The Fermion number processes as a functional central limit of quantum Hamiltonian models

The Fermion Number Processes as a Functional Central Limitof Quantum Hamiltonian Models

L.Accardi Y.G.Lu∗

Centro Matematico V.Volterra

Dipartimento di Matematica

Universita di Roma II1*On leave of absence from Beijing Normal University

ABSTRACT

In the present paper, we investigate, in the Fermion case, how the numberprocesses arise from a limit of a quantum Hamiltonian model. Our conclusionis that the time evolution of a certain quantum Hamiltonian model tendsto the solution of a quantum stochastic differential equation driven by theFermion number processes.

1

1

1 §1. Introduction

In the series papers [1, · · ·, 6, 16], we have investigated for the Boson case thelow density limit of a quantum Hamiltonian system and shown that the timeevolution of the quantum Hamiltonian system tends to a quantum stochasticprocess which satisfies a quantum stochastic differential equation diven byquantum Poisson processes.

The present paper is devoted to the Fermion analogue of [1] and for sakeof bravity we shall omit here the major part of the motivations of the problemand prefer to suggest readers refrence the section “Introduction” of [1, 6].

Following the pattern of [3], we formulate the problem for a general quasi-free state and we prove the convergence of the kinematical process of thecollective number vectors to Fermion Brownian motion in the general case.Starting from Section 3.) we restrict our attention to the Fock case.

Let H0 denote the system Hilbert space; H1 the one particle reservoirHilbert space and W (H1) the CAR-algebra on H1, i.e. the algebra generatedby the set

{A(f) : f ∈ H1} (1)

where, A(f) is the Fermion annihilation operator. Let H be a self-adjointbounded below operator on H1 and z, β positive real numbers interpretedrespectively as density of the reservoir particles and inverse temperature.Denote ϕ the Fock characterized by the condition:

ϕ(A+(f)A(g)) = < f, g > , ∀f ∈ H1 (2)

and let {H, π,Φ} be the GNS - triple of {W (H1), ϕ}, so that

< Φ, π(A+(f))π(A(g))Φ > = ϕ(A+(f)A(g)) (3)

We shall write A (resp. A+) for π ◦ A (resp. π ◦ A+). Let St be a unitarygroup on B(H1) (the one particle free evolution of the reservoir). Thatthe second quantization of St, denoted Γ(St), leaves ϕ invariant hence it isimplemented, in the GNS representation, by a unitary 1-parameter group Vtwhose generator HR is called the free Hamiltonian of the reservoir. As in [3]we assume that there exists a non zero subspace K of H1 (in all the examplesit is a dense subspace) such that∫

R

| < f, Stg > |dt <∞ , ∀f, g ∈ K (4)

2

Let be given a self-adjoint operator HS on the system space H0, called thesystem Hamiltonian. The total free Hamiltonian is defined to be

H(0) := HS ⊗ 1 + 1⊗HR (5)

We define the interaction Hamiltonian V as in [1] i.e., we fix two functionsg1, go ∈ K and define

V := i(D ⊗ A+(g0) · A(g1) − D+ ⊗ A+(g1) · A(g0)

)= i

∑ε∈{0,1}

Dε ⊗ A+(gε) · A(g1−ε) (6)

with the notationsD0 = D , D1 = −D+ (7)

and where D is a bounded operator on Ho satisfying

exp (−itHS) ·D · exp (itHS) = D (8)

Moreover we assume that g0 and g1 have disjoint energy spectra, i.e.

< g0, Stg1 > = 0 , ∀t ∈ R (9)

More general interactions will be discussed in subsequent papers.The condition (1.9) is natural and has already been used in the literature

on the weak coupling limit (cf. [8, 8a, 8b]). With the condition (1.9), thecondition (1.8) is also natural since a typical example for D in quantum opticsis D = |0 >< 1|, where |1 >, |0 > are eigenvectors of the system HamiltonianH0 (rotating wave appraximation). This corresponds to [H0, D] = (ω1 −ω0)D (ω1, ω0 are the eigenvalues). The condition (1.8) corresponds to takingω1 = ω0, but the choice ω1 6= ω0 results only in a trival shift in the oneparticle reservoir Hamiltonian (c.f. Section 5 in [6] for the detail). Alsofrom point of view of mathematics, the difference beween the condition (1.8)and the general condition (N-levels condition) is, as what we have shown in[3], only to applying (many times) Reimann-Lebesgue Lemma — of course adifferent quamtum process is obtained in N-levels case but the difference isnot fundamental (cf.[3], also for the weak coupling case [9]).

