Two-dimensional thermofield bosonization II: Massive fermions

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Annals of Physics 323 (2008) 2662–2684 www.elsevier.com/locate/aop

Two-dimensional thermofield bosonization II: Massive fermions R.L.P.G. Amaral a,*, L.V. Belvedere a, K.D. Rothe b Instituto de Fı´sica, Universidade Federal Fluminense, Av. Litoraˆnea S/N, Boa Viagem, Nitero´i, CEP 24210-340, Rio de Janeiro, Brazil b Institut fu¨r Theoretische Physik, Universita¨t Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany a

Received 7 January 2008; accepted 12 January 2008 Available online 26 January 2008

Abstract We consider the perturbative computation of the N-point function of chiral densities of massive free fermions at finite temperature within the thermofield dynamics approach. The infinite series in the mass parameter for the N-point functions are computed in the fermionic formulation and compared with the corresponding perturbative series in the interaction parameter in the bosonized thermofield formulation. Thereby we establish in thermofield dynamics the formal equivalence of the massive free fermion theory with the sine-Gordon thermofield model for a particular value of the sine-Gordon parameter. We extend the thermofield bosonization to include the massive Thirring model. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Bosonization; Thermofields; Massive Thirring

1. Introduction The bosonization of fermions has proven in the past to be a very useful technique for solving quantum field theoretic models in 1 + 1 dimensions [1]. In a recent paper [2] we have considered the operator formulation of the bosonic representation of massless free fermions at finite temperature (thermofield bosonization) with the *

Corresponding author. E-mail address: [email protected]ff.br (R.L.P.G. Amaral).

0003-4916/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2008.01.005

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thermofield dynamics formalism [3–9]. The well known two-dimensional Fermion– Boson correspondences at zero temperature are shown to hold also at finite temperature. In Ref. [2] we have also used the thermofield bosonization for obtaining the real time fermion N-point functions of the massless Thirring model at finite temperature. The equivalence of the massive Thirring model and the sine-Gordon theory at finite temperature in the imaginary time formalism has been the subject of a number of authors [10,11]. The discussions have been carried out predominantly from the functional point of view. The main objective of the present work is to extend the thermofield bosonization approach presented in Ref. [2] to the case of massive fermions. To this end we use strictly fermionic techniques on the one hand, and bosonization techniques on the other, in order to demonstrate within the Thermofield Dynamics approach the equivalence of the theory of massive free fermions at finite temperature with the sine-Gordon thermofield theory. The demonstration will be done for the case of the N-point functions of chiral densities, by working in the interaction picture of the respective formulations, with the mass term playing the role of the interaction Hamiltonian. The selection rule to be imposed in both formulations (fermionic and bosonic) in order to prove the equivalence emerges here in an interesting way. As a byproduct the role of the ‘‘tilde” fields in thermofield dynamics and the lower branch at t ¼ ib=2 in the real time formalism [7] emerges naturally in this calculation. The intimate relationship between the thermofield dynamics formalism and the algebraic formulation due to Haag–Hugenholtz–Winnink (HHW) of statistical mechanics has been established in Ref. [8]. In this reference the relevance of tilde objects to the modular conjugation appearing in the algebraic formulation of statistical mechanics in the HHW formalism based upon the KMS condition is clarified. This formulation of thermofield dynamics in a way consistent with the HHW formalism enable to extend it to gauge theories and becomes crucial in the treatment of the Faddeev-Popov ghosts [8]. To this end, a ‘‘new” version for the thermofield dynamics approach for fermions is presented in [8]. In this paper we shall follow this revised version for the thermofield dynamics approach for fermions since it corrects a mistake in our previous work [2] (this is discussed in Appendix) and as we shall see, becomes fundamental for the success of the thermofield bosonization scheme. The paper is organized as follows: We begin in Section 2 by considering the thermofield dynamics approach to the massive free fermion theory. We use the generalization of the perturbation theory to finite temperature [9] in order to compute in thermofield dynamics the N-point function of chiral densities of massive free fermions as a power series in the mass M of the fermion, with an explicit expression for the expansion coefficients as a ratio of temperature dependent polynomials. In Section 3 we repeat this calculation for the bosonized formulation and extend to include the massive Thirring model. The equivalence of the two formulations is thereby established, upon taking suitable account of the selection rule emerging in this calculation. In Section 4 we discuss the physical meaning of the selection rule. We conclude in Section 5 with some comments. In Appendix we discuss some modifications of the thermofield dynamics formulation for fermions of Ref. [8] due to Ojima, which shall play an important role in order to obtain the correct bosonized expression for the Fermi thermofields. This streamlines the presentation of Ref. [2].

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2. Massive free fermions In thermofield dynamics the construction of a field theory at finite temperature requires doubling the numbers of fields degrees of freedom by introducing ‘‘tilde” operators corresponding to each of the operators describing the system considered [3–9]. To this end, let us consider the total Lagrangian  of the  two-dimensional massive free Fermi field correw sponding to the fermion doublet ~y iw b ¼ L  L; e L

ð2:1Þ

e is the corresponding where L is the usual Lagrangian of a massive free fermion, and L ~ Lagrangian in terms of the field w and obtained from L by the tilde conjugation defined g ¼ c w. ~ Since we shall consider perturbation theory around the massless theory, by ðcwÞ b in the form we choose to write L bI b¼L b0 þ L L where

ð2:2Þ

1

~ l ol w; ~ b 0 ¼ wicl ol w þ wic L

ð2:3Þ

 w ~ wÞ: ~ b I ¼ Mðww L

ð2:4Þ

and

At finite temperature and within the thermofield approach, the perturbative computations can be performed using the generalization of the Gell–Mann–Low formula for T 6¼ 0 [9]. The vacuum expectation value of time-ordered products of Heisenberg operators U in the physical vacuum j 0ðbÞi at finite temperature [7–9] is given by  R  Q b I ðzÞ j0; bi h0; bjT k /k ðxk Þ exp i d2 z L Y  R  ; ð2:5Þ Uk ðxk Þ j 0ðbÞi ¼ h0ðbÞ j T b I ðzÞ j0; bi h0; bjT exp i d2 z L k where the right hand side is computed in the interaction picture. The thermal interaction picture vacuum is defined by j 0; bi ¼ U F ½hF  j ~ 0; 0i;

ð2:6Þ

where j ~ 0; 0i is the interaction picture vacuum at zero temperature, and U F ½hF  is the unitary operator (We shall adopt the ‘‘revised” thermofield dynamics approach for fermions as formulated in Ref. [8]) R1 ~ 1 ÞdðpÞþd y ðp1 Þd~y ðp1 ÞÞ i dp h ðjp1 j;bÞð~ bðp1 Þbðp1 Þþby ðp1 Þ~ by ðp1 Þþdðp U F ½hF  ¼ e 1 F ; ð2:7Þ 1

The conventions used are:    0 1 0 c0 ¼ ; c1 ¼ 1 0 1

1 0

 ; c5 ¼ c0 c1 ; e01 ¼ 1; g00 ¼ 1; x ¼ x0  x1 ; o ¼ o0  o1 :

For massless scalar field uðxÞ ¼ /ðxþ Þ þ /ðx Þ, and for the pseudo-scalar field /ðxÞ ¼ /ðxþ Þ  /ðx Þ.

