The multi-time correlation functions, free white noise, and the generalized Poisson statist

a r X i v :m a t h -p h /0701055v 1 22 J a n 2007The multi-time correlation functions,free white noise,and the generalized Poisson statistics in the low density limit A.N.Pechen Princeton University,Department of Chemistry,Frick Laboratory,Princeton,NJ 08544-1009E-mail:apechen@ Abstract.In the present paper the low density limit of the non-chronological multitime correlation functions of boson number type operators is investigated.We prove that the limiting truncated non-chronological correlation can be computed using only a sub-class of diagrams associated to non-crossing pair partitions and thus coincide with the non-truncated correlation functions of suitable free number operators.The independent in the limit subalgebras are found and the limiting statistics is investigated.In particular,it is found that the cumulants of certain elements coincide in the limit with the cumulants of the Poisson distribution.An explicit representation of the limiting correlation functions and thus of the limiting algebra is constructed in a special case through suitably defined quantum white noise operators.1.INTRODUCTIONThe reduced dynamics of a quantum open system interacting with a reservoir in certain physical regimes is approximated by Markovian master equations.These regimes include the weak system–reservoir interactions and dilute reservoirs and in the theoretical framework they are described by certain limits.For a weakly interacting system one considers the limit as the coupling constant goes to zero(Weak Coupling Limit,WCL) whereas for a dilute reservoir one considers the limit as the density of the reservoir goes to zero(Low Density Limit,LDL)and an appropriate time rescaling should be performed in order to get a non-trivial limit.The Markovian reduced dynamics in these limits is considered in the review papers by Spohn and Lebowitz1,2.The reduced dynamics in the LDL was considered in details later by D¨u mcke3using the method based on the quantum Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy.The total dynamics in these limits is governed by various quantum stochastic equations.There is a unique up to now approach,called the stochastic limit method, which allows an efficient derivation of the stochastic equations in the WCL.This approach is based on the quantum white noise technique and was developed by Accardi, Lu,and Volovich4.The convergence of the evolution operator of the total system in the LDL to a solution of a quantum stochastic equation was proved by Accardi and Lu5and by Rudnicki,Alicki,and Sadowski6.Recently the low density limit was investigated with the quantum white noise technique7,8.This technique,well developed for the WCL, was non-trivially modified to include the LDL and for this case was called the stochastic golden rule for the low density limit.This technique was applied to the derivation of the quantum stochastic equations in the LDL.An advantage of the obtained equations is that they,in contrast with the exact Schr¨o dinger equation,are explicitly solvable.At the same time they provide a good approximation of the exact dynamics.The approach of7,8uses the so called Fock-antiFock representation for the canonical commutation relations(CCR)algebra(this representation is unitary equivalent to the Gel’fand–Naimark–Segal representation).The difficulty with this approach is that the creation and annihilation operators in the Fock-antiFock Hilbert space do not describe creation and annihilation of physical particles and thus do not have direct physical meaning.To avoid this difficulty the investigation of the LDL directly in terms of the physicalfields was ing this approach the chronological correlation functions in the LDL were found and the corresponding stochastic equations derived.In the present paper we investigate the low density limit of the non-chronologically ordered correlation functions of boson number type operators.The investigation is related with ab initio derivations of quantum stochastic