On Quantum Field Theories in Operator and Functional Integral Formalisms

0.21 MB·English

by  Aba Teleki

APS/123-QED
On Quantum Field Theories in Operator and Functional Integral Formalisms
Aba Teleki∗
Department of Physics, Faculty of Natural Sciences,
Constantine the Philosopher University, SK-949 74 Nitra, Slovakia
Milan Noga†
Department of Theoretical Physics, Faculty of Mathematics, Physics and Computer Sciences,
Comenius University, SK-842 48 Bratislava, Slovakia
(Dated: February 1, 2008)
Relations and isomorphisms between quantum field theories in operator and functional integral
formalisms are analyzed from the viewpoint of inequivalent representations of commutator or an-
ticommutator rings of field operators. A functional integral in quantum field theory cannot be re-
gardedasaNewton-Lebesgueintegralbutratherasaformalobjecttowhichoneassociatesdistinct
6 numerical values for different processes of its integration. By choosing an appropriate method for
0 theintegration of agiven functional integral, onecan select a single representation out of infinitely
0 many inequivalent representations for an operator whose trace is expressed by the corresponding
2 functional integral. These properties are demonstrated with two exactly solvable examples.

n
a PACSnumbers: 03.70.+k,05.30.-d,11.10.-z,05.30.Ch,74.20.Fg,02.90.+p
J

9
I. INTRODUCTION integration,andthentouse anappropriateperturbation
1
expansion. In this method one divides the action func-

1 In quantum field theory based on the functional inte- tional S(a∗,a) into a sum of two terms
v gralformalism[1,2,3,4,5],thepartitionfunctionZ and
6 quantum expectation values hAi of physical observables S(a∗,a)=S0(a∗,a)+SI(a∗,a), (1.2)
3
1 A are given by the formal functional integrals where S0 has a bilinearformin the fields a∗ anda.Thus
1 the corresponding partition function
0 Z = e−S(a∗,a)D(a∗,a) (1.1a)
06 Z Z0 = e−S0(a∗,a)D(a∗,a) (1.3)
/ and Z
h
is a Gaussian integral and can be exactly and explicitly
t
p- hAi= 1 A(a∗,a)e−S(a∗,a)D(a∗,a), (1.1b) evaluated. The total partition function Z can be ex-
Z pressed in the form
e
Z
h
v: respectively, where S(a∗,a) is an action functional of Z =Z0 exp(−SI) 0, (1.4)
fields a∗ and a, A(a∗,a) is a function of a∗ and a, and
i where (cid:10) (cid:11)
X D(a∗,a) denotes a formal measure on a space of fields
r a∗ and a. All problems of quantum field theory are thus ∞ (−1)ν
a reduced to problems of finding a correct definition and exp(−SI) 0 = ν! hSIνi0 (1.5)
a computation method of the functional integrals (1.1). ν=0
(cid:10) (cid:11) X
However, the results of the integration of (1.1) are not
is the infinite perturbation series involving quantum ex-
unique and depend on the chosen method for carrying pectation values of hSνi evaluated with respect to the
outthecomputationofthesefunctionalintegrals. Thisis I 0
chosenactionS .Theseparation(1.2)oftheactionfunc-
0
whyintheirmonograph[6]themathematiciansKobzarev tionalS(a∗,a)asthesumoftermsS andS is,however,
0 I
andManinhaveexpressedthefollowingstatementabout
not unique. There are infinitely many ways to select S .
0
(1.1): “From a mathematician’s viewpoint almost every
To each selected S one obtains a different result for the
0
such computation is in fact a half-baked and ad hoc def-
perturbation series (1.5). Thus the results of functional
inition, but a readiness to work heuristically with such a
integrations of (1.1a) as expressed by (1.4) seem to be
prioriundefinedexpressionsas(1.1)is amustinthis do-
indeed as “half-baked and ad hoc definitions” from the
main.”Themoststandardmethodfortheintegrationsof
mathematician’s viewpoint.
