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Correlation diagrams and perturbation theory in thermal coherent states
PHYSICS LETTERS A
Physics Letters A 165 (1992) 396—400 North-Holland
Correlation diagrams and perturbation theory in thermal coherent states Peter A. Henning Institutfljr Kernphysik der THDar,nstadt and GesellschaflflJr Schwerionenforschung GSJ, W-6100 Darmstadt, Germany Received 16 December 1991; revisedmanuscript received 30 March 1992; accepted for publication 31 March 1992 Communicated by A.R. Bishop
Thermal quantum theories employ a free parameter ae (0,11. It has been demonstrated recently that a standard perturbation expansion exists only for a = 0 or a 1. In this work it is shown how this limitation can beovercome in thermal coherent states at the cost of increased complexity.
1.Introduction
Tr[WI~!2Wa] <
Many scenarios of modem physics concern hot quantum systems with rapidly changing field expectation values, like the early universe in its inflationary phase, or relativistic heavy-ion collisions. The real time Green function method [1,2] and its operator equivalent thermo field dynamics (TFD) [3,4] are tailored to describe non-equilibrium systems, and could therefore in principle be applied to such scenarios. We assume for the purpose of the present paper that within such a framework a perturbative treatment of the system, e.g. in terms of Feynman diagrams, is desirable. Furthermore we ignore the problem of quasi-particle lifetimes in thermal states as touched by the Narnhofer—Thirring theorem. At every instant of the time evolution, the system is characterized by a statistical operator (density matrix) W, time dependent in the interaction picture. Observables are calculated as a trace over the Fock space of the system, weighted with this statistical operator. In any formulation ofa thermal quanturn theory we then encounter a problem with the socalled a degree offreedom: the statistical operator is a positive operator, and thus can be raised to noninteger powers. Hence, neglecting Fock space equivalence problems, an expectation value can be calculated as 396
>
(1)
Tr[W]
where W~ and W’ are powers of the statistical operator and a is a real parameter, ae [0,1]. It has been shown recently within the framework of TFD that the usual (i.e. time ordered or anti-time ordered) perturbation expansion of a thermal quanturn theory exists only for two values of this pararneter, i.e. for a = 0 and for a=1 [5,6]. This raises the question of how to modify the usual perturbation theory in order to accommodate for different values of a. In view of their physical importance, this question is especiallybothering for the time dependent coherent states mentioned initially. Different values ofa are ofcourse desirable for technical reasons (note that e.g. a = ~ renders eq. (1) symmetric), but are also necessary to treat for consistency investigations of thermal field theories [6]. We therefore re-investigate the a-problem for a real scalar boson field in a thermal coherent state. For brevity we refer to the literature for a detailed introduction to TFD [3,5]. The Fock space of our systern is constructed from a set of momentum eigenstates, created and annihilated by operators obeying canonical commutation relations. The Liouville space 2 is the space of (bounded) operators acting on the Fock space of our quantum system, equipped with the trace as inner product.
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 165, number 5,6
PHYSICS LETFERS A
Naturally, two commuting representations of the canonical commutation relations arise in the Liouville space, denoted by the operators a,~,4 and ~h ~ [3]. Each pair of these also obeys canonical commutation relations, while the tildean and non-tildean operatom commute. a,,, and a,. annihilate the Liouville state 110, 0>>, which is the Fock space projection operator on the vacuum. Traces over the Fock space reduce to simple expectation values in this formulation, and eq. (1) reads
~uWa>> <>
<
< >=
(2)
since the statistical operator is a Liouvillevector. We assume that our system is quasi-free, i.e. that its Hamiltonian is a bilinear form of the fields. In the TFD formulation this gives
0=
+z~(ak—ã~)+zk(4—âk)]
.
