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Thermo field dynamics of a quantum algebra
Volume 103A, number 9
PHYSICS LETTERS
30 July 1984
THERMO FIELD DYNAMICS OF A QUANTUM ALGEBRA H. MATSUMOTO and H. UMEZAWA Theoretical Physics Institute, Department of Physics, The University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 25 May 1984
Thermo field dynamics (a real-time finite-temperature quantum field theory) is used to construct a reduction formula for the causal multipoint Green function. The method is applicable to the case in which the field operators form some algebra. The application to current problems in physics is suggested.
Quantum field theory provides us with an effective analytical procedure in many diverse areas of physics such as high energy physics, condensed matter physics, astrophysics, etc. However, the problems in these subjects frequently demand a further generalization with regard to the forms of the quantum field theory. For example, in the case of interacting quanta with a discrete energy spectrum, or in the case of localized particles, we cannot assume that the operators may be constructed in the Fock space representation from the renormalized free field in which the unperturbed hamiltonian has the usual bilinear form. This unusual state of affairs occurs in many areas of physics. A notable example of this situation is provided by the Anderson model of valence-fluctuating systems [ 1]. The extension of the conventional zero temperature quantum field theory to such a case is by no means a straightforward task within the context of conventional zero-temperature field theory. It is therefore surprising that such an extension becomes relatively straightforward when we use the recently formulated real-time quantum field theory at finite temperature, thermo field dynamics (TFD) [2,3 ] * 1. The reason for this rather bizarre fact is the so-called tilde substitution law in TFD, which is the basis of the Kubo-Martin-Schwinger relation. In~ this article, we present this generalization of quantum field theory in a model-independent form because we feel that it may fred widespread application. Its appli-
cation to the Anderson model is now in progress and will be published elsewhere. Since the structure of TFD has been published in many forms [2], it will not be repeated here in order to save space. In TFD, to any operator (say A) is associated its tilde conjugate A"which is related to A through the following tilde-rules; A1A 2 = .~i,~2, (ClA 1 + c2A2)~ = e j , ~ + c~z~2, where c I and c 2 denote c-numbers and A = PA A with PA being a phase factor: +1(-1) for bosonic (fermionic) A. The tilde and non-tilde operators are mutually commuting (anti-commuting). The temperature appears in the theory through the tilde substitution law [2] which controls the relation between A and ~ ' t through their action on the temperature-dependent vacuum 10,/3): (0, f3IA(t - ~i/3) = (0,/31~z~t (t)a A ,
(1 a)
A ( t + ½i/3)10,/3) = O A A ~ (t) lO,/3),
(lb)
where A ( t ) - exp(if-lt)A exp(-i/~t) w i t h / t = H - / 7 and oA = ( - 1 ) F(F+I)/2, where F denotes the fermion number operator. (In ref. [3] use was made of a different convention which gives oA = 1 .) In the above, H is th~ usual hamiltonian for the non-tilde fields, while/~ is the so-called thermal hamiltonian. From (la, b) it follows that ~10,/3) = <0,/3 hO = 0. When/1 = /~0 +/~I, in which/t0 is the unperturbed part and/1I is the interaction, the causal Green functions in the interaction representation are given by
*t Ref. [2] contains most of the references up to 1982. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
405
Volume 103A, number 9 oq
PHYSICS LETTERS
otn
<0,/31TA] ( t l ) . . A n (tn)10, 3>~ = (0,/3lTexp(-i f
dt/4i(t) )
_oo
o/1
o~ n
× A1 (tl)...An (tn)10,3>,
(2)
where the operators on the left-hand side are those in the Heisenberg representation, while the operators on the right-hand side are the ones in the interaction representation. It is understood that//i(t) contains a suitable adiabatic factor. In (2), we used the thermal doublet notation A s; A 1 = A, A 2 = ~"t. The component a 1 = ct2 = ... = a n = 1 in (2) corresponds to the thermal average of the time-ordered product. We now define the "annihilation operators of the vacuum 10,/3)". When an operator (say s4 ) satisfies M I0, 3> = 0, M is said to be an annihilation operator of the vacuum 10, 3). The tilde substitution law (la, b) leads to a fundamental theorem in TFD which states that, to any operator A is associated an annihilation operator of the vacuum 10,/3). Indeed, (lb) leads to ~ a (t)10, 3) = 0 when
(t) A(t
+
it3)
-
oA3i (t).