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where the Bogoliubov parameter hF ðp; bÞ is implicitly defined by (p ¼j p1 j), 1 cos hF ðp; bÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ ebp

ð2:8Þ

ebp=2 sin hF ðp; bÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ ebp

ð2:9Þ

and

with the Fermi–Dirac statistical weight given by, N F ðp; bÞ ¼ sin2 hF ðp; bÞ ¼

ebp

1 : þ1

ð2:10Þ

We can pass the time independent unitary operator (2.7) through the time ordering operation, and rewrite the Gell–Mann–Low formula in terms of the zero temperature vacuum j~ 0; 0i and the transformed annihilation operators are given by [8] bðp; bÞ ¼ bðpÞ cos hF ðp; bÞ þ i~ by ðpÞ sin hF ðp; bÞ; ~ bðp; bÞ ¼ ~ bðpÞ cos hF ðp; bÞ  iby ðpÞ sin hF ðp; bÞ;

ð2:11Þ ð2:12Þ

~ For free massless fermions in 1 þ 1 and their adjoints, with similar expressions for d and d. dimensions the spinor components w1 ðw2 Þ are left (right) moving fields:   w1 ðxþ Þ wðxÞ ¼ ; ð2:13Þ w2 ðx Þ and similarly for the tilde fields. Following the approach given in Ref. [8], the corrected expression for the Fermi thermofield is given by (see Appendix) Z 1 n 1  wðx ; bÞ ¼ pffiffiffiffiffiffi dp fp ðx ÞðbðpÞ cos hF ðp; bÞ þ i~by ðpÞ sin hF ðp; bÞÞ: 2p 0 o ~ sin hF ðp; bÞÞ ; ð2:14Þ þfp ðx Þðd y ðpÞ cos hF ðp; bÞ  idðpÞ Z 1 n ~  ; bÞ ¼ p1ffiffiffiffiffiffi wðx dp fp ðx Þð~ bðpÞ cos hF ðp; bÞ  iby ðpÞ sin hF ðp; bÞÞ 2p 0 o þfp ðx Þðd~y ðpÞ cos hF ðp; bÞ þ idðpÞ sin hF ðp; bÞÞ ; ð2:15Þ where fp ðxÞ ¼ eipx :

ð2:16Þ

2.1. N-point function of chiral densities from fermionic point of view In order to compactify the calculation it proves convenient to introduce the following notation for the Fermi fields. We label them by an upper index g taking the values zero and one, with the identifications wg ¼ w; for g ¼ 0; ~ y ; for g ¼ 1: wg ¼ w

ð2:17Þ

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Now, let us consider the chiral densities of the massive free Fermi field in terms of the spinor components y Jg¼0 þ1 ðxÞ ¼: w1 ðxÞw2 ðxÞ :;

g¼0 J1 ðxÞ ¼: wy2 ðxÞw1 ðxÞ :;

~ ~y Jg¼1 þ1 ðxÞ ¼: w1 ðxÞw2 ðxÞ :;

g¼1 ~ 2 ðxÞw ~ y ðxÞ : : J1 ðxÞ :¼: w 1

ð2:18Þ

In a compact notation these expressions become g Jgþ ðxÞ ¼: wgy 1 w2 :;

g Jg ðxÞ ¼: wgy 2 w1 : :

ð2:19Þ

Here the notation for the lower index (chiral) has been chosen such as to facilitate later comparison with the corresponding bosonized expressions. In terms of the chiral densities the mass term reads, X X g ~ wÞ ~ ¼M  w Jk : ð2:20Þ Mðww k¼1 g¼0;1

Note that the minus sign on the left hand side for the tilded fields has been taken into account by reordering the fields on the rhs, taking account of the definitions (2.17) and of Fermi statistics. With the identification (2.4) for the interaction Lagrangian we have, setting J ¼ J0 , h0ðbÞjT Jþ1 ðx1 Þ    Jþ1 ðxN ÞJ1 ðy 1 Þ    J1 ðy N 0 Þj0ðbÞi ¼

n 1 X 1 ðiMÞ X N F ðbÞ n¼0 n! ðk ;g Þ k

k

h0; ~ 0jTJ þ1 ðx1 ; bÞ    J þ1 ðxN ; bÞJ 1 ðy 1 ; bÞ    J 1 ðy N 0 ; bÞ

Z

dZ n

n Y g ðJ kkk ðzk ; bÞÞj~0; 0i; ð2:21Þ k

where we transfered the temperature dependence of the ground state to the densities by commuting the time independent unitary operator U F ½hF  through the time ordering operation. J 1 ðx; bÞ are the thermal chiral densities of the free massless Fermi field y þ  J þ1 ðx; bÞ ¼ U F ½hF J þ1 ðxÞU 1 F ½hF  ¼: w1 ðx ; bÞw2 ðx ; bÞ :;

ð2:22Þ

y  þ J 1 ðx; bÞ ¼ U F ½hF J 1 ðxÞU 1 F ½hF  ¼: w2 ðx ; bÞw1 ðx ; bÞ :;

ð2:23Þ

R the integral dZ n is short hand for Z n Z 1 Y dZ n ¼ d2 zk ; k¼1

ð2:24Þ

1

and N F ðbÞ represents the contribution of the vacuum graphs, Z n 1 n X Y ðiMÞ X g dZ n h0; ~ N F ðbÞ ¼ 0jT ðJ kkk ðzk ; bÞÞj0; ~0i: n! k n¼0 ðk ;g Þ k

k

ð2:25Þ

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2667

We can thus apply Wick’s theorem to compute each term in the expansion in powers of M. The two-point functions for the fermion thermofield have been computed in Ref.[2] and adopting the notation given in (2.17) the fermionic propagators are given by y h0; ~ 0 j T w0 ðx ; bÞw0 ðy  ; bÞ j 0; ~ 0i ¼

2ib sinh y 0i ¼ h0; ~ 0 j T w1 ðx ; bÞw1 ðy  ; bÞ j 0; ~

2ib sinh

1

h

p ðx b

i;  y   ieðx0  y 0 ÞÞ

ð2:26Þ

1

h

p ðx b

i;  y  þ ieðx0  y 0 ÞÞ

ð2:27Þ

and y

y

0i ¼  h0; ~ 0 j T w1 ðx ; bÞw0 ðy  ; bÞ j 0; ~0i h0; ~ 0 j T w0 ðx ; bÞw1 ðy  ; bÞ j 0; ~ i h i: ¼ p 2b cosh b ðx  y  Þ

ð2:28Þ

These functions can be collected in a compact notation by using coshðxÞ ¼ i sinhðx  ip=2Þ

ð2:29Þ

so that we can write gj y  ~ h0; ~ 0 j T wgi ðx i ; bÞw ðy j ; bÞ j 0; 0i

¼ 2ib sinh

h  p b

ðiÞgi þgj bgi  x i  i 2  yj þ i

bgj 2

g

 ieð1Þ j ðx0i  y 0j Þ

i :

ð2:30Þ y

In applying the Wick’s theorem only terms with equal number of wg1 and wg1 survive irrespective of their g values. The same is true for the second spinor component. In terms of the chiral densities J gk this leads to the selection rule requiring the sum of all values of k have to vanish, irrespective of the g values. There are thus equal number of positive and negative values of k. We shall denote the space-time coordinates of the fields associated to k ¼ 1 by xi , while the ones associated to k ¼ 1 are denoted by y j . The values of the g upperscript are accordingly splited as gi and gj . The result of the computation can be written as h0; ~ 0jT

n Y

g

J kii ¼þ1 ðxi ; bÞ

i¼1

¼ det

n Y

g J kjj ¼1 ðy j ; bÞ j 0; ~ 0i

j¼1

ðiÞgi þgj  xððx i  igi b=2Þ  ðy j  igj b=2ÞÞ g þg

 det

ðiÞ i j ; þ xððxi  igi b=2Þ  ðy þ j  igj b=2ÞÞ

ð2:31Þ

where p g xðxi  y j Þ ¼ ð2ibÞ sinh ðxi  xj  ið1Þ j eðx0i  y 0j ÞÞ: b

ð2:32Þ

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Note that the factors ðiÞ i j can be factorized out of the determinant computation resulting in an overall sign depending on the total numbers of tilde fields w1 ~ m and of corresponding tilde fields w2 . Both contributions lead to a factor ð1Þ , with ~¼ m

n X

gi þ

n X

i

ð2:33Þ

gj

j

representing the total number of e J  in the string. Now using the factorization formula [12] for the determinant of the n  n matrix 1=xðxi  y j Þ Q Q 0 1 i
n Y i¼1

g

i J þ1 ðxi Þ

n Y

g

j J 1 ðy j Þ j 0; ~ 0i

j¼1

Qn þ þ þ þ i
ð2:35Þ Here we introduced the function XðxÞ ¼ xðxþ Þxðx Þ;