(1.1) is their reduction to Gaussianintegrals,whose the-
If quantum field theory based on the functional inte-
ory is the only developed chapter of infinite-dimensional
grals(1.1)isequivalenttothesamequantumfieldtheory
formulated in the operator formalism, then the relations
∗Electronicaddress: [email protected] e−S(a∗,a)D(a∗,a)=Tre−βH(a+,a) (1.6a)
†Electronicaddress: [email protected]
Z
2
and andthe statisticalaveragevalues correspondingto phys-
ical observables associated with the operators A(a+,a)
A(a∗,a)e−S(a∗,a)D(a∗,a) are given by
Z 1
=Tr A(a+,a)e−βH(a+,a) (1.6b) hAi= Z Tr A(a+,a)e−βH . (1.11b)
n o (cid:8) (cid:9)
mustbesatisfied,whereH(a+,a)istheHamiltonianop- As long as the number of the operators ak,σ, a+k,σ en-
eratorcorrespondingto the actionS(a∗,a), the operator tering the commutator or anticommutator ring (1.7) is
A(a+,a) corresponds to its normal symbol A(a∗,a), a+ finite,thereisonlyoneinequivalentrepresentationofthe
and a are field operators, and β = 1/T is the inverse relations(1.7)and(1.8). However,inquantumfieldtheo-
temperature. riesdescribingsystemswithaninfinitenumberofdegrees
From the last relations it follows that the functional of freedom, the algebraic structure (1.7) has infinitely
many inequivalent representations[7]. Intuitively speak-
integrals(1.1)cannotbedefinedinsuchawayastogivea
uniqueresultafteraprocessoftheirintegrationsbecause ing,onecansaythatthereexistsinfinitelymanydifferent
matrix realizations of the operators a and a+ satis-
the right-handsides ofthe relations(1.6)are distinct for k,σ k,σ
each inequivalent representation of the commutator or fying the same algebraic structure (1.7). The situation
anticommutatorringoffieldoperatorsa+ andaentering reminds us distantly of a Lie algebra of a non-compact
the Hamiltonian H(a+,a) and the operator A(a+,a). group which has infinitely many unitary irreducible rep-
The reasons are given by the following arguments. resentations realized by sets of infinite-dimensional ma-
trices. Thus, to each inequivalent representation of the
In the operator approach one assumes to have a com-
plete set of annihilation and creationoperatorsa and commutator or anticommutator ring (1.7) one has to as-
k,σ
sociate the correspondingrepresentationof the Hamilto-
a+ of particles in quantum states denoted by quantum
k,σ nian (1.9) and the density matrix (1.10).
numbers(k,σ),as,forexample,bythemomentumkand
Intuitively speaking,one cansaythatthe matrix form
the spin σ. In any quantum field theory the number of
of the Hamiltonian (1.9) and the density matrix (1.10)
the pairs of the operators (a ,a+ ) is infinite. These
k,σ k,σ are distinct for each inequivalent representation of (1.7).
operatorssatisfy the canonicalcommutationoranticom- This implies that the partitionfunction Z andthe aver-
mutation relations
age values hAi as given by (1.1) can give rise to various
results depending on the choseninequivalent representa-
{ak,σ,a+k′,σ′} = δkk′δσσ′, (1.7a) tions of the ring (1.7). This non-uniqueness of the value
{a ,a } = {a+ ,a+ }=0. (1.7b) of the partition function Z and of the average values
k,σ k′,σ′ k,σ k′,σ′
hAiassociatedwiththegivenHamiltonianpresentinthe
TheoperatorsactonstatevectorsΨwhichspanaHilbert operator formalism (1.7)-(1.11) should be preserved in
space H. In order to achieve a unique specification of quantum field theory based on the functional integrals
thecommutatororanticommutatorringoftheoperators (1.1)ifthesetwoapproachesareequivalent. Theyshould
(1.7), one requires in addition to (1.7) the existence of a be equivalent because the integrand exp{−S(a∗,a)} en-
vacuum state Φ for which tering the functional integral (1.1a) is in fact the kernel
0
of the density matrix ρ=exp{−βH} [1, 2, 3, 4, 5]. For
a Φ =0 (1.8) thesereasons,thefunctionalintegral(1.1a)or(1.1b)can
k,σ 0
be as well defined as is permitted by a freedom present
for all (k,σ). in the operator approach to quantum field theory. This
Inthiscase,theHilbertspaceH isthespaceforarep- freedom is associated with the existence of the inequiva-
resentation of the commutator or anticommutator ring lent representations of the commutator or anticommuta-
of the operators (1.7) with the auxiliary condition (1.8). tor ring (1.7) of field operators.
Since the operators a and a+ form a complete set For a detailed understanding of the fact that the com-
k,σ k,σ
of operators,a Hamiltonian H of a given system can be putation of the functional integrals (1.1) can give rise to
expressed as a given function of a and a+ , i.e. various different results, we will analyze the process of
k,σ k,σ
their integrations from three different aspects. Firstly,
H =H(a+,a). (1.9) weanalyzeacertainclassofinequivalentrepresentations
of the anticommutator ring (1.7) and show that to each
The partition function Z of the system is expressed as inequivalentrepresentationonehastoassociateadistinct
the trace of the density matrix action functional S(a∗,a). Thus, even the explicit form
oftheactionfunctionalS(a∗,a)correspondingtoagiven
ρ=e−βH, (1.10) Hamiltonian H is not unique, but depends on the cho-
sen inequivalent representation of the anticommutator
i.e., ring of field operators. Secondly, we study perturbation
series in both the operator and functional integral for-
Z =Tre−βH, (1.11a) malisms of quantum field theories in order to show that
3
by selecting an unperturbed part of a Hamiltonian and of the operator (2.4) in which every operator a and
k,σ
the corresponding unperturbed action functional, one in a+ isreplacedbytheGrassmanngeneratorsa (τ)and
k,σ k,σ
fact selects one inequivalent representation of the com- a∗ (τ), respectively [1, 2, 3, 4, 5]. These generators
mutator or anticommutator ring (1.7) of field operators. k,σ
are thus enumeratedalso by the continuous parameterτ
Thirdly, we demonstrate explicitly the properties of the
in addition to the quantum numbers (k,σ). The action
functional integrals mentioned above on a toy model of functional S =S(a∗,a) is then defined by the integral
quantumfieldtheory. Weconstructasimpleactionfunc-
tionalS(a∗,a) which permits the exactevaluationof the
β
functional integral (1.1a), but nonetheless it leads to in- S(a∗,a) = dτ lim a∗ (τ)a˙ (τ)
k,σ k,σ
finitely many different results correspondingto infinitely Z0 N→∞(cid:26) k,σ
X
many perturbation series.
