Exact solution for the Schwinger model at finite temperature

Volume 180, number 1,2

PHYSICS LETTERS B

6 November 1986

EXACT SOLUTION FOR THE SCHWINGER MODEL AT FINITE TEMPERATURE F. R U I Z R U I Z and R.F. A L V A R E Z - E S T R A D A 1,2

Departamento de Ftsica Tebrica, Facultad de Ciencias l:~tsicas, Universidad Complutense, 28040 Madrid, Spain Received 15 May 1986

An exact solution for the Schwinger model at finite temperature is given. Path integral methods and thermo-field dynamics are used. The explicit dependence of the complete propagators with temperature is found.

Quantum field theory at non-zero temperature is playing an increasingly important role since it is a necessary tool to study the early universe, in which the temperature was very high. To quote some of the contributions in this area let us mention the analysis of effective potentials and phase transitions at finite temperature [ 1 ], the evaluation of thermal corrections to physical parameters (mass, electric charge, magnetic moment) [2], and a quantum study of plasmas [3]. Here, an exact solution for the Schwinger model [4] at finite temperature is given. Among the various formalisms proposed to describe a quantum field theory at finite temperature we wiU use the one called "thermo-field dynamics" (TFD), due to Takahashi and Umezawa [5]. Ojima [6] has adapted it to gauge theories. In TFD, a temperature-dependent vacuum 10(13)) is introduced, which has the property that the usual statistical average for any operator B can also be expressed as (0(13) 1BI 0(/3)): see refs. [5,6]. Using this formalism for the Schwinger model, (1 + 1)-dimensional QED with massless fermions, we get the thermal lagrangian density

=

a=l,2

{raa[- ~ F~uF ~ - (1/2a) (0vA~) 2 - i (~vr~a) (aUr~a)] + ~a (~i~-" - e4ta) ¢/a},

(1)

where f l l =-7"22 = 1,7"12 =7"21 = 0 ,

A ~ = A u,

fa = f ,

rla=77,

A~=-X"*~, fa=~,

~a =¢/a

%=~,

fora=l,

~=i(~t) t fora=2,

(2)

and the superscript t means matrix transposition. A v and ff denote, respectively, the standard EM and fermion fields. The fictitious tilde fields ~'u, ~" are characteristic of TFD: see ref. [6] for their definition and properties. Ghosts fields fa and % are introduced to eliminate unphysical negative-norm states (as for the zero temperature case), so the expectation value of an operator in 10(13)) coincides with its statistical average, in which only physical states enter. They satisfy £r/a (X), f/b@)} = {fa(x), fib@)} = {~a(X), l"/b(Y)} = 0 (for a detailed discussion see ref. [6]). Let ac,a c+ (bc, b+), c = 1 2, be boson (fermion) annihilation and creation operators corresponding to the various fields appearing in eqs. (1), (2). Next, and following TFD, new annihilation operators, denoted by a superindex 13, are defined by means of Bogoliubov transformations 1 Address for the academic year 1985-1986: Theoretical Physics Group, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA. 2 Senior Fulbright Fellow. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)

153'

Volume 180,number 1,2

aS(k)=

PHYSICS LETTERSB

V ad(k),

d=l,2

bcqk)=

6 November 1986

d=l,2

and so on for creation operators. The doublets ac, a~c,bc and b~c (c = 1,2) are defined in the same way as in eq. (2); for instance, for an electron, b2(k , ;k) = ib+ (k, X),'with X the polarization index. The matrices UB and UF are given by =

cosh 0 B (3)

sinh' 0 a (3) //,

sinh~0B (3)

cosh"0 B (3) ]

sinh20F (3) = [exp(3l ku[) - 1 ] - 1 ,

uF = ¢cos'0F (3) --sin~0F (fl)

sin OF (3) //, cos OF (/3)/

sin20F (/3) = [exp(3[ ku[) + 1] -1 ,

u v being the velocity of the thermal fluid and/3 a Lorentz invariant parameter related to the temperature [7], and where we have assumed zero chemical potential. The new annihilation operators annihilate 10(/3)). The free propagators are the expectation values of the chronological product ordered with respect to the Lorentz invariant time xVuv of the field doublets of eq. (2), that is (e ~ 0 +) 2