ð2:36Þ

Returning the i prescription means that in the denominator of Eq. (2.35) the change should be performed     p þ p   ðxi  y þ ðx sinh Þ sinh  y Þ j j b b i     p þ p  þ 0 0  0 0 ðx  y j  iej ðxi  y j ÞÞ sinh ðx  y j  iej ðxi  y j ÞÞ ; ! sinh ð2:37Þ b i b i where ej ¼ ð1Þgj e. Returning now to the expression (2.21) we reorganize the terms in the expansion (2.21) according to chirality, with n being the total number of internal currents with kk ¼ 1 and n0 the same for kl ¼ 1. To the former we associate the variables zk and to the last ones z0l . The contributions can now be collected into the expression

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684 N Y

h0ðbÞjT 

Z

Jþ1 ðxi Þ

N0 Y

! J1 ðy j Þ j0ðbÞi ¼

j¼1

i¼1

1 N ðbÞ

2669

nþn0 X Z 1 X  ðiMÞ dZ n 0! n!n nþn0 ¼0 ðg ;g Þ k

l

n0

dZ h0; ~ 0jTJ þ1 ðx1 ; bÞ    J þ1 ðxN ; bÞJ 1 ðy 1 ; bÞ    J 1 ðy N 0 ; bÞ

n n0 Y Y gk gl ðz0l ; bÞÞj~ 0; 0i;  ðJ þ1 ðzk ; bÞÞ ðJ 1 k

ð2:38Þ

l

where the notation now reads

P

means that the sum obeys the chiral conservation condition that

N  N 0 þ n  n0 ¼ 0;

ð2:39Þ

n

and dZ refers to integration over the variables fz1 ;    ; zn g. Each term of this expansion is obtained directly from (2.35) leading to ! N N0 Y Y h0ðbÞ j T Jþ1 ðxi Þ J1 ðy j Þ j 0ðbÞi j¼1

i¼1

0 Z Z ðiMÞnþn X 0 ~ m n dZ dZ n ð1Þ I ðx; y; z; z0 Þ; 0 n!n ! 0 nþn ¼0 ðg ;g Þ

1 X

1 ¼ N F ðbÞ



k

ð2:40Þ

j

with I ðx; y; z;z0 Þ ¼

QN 0 0 i
QN

QN Qn k¼1 Xðxi  ðzk  igk b=2ÞÞ Qi¼1 N Qn 0 i¼1 l¼1 Xðxi  ðzl  ibgl =2ÞÞ

QN 0 Qn0 j¼1

0 l¼1 Xðy j  ðzl  igl b=2ÞÞ

k¼1

j¼1 Xðy j  ðzk

 Q n QN 0 Qn 

k
 igk b=2ÞÞ

Q0  igk b=2Þ  ðzk0  igk0 b=2ÞÞ nl
It is interesting to note, that the shift in the argument of the two-point function involving tilde field, (2.30), (2.31), can be understood in the context of the real time formalism as the tilde field living on the lower branch of the integration contour localized at ImðtÞ ¼ ib=2 in the complex time plane. This feature appears further in expression (2.41) since to all integrated variables zk are associated fields with g labels and these labels are summed for g ¼ 0 and g ¼ 1. In the real time formalism this corresponds to contributions from the upper branch, the lower branch and functions from mixed branches. This has been observed to be necessary to satisfy the KMS conditions [4]. Note that all expressions in the numerator link currents of the same chirality, whereas all terms in the denominator link currents of opposite chirality. We thus associate with each coordinate ni the chirality ki . We further reorganize the terms within the products by separating the integration variables into those which belong to ordinary currents and tilde currents (i.e. zk and ~zk , respectively), and denote the respective chiralities by k‘ and ~ kk . The assignment of ~ kk to the tilde currents chirality is due to the definition

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Eq. (2.18). With this we can write expression (2.41) in the following form, suitable for later comparison: ! N N0 Y Y Jþ1 ðxi Þ J1 ðy j Þ j 0ðbÞi h0ðbÞ j T j¼1

i¼1

¼

Z Y Z Y ~ m X m X dmþm;n  2 ~ ~ m ð1Þ d z‘ d2~zk ~ m! m! ~ m;m n¼0 ‘¼1 fkl ;~ k¼1 kk g QN 0 N m YY 0 i0 >i ½Xðxi  xi ; bÞ j0 >j ½Xðy j  y j0 ; bÞ ½Xðxi  z‘ ; bÞk‘ QN QN 0 i¼1 ‘¼1 i¼1 j¼1 ½Xðxi  y j ; bÞ

1 N F ðbÞ QN 

1 X

ðiMÞ

n

0

~ N Y m Y

N Y m ib ~ Y  ½Xðxi  ~zk  Þkk ½Xðy j  z‘ ; bÞk‘ 2 j¼1 ‘¼1 i¼1 k¼1 ~kk Y ~  ~ N0 Y m m m Y Y ib kk e zk  ~zk0 ; bÞ~kk ~kk0  Xðy j  ~zk  Þ ½Xðz‘  z‘0 ; bÞ ‘ ‘0 ½ Xð~ 2 j¼1 k¼1 k 0 >k ‘0 >‘   ~ k k ~ m Y m ‘ k Y ib  Xðz‘  ~zk  Þ ; 2 k ‘

where

ð2:42Þ



   p þ þ p   0 0 0 0 Xðxi xj ;bÞ¼ð2ibÞ sinh ðxi xj ieðxi xj ÞÞ sinh ðxi xj ieðxi xj ÞÞ ; ð2:43Þ b b     p þ þ p   2 0 0 0 0 e Xðxi xj ;bÞ¼ð2ibÞ sinh ðxi xj þieðxi xj ÞÞ sinh ðxi xj þieðxi xj ÞÞ ; ð2:44Þ b b         ib p þ þ ib p   ib ¼ð2ibÞ2 sinh xi xj  sinh xi xj  ; ð2:45Þ X xi xj  2 b 2 b 2 2

and the sum over the chiralities k‘ and ~ kk respects the total chirality condition X X ~ N  N0 þ kk ¼ 0: k‘  ‘

ð2:46Þ

k

This concludes the computation of the N-point function of the chiral densities in the fermionic version. In the next Section we shall consider the computation of these N-point function within the bosonized version of the theory. 3. Thermofield Bosonization point of view The next step is to compute the N-point function of the chiral densities of the massive free fermion theory from the thermofield bosonization point of view. Since at T ¼ 0 the bosonized theory describing both the massive free fermion theory and the massive Thirring model is the sine-Gordon theory [13,14] with distinct values of the sine-Gordon parameter j (different values of the scale dimension of the mass operator), we shall consider the perturbative the perturbative computation of the N-point function of chiral densities of the bosonized massive Thirring model from thermofield dynamics point of view.

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

2671

e whose To begin with, let us consider the sine-Gordon theory for the doublet (U; U), Lagrangian can be decomposed as b I ðxÞ; b b ð0Þ ðxÞ þ L LðxÞ ¼L

ð3:1Þ

with le b ð0Þ ðxÞ ¼ Lð0Þ ðxÞ  L e ð0Þ ðxÞ ¼ 1 : ol UðxÞol UðxÞ :  1 : ol UðxÞo e L UðxÞ :; 2 2 l b I ðxÞ ¼ M e ð: cosðjUðxÞÞ :  : cosðj UðxÞÞ :Þ; L p

ð3:2Þ ð3:3Þ

where l is the infrared regulator (IR) reminiscent of the free massless scalar theory. The scale dimension D of the mass operator is given in terms of the Thirring coupling parameter g as [14] D¼

j2 : 4p

ð3:4Þ

For the values j2 ¼ 4p, the sine-Gordon theory describes the massive free fermion theory discussed in the preceding section. Now, let us introduce the Mandelstam representation [13,15] for the Fermi field operator R1  l 12 p 5 0 1 1 ijc5 UðxÞþi2p j x1 o0 Uðx ;z Þdz WðxÞ ¼ ei4c : e 2 :; ð3:5Þ 2p R1  l 12 p 5 e 0 1 1 ijc5 e U ðxÞþi2p j x1 o0 U ðx ;z Þdz e i WðxÞ ¼ ei4c : e 2 :: ð3:6Þ 2p e is not obtained from W by the ‘‘tilde” conjugation Notice that the field operator W e should be an identical copy of W carrying operation. This follows from the fact that W the same charge and chirality quantum numbers. This is explained in detail in Section 4 and in the Appendix. The bosonized chiral densities J1 ðxÞ are given by l Jþ1 ðxÞ ¼  WðxÞ; ð3:7Þ 2p  l ð3:8Þ W ðxÞ; J1 ðxÞ ¼  2p where the Wick-ordered exponentials carrying opposite chirality are WðxÞ ¼: eijUðxÞ : : 