+ H a∗(τ),a(τ) , (2.6)
(cid:27)
(cid:0) (cid:1)
II. INEQUIVALENT REPRESENTATIONS
where a˙ (τ) denotes the “time” derivative of a (τ).
k,σ k,σ
ThepartitionfunctionZ ofthesystemisexpressedas
For the sake of simplicity, we start by considering a
complete set of annihilation and creation field operators
ak,σ and a+k,σ of the fermion type. Let the index k run Z =Tre−βH(a+,a) (2.7)
over integer numbers over the interval k ∈ [−N,N] and
2 2
σ denote spin 1 projection of a fermion, i.e., in the operator formalism, or as the integral
2
σ =↑,↓=+,−. Z = e−S(a∗,a)D(a∗,a) (2.8)
In order to have a quantum field theory, we take the Z
limit N → ∞. The field operators a and a+ satisfy in the functionalintegralformalismofthe quantum field
k,σ k,σ
the anticommutator ring theory. From the relations (2.7) and (2.8), one evidently
seesthatexp −S(a∗,a) isinfactthekerneloftheden-
{ak,σ,a+k′,σ′} = δkk′δσσ′ (2.1a) sity matrix exp −βH(a+,a) operator, and therefore
(cid:8) (cid:9)
{a ,a } = {a+ ,a+ }=0 (2.1b) the expressions (2.7) and (2.8) should give the same re-
k,σ k′,σ′ k,σ k′,σ′ (cid:8) (cid:9)
sult if these two approaches to the quantum field theory
with the subsidiary condition are equivalent.
Next, we study a class of inequivalent representations
ak,σΦ0 =0 (2.2) of the anticommutator ring (2.1). We start from the op-
erators a and a+ obeying (2.1) and introduce the
k,σ k,σ
onthe vacuumstateΦ0 forall(k,σ). Therepresentation “unitary” transformations
space for the anticommutator ring (2.1) with the sub-
sidiary condition (2.2) can be chosen to be the Hilbert c =eiQa e−iQ, c+ =eiQa+ e−iQ, (2.9)
space H spanned by the basis vectors Ψ defined k,σ k,σ k,σ k,σ
{nk,σ}
by the formula where Q is the Hermitian operator
Ψ{nk,σ} d=efNl→im∞Yk,σ(cid:0)a+k,σ(cid:1)nkσΦ0, (2.3) Q = Nl→im∞Xk αkTk, (2.10a)
T = i(a+ a+ −a a ), (2.10b)
wherenk,σ =0,1aretheoccupationnumbersoffermions k k,+ −k,− −k,− k,+
in states (k,σ), and {n } denotes an array with an
k,σ
and α are arbitrary real parameters. The anticommu-
infinitenumberofitems0and1.Eachsuchinfinitearray k
tation relations for the transformed operators c and
{nk,σ} specifies one of the basis vectors of the Hilbert k,σ
space H. c+ are,ofcourse,thesameasthosegivenby(2.1). The
k,σ
The Hamiltonian H governing a physical system in a operator eiQ can be expressed as the infinite product
quantum field theory is a function of the operators a
k,σ
and a+ . Let its normal form be denoted by N/2
k,σ eiQ = lim 1+iT sinα −T2(1−cosα ) ,
k k k k
N→∞
H =H(a+,a). (2.4) k=Y−N/2(cid:2) (cid:3)
(2.11)
To the normal form of H(a+,a) one assigns the normal where
symbol
T2 =2a+ a a+ a −a+ a −a+ a +1.
k k,+ k,+ −k,− −k,− k,+ k,+ −k,− −k,−
H =H a∗(τ),a(τ) (2.5) (2.12)
(cid:0) (cid:1)
4
The transformations (2.9), if elaborated with (2.11), are butthereisnoproperunitarytransformationconnecting
similar to the well-known Bogoliubov-Valatin transfor- these two operatorsystems. In other words,they belong
mations [8, 9] to inequivalent representations of the same anticommu-
tator ring (2.1) or (2.17) of the field operators. Each
ck,+ = ukak,++vka+−k,−, (2.13a) inequivalent representation of (2.17) is specified by the
c = u a −v a+ , (2.13b) chosen infinite set of parameters αk entering the trans-
k,− k k,− k −k,+ formations(2.9)-(2.13). Thus,thenumberoftheinequiv-
c+k,+ = uka+k,++vka−k,−, (2.13c) alent representations of (2.17) is infinite.