(3)
This theorem is remarkable because there is no corresponding operator in the conventional zero-temperature formalism. An application of this theorem to the response theory was given in ref. [4]. In this paper, we take full advantage of this theorem in formulating generalized Feynman rules in an extended form of quantum field theory. Let us now define the extended framework of quantum field theory which we wish to study in this paper. We assume that H 0 and H I consist of certain fundamental fields ~ and qjt. Let ¢ stand for those operators which satisfy [H0, ¢] = - E ¢ .
(4)
These operators will be called the eigen operators. (The concept of eigen operator is a generalization of products of creation and annihilation operators of the physical particles in the conventional quantum field theory.) We assume that some basic countable set of eigen operators which we denote by (0n} can be constructed such that any Heisenberg operator O which can be constructed from the Heisenberg fields may be expressed as O = ~ , n C n e n . ( T h i s is a generalization o f the Lehmann-Symanzik-Zimmermann expansion rule in conventional quantum field theory.) 406
30 July 1984
If we now introduce an irreducible subset of (¢n) which we will denote by {Ca} such that every element of en may be written in terms of products of Ca while any member of {Ca} cannot be expressed in terms of the other members of {Ca}" The irreducible subset {¢a} together with its successive (anti-) commutators: [¢, ¢]+ = ¢(2), [¢, ¢(2)]= ¢(3) ... [¢(2), ¢(2)I = ¢(4) ... (the anticommutator is only used among the fermionic operators) define the minimal algebra which we denote by ~. The operators in - are denoted by ~i and will be referred to as minimal eigen operators. From their definition they satisfy, in the thermal doublet notation [H0, ~ ] = - ei~ ~ ,
(S) (6)
=
k
where the anticommutators are used only when both ~i and ~/. are fermionlc. In (6) we have used the following definition: T÷ = unit matrix and
0)
From (6) we have that e i + e] = e k, while from (5) we obtain ~ ( t ) = exp(i/~0t)~ ~ exp(-i/~ot) = e x p ( - i e i t ) ~ .
(7)
The set {~i} may contain a set of zero energy eigen operators {na}: [H0, n a] = O, which form a Lie algebra [na, rib] = ifabcn c .
(8)
This is the Lie algebra of the invariant subgroup of Our task is now to construct the generalized Feynman rules for calculating the quantity in (2) when the algebra (6) is given. Since HI and AS(t) on the r.h.s, of (2) can be written in terms of the set { ~ } we need to calculate only the vacuum expectation values of the T products of the ~ . It should be noted that the general nature o f out considerations arises from the algebra o f the field operators given in (6). In the case where the (anti-) commutators appearing in (6) are given by c-numbers then the results obtained here are reduced to the normal Feynman rules. Suppose ~i is a non-zero energy eigen operator (e i d: 0). One can then construct by means of (3) the
Volume 103A, number 9
PHYSICS LETTERS
annihilation operators of the vacuum I0,/3 ), a i and a'i, as follows
ai(t ) = [exp(flei) - Pi] - 1/2 X [exp(½ [3ei)~i(t ) - ai'(~i (t)] ,
(9a)
a~t(0 = [exp(flei) - Pi]-1/2
X [-Piai~i(t) + exp(½ ~ei)~ t (t)] .
(9b)
In terms of the thermal doublet notation, this takes the simple form
a~(t) = ( u i l ) ~ / ~ 7 ( t ) ,
~ ( t ) = (Ui)a~aT(t) ,
(10)
which depends on t as exp(-ieit). In (10) U/= [exp(/~ei) - Pi]-1/2 X
( exp(½ [Jei) cri
PiOi
(1 1)
exp(l [Jei)] "
By means of (6) and (10) we obtain
[a~(t), ~(t')] + = (U/-/1T+_)~ X ~ Ci/k~(t' ) e x p [ - i e i ( t k
t')].