W ðxÞ ¼: e

ijUðxÞ

ð3:9Þ ::

ð3:10Þ

The interaction picture vacuum is now given by j 0; bi ¼ U B ½hB  j ~ 0; 0i; where the unitary operator taking one to the thermofields is given by (j p1 j¼ p) R þ1 1 1 1 y 1 y 1  dp ð~ aðp Þaðp Þa ðp Þ~ a ðp ÞÞhB ðp;bÞ U B ½hB  ¼ e 1 ;

ð3:11Þ

ð3:12Þ

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R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

and the Bogoliubov parameter hB ðp; bÞ is implicitly defined by bp

e 2 sinh hB ðp; bÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1  ebp 1 cosh hB ðp; bÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1  ebp

ð3:13Þ ð3:14Þ

with the Bose–Einstein statistical weight given by N B ðp; bÞ ¼ sinh2 hB ðp; bÞ ¼

1 : ebp  1

ð3:15Þ

Following the same procedure of the preceding section, in the bosonized theory the N-point function of the chiral densities is given by ! N N0 Y Y h0ðbÞ j T Jþ1 ðxi Þ J1 ðy j Þ j 0ðbÞi j¼1

i¼1

h0; ~ 0jT ¼

N Q i¼1

J þ1 ðxi ; bÞ

N0 Q

! n R o 2 b J 1 ðy j ; bÞ exp i L I ðz; bÞd z j ~0; 0i

j¼1

n R o b I ðz; bÞd2 z j ~0; 0i h0; ~ 0 j T exp i L

;

ð3:16Þ

where the interaction Lagrangian (in the interaction picture) at finite temperature is   b I ðzÞU 1 ½hB  ¼ M l e2Dzðb;l0 Þ b I ðz; bÞ ¼ U B ½hB  L L B p ~ bÞÞ :Þ;  ð: cosðj/ðz; bÞÞ :  : cosðj/ðz; ð3:17Þ J 1 ðx; bÞ are the thermal chiral densities in the interaction picture [2] l  0 e2Dzðb;l Þ W ðxi ; b; kxi Þ; J þ1 ðxi ; bÞ ¼ U B ½hB J þ1 ðxi ÞU 1 B ½hB  ¼  2p l  0 1 e2Dzðb;l Þ W  ðy j ; b; ky j Þ; J 1 ðy j ; bÞ ¼ U B ½hB J 1 ðy j ÞU B ½hB  ¼  2p

ð3:18Þ ð3:19Þ

with the thermal Wick-ordered exponentials carrying opposite chirality given by (in our convention kxi ¼ ky j ¼ 1) W ðxi ; b; kxi Þ ¼: eijkxi /ðxi ;bÞ :; W  ðy j ; b; ky j Þ ¼: e

ijky j /ðy j ;bÞ

ð3:20Þ :;

ð3:21Þ

/ðx; bÞ is a free massless pseudo-scalar thermofield [2] and zðb; l0 Þ is the infrared divergent integral [2] Z 1 dp : ð3:22Þ zðb; l0 Þ ¼ bp  1Þ pðe 0 l Expanding the exponential in powers of M, introducing the tilde Wick-ordered exponential ~ e  ðz; b; kz Þ ¼: eijkz /ðz;bÞ W :;

ð3:23Þ

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

2673

~ bÞ commute [2], we get and using the fact that the fields /ðx; bÞ and /ðy;

ei

R

d2 zb L ðzÞ

l 2Dz pe

iM ð

nR o R 2 2 ~ Þ d z:cos j/ðz;bÞ: d ~z:cos j/ð~z;bÞ:

¼e Z n Z n 1 X ðiMÞ l 2Dz n 2 2 ~ e d z : cos j/ðz; bÞ :  d ~z : cos j/ð~z; bÞ : ¼ ðn!Þ p n¼0 Z m n 1 X ðiMÞ l 2Dz n X ðn!Þdmþm;n ~ ~ m e d2 z : cos j/ðz; bÞ : ð1Þ ¼ ~ ðn!Þ p m!m! m;m ~ n¼0 Z m~ X 1  n X d ~ mþm;n n l 2Dz m ~ ~ z; bÞ : e ð1Þ  d2~z : cos j/ð~ ¼ ðiMÞ ~ p m! m! ~ m;m n¼0 Z Y Z Y ~ ~ m m m m Y Y ~ zk ; bÞ :  d2 z‘ d2~zk : cos j/ðz‘ ; bÞ : : cos j/ð~ k¼1

‘¼1

k¼1

‘¼1

Z Y Z Y m m ~ n X d ~ mþm;n n m ~ e2Dz ð1Þ ¼ ðiMÞ d 2 z‘ d2~zk ~ 2p m! m! ~ m;m n¼0 ‘¼1 k¼1 l

1 X



m X X Y

W ðz‘ ; b; kj Þ

fk‘ gm f~ kk gm~ ‘¼1

~ m Y

f ð~zk ; b; ~kk Þ; W

ð3:24Þ

k¼1

P P where k‘ ; ~ kk ¼ 1, and runs over all possibilities in the set fk‘ gm ð f~ kk gm~ Þ fk1 ; . . . ; km gðf~ k1 ; . . . ; ~ km~ gÞ. Denoting by N B ðbÞ the normalization factor, the Green’s function (3.16) can be written as h0ðbÞ j T

Y N

Jþ1 ðxi Þ

1 N B ðbÞ



Z Y ~ m

1 X

0

ð1ÞðNþN Þ ðiMÞn

l

n¼0

d2~zk 

2p

0

e2Dzðb;l Þ

ðnþN þN 0 Þ X d

~ mþm;n

~ m;m

~ m!m!

~

ð1Þm

Z Y m

d2 z‘

‘¼1

XX fk‘ gm f~ kk gm~

k¼1

h0; ~ 0jT

 J1 ðy j Þ j 0ðbÞi

j¼1

i¼1

¼

N0 Y

N Y i¼1

W ðxi ;b;kxi Þ

N0 Y



W ðy j ;b;ky j Þ

j¼1

m Y

W ðzl ;b;kl Þ

~ m Y

  ~ f W ð~zk ;b; kk Þ j ~0;0i:

k¼1

l¼1

ð3:25Þ

The time-ordered product of two Wick-ordered exponentials is defined by, T ðW ðx;b;kx ÞW ðy;b;ky ÞÞ ¼: W ðx;b;kx ÞW ðy;b;ky Þ : ðhW ðx;b;kx ÞW ðy;b;ky Þihðx0  y 0 Þ þ hW ðy;b;ky ÞW ðx;b;kx Þihðy 0  x0 ÞÞ : W ðx;b;kx ÞW ðy;b;ky Þ : hT ðW ðx;b;kx ÞW ðy;b;ky ÞÞi;

ð3:26Þ

where 2 h0;~ 0j/ðx;bÞ/ðy;bÞj~ 0;0i

hW ðx; b; kx ÞW ðy; b; ky Þi ¼ ekx ky j

:

ð3:27Þ

2674

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

Using the identity (/ðxi Þ ¼ /i , hðx0i  y 0j Þ ¼ hij ) eh/i /j i hij þ eh/j /i i hji eh/i /j ihij þh/j /i ihji ¼ ehT /i /j i ;

ð3:28Þ

we can write 2 hT /ðx;bÞ/ðy;bÞi

T ðW ðx; b; kx ÞW ðy; b; ky ÞÞ ¼: W ðx; b; kx ÞW ðy; b; ky Þ : ekx ky j

:

ð3:29Þ

The Wick’s theorem can be extended to generalized time-ordered product of Wickordered exponentials and we obtain hT

Y ‘

W ðxj ; b; kj Þ

 e ð~xk ; b; ~ W kk Þ i

k¼1

j¼1

¼

‘~ Y

‘ Y

hT ðW ðxj ; b; kj ÞW ðxj0 ; b; kj0 ÞÞi

j>j0



‘~ Y

e ð~xk ; b; ~kk Þ W e ð~xk0 ; b; ~kk0 ÞÞi hT ð W

k>k 0

‘;‘~ Y e ð~xk ; b; ~ hT ðW ðxj ; b; kj Þ W kk ÞÞi:

ð3:30Þ

j;k

In this way one gets for the Fock vacuum expectation value of T-ordered product of Wick exponentials in (3.25), ! ~ N N0 m m Y Y Y Y f ð~zk ;b; ~kk Þ j ~0;0i W h0; ~ 0jT W ðxi ;b;ki Þ W  ðy ;b; kj Þ W ðz‘ ; b;k‘ Þ j

j¼1

i¼1

¼

N0 Y

i0 >i

j0 >j

hT ðW ðxi ; b;ki ÞW ðxi0 ;b;ki0 ÞÞi



N Y N0 Y

k¼1

‘¼1

N Y

hT ðW  ðy j ;b; kj ÞW  ðy j0 ; b;kj0 ÞÞi

hT ðW ðxi ;b; ki ÞW  ðy j ;b; kj ÞÞi

i¼1 j¼1



N Y m Y

hT ðW ðxi ;b; ki ÞW ðz‘ ;b; k‘ ÞÞi

i¼1 ‘¼1



N0 Y m Y

~ N Y m Y e  ð~zk ;b; ~kk ÞÞi hT ðW ðxi ;b;ki Þ W i¼1 k¼1

~ N0 Y m Y f ð~zk ;b; ~kk ÞÞi hT ðW ðy j ;b;kj ÞW ðz‘ ;b; k‘ ÞÞi hT ðW  ðy j ; b;kj Þ W 

j¼1 ‘¼1



m Y

hT ðW ðz‘ ;b; k‘ ÞW ðz‘0 ;b; k‘0 ÞÞi

0

k >k

m Y m ~ Y ‘

f ð~zk ;b; ~kk Þ W f ð~zk0 ;b; ~kk0 ÞÞi hT ð W

0

‘ >‘



j¼1 k¼1 ~ m Y

f ð~zk ;b; ~ hT ðW ðz‘ ;b;k‘ Þ W kk ÞÞi:

ð3:31Þ

k

The propagators of the scalar thermofields can be written in terms of the propagators of the Fermi thermofields as follows [2]

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

1 1 l 1 h0; ~ 0 j T /ðx;bÞ/ðy; bÞ j ~ 0;0i ¼ zðl0 ;bÞ  ln  ln Xðx  y;bÞ; p 2p 2p 4p  i 1 1 l 1 ~ ~ bÞ j ~ e  y;bÞ h0; ~ 0 j T /ðx;bÞ /ðy; 0;0i ¼  þ zðl0 ;bÞ  ln  ln Xðx 2 p 2p 2p 4p   1 1 1 ib ~ bÞ j ~ h0; ~ 0 j T /ðx;bÞ/ðy; 0;0i ¼  f ðl00 ;bÞ  lnð2bÞ þ ln X x  y  ; 2p 2p 4p 2

2675

ð3:32Þ ð3:33Þ ð3:34Þ

where the X’s are defined in (2.43)–(2.45). The dependence on the infrared cut-offs l0 and l00 are given by [2] Z 1 dp bp 1 0 ðe  1Þ ; zðl ; bÞ ¼ ð3:35Þ p 0 l Z 1 dp 1 00 ; f ðl ; bÞ ¼ ð3:36Þ p 00 sinh bp2 l and the corresponding asymptotic behavior are zðl0 0; bÞ !

1 1 þ lnðbl0 Þ; bl0 2

f ðl00 0; bÞ !

ð3:37Þ

2 : bl00

ð3:38Þ

The N-point function is then given by (kxi ¼ ky j ¼ 1) ! N N0 1 Y Y 0 1 X Jþ1 ðxi Þ J1 ðy j Þ j 0ðbÞi ¼ ð1ÞðN þN Þ ðiMÞn h0ðbÞ j T N B ðbÞ n¼0 j¼1 i¼1 

X dmþm;n ~ ~ m;m



~ m!m!

Z Y m

2

d z‘

ð1Þ



X X fk‘ gm f~ kk gm~

Z Y m ~

QN

2

d ~zk

Dki k‘

½Xðxi  z‘ ; bÞ

i¼1 ‘¼1

~ ~ F ðm;mÞ ðl; l0 ; l00 ; bÞGðm;mÞ sðl; l0 ; bÞ

Q 0  xi0 ; bÞDki ki0 Nj0 >j ½Xðy j  y j0 ; bÞDkj kj0 QN QN 0 Dki kj j¼1 ½Xðxi  y j ; bÞ i¼1

i0 >i ½Xðxi

k¼1

‘¼1

N Y m Y

m ~

Dki ~kk Y N Y m ~   N0 Y m Y ib Dk k X xi ~zk  ½Xðy j  z‘ ;bÞ j ‘ 2 j¼1 ‘¼1 i¼1 k¼1

Dkj ~kk Qm ~ Dk‘ k‘0 Qm N0 Y m ~   e zk ~zk0 ;bÞD~kk ~kk0 Y 0 ib ‘0 >‘ ½Xðz‘  z‘ ;bÞ k 0 >k ½ Xð~  X y j ~zk  K;

Dk‘ ~kk Qm Qm~  2 j¼1 k¼1 X z ~z  ib ‘

k

‘

k

2

ð3:39Þ where the phase

2ipD

K¼e

m ~ P ~ ~

kk kk 0

k 0 >k

;

ð3:40Þ

2676

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

is the identity for integer values of the scale dimension D and the cut-off dependence is given by lð1DÞðN þN 0 þnÞ h l iD 0 ~ e2zðl ;bÞ Gðm;mÞ ðl; l0 ; bÞ ¼ p 2p



N N 0 þ

m P

k‘ 

2 ~ m P ~ kk

k¼1

‘¼1

ð3:41Þ

;

m~ m "  # P ~kk N N 0 þP k‘ 2D ‘¼1 k¼1 bl 00 0 ~ F ðm;mÞ ðl; l0 ; l00 ; bÞ ¼ e2Dðf ðl ;bÞ2zðl ;bÞÞ p

m~ m   # P ~kk N N 0 þP k‘ "  2D 4D 1 1 ‘¼1 k¼1 l 00  0 : ¼ eb l l 0 pl

ð3:42Þ

In arriving at these expressions we collected the cut-off dependent terms, noting that N X

ki k þ i0

N0 X

i0 >i

¼

N N0 N m N0 m m m ~ X X X X X X X X ~kk ~kk0 kj k  ki kj þ ki k‘  kj k‘ þ k‘ k‘0 þ j0

j0 >j

i

m m ~ X X 1 ~ N  N0 þ kk k‘  2 ‘¼1 k¼1

j

!2

i

‘

j

‘

‘0 >‘

!

k 0 >k

m ~ m X X 1 ~kk N  N 0 þ  ð N þ N 0 þ nÞ þ k‘ ; 2 k ‘

ð3:43Þ

with (ki ¼ kj ¼ 1) N X

ki ¼ N ¼

N X

i¼1 N0 X j¼1

2

ðki Þ ;

ð3:44Þ

i¼1 0

kj ¼ N ¼

N0 X

2

ðkj Þ ;

ð3:45Þ

j¼1

and made use in (3.42) of the asymptotic behavior (3.37) and (3.38). Now, let us consider the free massive fermion theory (D ¼ 1), which is an infrared cutoff independent scale non-invariant theory. Following the procedure introduced in Ref. [2] (see Appendix), the theory of the free ~ bÞ can be constructed as the zero mass limit massless scalar thermofields /ðx; bÞ and /ðx; e bÞ. In this way, the infrared reguof the massive free scalar thermofields Rðx; bÞ and Rðx; lator l of the zero temperature two-point function should be identified with the infrared cut-offs l0 and l00 of the temperature-dependent contributions zðl0 ; bÞ and f ðl00 ; bÞ, l ð3:46Þ l00 ¼ l0 ¼ ; p such that ~ F ðm;mÞ ðl; l0 ; l00 ; bÞjl00 ¼l0 ¼lp ¼ 1:

ð3:47Þ

In this way, the only non zero contributions in the expansion (3.39) are those which satisfy the selection rule

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

N  N0 þ

m X ‘¼1

k‘ 

~ m X

~ kk ¼ 0:

2677

ð3:48Þ

k¼1

The selection rule (3.48) is in agreement with our previous computations (Eq. (2.39)) for the massive free fermionic theory (D ¼ 1). Indeed, for D ¼ 1 the bosonized version of the perturbative expansion given by (3.39) coincide with our previous result in the fermionic n formulation obtained in Section 2. In order to recover the factor ð1Þ that appears in (2.42), P we shall make use of the selection rule (3.48). Taking into account the summations P fk‘ gm f~ kk gm~ in (3.39), one can write 0

0

0

ð1ÞðN þN Þ ðiMÞn ¼ ð1ÞðN þN þnÞ ðiMÞn ð1ÞðN N þnÞ ðiMÞn   ~ m m P P 0 ~ N N þ

ð1Þ

 N N 0 þ

ð1Þ

k‘ þ

m P

kk

k¼1

‘¼1

k‘ 

‘¼1

m ~ P ~

kk

k¼1



ðiMÞ

n

n

n

ðiMÞ ¼ ðiMÞ :

ð3:49Þ

In this way, for D ¼ 1, the bosonized N-point function (3.39) can be written as h0ðbÞ j T

N Y i¼1

Jþ1 ðxi Þ

N0 Y

! J1 ðy j Þ j 0ðbÞi

j¼1

Z Y ~ 1 m m X dmþm;n X XZ Y 1 X ~ ~ n m ð1Þ ðiMÞ d 2 z‘ d2~zk ~ N B ðbÞ n¼0 m! m! m;m ~ ‘¼1 k¼1 fk‘ gm f~ kk gm~ QN 0 QN 0 i0 >i ½Xðxi  xi ;bÞ j0 >j ½Xðy j  y j0 ;bÞ  Q N QN 0 i¼1 j¼1 ½Xðxi  y j ;bÞ ~kk N m N Y m ~   YY Y ib k  ½Xðxi  z‘ ;bÞ ‘ X xi ~zk  2 i¼1 ‘¼1 i¼1 k¼1   ~kk 0 0 N Y m N Y m ~ Y Y ib k‘  ½Xðy j  z‘ ;bÞ X y j ~zk  2 j¼1 ‘¼1 j¼1 k¼1 k‘ ~kk ~ ~   m m Y m m Y Y Y ib kk e zk ~zk0 ;bÞ~kk ~kk0  ½Xðz‘  z‘0 ;bÞ ‘ ‘0 ½ Xð~ X z‘ ~zk  ; 2 k ‘ k 0 >k ‘0 >‘

¼

ð3:50Þ

in agreement with the corresponding function (2.42) obtained in the original fermionic version of the theory. This proves the formal equivalence of the massive free fermion thermofield theory with the sine-Gordon thermofield model for the particular value of the sine-Gordon parameter j2 ¼ 4p. 4. Physical meaning of the selection rule In this section we shall discuss the physical interpretation of the bosonized expression e and its connection with the interpretation of the selection rule (3.48). (3.6) for the field W

2678

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

To begin   with, let us consider the massless free fermionic theory described by the doublet pffiffiffi w ~ y . The corresponding bosonized expressions are given by (3.5),(3.6) with j ¼ 2 p, iw and can be written as  l 12 pffiffi 5 : ei pfc /ðxÞþuðxÞg :; ð4:1Þ wðxÞ ¼ 2p  l 12 pffiffi 5 ~ ~ iwðxÞ ¼ : ei pfc /ðxÞþeu ðxÞg :; ð4:2Þ 2p ~ is described by the total Lagrangian with lm om u ¼ ol /. The bosonic doublet ð/; /Þ ~ 2: b ¼ 1 ðol /Þ2  1 ðol /Þ L 2 2

ð4:3Þ

~ is The physical interpretation for the fact that the bosonized expression (4.2) for the field w not obtained from the corresponding bosonized expression (4.1) for the field w by the ‘‘tilde conjugation operation” is the following: In the Thermofield Dynamics formalism, the fictitious ‘‘tilde” system should be an identical copy of the system under consideration, ~ should be an identical copy of w, i. e., carrying the same which implies that the field w charge and chirality quantum numbers. At T ¼ 0, we obtain the following equal-time commutation relations for the free scalar fields2 ½/ðxÞ; o0 /ðyÞ ¼ idðx1  y 1 Þ; ~ ~ ¼ idðx1  y 1 Þ: ½/ðxÞ; o0 /ðyÞ

ð4:4Þ ð4:5Þ

Within the Thermofield Dynamics approach the above algebraic relations are retained at finite temperature. The computation of the bosonized expression for the fermionic cur~ lw ~ of the free massless theory were performed in Ref. [2]. e l ¼ wc rents J l ¼ wcl w and J Using the bosonized expressions for the free massless fermion fields (4.1), (4.2), one finds 1 J l ðx; bÞ ¼  pffiffiffi ol uðx; bÞ; p 1 e l ðx; bÞ ¼ þ pffiffiffi ol u ~ ðx; bÞ: J p 2

The corresponding canonical momenta are PðxÞ ¼ @ 0 /ðxÞ; ~ e PðxÞ ¼ @ 0 /ðxÞ;

in such a way that the canonical equal-time commutation relations are given by ½/ðxÞ; PðyÞ ¼ idðx1  y 1 Þ; ~ e ½/ðxÞ; PðyÞ ¼ idðx1  y 1 Þ: The dynamical equations are b ; @ 0 / ¼ i½/ðxÞ; H  ¼ i½/ðxÞ; H ~ ¼ i½/ðxÞ; ~ ~ e  ¼ i½/ðxÞ; b ; H H @0/ b ¼HH e is the generator of time evolution of the combined system. where the total Hamiltonian H

ð4:6Þ ð4:7Þ

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

2679

The axial-vector currents are (c5 cl ¼ lm cm ) 1 J 5l ðx; bÞ ¼  pffiffiffi ol /ðx; bÞ; p ~ bÞ: e 5 ðx; bÞ ¼ þ p1ffiffiffi ol /ðx; J l p

ð4:8Þ ð4:9Þ

Introducing the corresponding charges Z þ1 Z þ1 e e J 0 ðz; bÞdz1 ; QðbÞ ¼ J 0 ðz; bÞdz1 ; QðbÞ ¼ 1 1 Z þ1 Z þ1 5 5 1 5 e ðbÞ ¼ e J 50 ðz; bÞdz1 ; J 0 ðz; bÞdz ; Q Q ðbÞ ¼ 1

ð4:10Þ ð4:11Þ

1

and using (4.4),(4.5) we get ½QðbÞ; wðx; bÞ ¼ wðx; bÞ; 5

ð4:12Þ

5

½Q ðbÞ; wðx; bÞ ¼ c wðx; bÞ; ~ bÞ ¼ wðx; ~ bÞ; e ½ QðbÞ; wðx; e5

ð4:13Þ ð4:14Þ

5~

~ bÞ ¼ c wðx; bÞ; ½ Q ðbÞ; wðx;

ð4:15Þ

~ carry the same charge and chirality quantum numbers. The total implying that w and w charge operators corresponding to the fermionic doublet of the combined system are given by b e QðbÞ ¼ QðbÞ  QðbÞ; b 5 ðbÞ ¼ Q5 ðbÞ  Q e 5 ðbÞ: Q