With these notations, the total Hamiltonian is

Htotal := HS ⊗ 1 + 1⊗HR + V (10)

3

and the wave operator at time t is defined by

Ut := exp(−itH(0)

)· exp (itHtotal) (11)

Therefore we have the equation

d

dtUt =

1

iV (t)Ut ; U(0) = 1 (12)

where,

V (t) := exp (−itHS ⊗ 1) V exp (itHS ⊗ 1) = i∑

ε∈{0,1}

Dε⊗A+(Stgε)A(Stg1−ε)

(13)Moreover the solution of (1.12) is given by the iterated series

Ut =∞∑n=0

∫ t

0

dt1 · · ·∫ tn−1

0

dtn(−i)nV (t1) · · ·V (tn) (14)

which is norm convergent since the field operator are bounded.An important role in the present paper will be played by the collective

number vectors defined by

u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N) (15)

where u ∈ Ho, and for each n ∈ N, f1, · · · , fn ∈ H1

Φn(fk, k ∈ {1, · · · , n}) := A+(f1) · · ·A+(fn)Φ (16)

From Lemma (3.2) of [10], we know that the assumption (1.4) implies thatthe sesquilinear form (·|·) : K ×K −→ C defined by

(f |g) :=

∫R

< f, Stg > dt , f, g ∈ K (17)

defines a pre-scalar product on K. We denote {K, (·|·)}, or simply K, thecompletion of the quotient of K by the zero (·|·) -norm elements. The analogy

with the new techniques, developed in [13], for the weak coupling limit,suggests to consider the limit, as z → 0 of expressions of the form

< u, ( · )v > ⊗ϕQz(1⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

4

v ⊗ Ut/z2(X ⊗ 1)U+t/z2 1⊗ ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′)

)(18)

In analogy with the strategy of [1], the first step in our investigation willbe to control the following limit:

limz→0

< u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

Ut/z2v ⊗ ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) >

= limz→0

< u⊗ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),∞∑n=0

∫ t

0

dt1 · · ·∫ tn−1

0

dtn(−i)nV (t1) · · ·V (tn)

ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (19)

Let first outline the common and different points between the presentwork and [1]:

Since the Wick ordered form of the product

A+(f1)A(g1) · · ·A+(fn)A(gn) (20)

(this type of product is the main subject to be considered in the low densitylimit) has, both in the Boson and the Fermion cases, the effectively sameform–the only difference is some power of (–1) which is due to the differentcommutation relations, therefore one can hope that

1. The negligible terms will be similar to th Boson case.

2. The uniform estimate theorem in [1] can be used directly to the presentsituation.

3. The limit of the non–negligible terms is similar to the Boson case.

Exactly as in the Boson case, the estimates needed to solve this problemwill allow, with minor modifications, to control more general situations (cf.[1]). In order to formulate our result, let us recall from [13] the definition ofthe Fermion Brownian Motion:

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DEFINITION(1.1) Let K be a Hilbert space, T an inteval in R. Let0 ≤ Q ≤ 1 be a self-adjoint operator on K and let

{HQ, πQ,ΦQ} (21)

denote the GNS representation of the CAR algebra over L2(T, dt;K) withrespect to the quasi-free state ϕQ on W (L2(T, dt;K)) characterized by

ϕQ(A+(ξ)A(ξ′)) =< ξ,1⊗ (1−Q)

2ξ′ > ; ξ, ξ′ ∈ L2(T, dt;K) (22)

The quantum stochastic process{Γ(L2(T, dt;K)) , A(χ(s,t] ⊗ f) , A+(χ(s,t] ⊗ f) ; (s, t] ⊆ T , f ∈ K

}(23)

where A(·), A+(·) denote respectively the annihilation and creation fields inthe representation (1.23), is called the Q-Fermion Brownian Motion onL2(T, dt;K). The Fock Fermion Brownian Motion corresponds to the choiceof Q = 1.