(3)
The z’s are complex valued parameters, time independent for now. Such a Hamiltonian can be considered the first approximation to a coupled system ofbosons and fermions, when the fermionic part has been integrated out [71. The Hamiltonian is diagonalized by operators
[~, ~ =ö(k—k’), [~, ~~1] =o(k—k’)
(7)
(tildean and non-tildean operators commute as before). We think of the ~-operatorsas creating and annihilating the quanta of the physical representation, i.e. the quasi-particles of the system. In case of an equilibrium system at finite temperature, they are determined by requiring the causality of the (quasi-)particle propagation, e.g. in terms of the well-known boundary condition Green functions. This KMS condition specifies how theforthermal quasi-particle operators ~, ~ are related to the bare operators. Expressed in terms of the matrix ~ from eq. (5) this reads for an equilibrium state at inverse temperature fi and zero chemical potential 1 —f~\ .~k exp(sr3) (_fJc_a 1 (8)
)~
wherefk=exp(—flwk) and s is a free parameter. With
this transformation matrix one always has ~kIlwa>> =0, ~fl wa>> =0, <
(ak\
I,~j=_~)—A~~(~), T fa~\T fb~\ ~ = ‘s..ak) —A
limits off—~0and a—~0,1 do not commute.
T
,
(4)
2. Thermal coherent state
(5)
Let us set a = 1 for this section to derive a simple result. The generalized thermal equilibrium state of the b-particles is given by [8,9]
where 4k
struction, the c-operators have the commutation relations
______
~ [wk(a~ak—ã)~âk) k
fbk\
1 June 1992
—Zk/O)k,
A~
—4/Wk.
With this substitution, the Hamiltonian exhibits a symmetry, i.e. it is invariant under the replacement of the b’s by operators determined from / ~k ~
/bk)
i W~q>>=
fl
CXP
(J,,bj~6~) I Ob, 0~>>,
(9)
k
where the rightmost factor is the Liouville vector associated with the vacuum of the b-particles, i.e. IIOb,Or>>= flexp(A,.bt+A~E~+lA,.I2)lI0,0>> k
bZT r~’r~, (6) is a real 2 x 2 matrix. This Bogoliubovsym-
=flexp(A,.a~+A~a~—IA,.I2)Il0,0>>. (10) Now assume that z=z(t) is explicitly time depen-
metry is discussed in more detail in ref. [8]. By con-
dent. We can then hope that starting from the equi-
(~) =(~) ~fT
where 3k
397
Volume 165, number 5,6
PHYSICS LET1’ERS A
1 June 1992
librium state given in (9) the time evolution of the system will not change this state provided the z’s change sufficiently slowly. This is indeed true, since for arbitrary time dependence of the z’s one obtains the exact result [7]
with time over a period te [0, T]. For convenience (and without loss of generality) we set z(0) = z( T) = 0. This ensures that at times 0 and T the Hamiltonian (3) is diagonalized by the bare a-operators. The functional character of the coefficients
lI~”ex>> flexp[f,,a~a~
4 leads to a0 difference of [z] the however system at times and T. between the states 12k l2)]ll0,
+ (1 —fk)(Aka~+)~ã~—
0>>
.
(11)
T, the state of the system is still an eigenstate of the c’s, but with non-zero eigenvalue in general. If we abbreviate 1—f a
The coefficients are given by Ak[z] = —i
$
drz,.(r)
z(t) =——~——+
t = 0. At times later than
exp[lwk(r—t)]
3~=exp(s)
1(r)
J dr—~---—exp[iwk(r.--t)] I’
=Ak(t)+4k[z]
,
,
(14)
(12) we obtain for these eigenvalues c~kIIW~>>S~Ak[z]IIW~>>,
,,
~ ‘.k
~
,,
~We
i—a
—
A I
~k~kLZ
~ wi_a
ex
oft
I
~_4*
WI_a =
ex
I
k
15
k [z
Hence in general the eigenvalues of ~ and ~ are not mutually complex conjugate. This implies that the quasi-particle field operators 0 [t~k, ~t] and ~[i,,, ~] have complex expectation values. Complex scalar fields however are quantized with two sets of creation and annihilation operators, cor-
2)]
W~q(t)>>
(13) where the vector on the right is the state defined in (9), but with time dependent b-operators. Hence, at every time, the exact solution is a coherent state of the b-particles. The kth Fourier component of the amplitude for this coherent state is given by (1 —fk)Ak [z]. Eq. (13) is indeed a result expected classically: a time dependent change of a source density gives rise to a classical radiation field.