ð4:16Þ ð4:17Þ

For the fermionic charge of the combined system we obtain the following selection rules n Y

b ½ QðbÞ;

wðxi ; bÞ

i¼1

~ ; bÞ ¼ ðn  ~ wðy nÞ j

wðxi ; bÞ

i¼1

~ n Y

n Y

wðxi ; bÞ

i¼1

j¼1

n Y

b ½ QðbÞ;

~ n Y

~ y ðy ; bÞ ¼ ðn þ ~ w nÞ j

n Y

~ n Y

wðxi ; bÞ

i¼1

j¼1

~ ; bÞ; wðy j

ð4:18Þ

j¼1 ~ n Y

~ y ðy ; bÞ: w j

ð4:19Þ

j¼1

This implies that in the massless free fermionic theory one finds the following off-diagonal 2n-point functions h0; ~ 0j

n Y

wðx i ; bÞ

i¼1

 ¼

n Y

~  ; bÞ j ~ wðy 0; 0i j

j¼1

n Q

ð2ibÞ sinh pb ðx i0 0 i
n Q i;j



x i Þ p

"

n Q

j0
# ð2ibÞ sinh pb ðy  j0

 ð2ibÞ sinh b x i  yj 

i b2

 i

#



y j Þ ;

ð4:20Þ

2680

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

and h0; ~ 0j

n Y

wðxi ; bÞ

i¼1

n Y

~ y ðy ; bÞ j ~ w 0; 0i ¼ 0: j

ð4:21Þ

j¼1

For the axial charge we obtain the selection rules (here J ¼ wy1 w2 ¼: e2i pffiffi ~ y~ 2i p/ ~ e J ¼ w1 w2 ¼: e :) b 5 ðbÞ; ½Q

n Y

J ðxi ; bÞ

i¼1

b 5 ðbÞ; ½Q

n Y

~ n Y

e J ðy j ; bÞ ¼ 2ðn  ~ nÞ

i¼1

~ n Y

J ðxi ; bÞ

i¼1

j¼1

J ðxi ; bÞ

n Y

e J y ðy j ; bÞ ¼ 2ðn þ ~ nÞ

j¼1

n Y i¼1

~ n Y

e J ðy j ; bÞ;

pffiffi p/

:;

ð4:22Þ

j¼1

J ðxi ; bÞ

~ n Y

e J y ðy j ; bÞ:

ð4:23Þ

j¼1

This provides a clear understanding of the physical meaning of the selection rule (3.48), that is, the only non zero contributions in the expansion (3.39) are those with zero total b 5 ðbÞ chirality Q Xm Xm~ ~ kk Þ ¼ 0 k ðN  N 0 þ ‘Þ  ð ‘¼1 k¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflffl ffl} Q5 e Q5

ð4:24Þ

of the combined system. 5. Concluding remarks The main objective of this paper was to prove in thermofield dynamics the equivalence of the theory of massive free Fermi fields to the the sine-Gordon theory for a particular value of the sine-Gordon parameter j2 ¼ 4p. Approaches of other authors [10,12] differ from ours in two respects: (i) they make use of the imaginary time formalism, and ii) treat the fermionic side in a hybrid way. We have treated the fermionic side of the problem strictly from the fermionic point of view. On the bosonic side we have used the thermofield bosonization [2] in order to compute the n-point functions of the thermal chiral densities. On the bosonic side we first obtained the n-point functions of the chiral densities from a generalized Mandelstam operator (with non-canonical scale dimension) for the corresponding bosonized expressions at finite temperature, and then recovered from there the corresponding n-point functions of the free theory as a limiting case. A gratifying byproduct of our analysis was the observation, that these n-point functions showed in a natural way, that the tilde fields of the thermofield dynamics could be regarded as living on the lower branch of the integration contour in the complex time plane, displaced from the real time axis by i b2, in accordance with the work of ref. [7]. We recognize that the ‘‘revised version” of thermofield dynamics formulation for fermions introduced in Ref. [8] enables to obtain the correct bosonized expression for the field ~ and becomes crucial in order to establish the two-dimensional Fermion-Boson mapping w within the thermofield dynamics approach, as well as, to obtain a clear understanding of the chiral selection rule of the combined system.

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Acknowledgment We are grateful to Brazilian Research Council (CNPq) for partial financial support and to the FAPERJ(E-26/170.949/2005)-DAAD scientific exchange program which make this collaboration possible. Appendix. Thermofield Bosonization of the free massless fermion field revised In this Appendix we shall consider the revised thermofield dynamics approach for fermions presented in Ref. [8] in order to correct a mistake in our previous paper [2] and give the revised thermofield bosonization prescription for the free massless Fermi field doublet ~ y Þ. Although in the fermionic formulation the use of the new version does not ðw; i w change the computation of the diagonal two-point functions of the free massless fields ~  ; bÞw ~ y ðy  ; bÞ j ~0; 0i, it corrects a ‘‘sign” in h0; ~ 0 j wðx ; bÞwy ðy  ; bÞ j ~ 0; 0i and h0; ~ 0 j wðx ~  ; bÞ our previous computation [2] of the off-diagonal contribution h0; ~0 j iwðx  ~ wðy ; bÞ j 0; 0i, and as we shall see, gives a new insight into the thermofield bosonization scheme. In the ‘‘old version”, the operator taking one to the Fermion thermofields is given by [3–5,8]  Z 1  ~ 1 ÞdðpÞ  d y ðp1 Þd~y ðp1 ÞÞ ; U F ðhF Þ ¼ exp  dphF ðj p1 j; bÞð~ bðp1 Þbðp1 Þ  by ðp1 Þ~ by ðp1 Þ þ dðp 1

ðA:1Þ

and the corresponding transformed annihilation operators are given by ~y ðpÞ sin hF ðp; bÞ; bðp; bÞ ¼ bðpÞ cos hF ðp; bÞ  b ~ bÞ ¼ ~ bðp; bðpÞ cos hF ðp; bÞ þ by ðpÞ sin hF ðp; bÞ;

ðA:2Þ ðA:3Þ

~ As stressed in Ref. [8], the requirement for (A.2) and with similar expressions for d and d. (A.3) to be consistent with each other implies the following ‘‘tilde substitution rule” for fermions (b bðpÞ) e ~ b ¼ b:

ðA:4Þ

In the ‘‘new version” [8], the vacuum state j 0ðbÞi is obtained from the Fock vacuum j~ 0; 0i by the ‘‘modified” unitary operator U F ðhF Þ, which can be formally obtained from the old one (A.1) by the substitution ~ b ! i~ b; y ~ ! i~ b by ;

ðA:5Þ ðA:6Þ

~ i.e., with similar substitution for d,  Z U F ðhF Þ ¼ exp i

1

 ~ 1 ÞdðpÞ þ d y ðp1 Þd~y ðp1 ÞÞ : dphF ðj p1 j;bÞð~ bðp1 Þbðp1 Þ þ by ðp1 Þ~by ðp1 Þ þ dðp

1

ðA:7Þ

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The revised fermionic transformations are given by [8] by ðpÞ sin hF ðp; bÞ; bðp; bÞ ¼ bðpÞ cos hF ðp; bÞ þ i~ ~ bðp; bÞ ¼ ~ bðpÞ cos hF ðp; bÞ  iby ðpÞ sin hF ðp; bÞ;

ðA:8Þ ðA:9Þ

~ This procedure replace the ‘‘tilde substitution rule” with similar expressions for d and d. (A.4) by the ‘‘tilde conjugation operation” g ¼ i~ ðibÞ b;

ðA:10Þ

analogous to the case of boson operators. Taking this into account, the revised expression for the two-dimensional free massless Fermi thermofields are given by Z 1 1 dpffp ðx ÞðbðpÞ cos hF ðp; bÞ þ i~by ðpÞ sin hF ðp; bÞÞ wðx ; bÞ ¼ pffiffiffiffiffiffi 2p 0 ~ þ fp ðx Þðd y ðpÞ cos hF ðp; bÞ  idðpÞ sin hF ðp; bÞÞg; ðA:11Þ Z 1 ~  ; bÞ ¼ p1ffiffiffiffiffiffi wðx dpffp ðx Þð~ bðpÞ cos hF ðp; bÞ  iby ðpÞ sin hF ðp; bÞÞ 2p 0 þ fp ðx Þðd~y ðpÞ cos hF ðp; bÞ þ idðpÞ sin hF ðp; bÞÞg; ðA:12Þ where fp ðxÞ ¼ eipx :