equations describing quantum dynamics of a test particle interacting with a dilute gas.Wefind the limiting truncated correlation functions of the number type operators and show that they can be computed by representing the number operators through creation and annihilation operators and then considering only a sub-class of diagrams associated to non-crossing pair partitions.This fact allows to represent the limiting truncated correlation functions as the non-truncated correlation functions of number operators of a free quantum white noise thus making a connection with the Voiculescu free probability theory.Wefind the limiting statistics and show that the cumulants of certain elements coincide in the limit with the cumulants of the Poisson distribution.The free probability theory was developed by Voiculescu around1985as a way to deal with von Neumann algebras of free groups.Then the theory was separated from this special context and began to develop as an independentfield.In particular, applications of the free independence theory to random matrices were found.The details of free probability theory and its applications to random matrices could be found,for example,in references10,11.Expectations of free random variables are characterized by diagrams associated to non-crossing pair partitions.The vanishing of crossing diagrams in the stochastic weak coupling limit for nonrelativistic QED and for the Anderson model was found in Refs4and12,respectively,thus making a connection between the WCL and free probability.The WCL is typically described by the quantum Boltzmann statistics4.In Ref13a generalized version of Boltzmann commutation relations,the so called entangled commutation relations,was found in the weak coupling limit for nonlinear interactions and possible applications to photon splitting cascades were discussed.The investigation of the multitime non-chronologically ordered correlation functions could have a connection with the behavior offluctuations in certain asymptotic regimes. The latter is described in the review paper by Andries,Benatti,De Cock and Fannes14. In that approach the limiting statistics is defined in terms of ground state distribution determined by non trivial pair partitions.The authors conjecture the appearance of exotic statistics in certain asymptotic regimes.The asymptoticfluctuations are the limiting correlation functions of appropriate centered elements and thus the results of the present paper could be applied to study thefluctuations in the low density limit.In Sec.II the truncated non-chronologically ordered correlation functions are defined and their low density limit is established(Theorem1).In Sec.III the irreducible diagrams(pair partitions)which contribute to the limiting correlation functions are found(Theorem2).In Sec.IV the limiting truncated correlation functions are represented as correlation functions of a suitable free white noise.In Sec.V we identify the independent in the limit subalgebras(Theorem4)and calculate the limiting cumulants which for some elements coincide with the cumulants of the Poisson distribution(Theorem5).In Sec.VI an explicit representation of the limiting correlation functions and thus of the limiting algebra is constructed for a special case by using suitable quantum white noise operators.2.THE CORRELATION FUNCTIONS IN THE LDLWe begin this section with construction of a general class of non-commutative probability spaces relevant for the investigation of the low density limit.The framework of a∗-probability space is used.A relation between the objects defined in this section and the model of a test particle interacting with a dilute gas is given in Appendix A.Definition1A∗-probability space is a pair(A,ω),where A is a unital∗-algebra over C andω:A→C is a state,i.e.,a linear normalized,ω(1A)=1,and strictly positive functional.Let H be a Hilbert space with inner product denoted by ·,· (called as one particle Hilbert space),{S t}t∈R a one parameter unitary group in H(a one particle free evolution),ˆn a bounded positive operator in H(density