Our main result in this paper is to prove that, the limit (1.19) exists andis equal to

< u⊗ΨN(χ[Sk,Tk]⊗fk, k = 1, · · · , N), U(t)·v⊗ΨN ′(χ[S′k,T′k]⊗f ′k, k = 1, · · · , N ′)

(24)where {H, A,A+,Ψ} is the Fock Brownian motion on L2(R, dt;K) ∼= L2(R)⊗K and U(t) satisfies a quantum stochastic differential equation driven bypurely discontinuous noises in the sense of [14] and [15], whose form is givenby (5.28). §2. The noise space

We know from [1] that for each S, T, S ′, T ′ ∈ R, and f, f ′ ∈ K satisfying(1.6), one has

limz→0

< z

∫ T/z2

S/z2

Sufdu, z

∫ T ′/z2

S′/z2

Suf′du >=< χ[S,T ], χ[S′,T ′] >L2(R) ·(f |f ′)

(25)Moreover, the limit is uniform for S, T, S ′, T ′ in a bounded set in R.

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THEOREM(2.1) For eachN,N ′ ∈ N, f1, · · · , fN , f ′1,· · ·,f ′N ′ ⊂ K, {Sh, Th}Nh=1,

{S ′h, T ′h}N′

h=1 ⊂ R

limz→0

< ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) >

=< ΨN(χ[Sk,Tk] ⊗ fk, k = 1, · · · , N),ΨN ′(χ[S′k,T′k] ⊗ f ′k, k = 1, · · · , N ′) > (26)

PROOF The proof is similar to that of Lemma (2.1) of [1]. The onlydifference being that now we have number rather then coherent collectivevectors By expanding the scalar product in the left hand side of (2.2) andusing the CAR, one finds

δN,N ′∑

σ,ε∈SN

(−1)‖ε‖+‖σ‖N∏k=1

z2

∫ Tσ(k)/z2

Sσ(k)/z2

du

∫ T ′ε(k)

/z2

S′ε(k)

/z2

dv < SuQ+fσ(k), SvQ+f′ε(k) >

(27)which, as z → 0, (2.7), by formula (2.1), tends to

δN,N ′∑

σ,ε∈SN

(−1)‖ε‖+‖σ‖N∏k=1

< χ[Sσ(k),Tσ(k)], χ[S′ε(k)

,T ′ε(k)

] >L2(R) ·(fσ(k)|f ′ε(k))

=< ΨN(χ[Sk,Tk] ⊗ fk, k = 1, · · · , N),ΨN ′(χ[S′k,T′k] ⊗ f ′k, k = 1, · · · , N ′) > (28)

Since the limit (2.1) corresponds to the 0–th term in the expansion (1.14),Theorem (2.1) shows that our limit processes, if it exists, lives on the Hilbertspace Γ(L2(R, dt) ⊗ K))–the Fermi Fock space over L2(R) ⊗ K, i..e. thespace of the Fermi Fock Brownian Motion. §3 The Collective Terms and

the Negligible Terms

Starting from the iterated series (1.14) and using (1.13), one has

(−i)nV (t1) · · ·V (tn) =∑

ε∈{0,1}nDε(1) · · ·Dε(n)⊗

7

⊗A+(St1gε(1))A(St1g1−ε(1)) · · ·A+(Stngε(n))A(Stng1−ε(n)) (29)

In the right hand side of (3.1) the operator on the system space is rathersimple and the most important thing is to know what is the contribution ofthe product of creation and annihilation operators. In order to do this, asusual, we shall bring that product to the normal ordered form. This is donein the following:

THEOREM(3.1) For each n ∈ N, the normal ordered form of theproduct

A+(St1gε(1))A(St1g1−ε(1)) · · ·A+(Stngε(n))A(Stng1−ε(n)) (30)

is equal to

n−1∑m=0

∑2≤q1<···<qm≤n

(−1)(n,{qh}mh=1)

m∏h=1

< Stqh−1g1−ε(qh−1), Stqhgε(qh) >

∏α∈{1,···,n}\{qh}mh=1

A+(Stαgε(α)) ·∏

α∈{1,···,n}\{qh−1}mh=1

A(Stαgε(α))

+n−1∑m=0

∑2≤q1<···<qm≤n

′∑(p1,q1,···,pm,qm)

ϑm∏h=1

< Stphg1−ε(ph), Stqhgε(qh) >

∏α∈{1,···,n}\{qh}mh=1


α∈{1,···,n}\{ph}mh=1

A(Stαgε(α))

=: In(ε) + IIn(ε) (31)

where, ϑ = ±1 and (n, {qh}mh=1) is defined as∑α∈{1,···,n−1}\{qh−1}mh=1

|{j, j > α, j ∈ {1, · · · , n} \ {qh}mh=1| (32)

The sum∑′

(p1,q1,···,pm,qm) means the sum for all 1 ≤ p1, · · · , pm ≤ n satisfying

|{ph}mh=1| = m ( the cardinality of the set {ph}mh=1 is equal to m), ph < qh forany h = 1, · · · ,m and ph < qh − 1 for some h = 1, · · · ,m.

REMARK In the second term of (3.3) (type II), the value of ϑ is notrelevant because we shall majorize the modulus of the sum with the sum of

8

the moduli (for which the value of ϑ is irrelevant) and then we prove thatthe latter tends to zero.

PROOF The only difference between the proof of this Lemma and thatof Lemma (3.1) in [6] is the precise computation of the exponent of (–1) in thetype I term, i.e. of the quantity (3.4). This is achived as follows: by bringingto normal form the products of the creation and annihilation operators in(3.1) and arguing as in Lamma (3,1) of [6], one arrives to an expression whichdiffers from (3,2) only by the replacement of the power of (–1) and by anunknown factor ϑ.

In order to compute this factor, denote {βh}n−mh=1 = {1, · · · , n}\{qh−1}mh=1

withβ1 < · · · < βn−m (33)

the indices which label the annihilators which have not been used to producescalar products. Then notice that to move A(Stβn−mg1−ε(βn−m)) to the right

hand side of A+(Stngε(n)) one needs to exchange A(Stβn−mg1−ε(βn−m)) with thecreators which are the right hand side of it and have not not used to producescalar products, i.e. A+(Stjgε(j)) for j > βn−m, and j ∈ {1, · · · , n} \ {qh}mh=1,so one gets a factor

(−1)|{j,j>βn−m, j∈{1,···,n}\{qh}mh=1| (34)

The same argument shows that to move A(Stβn−m−1g1−ε(βn−m−1)) to the left

hand side ofA(Stβn−mg1−ε(βn−m)) one needs to exchangeA(Stβn−m−1g1−ε(βn−m−1))

with A+(Stjgε(j)) for j > βn−m−1, and j ∈ {1, · · · , n} \ {qh}mh=1, So, one getsa factor

(−1)|{j,j>βn−m−1, j∈{1,···,n}\{qh}mh=1| (35)

Repeating the argument n −m times (i.e. once for each of the βj in (3.5)),the factor (–1) to the power (3.4) arises.

Now let investigate the contributions of terms In(ε) and IIn(ε). Firstof all we have: THEOREM(3.2) For each N,N ′ ∈ N, n ∈ N, f1, · · ·,fN ,

f ′1,· · ·,f ′N ′ ∈ K, {Sh,Th}Nh=1, {S ′h, T ′h}N′

h=1 ⊂ R, ε ∈ {0, 1}, let the term IIn(ε)be defined by (3.3). Then

limz→0

< ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

9

IIn(ε)ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) >= 0 (36)

PROOF By the definition of IIn(ε) (see (3.3)) and letting the creation,annihilation operators in IIn(ε) act on the number vectors in the left handside of (3.8), one shows that the module of the scalar product in the left handside of (3.8) is dominated by

n−1∑m=0

∑2≤q1<···<qm≤n

′∑(p1,q1,···,pm,qm)

z2(n−m)