ðA:13Þ

Now, let us compute the off-diagonal two-point function within the fermionic version. Using (A.11) and (A.12), one gets3 Z 1 1 cos pðx  y  Þ ~  ; bÞwðy  ; bÞ j ~ h0; ~ 0 j iwðx 0; 0i ¼  dp 2p 0 cosh bp2 1 ; ðA:14Þ ¼  2b cosh pb ðx  y  Þ which can be written as ~  ; bÞwðy  ; bÞ j ~ h0; ~ 0 j iwðx 0; 0i ¼ 

1 2ib sinh pb ðx

 y   i b2Þ

:

ðA:15Þ

In order to establish the Fermion–Boson mapping, let us compute the two-point function above from the bosonized point of view. To this end, we define  l 12 pffiffi 0  wðx ; bÞ ¼ ezðl ;bÞ : e2i p/ðx ;bÞ :; ðA:16Þ 2p  l 12 pffiffi ~  0 ~  ; bÞ ¼ ðiÞ wðx ezðl ;bÞ K b : e2ic p/ðx ;bÞ :; ðA:17Þ 2p 3 The Eq. (4.23) of Ref. [2] is correct, but it cannot be obtained from (4.14) and (4.13) (where we have used the old version with (A.2) and (A.3)), which gives Z 1 bp i sin pðx  y  ÞN F ðb; pÞe 2 dp; p 0

instead of cos pðx  y  Þ in the integrand.

R.L.P.G. Amaral et al. / Annals of Physics 323 (2008) 2662–2684

2683

where K b is a Klein factor [16] which ensures normal anticommutativity between tilde fermion thermofield components and non-tilde ones [2], and the value of c ¼ 1 will be fixed at the end of the calculation. Using that [2] ~  ; bÞ j ~ h0; ~ 0 j /ðx ; bÞ/ðy 0; 0i ¼

   1 1 p f ðl00 ; bÞ þ ln cosh ðx  y  Þ ; 4p 4p b

ðA:18Þ

we obtain (the global minus sign arises from the Klein factor)  c l 2zðl0 ;bÞþcf ðl00 ;bÞ p     ~ ~ ~ h0; 0 j iwðx ; bÞwðy ; bÞ j 0; 0i ¼  e cosh ðx  y Þ : 2p b

ðA:19Þ

Using the asymptotic behavior 1 1 þ lnðbl0 Þ; bl0 2 2 f ðl00 0; bÞ ! 00 ; bl zðl0 0; bÞ !

ðA:20Þ ðA:21Þ

we get

 2 c 1  l 1 00  0 ~ bÞwðy; bÞ j ~ h ic : h0; ~ 0 j iwðx; 0; 0i ¼  eb l l pl0 p 2b cosh ðx  y  Þ 

ðA:22Þ

b

In order to recover the same space–time dependence as in (A.14) we must require that c ¼ 1:

ðA:23Þ

In order to obtain an infrared cut-off independent two-point function, the free massless scalar thermofield theory should be considered as the zero mass limit of the massive free scalar thermofield h0; ~ 0 j Rðx; bÞRðy; bÞ j ~ 0; 0im!0 ! h0; ~ 0 j /ðx; bÞ/ðy; bÞ j ~0; 0i;

ðA:24Þ

~ bÞ/ðy; ~ bÞ j ~0; 0i; e bÞ Rðy; e bÞ j ~ h0; ~ 0 j Rðx; 0; 0im!0 ! h0; ~ 0 j /ðx;

ðA:25Þ

~ bÞ j ~0; 0i: e bÞ j ~ h0; ~ 0 j Rðx; bÞ Rðy; 0; 0im!0 ! h0; ~ 0 j /ðx; bÞ/ðy;

ðA:26Þ

In this way, the infrared regulator l of the zero temperature two-point function should be identified with the infrared cut-offs l0 and l00 of the temperature-dependent contributions zðl0 ; bÞ and f ðl00 ; bÞ, i. e., l ðA:27Þ l00 ¼ l0 ¼ ; p and we get, ~ bÞwðy; bÞ j ~ h0; ~ 0 j iwðx; 0; 0i ¼ 

1 ; 2b cosh pb ðx  yÞ

ðA:28Þ

in accordance with (A.14). Since the off-diagonal selection rule carried by the Wickordered exponential requires that c ¼ 1, one concludes that the bosonized expression

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~ e ) of the correfor wðxÞ is not obtained just by the tilde conjugation operation ( W sponding Wick-ordered exponential W defining wðxÞ. Besides a multiplicative factor ~ ðiÞ and Klein factors, the bosonized version of the field wðxÞ is obtained only by the tilde conjugation of the creation and annihilation components of the field /ðxÞ ~ in terms of the Wick exponenat the exponent, which can be achieved by defining w  f tial W , 1 ~ bÞ ¼ U B ðhB ÞiwðxÞU ~ iwðx; B ðhB Þ;

ðA:29Þ

with  1  D 12 pffiffi ~ l : lD 2 f ~ iwðxÞ¼ K W ðxÞ ¼ K : ei2 p/ðxÞ : : 2p 2p

ðA:30Þ

One concludes that the revised version of the thermofield dynamics formulation for fermions introduced in Ref. [8] plays an important role in order to establish the twodimensional Fermion-Boson mapping within the thermofield dynamics approach. As a byproduct, according with Eq. (A.15), this formulation enable to regard the tilde fields of the thermofield dynamics as living on the lower branch of the integration contour in the complex time plane, displaced from the real time axis by i b2, in accordance with the work of ref. [7]. References [1] E. Abdalla, M.C.B. Abdalla, K.D. Rothe, Non-Perturbative Methods in Two Dimensional Quantum Field Theory, World Scientific, Singapore, 1991; E. Abdalla, M.C.B. Abdalla, K.D. Rothe, Non-Perturbative Methods in Two Dimensional Quantum Field Theory, second ed., World Scientific, Singapore, 2001. [2] L.R.P.G. Amaral, L.V. Belvedere, K.D. Rothe, Ann. Phys. 320 (2005) 399. [3] (a) L. Leplae, F. Mancini, H. Umezawa, Phys. Rep. 10 C (1974) 151; (b) Y. Takahashi, H. Umezawa, Collet. Phenomena 2 (1975) 55; (c) H. Matsumoto, Fortschr. Physik 25 (1977) 1. [4] H. Umezawa, H. Matsumoto, M. Tachiki, Thermo Field Dynamics and Condensed States, Nort-Holland, Amsterdam, 1982. [5] A. Das, Finite Temperature Field Theory, World Scientific, 1997. [6] R. Haag, N.M. Hugenholtz, Winnink, Comm. Math. Phys. 5 (1967) 215. [7] H. Matsumoto, Y. Nakano, H. Umezawa, J. Math. Phys. 25 (1984) 3076. [8] I. Ojima, Ann. phys. 137 (1981) 1. [9] H. Matsumoto, I. Ojima, H. Umezawa, Ann. Phys. 152 (1984) 348. [10] D. Delee´pine, R. Gonza´lez, J. Weyers, Phys. Lett. 419 (1998) 296. [11] (a) A. Gomez Nicola, D.A. Steer, Nucl. Phys. B549 (1998) 409; (b) A. Gomez Nicola, R.J. Rivers, D.A. Steer, Nucl. Phys. B570 (2000) 475. [12] A. Liguori, M. Mintchev, L. Pilo, Nucl. Phys. B 569 (2000) 577. [13] S. Mandelstam, Phys. Rev. D 11 (1975) 480. [14] S. Coleman, Phys. Rev. D 11 (1975) 2088. [15] K.D. Rothe, J.A. Swieca, Phys. Rev. D 15 (1977) 1675. [16] (a) O. Klein, J. Phys. Radium 9 (1938) 1; (b) G. Lu¨ders, Z. Naturforsch. A 13 (1958) 254; (c) H. Araki, J. Math. Phys. 2 (1961) 267.