c+ = u a+ −v a , (2.13d) For each inequivalent representation of (2.17), we can
k,− k k,− k −k,+ construct the transformed Hamiltonian
where H˜(c+,c)=eiQH(a+,a)e−iQ (2.18)
u =cosα and v =sinα . (2.13e) in its normal form by employing the anticommutation
k k k k
relations (2.17). The corresponding partition function
In the limit N → ∞, the operator Q given by (2.10) is
not a proper operator, but transforms every vector Ψ of Z˜=Tre−βH˜(c+,c) (2.19)
theHilbertspaceH intoΨ′ =eiQΨoftheHilbertspace
can be different from that given by (2.7) because the
H′ with unexpected properties.
operatorsH(a+,a)andH˜(c+,c)actin differentHilbert
Let us denote by ϕ any basis vector of H given
{nk} spaces H and H′, respectively.
by the formula
Inthefunctionalformalismofquantumfieldtheory,we
construct the normal symbol H˜(c∗(τ),c(τ)) of the oper-
N/2
ϕ = lim a+ a+ nkΦ , (2.14) ator H˜(c+,c) and the action functional
{nk} N→∞ k,+ −k,− 0
k=Y−N/2(cid:0) (cid:1) S˜(c∗,c) = βdτ lim c∗ (τ)c˙ (τ)
k,σ k,σ
where nk = nk,+ = n−k,− = 0,1. All the basis vectors Z0 (cid:26)N→∞k,σ
ϕ form a subspace of H. The transformation eiQ X
{nk}
transforms every basis vector ϕ into one ϕ′ = + H˜ c∗(τ),c(τ) (2.20)
{nk} {nk}
eiQϕ of H′, given by the formula (cid:27)
{nk} (cid:0) (cid:1)
foreachinequivalentrepresentationofthe anticommuta-
N/2 tor ring (2.17) of the field operators c and c+ . The
k,σ k,σ
ϕ′{nk} = Nl→im∞ δnk,1−(a+k,+a+−k,−)nk+1 sinαk corresponding partition function Z˜ in the functional in-
k=Y−N/2n(cid:2) (cid:3) tegral form
+ (a+k,+a+−k,−)nkcosαk Φ0. (2.15) Z˜= e−S˜(c∗,c)D(c∗,c) (2.21)
o Z
The result is that the scalar product Ψ,eiQϕ{nk} of gives the distinct result for each inequivalent represen-
every basis vector Ψ given by (2.3) of H is either iden- tation. One can be easily convinced by studies of con-
(cid:0) (cid:1)
tically equal to zero or equal to the infinite product crete Hamiltonians with interactions between fields that
the functionalintegral(2.21)cannotbetransformedinto
Ψ{n′k′},eiQϕ{nk} =Nl→im∞ Dssinαk+Dccosαk , that of (2.8) by the transformations (2.13),
(cid:0) (cid:1) kY,k′(cid:16) (2.16(cid:17)a) ck,+(τ) = ukak,+(τ)+vka∗−k,−(τ), (2.22a)
where ck,−(τ) = ukak,−(τ)−vka∗−k,+(τ), (2.22b)
c∗ (τ) = u a∗ (τ)+v a (τ), (2.22c)
k,+ k k,+ k −k,−
Ds = δ1,nkδ0,n′k′ −δ0,nkδ1,n′k′δk,k′, (2.16b) c∗k,−(τ) = uka∗k,−(τ)−vka−k,+(τ), (2.22d)
Dc = δnk,n′k′δk,k′, (2.16c) of the integration variables.
WeconcludethissectionbystatingthatagivenHamil-
which also diverges to zero in the limit N → ∞ for any
tonian H of a system leads to distinct results for the
suitable parameters α . Thus the Hilbert space H ′ con-
k partition function Z, and for statistical average values
tains a subspace of the state vectors ϕ′ which are or-
{nk} hAi depending onthe choseninequivalentrepresentation
thogonaltoeveryvectorΨofH.Tomaketheconclusion
ofthe anticommutatorringoffieldoperatorsinboththe
as in Haag’s paper [7], c and c+ given by (2.9) are
k,σ k,σ operator and the functional integral approach to quan-
operators satisfying the same canonical anticommutator
tumfieldtheory. Ouranalysisofinequivalentrepresenta-
ring as (2.1), i.e.,
tions of the anticommutator ring of field operators (2.1)
presented above can be generalized in a straightforward
{ck,σ,c+k′,σ′} = δkk′δσσ′ (2.17a) way to any set of quantum numbers (k,σ) and to many
{c ,c } = {c+ ,c+ }=0, (2.17b) other classes of inequivalent representations.