(12)
With the aid of (12) we now obtain the following reduction formula for an eigen operator ~ such that e i 4=0: Ot n
10, ~ IT ~ ( t ) ~ 1(tl) ... Gin (tn)I0,13)
= ~k e k [~x,(t - tk)(T/r+_)l ~ k Ciigl
30 July 1984
The proof of (13)is relatively straigtforward; we choose a particular time ordering and write the field ~i in terms of the annihilation operators by means of (10) and then move ali(=ai) gradually tt~ the right and a2( = ~ to the left by means of the (anti-) commutation relation given in (12) where they annihilate the vacuum as ailO,/3) = ( 0 , / 3 ~ = 0. If the generalization to arbitrary times is made, then the result is the reduction formula (13). Using the reduction formula (13), we can express the vacuum expectation value of any time ordered product o f ~ in terms of A~. The procedure proceeds as follows. When a n-point time ordered product is given, identify a non-zero energy eigen operator and eliminate it by the use of the reduction formula (13). By repeating this process, we finally reach the sum of time ordered products of na only. Since the zero energy operators n a are sums of products of the irreducible eigen operators ~a, and since ~a belong to the set (~i),; we can apply the reduction formula (13) to Ca" Finally all operators are eliminated and the vacuum expectation value of the time ordered product of ~ is expressed in terms of the combination of A. In this way we can obtain the generalized Feynman rules for the quantity in (2). Practical applications of this formalism will be presented elsewhere. This work was partly supported by the Natural Science and Engineering Research Council, Canada and by the Dean of Science, The University of Alberta, Edmonton, Alberta, Canada.
References
x (0, #IT~/~ (T1)... ~?k(tk)... ~?,"(tn)10, ~>, (13) where T_+comes from (12) with/" -->ik and k ~ l, T/ m unit matrix or r for fermionic or bosonic ~i respectively, Pk = ( - 1 ) mk with m k being the number of anti-commutation needed in bringing ~ next to ~t¢. In the above, A is defined as i f d ¢ o exp[-iw(t - t')] A?g(t - t ' ) = ~'n X [U/T/(6o - e i + i S r ) - l u ~ ~ .
(14)
[1] P.W. Anderson, Phys. Rev. 124 (1961) 41; J. Hubbard, Proc. R. Soc. A277 (1964) 237; N. Grewe and H. Keiter, Phys. Rev. B24 (1981) 4420; M. Robert and K.W.H. Stevens,J. Phys. C13 (1950) 5941; E.V. Anda, J. Phys. C14 (1981) L1037. [2] H. Umezawa, H. Matsumoto and M. Tachiki,Thermo field dynamics and condensed states (North-Holland, Amsterdam, 1982). [3] I. Ojima, Ann. Phys. 137 (1981) 1. [4] H. Matsumoto, Y. Nakanao and H. Umezawa,Phys. Lett. 100A (1984) 124.
407
PHYSICS LETTERS
30 July 1984
THERMO FIELD DYNAMICS OF A QUANTUM ALGEBRA H. MATSUMOTO and H. UMEZAWA Theoretical Physics Institute, Department of Physics, The University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 25 May 1984
Thermo field dynamics (a real-time finite-temperature quantum field theory) is used to construct a reduction formula for the causal multipoint Green function. The method is applicable to the case in which the field operators form some algebra. The application to current problems in physics is suggested.
Quantum field theory provides us with an effective analytical procedure in many diverse areas of physics such as high energy physics, condensed matter physics, astrophysics, etc. However, the problems in these subjects frequently demand a further generalization with regard to the forms of the quantum field theory. For example, in the case of interacting quanta with a discrete energy spectrum, or in the case of localized particles, we cannot assume that the operators may be constructed in the Fock space representation from the renormalized free field in which the unperturbed hamiltonian has the usual bilinear form. This unusual state of affairs occurs in many areas of physics. A notable example of this situation is provided by the Anderson model of valence-fluctuating systems [ 1]. The extension of the conventional zero temperature quantum field theory to such a case is by no means a straightforward task within the context of conventional zero-temperature field theory. It is therefore surprising that such an extension becomes relatively straightforward when we use the recently formulated real-time quantum field theory at finite temperature, thermo field dynamics (TFD) [2,3 ] * 1. The reason for this rather bizarre fact is the so-called tilde substitution law in TFD, which is the basis of the Kubo-Martin-Schwinger relation. In~ this article, we present this generalization of quantum field theory in a model-independent form because we feel that it may fred widespread application. Its appli-