3. Perturbation theory in thermal coherent states We now assume that the driving terms z(t) change 398
I—a
~j~=exp(—s)
flk exp [(1 —fk)
x (4[z]b~ +A~[z]5~+ lAk[Z]
,—~-,
Vlfk 1
where z is the time denvative of z and to is the time when z=0 (starting from a non-zero constant z is a trivial generalization). Hence the generalized equihbnurn state in (9) will not change, whenthe second partin (l2),A,.[Z], canbeneglected.Thisisthecase, when the typical time scale for a change in the z’s is much greater than ~ The notation tk [ z], 4k [z] was chosen to indicate that these coefficients are functionals rather than functions of z. Expressed in terms of A,. [z], one can rewrite the exact solution from eq. (11) as
I Wex>> =
The thermal quasi-particle operators state ~ have chosen to annihilate the corresponding at been time
respondingto a doublet of particles. For ourcase this means that in general all four c-operators are necessary to construct the Fock space of the theory, i.e. the corresponding fields are coupled. On the other hand, one has decoupled scalar fields at zero temperature. The physical quasi-particle representation at finite temperature is therefore distinguished in having two decoupled fields, which was pointed out to be equivalent to real expectation values. To construct this representation we note that in the Hamiltonian (3) the decoupling is expressed as symmetry of the linear interaction terms: they exhibit a sign change under the exchange ak~-~ã~, a)~—~ak.This is due to the time local character of a linear interaction, i.e. it acts similarly in the tildean
Volume 165, number 5,6
PHYSICS LETTERS A
and non-tildean sector of the Liouville space. One now assumes that the lineardriving terms have been absorbed into the quasi-particle operators only partially. Since L ~ + ~ ak—ak=uk—uk=Fkc,,,—~kck , (16, -~
—
a preservation of the decoupling also for quasi-partides then requires = .~, equivalent to
1 June 1992
theorem. To this end one can apply the method described in ref. [1.0]. Its detailed description would go beyond the central idea ofthis paper, let us therefore summarize it with a few remarks. In the coherent state under consideration here, the expectation value ofan L-fold time ordered product offields Ø[~’ ~] is obtained as
.~
exp(2s)=
(19)= ~ c~~cD’~2~(iD—ço2)’ (space coordinates suppressed). D is the causal two-
I—a
lfk
—J k +
=
=
,~/T7~/l—f
),,—‘°
(17)
,
~
point factorsfunction cL7~areof the field, and the combinatorical
which is well defined provided a ~ 0, 1. Obviously this results in eigenvalues for ~k, t~j~and which are mutually complex conjugate, and in real expectation values for the quasi-particle field operators, e.g.
~
<0 ~ ~~]>=
grams). They are linked to each other and to vertices stemming from (possible) other interaction monomials by two-point functions. It is obvious that by this procedure the number of diagrams is enlarged
i,,,
=
~ v1~7~Jl_f),,
~
a
k
x [A~exp (
—
ikv) +Ak exp (ikx)]
.
(18)
q’~is zero for arbitrary well-behaved z, if a = 0 or a = I. However, as mentioned before, the thermal quasi-particle transformation (6) is singular at these points, and no perturbation theory in terms of the quasi-particles is possible. We therefore exclude the values a = 0 and a = I in the following. For all other values of a, we have thus shown that it is possible to choose a physical quasi-particle picture, i.e. field operators with an observable expectation value, that annihilate the thermal state at T= 0. In view of the scenarios we mentioned in the introduction, it is then meaningful to formulate a perturbation theory in terms ofthe particles created and annihilated by the c’s. We have in mind a system with a Hamiltonian, to which (3) is only a lowest order approximation. This approximation then leads to a time evolution problem that can be solved non-per;~r~~~yely, while additional interactions can be rated perturbatively. Such a mixture of perturbative and non-perturbative description seems supenor to a purely perturbative treatment. The remaining problem for this implementation of a perturbation expansion is to include the non-zero expectation values in a generalization of Wick’s
= ‘
L!