operator)such that∀t∈R, S−tˆn S t=ˆn,and B a countable set of real numbers.LetΓ(H)be the symmetric Fock space over H.For any trace class self-adjoint operator T acting in H we denote by N(T)≡dΓ(T)its second quantization operator inΓ(H)and extend this definition by complex linearity to the set of all trace class operators T(H).For any T∈T(H),ω∈B,and a positive numberε>0we define the following operator inΓ(H):e−i tω/εN T,ω,ε(t):=We will use for the correlation functions(2)and(3)also the shorter notations Wε(t1,...,t n)and Wε(ϕ1,...,ϕn).The reason for introducing the averaged operators N T,ω,ε(ϕ)and the averaged correlation functions(3)is that,as we will show below,the non-averaged operators N T,ω,ε(t)and the correlation functions(2)in the limit asε→0 become singular distributions.Clearly,one has the relationWε(ϕ1,...,ϕn)= d t1...d t n Wε(t1,...,t n)ϕ1(t1)...ϕn(t n)Definition2The truncated correlation functions W Tε(t1,...,t n)are defined for n=1 by W Tε(t1):=Wε(t1)and for n>1by induction through the relation:Wε(t1,...,t n)=W Tε(t1,...,t n)+nl=2′W Tε(t i1,...,t i k1)×W Tε(t i k1+1,...,t i k2)...W Tε(t i k l,...,t i n) where ′is the sum over i1<i2<...<i k1,i k1+1<...<i k2,...,i k l+1<...<i n.The truncated correlation functions are often used in quantumfield theory and in quantum kinetic theory15.They entirely determine the corresponding non-chronological correlation functions.Thus the investigation of the limit of the non-chronological correlation functions can be reduced to the investigation of the limit of the truncated correlation functions.We define the”projection”P E:=(2π)−1 d tS t e−i tE[it has the property P E P E′=δ(E−E′)P E]and for any k=1,2,...,n denote˜ωk=ωn+...+ωk.The following theorem states the low density limit of the truncated correlation functions.Theorem1One has the limit in the sense of distributions in variables t1,...,t n:lim ε→0W Tε,ˆn,T1,ω1,...,T n,ωn(t1,...,t n)=(2π)n−1δ(t2−t1)...δ(t n−t n−1)×δ˜ω1,0d E Tr ˆn P E+˜ω1T1P E+˜ω2T2...P E+˜ωn T n (4)where Tr denotes trace andδ˜ω1,0is the Kronecker delta symbol.The theorem is a corollary of Theorem2from Section3.3.THE NON-TRIVIAL DIAGRAMSIn the present section we investigate the low density limit of the non-chronologically ordered correlation functions for the particular case of operators of the form T l=|f l g l| andfind the diagrams which are non-trivial in the low density limit.In order to simplify the notations we will use the following energy representation for the creation and annihilation operators:A+ l :=e i t l E l/εεA+(P Elf l);A l:=1εA(S tl/εg l)(a slightly different version of the energy representation was introduced in7).One hasN Tl,ωl,ε(t l)=e−i t lωl/ε d E l A+l A l.Notice that the operator A+l is not the adjoint of A l.The symbols A l,A+lare used only to simplify the notations below.A multitime correlation function can be expressed using Gaussianity of the state ωˆn and the energy representation for the creation and annihilation operators asWε,ˆn,T1,ω1,...,T n,ωn(t1,...,t n)=exp −i n l=1ωl t l/ε ′ d E1...d E nωεˆn(A+i1A j1)...×ωεˆn(A+i k A j k)ωεˆn(A j k+1A+i k+1)...ωεˆn(A j n A+i n)(5) where ′is the sum over k=1,...,n,1=i1<i2<...<i k,j k+1<...<j n,i l≤j l for l=1,...,k and j l<i l for l=k+1,...,n.The sum contains terms of the formωεˆn(A+i1A j1)...ωεˆn(A+ikA jk)ωεˆn(A jk+1A+ik+1)...ωεˆn(A jnA+in)(6)To each such term we associate a diagram by pairing in the string A+1A1A+2A2...A+n A nthe operators A+il and A jlfor l=1,2,...n.Definition3We say that the expression(6)corresponds to a reducible diagram if there exists a nonempty subset I⊂{1,...,n}(strict inclusion)such that i l∈I⇔j l∈I. Otherwise we say that the expression(6)corresponds to an irreducible diagram.An important property of the truncated correlation functions(Def.2)is that they keep only all irreducible diagrams.The following are the examples of irreducible(first) and reducible(second)diagrams for n=2:A+1A1A+2A2A+1A1A+2A2(7) Given an reducible diagram,one can represent the set{1,...n}as a union of several disjoint subsets I1,...,I l such that the diagram contains only pairings between