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 realtime approach to computing the entanglement between spatial regions for Gaussian states in quantum entropy is characterized in terms of local correlation functions on spacelike 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 spacetime 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; QuarkGluon

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 nsheeted 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 realtime 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 twopoint 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
EinsteinPodolskyRosen 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 spacetime 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 electronpositron 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 thermallike 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 subregion, 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 closetoequilibrium distribution of hadron ratios as found experimentally. We
{ 3 {
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with
a temperature
describing
spectrum
that
decays
of particles.
with
proper time, T
= ~=(2
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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 electronpositron 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
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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
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tunneling processes via the Schwinger mechanism. In the standard implementation of the Lund model, as
variant of the
Glaubernoted in ref. [40], the thermallike 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].
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most in eq. 2.3, discuss
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by
explain
certain
this question
characterized
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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 electronpositron 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
purestate 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 quasiprobability 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 semide nite. (The density operator should of course
be hermitian and positive semide nite.) Note that eq. (2.9) is closely related (although
not identical) to the GlauberP 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 straightforward 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 semide nite because a
b and a + b are positive semide 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 ndimensional 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 positivede 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 antisymmetric 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 antisymmetric 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 antisymmetric 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 eveneven (ss) (m; n even) and oddodd (aa) (m; n odd) components
The products of symmetric and antisymmetric 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 crossterms.
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 paxis; 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 paxis 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 parityeven modes and antiperiodic boundary conditions for parityodd
modes. It corresponds to a pure state without entanglement. Indeed, keeping only (3.16)
together with the corresponding (poles only) approximation for the momentummomentum
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+1dimensions.
As noted previously, this is a popular model whose dynamics is observed to describe key
features of particle production in elementary electronpositron 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 vectorlike QED in 1 + 1 spacetime 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 nontrivial 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 fermionantifermion 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 quarkantiquark pair produced in
electronpositron 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 nonvanishing
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 energymomentum 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 equaltime 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 antisymmetric 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 timeindependent 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) (dotdashed 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 cfunction 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 cfunction 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 cfunction 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+1dimensional 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 onepoint expectation values and twopoint 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 socalled inverse temperature vector given by
the
uid velocity u (pointing here in direction) and the temperature is T = 1=(2
).
Moreover, T
is the energymomentum 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 socalled 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 energymomentum 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 spacetime position on the hypersuface we consider, q is the endpoint
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 rapidityindependent uid velocity
and temperature.
matrix
we have
The state
we are considering here is vacuumlike 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 fourmomentum 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 twopoint 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, twodimensional 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 quarkantiquark pair after an
electronpositron 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 KleinGordon 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 SineGordon 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 Bjorkentype expanding geometry with additional transverse
dimensions. If this is the case, quantum entanglement may be an important feature of
early time dynamics in heavyion collisions. We plan to explore this exciting possibility in
future work.
Similar considerations also apply to nonrelativistic condensed matter systems such as
ultracold quantum gases whose \horizon" is set by the speed of sound. Entanglement in
such manybody systems can be e ciently explored with tabletop 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 KullbackLeibler
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
zeropoint 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 vectorlike 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 massless 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 nontrivially 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 .
Acknowledgments
This work is part of and supported by the DFG Collaborative Research Centre \SFB 1225
(ISOQUANT)". R. V.'s research is supported by the U. S. Department of Energy O ce of
Science, O ce of Nuclear Physics, under contracts No. DESC0012704. R. V. would like
to thank ITP Heidelberg and the Alexander von Humboldt Foundation for support, and
ITP Heidelberg for their kind hospitality.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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