(20)
(L— 2j)!j! ( ±2)’
Hence, the field expectation values appear as vertices in a perturbation expansion (correlation dia-
tremendously as compared to standard perturbation theory. 4. Summary and conclusions Motivated by the physical importance of (arbitrarily fast) time dependent thermal coherent states, we have re-investigated the a-problem for this special case. This investigation was based on the idea that the existence of a perturbation theory requires the existence of asymptotic states and their proper ladder operators in Fock space. We found that the thermal quasi-particles for a time dependent bilinear Hamiltonian (i.e. for a system with a time dependent source density for the field) do not play this role in general: their eigenvalues in the thermal state are not complex conjugate. Only if a certain choice of a is accompanied by a choice of the parameter s as present in eq. (8), i.e. if the thermal quasi-particles are chosen in a special way, they resemble a proper foundation for a perturbation theory. One method to formulate this was quoted shortly, i.e. the use ofcorrelation diagrams in the perturbation expansion, which appear as additional vertices. Because of its technical complexity, 399
Volume 165, number 5,6
PHYSICS LETTERS A
this might however only be ofprincipal but not pradtical importance. Although the above considerations apply strictly only to the coherent states (or bilinear Hamiltonians), we expect them to hold also for thermo field dynamics in general. Thus we conclude that the formal way to extend TFD perturbation theory to general values of a is the inclusion of correlation diagrams. Finally we point out that the problems discussed here are not unlike those from gauge theories: certain perturbative methods only work in fixed gauges. In this context it seems worthwile to note that a choice of a (and s) can be understood as the fixing of a gauge parameter [81.
1 June 1992
References [1] 5. Mrówczyñski and P. Danielewicz, Nucl. Phys. B 342 (1990) 345. [21 J.E. Davis and Perr3,, Phys. Rev. C 43 Phys. (1991) [3] T. Arimitsu andR.J. H. Umezawa, Prog. Theor. 77 1893. (1987) 32, 53. [4]
K.L. Kowalski, N.P. Landsman and Ch.G. van Weert, eds.,
Proc. Workshop on Thermal QFT and their applications, PhysicaA 158 (1989). [5] T.S. Evans, I. Hardman, H. Umezawa and Y. Yamanaka, A time.dependent non-equilibrium calculational scheme
towards the study of temperaturefluctuations, University of Alberta Preprint (1990).
[6] T.S. Evans, I. Hardman, H. Umezawa and Y. Yamanaka,
Heisenberg and interaction representations in TFD, University of Alberta Preprint (1991), J. Math. Phys., in press.
[7] P.A. Henning, Ch. Becker, A. Lang and U. Winkler, Phys. Lett. B 217 (1989) 211.
Acknowledgement I wish to thank M. Herrmann for useful comments.
[8] P.A. Henning, F. MatthSus and M. Graf, Physica A 182 (1992) 489. [9] A. Mann, M. Revzen, K. Nakamura, H. Umezawa and Y. Yamanaka,J. Math. Phys. 30(1989)2883. [10] P.A. Henning, NucI. Phys. B 337 (1990) 547; Phys. Lett. A 145 (1990) 329.
400
Physics Letters A 165 (1992) 396—400 North-Holland
Correlation diagrams and perturbation theory in thermal coherent states Peter A. Henning Institutfljr Kernphysik der THDar,nstadt and GesellschaflflJr Schwerionenforschung GSJ, W-6100 Darmstadt, Germany Received 16 December 1991; revisedmanuscript received 30 March 1992; accepted for publication 31 March 1992 Communicated by A.R. Bishop
Thermal quantum theories employ a free parameter ae (0,11. It has been demonstrated recently that a standard perturbation expansion exists only for a = 0 or a 1. In this work it is shown how this limitation can beovercome in thermal coherent states at the cost of increased complexity.
1.Introduction
Tr[WI~!2Wa] <
Many scenarios of modem physics concern hot quantum systems with rapidly changing field expectation values, like the early universe in its inflationary phase, or relativistic heavy-ion collisions. The real time Green function method [1,2] and its operator equivalent thermo field dynamics (TFD) [3,4] are tailored to describe non-equilibrium systems, and could therefore in principle be applied to such scenarios. We assume for the purpose of the present paper that within such a framework a perturbative treatment of the system, e.g. in terms of Feynman diagrams, is desirable. Furthermore we ignore the problem of quasi-particle lifetimes in thermal states as touched by the Narnhofer—Thirring theorem. At every instant of the time evolution, the system is characterized by a statistical operator (density matrix) W, time dependent in the interaction picture. Observables are calculated as a trace over the Fock space of the system, weighted with this statistical operator. In any formulation ofa thermal quanturn theory we then encounter a problem with the socalled a degree offreedom: the statistical operator is a positive operator, and thus can be raised to noninteger powers. Hence, neglecting Fock space equivalence problems, an expectation value can be calculated as 396
>
(1)
Tr[W]
where W~ and W’ are powers of the statistical operator and a is a real parameter, ae [0,1]. It has been shown recently within the framework of TFD that the usual (i.e. time ordered or anti-time ordered) perturbation expansion of a thermal quanturn theory exists only for two values of this pararneter, i.e. for a = 0 and for a=1 [5,6]. This raises the question of how to modify the usual perturbation theory in order to accommodate for different values of a. In view of their physical importance, this question is especiallybothering for the time dependent coherent states mentioned initially. Different values ofa are ofcourse desirable for technical reasons (note that e.g. a = ~ renders eq. (1) symmetric), but are also necessary to treat for consistency investigations of thermal field theories [6]. We therefore re-investigate the a-problem for a real scalar boson field in a thermal coherent state. For brevity we refer to the literature for a detailed introduction to TFD [3,5]. The Fock space of our systern is constructed from a set of momentum eigenstates, created and annihilated by operators obeying canonical commutation relations. The Liouville space 2 is the space of (bounded) operators acting on the Fock space of our quantum system, equipped with the trace as inner product.