operators with indices from the same subsets.In this sense a general reducible diagram can be represented as a union of mutually disjoint irreducible diagrams.Examples of the truncated correlation functions,the corresponding irreducible diagrams,and their limits asε→0for n=1,2,3are given below.Example1n=1.The invariance of the state under the free evolution leads to the identity W Tε(t)≡Wε(t)≡Wε(0)= g1,ˆn f1 .Example2n=2.One hasW Tε(t1,t2)=Wε(t1,t2)−Wε(t1)Wε(t2)= d E1d E2ωεˆn(A+1A2)ωεˆn(A1A+2)= d E1d E2e i(t2−t1)(E2−E1)/εExample 3n =3.One hasW T ε(t 1,t 2,t 3)=d E 1d E 2d E 3 ωεˆn (A +1A 3)ωεˆn (A 1A +2)ωεˆn (A 2A +3)+ωεˆn (A +1A 2)ωεˆn (A 1A +3)ωεˆn (A +2A 3) This expression corresponds to the sum of the two irreducible diagrams:A +1A 1A +2A 2A +3A 3+A +1A 1A +2A 2A +3A 3In this case only the first diagram is non-zero in the limit and Lemma 1giveslimε→0W T ε(t 1,t 2,t 3)=(2π)2δ(t 3−t 2)δ(t 2−t 1) d E g 3,P E ˆn f 1 g 1,P E f 2 g 2,P E f 3 The case of arbitrary n is described by the following theorem.Theorem 2Let T l =|f l g l |,where f l ,g l ∈H for l =1,2,...,n .One has the limit in the sense of distributions in variables t 1,...,t n :lim ε→0W T ε,ˆn ,T 1,ω1,...,T n ,ωn (t 1,...,t n )=(2π)n −1δ(t 2−t 1)...δ(t n −t n −1)×δ˜ω1,0 d E g n ,P E ˆn f 1 g 1,P E +˜ω2f 2 ... g n −1,P E +˜ωn f n (9)For each n only the following irreducible diagram is non-zero as ε→0:εg j β,(1+εˆn )f i βone can write (6)as1εnexp i t n (E n −E αn )+...+t 1(E 1−E α1) ε εk F (E )+O (εk +1)and with the notations Ωl (E )=E n +...+E l −E αn −...−E αl for l =2,...,n as e i(t n −t n −1)Ωn (E )/εε εk −1F (E )+O (εk ) (12)If the expression(6)corresponds to an irreducible diagram then the functionsΩl(E) are linearly independent and,since they are linear in their arguments,the convolution δ(Ω2(E))...δ(Ωn(E))is well defined.In the case k>1,since for any l=2,...,n(see Lemma1):limε→0e i(t l−t l−1)Ωl(E)/εε...e i(t2−t1)Ω2(E)/εε...e i(t2−t1)(Ω2(E)−˜ω2)/εLet A be the algebra generated by the free white noise operators N T(t)and letφˆn be the state on A characterized byφˆn(N T(t))=Tr(ˆn T).Theorem3One has the equalitylim ε→0W Tε,ˆn,T1,0,...,T n,0(t1,...,t n)=φˆn(N T1(t1)...N Tn(t n))(16)Proof.By direct calculations using the Eq.(4)and the relation(15).The existence of the representation of the limiting truncated correlation functions by the free white noise number operators is related to the fact that only a sub-class of the non-crossing irreducible diagrams survives in the low density limit.We emphasize however,that the l.h.s.of Eq.(16)is the limit of a truncated correlation function whereas the r.h.s.contains the complete correlation function.5.INDEPENDENCE AND THE GENERALIZED POISSON STATISTICS IN THE LDLThe fact that the limiting truncated correlation functions are the distributions in variables t1,...,t n with support at t1=...=t n leads to the appearance of independent subalgebras in the low density limit.In the beginning of this section we remind the basic notions of independent subalgebras and of cumulants.Then wefind the asymptotically independent subalgebras of Aεand discuss the limiting statistics.We show that the cumulants and the moments of certain elements in the algebra Aεin the low density limit coincide with the cumulants and the moments of the Poisson distribution.Definition5Let(A,ω)be a∗-probability space.A family of unital∗-subalgebras {A i}i∈I,A i⊂A,is called independent ifω(a1...a n)=0whenever a l∈A i l,ω(a l)=0,and k=l implies i k=i l.Definition6Let(A,ω)be a∗-probability space.Cumulants of the space(A,ω)are the multilinear functionalsκn:A n→C,n≥1,uniquely determined byκ1(a):=ω(a),a∈A,and for n>1by induction through the relation:ω(a1...a n)=π,π=:{A1,...,A k}κ|Aj|((a1,...,a n)|A j)where the sum is over all partitionsπof the set{1,...,n}and”(a1,...,a n)|A”designates the set of a i with i∈A.Remark3The cumulantsκ(ε)n for a∗-probability space(Aε,ωεˆn)are directly related tothe truncated correlation ly,if a1=N T1,ω1,ε(ϕ1),...,a n=N Tn,ωn,ε(ϕn),thenκ(ε)n(a1,...,a n)=W Tε,ˆn,T1,ω1,...,T n,ωn(ϕ1,...,ϕn).For the analysis of independence in the