D~(x - y ) = ( d Z k exp [ - ik(x - y ) ] D ~ ( k ) , Da~(k)= ~

UBacF _2roe

c=1,2

(g'.

kk 2 ÷ ircce

kVkU ]U B .kV_k_~ ~ ~_ircc a k 2 + ircce] (k 2 + i%--'-ce)2j cb

(3)

for photons,

~ a b ( x - y ) = ( d2k e x p [ - i k ( x - y)] O ab(k), o (21r)2

Tee

ab(k) = - G U~ae k-7-------- UBcb c=1,2 + ircce

(4)

for ghosts, and

Sab(X-y) ( dZk exp[_ik(x_y)]Sab(k) '

Sab(k) = i~ ~

1

Ft

UaF k ~ - (U )cb c=1,2 + ircce

(5)

for fermions. In these equations we have akeady particularized for (1 + 1) dimensions and massless fermions. Only fields of type 1 are physical. Degrees of freedom of type 2 are fictitious and are introduced to construct the state 10(/3)). Fields of type 2 are also responsible, within the framework of the perturbative method, for the absence of powers of Dkac delta functions [8]. Semenoff and Umezawa [9] have proved that the generating functional for real-time Green functions at finite temperature is constructed in the same way as at zero temperature but taking as starting point the thermal lagrangian density provided by TFD. Therefore, for the Schwinger model we have the following generating functional:

Z[j,~,g]=c~_lexp(_ie

~ fd2 x o= ,2

Xexp( -1

8 .ru S____~___ ~ ) iBm(x)

i -Ax)

~ ;d2xd2y[Ja(x)D~vb(x-y)J~(y)+2~a(x)Sab(x-Y)~b(y)] 2a,b=l,2 ~

),

where q~ is a normalization constant, and where we have not written the ghosts part since, according to eq. (1), they do not couple to any other fields. Using standard path integrals properties [10] we get

154

Volume 180, number 1,2

PHYSICS LETTERS B

6 November 1986

b /.t Z [ J , ~ , ~ ] = q Z - l e x p ( -12a,b=l,2~ fd2xd2yjff(x)D~v°(x-y)J~(.y))

Xexp ~ E

Za, b=l,2

fd2xd2y8Aa(Xi DV,~(x-y) ao (6)

~ fd2xd2y~a(X) Gab(X,yleA)~b(Y)+L[A]) , a,b=l,2"

X exp(-

where A~ is the following doublet of external fields:

A:(x) =b ~ 2 f d2y iDff~(x - y)d~ (y), Gab (x,

y leA) is the solution of the integral equation

Gab(X,yleA ) =Sab(x- y ) + ie ~ f d2z Sac(X- z).~c(Z ) acb(Z,yleA),

(7)

c=1,2

and the closed fermion-loop functional L [A] is given by e

ttA]=i

~ fde'fd2xtr.#a(X) a=l'2

li,rn Gaa(X,x'leA)ex p i e x-x

0

0

dlvAVa , x'

J

with the limit taken symmetrically and preserving causality. Acting on both sides of eq. (7)with ~ - e4la and solving the resulting differential equation we get

Gab(X'yleA)=exp{ie[¢a(X)-¢b(Y)]} Sab(x-Y)'

Ca(X)=- b=l,2 ~ fd2y

Q)ab(X--Y)rbb~,(y)~b(y).

(8)

Notice that Ca does depend on ~ through CDab, see eq. (4). Eq. (8) allows us to write the closed fermion-loop functional L [A] as

i =1,2fd2x L[A] = 2a,

d2yA:,(x)

rI[~b(X

IIa~(X - Y) = e2 -~-f/ (-~d2k exp[- ik(x - y ) ] [g~Vrab +kVk"raac3 ab(k) rbb ]

(9)

From eq. (4) it follows that Ilffff(x- y ) is gauge invariant. Introducing eq. (9) into eq. (6) we obtain