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

m∏h=1

| < Stphg1−ε(ph), Stqhgε(qh) > |∑

1≤α1,···,αm≤N, 1≤β1,···,βm≤N ′σ({αj, βj}, N,N ′,m)

n−m∏h=1

∫ Tαh/z2

Sαh/z2

| < Sufαh , Stαhgε(αh) > | ·n−m∏h=1

| < Stβhg1−ε(αh),

∫ T ′βh/z2

S′βh/z2

Suf′βh> |

(37)where, σ({αj, βj}, N,N ′,m) is the modulus of the scalar product of a pairof collective number vectors, i.e.

|< ΦN−m(z

∫ Tα/z2

Sα/z2

Sufαdu, α ∈ {1, · · · , N} \ {αh}mh=1),

ΦN ′−m(z

∫ T ′β/z2

S′β/z2

Suduf′β, β ∈ {1, · · · , N ′} \ {βh}mh=1) > |

hence, by Theorem (2.1) a convergent, and therefore bounded quantity, asz → 0.

The factor in the last line of (3.9) is majorized by(max

F∈{fh}Nh=1∪{f′h}N′h=1,G∈{g0,g1}

∫ ∞−∞| < F, St G > |dt

)2(n−m)

(38)

The factor given by the first two lines in (3.9), up to a constant, is the sameas the right hand side of (3.16) of [1] and there we have proved that it tends,as z → 0, to zero. Thus the thesis follows.

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In order to compute the limit of the type I terms we rewrite the term In(ε)in another form in which the exponent of the factor −1 has an expressionmuch clearer than formula (3.3).

LEMMA(3.3) For each n ∈ N, ε ∈ {0, 1}n

In(ε) =n∑

m=1

∑1=q1<q2<···<qm≤n

(−1)12m(m−1)

∏α∈{1,···,n}\{qh}mh=1

< Stα−1g1−ε(α−1), Stαgε(α) >

m∏h=1

A+(Stqhgε(qh)) ·m∏h=1

A(Stqh+1−1g1−ε(qh+1−1)) (39)

where, qm+1 := n+ 1.

PROOF It is clear that when we bring the product (3.2) to the normalordered form, there will exist m (≤ n) creators not used to produce the scalarproducts with annihilators. Moreover in the product (3.2), A+(St1gε(1)) is inordered position, so m ≥ 1. Label the remaining creators with {qh}mh=1. Thismeans that the creators

{A+(Stαgε(α)); α ∈ {1, · · · , n} \ {qh}mh=1} (40)

have been used to produce scalar products with the annihilators

{A(Stα−1g1−ε(α−1)); α ∈ {1, · · · , n} \ {qh}mh=1} =

= {A(Stαg1−ε(α)); α ∈ {1, · · · , n} \ {qh}mh=1 − 1} (41)

i.e. the remaining annihilators are{A(Stαg1−ε(α)); α ∈ {1, · · · , n} \ {{1, · · · , n} \ {qh}mh=1 − 1}

}= {A(Stqh+1−1g1−ε(qh+1−1))}mh=1 (42)

The factor (−1)12m(m−1) comes from the following exchanges:

A(Stqm−1g1−ε(qm−1)) with A+(Stqmgε(qm)) (this gives the factor −1);A(Stqm−1−1g1−ε(qm−1−1)) with A+(Stqm−1

gε(qm−1))A+(Stqmgε(qm)) (this gives

the factor (−1)2); ... ;A(Stq2−1g1−ε(q2−1)) with A+(Stq2gε(q2)) · · ·A+(Stqmgε(qm)) (this gives the

factor (−1)m−1).