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 165, number 5,6
PHYSICS LETFERS A
Naturally, two commuting representations of the canonical commutation relations arise in the Liouville space, denoted by the operators a,~,4 and ~h ~ [3]. Each pair of these also obeys canonical commutation relations, while the tildean and non-tildean operatom commute. a,,, and a,. annihilate the Liouville state 110, 0>>, which is the Fock space projection operator on the vacuum. Traces over the Fock space reduce to simple expectation values in this formulation, and eq. (1) reads
~uWa>> <
<
< >=
(2)
since the statistical operator is a Liouvillevector. We assume that our system is quasi-free, i.e. that its Hamiltonian is a bilinear form of the fields. In the TFD formulation this gives
0=
+z~(ak—ã~)+zk(4—âk)]
.
(3)
The z’s are complex valued parameters, time independent for now. Such a Hamiltonian can be considered the first approximation to a coupled system ofbosons and fermions, when the fermionic part has been integrated out [71. The Hamiltonian is diagonalized by operators
[~, ~ =ö(k—k’), [~, ~~1] =o(k—k’)
(7)
(tildean and non-tildean operators commute as before). We think of the ~-operatorsas creating and annihilating the quanta of the physical representation, i.e. the quasi-particles of the system. In case of an equilibrium system at finite temperature, they are determined by requiring the causality of the (quasi-)particle propagation, e.g. in terms of the well-known boundary condition Green functions. This KMS condition specifies how theforthermal quasi-particle operators ~, ~ are related to the bare operators. Expressed in terms of the matrix ~ from eq. (5) this reads for an equilibrium state at inverse temperature fi and zero chemical potential 1 —f~\ .~k exp(sr3) (_fJc_a 1 (8)
)~
wherefk=exp(—flwk) and s is a free parameter. With
this transformation matrix one always has ~kIlwa>> =0, ~fl wa>> =0, <
(ak\
I,~j=_~)—A~~(~), T fa~\T fb~\ ~ = ‘s..ak) —A
limits off—~0and a—~0,1 do not commute.
T
,
(4)
2. Thermal coherent state
(5)
Let us set a = 1 for this section to derive a simple result. The generalized thermal equilibrium state of the b-particles is given by [8,9]
where 4k
struction, the c-operators have the commutation relations
______
~ [wk(a~ak—ã)~âk) k
fbk\
1 June 1992
—Zk/O)k,
A~
—4/Wk.
With this substitution, the Hamiltonian exhibits a symmetry, i.e. it is invariant under the replacement of the b’s by operators determined from / ~k ~
/bk)
i W~q>>=
fl
CXP
(J,,bj~6~) I Ob, 0~>>,
(9)
k
where the rightmost factor is the Liouville vector associated with the vacuum of the b-particles, i.e. IIOb,Or>>= flexp(A,.bt+A~E~+lA,.I2)lI0,0>> k
bZT r~’r~, (6) is a real 2 x 2 matrix. This Bogoliubovsym-
=flexp(A,.a~+A~a~—IA,.I2)Il0,0>>. (10) Now assume that z=z(t) is explicitly time depen-
metry is discussed in more detail in ref. [8]. By con-
dent. We can then hope that starting from the equi-
(~) =(~) ~fT
where 3k
397
Volume 165, number 5,6
PHYSICS LET1’ERS A
1 June 1992
librium state given in (9) the time evolution of the system will not change this state provided the z’s change sufficiently slowly. This is indeed true, since for arbitrary time dependence of the z’s one obtains the exact result [7]
with time over a period te [0, T]. For convenience (and without loss of generality) we set z(0) = z( T) = 0. This ensures that at times 0 and T the Hamiltonian (3) is diagonalized by the bare a-operators. The functional character of the coefficients
lI~”ex>> flexp[f,,a~a~
4 leads to a0 difference of [z] the however system at times and T. between the states 12k l2)]ll0,
+ (1 —fk)(Aka~+)~ã~—
0>>
.