low density limit we introduce the notion of asymptotically independent subalgebras for a∗-probability space(Aε,ωεˆn).Definition7Let(Aε,ωεˆn)be a∗-probability space for the LDL.We say that a family of subalgebras A1,ε,...,A l,εof Aεis asymptotically independent iflimε→0ωεˆn(a1,...,a n)=0whenever a l∈A i l,ε,ωεˆn(a l)=0,and k=l implies i k=i l.The next theorem identifies asymptotically independent subalgebras of Aε. Theorem4LetΛ1,...,Λl be a family of disjoint open subsets in R.Then the familyof subalgebras AΛ1,ε,...,AΛl,εis asymptotically independent.The proof follows from the fact that the truncated correlation functions become in the limit asε→0distributions in variables t1,...,t n with support at t1=t2=...=t n.Now let us analyze the statistics which appears in the low density limit.From Theorem1and the relation between the cumulants and the truncated correlation functions it follows that l-th cumulant for the element a=N T,ω,ε(ϕ)in the limit has the formκl(a,...,a)=limε→0W Tε,ˆn,T,ω,...,T,ω(ϕ,...,ϕ)=1|k||k′|)−1χ[0,√λ](|k′|),whereλis a positive number andχ[0,√λ].Letϕ0(t)=(2π)−1χ[0,2π](t).Theorem5Let aλ=N Tλ,ω,ε(ϕ0),where Tλandϕ0are defined as above.Then for anyl∈N one hasκl(aλ,...,aλ)=λδω,0or equivalently,the cumulants of the element aλwithω=0coincide in the low density limit with the cumulants of the Poisson distribution with expectation equal toλ. Proof.The proof of the theorem is based on the direct calculation of the cumulants using Eq.(17).One has1E,√λ](√Moments of the element aλwithω=0in the low density limit are equal to the sum over all partitions of the limiting cumulants and given by Touchard polynomials:lim ε→0ωεˆn(a nλ)=nk=1S(n,k)λkwhere S(n,k)is a Stirling number of the second kind,i.e.,the number of partitions of a set of size n into k disjoint non-empty subsets.The limiting moments coincide with the moments of the Poisson distribution with expectation equal toλ.For a1one has limε→0ωεˆn(a n1)=B nwhere B n is the n-th Bell number,i.e.,the number of partitions of a set of size n.The Bell numbers are the moments of the Poisson distribution with expectation equal to1.6.AN OPERATOR REPRESENTATION OF THE LIMITING CORRELATION FUNCTIONSIn the present section we explicitly realize the limiting correlation functions as correlation functions of certain operators acting in a suitable Hilbert space.Presence of delta functions in the limiting correlation functions suggests that they can be represented as correlation functions of certain white noise operators.Here such a representation is constructed in the special case using the results of7.Let g0,g1∈H satisfy the condition g0,S t g1 =0for any t∈R.Define for n,m=0,1the Hilbert space K nm:=L2(Spec H1,dµnm),where Spec H1⊂R is the spectrum of H1and dµnm:= g n,P E g n g m,P Eˆn g m d E.Let K:= n,m=0,1K nm and let H W N:=Γ(L2(R,K))be the symmetric Fock space over the Hilbert space of square integrable K-valued functions on R(abbreviation WN here stands for White Noise). Using the natural decomposition H W N= n,m=0,1Γ(L2(R,K nm))one can define the creation and annihilation operator valued distributions B±m,n(E,t)acting in H W N and satisfying the canonical commutation relations:[B−m,n(E,t),B+m′,n′(E′,t′)]=2πδ(t′−t)δ(E′−E) g m,P E g m′ g n′,P Eˆn g n (18) The operator valued distributions B±m,n(E,t)are called time-energy quantum white noise due to the presence ofδ(t′−t)δ(E−E′)in(18).Let define the number operators˜Nm,n(E,t):= n′=0,11Proof.r.h.s.of(19)has the formΩ,N g m1,g n1(t1)...N g m k,g n k(t k)Ω =d E1...d E k Ω,[˜N m1,n1(E1,t1)+B−n1,m1(E1,t1)+B+m1,n1(E1,t1)]...[˜N mk,n k(E k,t k)+B−nk,m k(E k,t k)+B+mk,n k(E k,t k)]ΩLet us denote˜N m,n(t):= d E˜N m,n(E,t).The truncated correlation function corresponds to the termd E d E′ Ω,B−n1,m1(E,t1)˜N m2,n2(t2)˜N m3,n3(t3)...˜N m k−1,n k−1(t k−1)B+m k,n k(E′,t k)Ω (20)Notice that˜N m,n(t)Ω=0.Therefore(20)equals tod E d E′ Ω,[...[[B−n1,m1(E,t1),˜N m2,n2(t2)],˜N m3,n3(t3)]...˜N m k−1,n k−1(t k−1)]B+m k,n k(E′,t k)ΩThe commutators can be calculated by induction using the canonical commutation relations(18).The result is(2π)k−2δ(t2−t1)...