cation to the Anderson model is now in progress and will be published elsewhere. Since the structure of TFD has been published in many forms [2], it will not be repeated here in order to save space. In TFD, to any operator (say A) is associated its tilde conjugate A"which is related to A through the following tilde-rules; A1A 2 = .~i,~2, (ClA 1 + c2A2)~ = e j , ~ + c~z~2, where c I and c 2 denote c-numbers and A = PA A with PA being a phase factor: +1(-1) for bosonic (fermionic) A. The tilde and non-tilde operators are mutually commuting (anti-commuting). The temperature appears in the theory through the tilde substitution law [2] which controls the relation between A and ~ ' t through their action on the temperature-dependent vacuum 10,/3): (0, f3IA(t - ~i/3) = (0,/31~z~t (t)a A ,
(1 a)
A ( t + ½i/3)10,/3) = O A A ~ (t) lO,/3),
(lb)
where A ( t ) - exp(if-lt)A exp(-i/~t) w i t h / t = H - / 7 and oA = ( - 1 ) F(F+I)/2, where F denotes the fermion number operator. (In ref. [3] use was made of a different convention which gives oA = 1 .) In the above, H is th~ usual hamiltonian for the non-tilde fields, while/~ is the so-called thermal hamiltonian. From (la, b) it follows that ~10,/3) = <0,/3 hO = 0. When/1 = /~0 +/~I, in which/t0 is the unperturbed part and/1I is the interaction, the causal Green functions in the interaction representation are given by
*t Ref. [2] contains most of the references up to 1982. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
405
Volume 103A, number 9 oq
PHYSICS LETTERS
otn
<0,/31TA] ( t l ) . . A n (tn)10, 3>~ = (0,/3lTexp(-i f
dt/4i(t) )
_oo
o/1
o~ n
× A1 (tl)...An (tn)10,3>,
(2)
where the operators on the left-hand side are those in the Heisenberg representation, while the operators on the right-hand side are the ones in the interaction representation. It is understood that//i(t) contains a suitable adiabatic factor. In (2), we used the thermal doublet notation A s; A 1 = A, A 2 = ~"t. The component a 1 = ct2 = ... = a n = 1 in (2) corresponds to the thermal average of the time-ordered product. We now define the "annihilation operators of the vacuum 10,/3)". When an operator (say s4 ) satisfies M I0, 3> = 0, M is said to be an annihilation operator of the vacuum 10, 3). The tilde substitution law (la, b) leads to a fundamental theorem in TFD which states that, to any operator A is associated an annihilation operator of the vacuum 10,/3). Indeed, (lb) leads to ~ a (t)10, 3) = 0 when
(t) A(t
+
it3)
-
oA3i (t).
(3)
This theorem is remarkable because there is no corresponding operator in the conventional zero-temperature formalism. An application of this theorem to the response theory was given in ref. [4]. In this paper, we take full advantage of this theorem in formulating generalized Feynman rules in an extended form of quantum field theory. Let us now define the extended framework of quantum field theory which we wish to study in this paper. We assume that H 0 and H I consist of certain fundamental fields ~ and qjt. Let ¢ stand for those operators which satisfy [H0, ¢] = - E ¢ .
(4)
These operators will be called the eigen operators. (The concept of eigen operator is a generalization of products of creation and annihilation operators of the physical particles in the conventional quantum field theory.) We assume that some basic countable set of eigen operators which we denote by (0n} can be constructed such that any Heisenberg operator O which can be constructed from the Heisenberg fields may be expressed as O = ~ , n C n e n . ( T h i s is a generalization o f the Lehmann-Symanzik-Zimmermann expansion rule in conventional quantum field theory.) 406
30 July 1984
If we now introduce an irreducible subset of (¢n) which we will denote by {Ca} such that every element of en may be written in terms of products of Ca while any member of {Ca} cannot be expressed in terms of the other members of {Ca}" The irreducible subset {¢a} together with its successive (anti-) commutators: [¢, ¢]+ = ¢(2), [¢, ¢(2)]= ¢(3) ... [¢(2), ¢(2)I = ¢(4) ... (the anticommutator is only used among the fermionic operators) define the minimal algebra which we denote by ~. The operators in - are denoted by ~i and will be referred to as minimal eigen operators. From their definition they satisfy, in the thermal doublet notation [H0, ~ ] = - ei~ ~ ,
(S) (6)
=
k
where the anticommutators are used only when both ~i and ~/. are fermionlc. In (6) we have used the following definition: T÷ = unit matrix and
0)
From (6) we have that e i + e] = e k, while from (5) we obtain ~ ( t ) = exp(i/~0t)~ ~ exp(-i/~ot) = e x p ( - i e i t ) ~ .
(7)
The set {~i} may contain a set of zero energy eigen operators {na}: [H0, n a] = O, which form a Lie algebra [na, rib] = ifabcn c .