k,σ k′,σ′ k,σ k′,σ′
5
III. PERTURBATION SERIES where the unperturbed HamiltonianH˜ is not relatedto
0
H by any proper unitary transforamtion. In this case,
0
We begin with an elucidation of how one tacitly se- theeigenstatesΨ′µ ofH˜0 formagainacompletebasisofa
lects a single inequivalent representation of the commu- new Hilbert space H′ for anotherinequivalent represen-
tator or anticommutator ring (1.7) of field operators in tation of the commutator or anticommutator ring (1.7)
a practicalapplicationofquantumfieldtheory. Inquan- offieldoperators. Thus,inthisanotherinequivalentrep-
tum field theories with interactions between fields, there resentation one gets the partition functions
is not known even one physical example with an exact
solution. In all practical calculations, one divides the Z˜0 =Tre−βH˜0(a+,a) (3.6)
Hamiltonian H of a system into the sum
and
H =H (a+,a)+H (a+,a), (3.1)
0 I ∞ (−1)ν β ν
Z˜=Z˜ T dτ V˜(τ) , (3.7)
whereH0iscalledtheunperturbedHamiltonian,andthe 0ν=0 ν! (cid:28) (cid:18)Z0 (cid:19) (cid:29)0
remaining term H is called the perturbative part. The X
I
unperturbed Hamiltonian H0 is chosen in such a way in which are distinct from those given by (3.2) and (3.3)
order to be exactly diagonalized, and by this fact its ef- because the operators H (a+,a) and H˜ (a+,a) act in
0 0
fects are treated exactly. Its eigenstates Ψµ, where µ different Hilbert spaces, H and H′, respectively.
denotes an array with an infinite series of items, form a
Two different separations (3.1) and (3.5) of the same
complete basis of a Hilbert space H. This is the repre- Hamiltonian H correspond to two different divisions of
sentation space for a single representation of the com- the action functional S(a∗,a) as given by the formulas
mutator or anticommutator ring (1.7) of field operators
a anda+ enteringtheHamiltonianH.Thepartition S(a∗,a) = S (a∗,a)+S (a∗,a) (3.8a)
k,σ k,σ 0 I
function
S(a∗,a) = S˜ (a∗,a)+S˜ (a∗,a) (3.8b)
0 I
Z =Tre−βH0(a+,a) (3.2)
0 The corresponding partition functions
canbeexactlyevaluatedandistypicalforthechosenrep-
resentation. The total partition function Z is expressed Z = D(a∗,a)e−S0(a∗,a), (3.9a)
0
as the perturbation series Z
Z˜ = D(a∗,a)e−S˜0(a∗,a) (3.9b)
β 0
Z = Tre−βH =Z0 Texp − dτ V(τ) Z
(cid:28) (cid:26) Z0 (cid:27)(cid:29)0 are, of course, different because the functional integrals
= Z ∞ (−1)ν T βdτ V(τ) ν , (3.3) (3.9) containdifferent integrandswhichcannotbe trans-
0 ν! formed one into another by any substitution of the inte-
ν=0 (cid:28) (cid:18)Z0 (cid:19) (cid:29)0
X gration variables. In the same way, the different results
wherethe symbolTstandsforthetime-ortemperature- for the corresponding perturbation series
ordered product and
Z = D(a∗,a)e−S(a∗,a)
V(τ)=eτH0H e−τH0. (3.4)
I
Z
∞ (−1)ν
The individual terms of the perturbation series (3.3) are = Z Sν(a∗,a) , (3.10a)
in one-to-one correspondence with the perturbation se- 0 ν! I 0
ν=0
ries (1.4) and (1.5) in the functional integral method. X (cid:10) (cid:11)
Namely, exp{−S (a∗,a)} in (1.3) is the kernelof the un- Z˜ = D(a∗,a)e−S(a∗,a)
0
perturbed density matrixρ0 =exp{−βH0(a+,a)}oper- Z
β ν ∞ (−1)ν
ator. The statistical average values T dτ V(τ) = Z˜ S˜ν(a∗,a) (3.10b)
evaluated in the chosen single inequivale0nt representa0- 0 ν! I 0
tion correspond to the functional in(cid:10)teg(cid:0)raRls Sν(a∗,(cid:1)a)(cid:11) Xν=0 (cid:10) (cid:11)
I 0
in (1.5). Thus, the perturbation series (1.4), (1.5) and can be understood from the viewpoint of inequivalent
(3.3) in both the functional integral and t(cid:10)he operat(cid:11)or representations of the commutator or anticommutator
formalism of quantum field theory should give the same ring of field operators. The last two formulae may seem
results in the chosen inequivalent representation of the to represent a paradox. Namely, the result of the inte-
commutator or anticommutator ring of field operators. gration of the same functional integral depends on the
However, the splitting of the total Hamiltonian process of its integration by means of perturbation se-
H(a+,a)asgivenby(3.1)isnotunique. Onecanequally ries. For this reason, the mathematicians Kobzarev and
well divide the same Hamiltonian as Manin regard every such computations of functional in-
tegralsdone byphysicists as“half-bakedandad hoc def-
H =H˜ (a+,a)+H˜ (a+,a), (3.5) initions.”