(11)
T, the state of the system is still an eigenstate of the c’s, but with non-zero eigenvalue in general. If we abbreviate 1—f a
The coefficients are given by Ak[z] = —i
$
drz,.(r)
z(t) =——~——+
t = 0. At times later than
exp[lwk(r—t)]
3~=exp(s)
1(r)
J dr—~---—exp[iwk(r.--t)] I’
=Ak(t)+4k[z]
,
,
(14)
(12) we obtain for these eigenvalues c~kIIW~>>S~Ak[z]IIW~>>,
,,
~ ‘.k
~
,,
~We
i—a
—
A I
~k~kLZ
~ wi_a
ex
oft
I
~_4*
WI_a =
ex
I
k
15
k [z
Hence in general the eigenvalues of ~ and ~ are not mutually complex conjugate. This implies that the quasi-particle field operators 0 [t~k, ~t] and ~[i,,, ~] have complex expectation values. Complex scalar fields however are quantized with two sets of creation and annihilation operators, cor-
2)]
W~q(t)>>
(13) where the vector on the right is the state defined in (9), but with time dependent b-operators. Hence, at every time, the exact solution is a coherent state of the b-particles. The kth Fourier component of the amplitude for this coherent state is given by (1 —fk)Ak [z]. Eq. (13) is indeed a result expected classically: a time dependent change of a source density gives rise to a classical radiation field.
3. Perturbation theory in thermal coherent states We now assume that the driving terms z(t) change 398
I—a
~j~=exp(—s)
flk exp [(1 —fk)
x (4[z]b~ +A~[z]5~+ lAk[Z]
,—~-,
Vlfk 1
where z is the time denvative of z and to is the time when z=0 (starting from a non-zero constant z is a trivial generalization). Hence the generalized equihbnurn state in (9) will not change, whenthe second partin (l2),A,.[Z], canbeneglected.Thisisthecase, when the typical time scale for a change in the z’s is much greater than ~ The notation tk [ z], 4k [z] was chosen to indicate that these coefficients are functionals rather than functions of z. Expressed in terms of A,. [z], one can rewrite the exact solution from eq. (11) as
I Wex>> =
The thermal quasi-particle operators state ~ have chosen to annihilate the corresponding at been time
respondingto a doublet of particles. For ourcase this means that in general all four c-operators are necessary to construct the Fock space of the theory, i.e. the corresponding fields are coupled. On the other hand, one has decoupled scalar fields at zero temperature. The physical quasi-particle representation at finite temperature is therefore distinguished in having two decoupled fields, which was pointed out to be equivalent to real expectation values. To construct this representation we note that in the Hamiltonian (3) the decoupling is expressed as symmetry of the linear interaction terms: they exhibit a sign change under the exchange ak~-~ã~, a)~—~ak.This is due to the time local character of a linear interaction, i.e. it acts similarly in the tildean
Volume 165, number 5,6
PHYSICS LETTERS A
and non-tildean sector of the Liouville space. One now assumes that the lineardriving terms have been absorbed into the quasi-particle operators only partially. Since L ~ + ~ ak—ak=uk—uk=Fkc,,,—~kck , (16, -~
—
a preservation of the decoupling also for quasi-partides then requires = .~, equivalent to
1 June 1992
theorem. To this end one can apply the method described in ref. [1.0]. Its detailed description would go beyond the central idea ofthis paper, let us therefore summarize it with a few remarks. In the coherent state under consideration here, the expectation value ofan L-fold time ordered product offields Ø[~’ ~] is obtained as
.~
exp(2s)=
(19)
I—a
lfk
—J k +
=
=
,~/T7~/l—f
),,—‘°
(17)
,
~
point factorsfunction cL7~areof the field, and the combinatorical
which is well defined provided a ~ 0, 1. Obviously this results in eigenvalues for ~k, t~j~and which are mutually complex conjugate, and in real expectation values for the quasi-particle field operators, e.g.