(8)
This is the Lie algebra of the invariant subgroup of Our task is now to construct the generalized Feynman rules for calculating the quantity in (2) when the algebra (6) is given. Since HI and AS(t) on the r.h.s, of (2) can be written in terms of the set { ~ } we need to calculate only the vacuum expectation values of the T products of the ~ . It should be noted that the general nature o f out considerations arises from the algebra o f the field operators given in (6). In the case where the (anti-) commutators appearing in (6) are given by c-numbers then the results obtained here are reduced to the normal Feynman rules. Suppose ~i is a non-zero energy eigen operator (e i d: 0). One can then construct by means of (3) the
Volume 103A, number 9
PHYSICS LETTERS
annihilation operators of the vacuum I0,/3 ), a i and a'i, as follows
ai(t ) = [exp(flei) - Pi] - 1/2 X [exp(½ [3ei)~i(t ) - ai'(~i (t)] ,
(9a)
a~t(0 = [exp(flei) - Pi]-1/2
X [-Piai~i(t) + exp(½ ~ei)~ t (t)] .
(9b)
In terms of the thermal doublet notation, this takes the simple form
a~(t) = ( u i l ) ~ / ~ 7 ( t ) ,
~ ( t ) = (Ui)a~aT(t) ,
(10)
which depends on t as exp(-ieit). In (10) U/= [exp(/~ei) - Pi]-1/2 X
( exp(½ [Jei) cri
PiOi
(1 1)
exp(l [Jei)] "
By means of (6) and (10) we obtain
[a~(t), ~(t')] + = (U/-/1T+_)~ X ~ Ci/k~(t' ) e x p [ - i e i ( t k
t')].
(12)
With the aid of (12) we now obtain the following reduction formula for an eigen operator ~ such that e i 4=0: Ot n
10, ~ IT ~ ( t ) ~ 1(tl) ... Gin (tn)I0,13)
= ~k e k [~x,(t - tk)(T/r+_)l ~ k Ciigl
30 July 1984
The proof of (13)is relatively straigtforward; we choose a particular time ordering and write the field ~i in terms of the annihilation operators by means of (10) and then move ali(=ai) gradually tt~ the right and a2( = ~ to the left by means of the (anti-) commutation relation given in (12) where they annihilate the vacuum as ailO,/3) = ( 0 , / 3 ~ = 0. If the generalization to arbitrary times is made, then the result is the reduction formula (13). Using the reduction formula (13), we can express the vacuum expectation value of any time ordered product o f ~ in terms of A~. The procedure proceeds as follows. When a n-point time ordered product is given, identify a non-zero energy eigen operator and eliminate it by the use of the reduction formula (13). By repeating this process, we finally reach the sum of time ordered products of na only. Since the zero energy operators n a are sums of products of the irreducible eigen operators ~a, and since ~a belong to the set (~i),; we can apply the reduction formula (13) to Ca" Finally all operators are eliminated and the vacuum expectation value of the time ordered product of ~ is expressed in terms of the combination of A. In this way we can obtain the generalized Feynman rules for the quantity in (2). Practical applications of this formalism will be presented elsewhere. This work was partly supported by the Natural Science and Engineering Research Council, Canada and by the Dean of Science, The University of Alberta, Edmonton, Alberta, Canada.
References
x (0, #IT~/~ (T1)... ~?k(tk)... ~?,"(tn)10, ~>, (13) where T_+comes from (12) with/" -->ik and k ~ l, T/ m unit matrix or r for fermionic or bosonic ~i respectively, Pk = ( - 1 ) mk with m k being the number of anti-commutation needed in bringing ~ next to ~t¢. In the above, A is defined as i f d ¢ o exp[-iw(t - t')] A?g(t - t ' ) = ~'n X [U/T/(6o - e i + i S r ) - l u ~ ~ .
(14)
[1] P.W. Anderson, Phys. Rev. 124 (1961) 41; J. Hubbard, Proc. R. Soc. A277 (1964) 237; N. Grewe and H. Keiter, Phys. Rev. B24 (1981) 4420; M. Robert and K.W.H. Stevens,J. Phys. C13 (1950) 5941; E.V. Anda, J. Phys. C14 (1981) L1037. [2] H. Umezawa, H. Matsumoto and M. Tachiki,Thermo field dynamics and condensed states (North-Holland, Amsterdam, 1982). [3] I. Ojima, Ann. Phys. 137 (1981) 1. [4] H. Matsumoto, Y. Nakanao and H. Umezawa,Phys. Lett. 100A (1984) 124.
407