0 I
6
However, the different results (3.10a) and (3.10b) of tional
the same functional integral are not due to its a priori
undefined expression, but are due to the existence of in-
finitelymanyinequivalentrepresentationsofthecommu-
′
tator or anticommutator ring of field operators entering SN(a∗,a) = εa∗k,σak,σ
the quantum field theory. σ=± k
X X
(3.I1n0)thweithneaxttosyecatciotnionwfeundcetmioonnasltrSa(tae∗,tah)ewphriocpherpteiers- −Ng ′a∗k,+a∗−k,−a−k′,−ak′,+,(4.1)
k,k′
mits exact summation of perturbation series and gives X
rise to infinitely many different results.
where ε and g are real parameters, N is a given even
IV. TWO EXPLICITLY SOLVABLE EXAMPLES integer number, k and k′ are integers k,k′ ∈ − N,N
2 2
with the exclusion of k = k′ = 0 which is indicated by
For the purpose of demonstrating the conclusions of theprimeonthesummationsymbols( ′),and(cid:2)a ,a∗ (cid:3)
k,σ k,σ
the two previous sections, we consider the action func- are Grassmann variables. Next, we use the identity.
P
′ 1 ′ ′
SN(a∗,a) = εa∗k,σak,σ− γ2 γ2∆∗−γ2∆∗− a∗k,+a∗−k,+ γ2∆−γ2∆− a−k′,−a∗k′,−
σ=± k (cid:18) k (cid:19)(cid:18) k′ (cid:19)
X X X X
′ ′
= γ2∆∗∆+ εa∗ a − ∆a∗ a∗ +∆∗a a
k,σ k,σ k,+ −k,− −k,− k,+
σ k k
XX X (cid:0) (cid:1)
1 ′ ′
− γ2∆∗− a∗ a∗ γ2∆− a a , (4.2)
γ2 k,+ −k,− −k,− k,+
(cid:18) k (cid:19)(cid:18) k (cid:19)
X X
where The corresponding unperturbed partition function
Z can be exactly and explicitly calculated with the
0,N
N result
γ = (4.3)
s g
Z = E2Nexp{−γ2∆∆∗}, (4.6a)
0,N
and ∆, ∆∗ are arbitrary complex numbers. We separate E = (ε2+∆∆∗)1/2. (4.6b)
the action S (a∗,a) into a sum of two terms,
N
The total partition function Z is given by the infinite
S (a∗,a)=S (a∗,a)+S (a∗,a), (4.4) N
N 0,N I,N perturbation series
where the unperturbed action S is chosen to be
0N ∞ (−1)ν
Z =Z Sν (a∗,a) . (4.7)
S = γ2∆∆∗+ ′ ε(a∗ a +a∗ a ) N 0,N ν! I,N 0
0,N k,+ k,+ k,− k,− ν=0
X (cid:10) (cid:11)
k
X (cid:8)
− ∆a∗k,+a∗−k,−−∆∗a−k,−ak,+ , (4.5a) For the purpose of making an explicit calculation of ZN
it is convenient to introduce the generating action func-
and (cid:9) tional S (a∗,a;b∗,b) defined by
0,N
1 ′
S = − γ2∆∗− a∗ a∗ b∗ ′
I,N γ2(cid:18) k k,+ −k,−(cid:19) S0,N(a∗,a;b∗,b) = S0,N + γ γ2∆− a−k,−ak,+
X (cid:18) k (cid:19)
′ X
× γ2∆− a a (4.5b) b ′
−k,− k,+ + γ2∆∗− a∗ a∗ ,
(cid:18) k (cid:19) γ k,+ −k,−
X (cid:18) k (cid:19)
X
is the perturbative term which will be treated by the (4.8)
perturbationseries. Thuswehaveinfinitelymanyunper-
turbed actions (4.5a) enumerated by arbitrary complex where b∗ and b are two complex variables. With
numbers ∆ and ∆∗. S (a∗,a;b∗b) we define the generating partition func-
0,N
7
tion Z (b∗,b) by the functional integral With the restrictive condition for the parameter ∆∆∗ in
0,N
the form
Z (b∗,b)= D(a∗,a)exp −S (a∗,a;b∗,b) ,
0,N 0,N
Z (cid:8) ((cid:9)4.9) 1− g 1− g +2 g ∆∆∗ >0.
which is a Gaussian integral and can be explicitly calcu- E2 E2 E4
lated with the result (cid:16) (cid:17)(cid:16) (cid:17)
N
1 1
Z0,N(b∗,b) = ε2+ ∆+ b ∆∗+ b∗ At the end we define the “density of the grand canon-
γ γ
(cid:26) (cid:18) (cid:19)(cid:18) (cid:19)(cid:27) ical potential” Ω in the “thermodynamic limit” as
×exp{−γ(γ∆∗∆+b∆∗+b∗∆)}.