Z[J,~=g=O] =°t~-lexp (

-

-

l ~ =1,2fd2xd2yjav(x)v. 2a, Dab(x _y)jbu(y)) ,

vp (x -- y ) is where the exact thermal propagator Dab

Da~(X_ y)= f d2k exp[_ik(x_y)].O~a~(k), 3(270 2

c=1,2 ac k2 -- e2/---~+ircce qVU k 2 + ircce _ ircca (k 2 + ircce)2_JUBcb" 155

Volume 180, number 1,2

PHYSICS LETTERS B'

6 November 1986

Notice that it has the same matrix structure as the free one, eq. (3). The a-independent part of the photon propagator has acquired a mass e/V~-independent of the temperature. The propagator for the physical EM field (of type 1) is given by D V~(x - y ) = [ ( - i)2/Z] ~ 2Z/~ j1 (x) ~J~1 (Y)lsources = 0 = .D ~ (x - y).

(10)

For the fermion propagator, using eqs. (6), (8), (9), the analogue for fermions ofeq. (10) and standard path integral techniques, we find S(x-y)=exp

(-

~-~ fd2k{1-exp[-ik(x-y)]}

[A(k) cosh2OB(f3)-A*(k)senh2OB(k)])Sll(X-y),

(11)

where 1

A(k)

- - k2+ie

1

k 2-e2/lr+ie

+ol

e2

1

7r (k 2 + i e ) 2

.

The exponent ofeq. (11) can be written for the Landau gauge (a = 0) in the more explicit form

i fd2k{l_exp[_ik(x_y)]}(1 ,~Tr k 2 + ie

1 k2-e2/Ir + ie

2hi exp(~lkul)- 1

[8(k2)_~(k2_e2/zr)]) "

The results of eqs. (10), (11) would have been much more difficult to obtain if only fields of type 1 had been considered, as the real-time formulation of Dolan and Jackiw [I ] does. For zero temperature (/3 ~ oo), 0B (/3) and OF (/3) approach zero, the matrices U B and U F become the unit 2 × 2 matrix, and the EM and fermion propagators given in eqs. (10), (1 1) become those for the Schwinger model at zero temperature [4]. We conclude that temperature does not change the masses of the particles with respect to the zero temperature case. All thermal effects are included in the propagators through suitable matrix elements of U B, U F and 0 (/~)-factors. The 0 B (/3)dependence for the EM complete propagator is the same as for the free one. However, the fermion complete propagator contains an extra/3-dependence in the exponent of eq. (11) that the free one does not have. Notice that we have not calculated the thermodynamical partition function, which requires a detailed evaluation of the temperature-dependent constant c~ o f eq. (6); we have studied the dynamical response of the system. We acknowledge the partial support given by Plan Movilizador de Altas Energfas, Comisi6n Asesora de Investigaci6n Cient ffica y T6cnica (CAOCYT), Spain. One of us (R. F. A.-E.) is grateful to the Council for International Exchange of Scholars (CIES) for support through a Fulbright/MEC Grant, and to Professor B. Zumino for the kind hospitality extended to him at the Theoretical Physics Group, Lawrence Berkeley Laboratory.

References [1 ] L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 1686. [2] G. Peressutti and B.S. Skagerstam, Phys. Lett. B 110 (1982) 406; Y. Fujirnoto and J.H. Yee, Phys. Lett. B 114 (1982) 359; J.F. Donoghue and B.R. Holstein, Phys. Rev. D 28 (1983) 340. [31 A. Weldom, Phys. Rev. D 26 (1982) 1394; D 26 (1982) 2789. [4] J. Schwinger, Phys. Rev. 128 (1962) 2425. [5] Y. Takahashi and H. Umezawa, Collect. Phenom. 2 (1975) 55; H. Umezawa, H. Matsumoto and M. Tachiki, Thermo-field dynamics and condensed states (North-Holland, Amsterdam, 1982). [6] I. Ojima, Ann. Phys. 137 (1981) 1. [7] W. Israel, Ann. Phys. 100 (1976) 310. [8} Y. Fujimoto, R. Grigjanis and R. Kobes, Prog. Theor. Phys. 73 (1984) 434. [9] G.W. Semenoff and H. Umezawa, Nucl. Phys. B 220 [FS8] (1983) 196. [10] H.M. Fried, Functional methods and models in quantum field theory (M1T Press, Cambridge, MA, 1972). 156