~
<0 ~ ~~]>=
grams). They are linked to each other and to vertices stemming from (possible) other interaction monomials by two-point functions. It is obvious that by this procedure the number of diagrams is enlarged
i,,,
=
~ v1~7~Jl_f),,
~
a
k
x [A~exp (
—
ikv) +Ak exp (ikx)]
.
(18)
q’~is zero for arbitrary well-behaved z, if a = 0 or a = I. However, as mentioned before, the thermal quasi-particle transformation (6) is singular at these points, and no perturbation theory in terms of the quasi-particles is possible. We therefore exclude the values a = 0 and a = I in the following. For all other values of a, we have thus shown that it is possible to choose a physical quasi-particle picture, i.e. field operators with an observable expectation value, that annihilate the thermal state at T= 0. In view of the scenarios we mentioned in the introduction, it is then meaningful to formulate a perturbation theory in terms ofthe particles created and annihilated by the c’s. We have in mind a system with a Hamiltonian, to which (3) is only a lowest order approximation. This approximation then leads to a time evolution problem that can be solved non-per;~r~~~yely, while additional interactions can be rated perturbatively. Such a mixture of perturbative and non-perturbative description seems supenor to a purely perturbative treatment. The remaining problem for this implementation of a perturbation expansion is to include the non-zero expectation values in a generalization of Wick’s
= ‘
L!
(20)
(L— 2j)!j! ( ±2)’
Hence, the field expectation values appear as vertices in a perturbation expansion (correlation dia-
tremendously as compared to standard perturbation theory. 4. Summary and conclusions Motivated by the physical importance of (arbitrarily fast) time dependent thermal coherent states, we have re-investigated the a-problem for this special case. This investigation was based on the idea that the existence of a perturbation theory requires the existence of asymptotic states and their proper ladder operators in Fock space. We found that the thermal quasi-particles for a time dependent bilinear Hamiltonian (i.e. for a system with a time dependent source density for the field) do not play this role in general: their eigenvalues in the thermal state are not complex conjugate. Only if a certain choice of a is accompanied by a choice of the parameter s as present in eq. (8), i.e. if the thermal quasi-particles are chosen in a special way, they resemble a proper foundation for a perturbation theory. One method to formulate this was quoted shortly, i.e. the use ofcorrelation diagrams in the perturbation expansion, which appear as additional vertices. Because of its technical complexity, 399
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PHYSICS LETTERS A
this might however only be ofprincipal but not pradtical importance. Although the above considerations apply strictly only to the coherent states (or bilinear Hamiltonians), we expect them to hold also for thermo field dynamics in general. Thus we conclude that the formal way to extend TFD perturbation theory to general values of a is the inclusion of correlation diagrams. Finally we point out that the problems discussed here are not unlike those from gauge theories: certain perturbative methods only work in fixed gauges. In this context it seems worthwile to note that a choice of a (and s) can be understood as the fixing of a gauge parameter [81.
1 June 1992
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K.L. Kowalski, N.P. Landsman and Ch.G. van Weert, eds.,
Proc. Workshop on Thermal QFT and their applications, PhysicaA 158 (1989). [5] T.S. Evans, I. Hardman, H. Umezawa and Y. Yamanaka, A time.dependent non-equilibrium calculational scheme
towards the study of temperaturefluctuations, University of Alberta Preprint (1990).
[6] T.S. Evans, I. Hardman, H. Umezawa and Y. Yamanaka,
Heisenberg and interaction representations in TFD, University of Alberta Preprint (1991), J. Math. Phys., in press.
[7] P.A. Henning, Ch. Becker, A. Lang and U. Winkler, Phys. Lett. B 217 (1989) 211.
Acknowledgement I wish to thank M. Herrmann for useful comments.
[8] P.A. Henning, F. MatthSus and M. Graf, Physica A 182 (1992) 489. [9] A. Mann, M. Revzen, K. Nakamura, H. Umezawa and Y. Yamanaka,J. Math. Phys. 30(1989)2883. [10] P.A. Henning, NucI. Phys. B 337 (1990) 547; Phys. Lett. A 145 (1990) 329.
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