(4.10)
1
By introducing the ratio Ω=− lim lnZN. (4.15)
N→∞N
By using the last formula, we obtain the final result
Z (b∗,b)
W (b∗,b) = 0,N
N Z
0,N
= 1+ 1 g (γ∆∗b+γ∆b∗+b∗b) N Ω = −∆∆∗ 1− g 2 1− g +2 g ∆∆∗ −1
N E2 g E2 E2 E4
(cid:26) (cid:27) (cid:26)(cid:16) (cid:17) (cid:16) (cid:17)
×exp{−γ(∆∗b+∆b∗)}, (4.11)
−1 −2lnE, (4.16)
(cid:27)
we expressthe totalpartition function (4.7) in the form
∂2
Z = lim Z exp W (b∗,b). (4.12) whichexplicitly showsthatthe functionalintegral(1.1a)
N b,b∗→0 0,N ∂b∂b∗ N with the given action functional (4.1) gives infinitely
(cid:26) (cid:27)
many grand canonical potentials Ω enumerated by ar-
Now we are prepared to take the limit N → ∞ in order bitrarycomplexnumbers ∆and∆∗. This simple exactly
to get the results (4.6)-(4.12) for the functional integrals
solvable example demonstrates explicitly that functional
with infinitely many integration variables. In this limit,
integralsinquantumfieldtheoriescannotberegardedas
the asymptotic formula for the ratio (4.11) has the form
Newton-Lebesgueintegrals. Differentresultscorrespond-
ing to distinctprocessesoftheir integrationsofthe same
g
W (b∗,b) = exp γ −1 (∆b∗+∆∗b) functionalintegralshouldnotbe regardedasad hoc def-
N E2
(cid:26) (cid:18) (cid:19) initions for a priori undefined expressions as (1.1). The
g g
+ b∗b− (∆b∗+∆∗b)2 distinct results associated with the same functional in-
E2 2E4 tegral correspond to the existence of inequivalent repre-
+O N−1/2 . (4.13) sentations of the commutator or anticommutator ring of
fieldoperatorsintheoperatorapproachtoquantumfield
(cid:27)
(cid:0) (cid:1) theory.
Thelastresultis insertedinto(4.12),andbyinterchang-
ing the order of the limits N → ∞ and b → 0, b∗ → 0, For demonstration purposes presented in this section,
we get the asymptotic formula we have selected the toy action functional (4.1) which
reminds us of an oversimplified BCS model of supercon-
N g 2 ductivity[10]. Themethodpresentedinthischaptercan
Z = E2Nexp ∆∆∗ 1−
N g E2 be straightforwardlygeneralizedandapplied,e.g.,to the
(cid:26) (cid:20)(cid:16) (cid:17) realisticBCSmodelofsuperconductivitywiththeaction
g g −1
× 1− +2 ∆∆∗ −1 +O(1) (4.14) functional
E2 E4
(cid:16) (cid:17) (cid:21) (cid:27)
β
S(a∗,a) = dτ a∗ (τ)a˙ (τ)+ξ a∗ (τ)a (τ)
k,σ k,σ k k,σ k,σ
Z0 (cid:26)k,σ
X(cid:2) (cid:3)
g
− V a∗k,+(τ)a∗−k,−(τ)a−k′,−(τ)ak′,+(τ)θ ~ωD−|ξk| θ ~ωD−|ξk′| , (4.17)
k,k′ (cid:27)
X (cid:0) (cid:1) (cid:0) (cid:1)
where k is the wave vector, σ is the spin 1 projection of is the kinetic energy of an electron counted from the
2
an electron, chemical potential µ, ω is the Debye frequency, g is
D
~2k2
ξ = −µ
k
2m
8
the squared electron-phonon coupling constant, and V freeparametersonwhichΩ is dependent. Therefore,the
is the volume of the system of electrons. By the same second law of thermodynamics restricts the class of ad-
steps as presented by the relations (4.2)-(4.15), however, missible inequivalent representations by the condition
withmoreinvolvedcalculations,onecanderivetheexact
density for the grand canonical potential ∂Ω ∆∆∗ (1−D)2
= −
∂(∆∆∗) g (1−D−2∆∆∗C)2
1 (cid:18) (cid:19)T,µ
Ω(T,µ)=− lim lnZ (4.18) ∂C
V→∞Vβ × 3C+2∆∆∗ =0 (4.23)
∂(∆∆∗)
in the form (cid:0) (cid:1)
The last condition admits only two solutions
∆∆∗
Ω(T,µ,∆∆∗) = − (1−D)2(1−D−2∆∆2C)−1
g ∆∆∗ =0 (4.24a)
2 (cid:2) βE
−1 − (2π)3β d3k ln 2cosh 2k −βξk , and 1−D =0, i.e.