δ(t k−1−t k−2)d E d E′ g n1,P E g m2 ... g n k−2,P E g m k−1× Ω,B−n k−1,m1(E,t k−1)B+m k,n k(E′,t k)Ω (21) The last two-point correlation function can be calculated using the commutation relations(18).This gives for(21)the expression(2π)k−1δ(t2−t1)...δ(t k−t k−1)d E g n1,P E g m2 ... g n k−1,P E g m1 g n k,P Eˆn g m1which coincides with the r.h.s.of(9)in the caseω1=...=ωk=0. Remark4The limiting correlation functions could be represented as expectations of certain quantum white noise operators in the general case if one could construct a Hilbert space H W N,a vectorΩ∈H W N,and operator valued distributions B±f,g(E,t) and˜N f,g(E,t)in H W N with the property B−f,g(E,t)Ω=˜N f,g(E,t)Ω=0and satisfying the commutation relations[B−f,g(E,t),B+f′,g′(E′,t′)]=2πδ(t′−t)δ(E′−E) f,P E f′ g′,P Eˆn g (22)[B−f,g(E,t),˜N f′,g′(E′,t′)]=2πδ(t′−t)δ(E−E′) f,P E f′ B−g′,g(E,t)(23)[˜N f,g(E,t),˜N f′,g′(E′,t′)]=2πδ(t′−t)δ(E′−E)[ g,P E f′ N f,g′(E,t)− g′,P E f N f′,g(E,t)](24) Suppose there exist such operators.Define N f,g(t):= d E[˜N f,g(E,t)+B−g,f(E,t)+B+f,g(E,t)].Then one can prove exactly in the same way as in Theorem6thatlim ε→0Wε,ˆn,|f1 g1|,0,...,|f n g n|,0(t1,...,t n)= Ω,N f1,g1(t1)...N fn,g n(t n)ΩACKNOWLEDGEMENTSThe author is grateful to Luigi Accardi for useful discussions and for kind hospitality inthe Centro Vito Volterra of Rome University”Tor Vergata”where a significant part ofthis work was done.Special thanks to an anonymous referee for several suggestionsimproving the quality and content of the paper.The author acknowledges partialsupport from the grant RFFI-05-01-00884-a.APPENDIX AHere we make a connection between the objects defined in section II and the model ofa test particle interacting with a dilute Bose gas(see Ref.9for details).The one particle Hilbert space for this model has the form H≡L2(R3),where R3is the3-dimensional coordinate or momentum space.The one particle free evolutionis a unitary group S t≡e i tH1whose generator H1in the momentum representation is the multiplication operator by the functionω(k)=|k|2/2m,where m is the massof a gas particle.The test particle is characterized by its Hilbert space H S and itsfree Hamiltonian H S acting in H S which is assumed to have a discrete spectrum.Thediscrete set B is the set of all transition frequencies of the test particle,or equivalently,the spectrum of its free Liouvillean−i[H S,·].The dynamics of a test particle interacting with a gas is described by an evolutionoperator U(t)acting in H S⊗Γ(H)and satisfying in the interaction picture,after the time rescaling t→t/ε,the following Schr¨o dinger equationd U(t/ε)Lemma1One has the limit in S′(R)×S′(R)limε→0e i tx/εεf(x)φ(t)=limε→0 dτφ(ετ)d xe iτx f(x)=limε→0 dτφ(ετ)˜f(τ)Since˜f∈S(R),the functionφ(ετ)˜f(τ)satisfies the conditions of the Lebesgue lemma which allows to exchange the limit and integration in the last expression.Thus I=φ(0) dτ˜f(τ)=2πφ(0)f(0) 1H.Spohn and J.L.Lebowitz,Adv.Chem.Phys.38,109(1978).2H.Spohn,Rev.Mod.Phys.52,569(1980).3R.D¨u mcke,Comm.Math.Phys.97,331(1985).4L.Accardi,Y.G.Lu,and I.V.Volovich,Quantum Theory and Its Stochastic Limit (Springer,Berlin,2002).5L.Accardi and Y.G.Lu,Comm.Math.Phys.141,9(1991).6S.Rudnicki,R.Alicki,and S.Sadowski,J.Math.Phys.33,2607(1992).7L.Accardi,A.N.Pechen,and I.V.Volovich,J.Phys.A:Math.Gen.35,4889(2002). 8L.Accardi,A.N.Pechen,and I.V.Volovich,Infinite Dimen.Anal.,Quantum Probab.,Relat.Top.6,431(2003).9A.N.Pechen,J.Math.Phys.45,400(2004).10D.V.Voiculescu,K.J.Dykema,and A.Nica,Free Random Variables CRM monograph series1,American Mathematical Society,1992.11D.Voiculescu,Lectures on Free Probability Theory,Lecture Notes in Math.1738, 279–349(Springer,Berlin,2000).12L.Accardi,Y.G.Lu,and V.Mastropietro,Infinite Dimen.Anal.,Quantum Probab., Relat.Top.1,467(1998).13L.Accardi,I.Ya.Aref’eva,and I.V.Volovich,Proc.Steklov Inst.Math.228,106 (2000).14J.Andries,F.Benatti,M.De Cock,and M.Fannes,Rev.Math.Phys.12,921(2000). 15N.N.Bogoliubov,A.A.Logunov,A.I.Oksak,and I.T.Todorov,General Principles of Quantum Field Theory(Dordrecht,Boston,Kluwer Academic Publishers,1990).。