar particle with mass m = pe . The vacuum angle is here just an irrelevant overall factor and therefore drops out. For the massive Schwinger model the standard bosonization is more involved because the mass term transforms non-trivially under the axial gauge transformations. Order by order in a perturbative series in m one nds that the theory is equivalent to the bosonic theory [71], L = 1 2 1 e2 2 2 2 3=2 cos 2 p : (B.20) Here, is the Euler constant but note that the factor in front of the cosine term could be absorbed into a rescaling of the fermion mass m. This is now an interacting theory, and it depends on the vacuum angle . 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Jürgen Berges, Stefan Floerchinger, Raju Venugopalan. Dynamics of entanglement in expanding quantum fields, Journal of High Energy Physics, 2018, 145, DOI: 10.1007/JHEP04(2018)145