Z (cid:26) (cid:20) (cid:21) (cid:27)
(cid:3) (4.19) g d3k βE
1= tanh kθ ~ω−|ξ | , (4.24b)
(2π)3 2E 2 k
k
where Z
(cid:0) (cid:1)
because the expression
E = ξ2 +∆∆∗ θ ~ω −|ξ |
k k D k
∂C
+qξkθ |ξk|−~(cid:0)ωD (cid:1) (4.20) 3C+2∆∆∗∂(∆∆∗)
is the energy spectrum o(cid:0)f elementary(cid:1) excitations
is always negative.
g d3k βE The last relation is the well-known gap equation of
D = (2π)3 2E θ ~ωD−|ξk| tanh 2k (4.21) the BCS theory of superconductivity [11]. Its solution
Z k gives the gap functions ∆ and ∆∗ as certain functions
(cid:0) (cid:1)
and of the temperature T and the chemical potential µ at
T < T , where T is the critical temperature. Thus, the
c c
∂D
C = superconductingstateoftheelectronsystemdescribedby
∂(∆∆∗) the actionfunctional(4.17)is associatedateachvalueof
T with the corresponding inequivalent representation of
with the restrictive condition on the parameter ∆∆∗ in
the anticommutator ring of the field operators a and
the form k,σ
a+ specified by the set of the parameters α given by
k,σ k
(1−D)(1−D−2∆∆∗C)>0. (4.22).
The result of the functional integral (1.1a) with the
The result (4.19) shows again that the functional in- actionfunctional(4.17)asgivenby(4.19)isnottheonly
tegral (1.1a) with the given action (4.17) gives infinitely one. One can, in fact, also find for it another class of
many densities of the grand-canonical potential Ω(T,µ) inequivalent representations of the anticommutator ring
enumeratedbyarbitrarycomplexvalues∆and∆∗which of electron field operators as discussed in [12].
are called the gap functions.
Thegapfunctions∆and∆∗ determinetheparameters
αk in the transformations (2.9)-(2.13) by the relation V. CONCLUSIONS
1 ξ
sin2αk = 1− k θ ~ωD−|ξk| . (4.22) In quantum field theories, each operator A(a+,a), as
2(cid:18) ξk2 +∆∆∗(cid:19) a function of the field operatorsa+ and a satisfying
(cid:0) (cid:1) k,σ k,σ
The given infinite sept of the parameters α specifies the the commutator or anticommutator ring of the field op-
k
erators, is an abstract object. It can be represented in
corresponding inequivalent representation of the anti-
commutator ring (2.17) of the field operators a+ and infinitely many inequivalent representations of the com-
k,σ mutator or anticommutator ring of the field operators.
a . Thus, the values of the functional integral (1.1a)
k,σ Its trace TrA(a+,a), is a number which is distinct for
with the action functional (4.17) leading to the density
each inequivalent representation.
(4.19)enumeratedbygivengapfunctions ∆and∆∗ cor-
In the functional integral formalism of quantum field
respond to distinct inequivalent representations of the theories, one associates with each operator A(a+,a) its
anticommutator ring (2.17) of the field operators a+
k,σ kernel A˜(a∗,a). The trace of the operator A(a+,a) is
and a .
k,σ expressed by the functional integral [1]
The second law of thermodynamics, however, requires
the density Ω(T,µ) atgivenvalues ofthe thermodynam- TrA(a+,a)= A˜(a∗,a)e−a∗aD(a∗,a), (5.1)
icalvariablesT and µto be minimalwith respect to any
Z
9
which should be regarded as an abstract object. All resentationsforthe sameoperator. Fromthis viewpoint,
problems of quantum field theories based on the func- theunexpectedpropertiesoffunctionalintegralsinquan-
tional integral formalism can be thought of as problems tum field theories should not be associated as with their
offindingcorrectdefinitionsandcomputationalmethods apriori undefinedexpressions,butwiththefundamental
for the functional integrals of the type (5.1). By the se- structure of quantum field theories.
lection of a method for the evaluation of the functional
integral (5.1), one selects tacitly an inequivalent repre-
sentation of the commutator or anticommutator ring of
fieldoperators. Fromthisviewpoint,functionalintegrals
Acknowledgements
inquantumfieldtheoriescannotberegardedasNewton-
Lebesgueintegralsgivinguniquevaluesasoneexpectsin
ordinaryintegralcalculus. Distinct valuescorresponding The author (M.N.) is very grateful to Prof. C. Cron-
tothesamefunctionalintegralsinquantumfieldtheories str¨om for many stimulating discussions on the relations
reflectoneofthefundamentalpropertiesofsuchtheories, betweenfunctionalintegralsandinequivalentrepresenta-
namely,theexistenceofinfinitelymanyinequivalentrep- tions in quantum field theories.
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