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THEOREM(3.4) For each N,N ′ ∈ N, n ∈ N, f1, · · ·,fN , f ′1,· · ·,f ′N ′ ∈ K,

{Sh,Th}Nh=1, {S ′h, T ′h}N′

h=1 ⊂ R, ε ∈ {0, 1}, the limit

limz→0

< ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

In(ε)ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (43)

exists and is equal to

n∑m=1

∑1=q1<q2<···<qm≤n

m∏r=1

qr+1−1∏h=qr+1

(g1−ε(h−1)|gε(h))−

∫0≤tqm≤tqm−1≤···≤tq1≤t

dtqm · · · dtq1

∑1≤x1,···,xm≤N|{xh}mh=1|=m

m∏h=1

(−1)(xh,N)χ[Sxh ,Txh ](tqh)(fxh|gε(qh))

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(yh,N′)χ[S′yh

,T ′yh](tqh)(g1−ε(qh+1−1)|f ′yh)

< ΨN−m(χ[Sα,Tα] ⊗ fα, α ∈ {1, · · · , N} \ {xh}mh=1),

ΨN ′−m(χ[S′α,T′α] ⊗ f ′α, α ∈ {1, · · · , N ′} \ {yh}mh=1) > (44)

where Ψ is the vacuum vector of Γ(L2(R; dt,K)),

(xh, N) := |{1, · · · , xh} \ {xα}h−1α=1|

and for f, g ∈ K the half–scalar product (f |g)− is defined by

(f |g)− :=

∫ 0

−∞dt < f, Stg > (45)

12

PROOF Clearly, for each n ∈ N, m ≤ n, 1 = q1 < q2 < · · · < qm ≤ n,qm+1 := n+ 1

{1, · · · , n} \ {qh}mh=1 =m⋃r=1

{qr + 1, · · · , qr+1 − 1} (46)

So, one can rewrite the product of scalar products in the right hand side of(3.11) in the form:

∏α∈{1,···,n}\{qh}mh=1

< Stα−1g1−ε(α−1), Stαgε(α) >=m∏r=1

qr+1−1∏h=qr+1

< Sth−1g1−ε(h−1), Sthgε(h) >

(47)Using (3.11) and (3.18) in (3.15) one finds that the limit (3.15) is equal tothe limit of

n∑m=1

∑1=q1<q2<···<qm≤n

(−1)12m(m−1)z2m

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

m∏r=1

qr+1−1∏h=qr+1


< ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

m∏h=1

A+(Stqhgε(qh)) ·m∏h=1

A(Stqh+1−1g1−ε(qh+1−1))

ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (48)

Letting the creators in (3.19) act on the number vectors in the left hand sideof the scalar product, one has

1∏h=m

A+(Stqhgε(qh))ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N) =

=∑

1≤x1,···,xm≤N|{xh}mh=1|=m

m∏h=1

(−1)(xh,N)

∫ Txh/z2

Sxh/z2

< Sufxh , Stqhgε(qh) > du

13

ΦN−m(z

∫ Tα/z2

Sα/z2

Sufαdu, α ∈ {1, · · · , N} \ {xh}mh=1) (49)

where we have used the symbol∏1

h=m to denote the product of operatorswith decreasing time-indices. Similarly, letting the annihilators in (3.19) acton the ΦN ′–number vectors and changing their order, using the CAR, so toobtain a sequence of decreasing time-indices, we obtain the expression

(−1)m(m−1)

2

1∏h=m

A(Stqh+1−1g1−ε(qh+1−1)) (50)

acting on the ΦN ′–number vectors in (3.19). Now we can apply (3.20) andthis leads to the result:

(−1)12m(m−1)

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(yh,N′)

∫ T ′yh/z2

S′yh/z2

< Stqh+1−1gε(qh+1−1), Svf′xh> dv

ΦN ′−m(z

∫ T ′β/z2

S′β/z2

Suf′βdu, β ∈ {1, · · · , N ′} \ {yh}mh=1) > (51)

Summing up, (3.19) is equal to

n∑m=1

∑1=q1<q2<···<qm≤n

(−1)12m(m−1)z2m

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

m∏r=1

qr+1−1∏h=qr+1


∑1≤x1,···,xm≤N|{xh}mh=1|=m

m∏h=1

(−1)(xh,N)

∫ Txh/z2

Sxh/z2

< Sufxh , Stqhgε(qh) > du

(−1)12m(m−1)

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(yh,N′)

∫ T ′yh/z2

S′yh/z2

< Stqh+1−1gε(qh+1−1), Svf′xh> dv

< ΦN−m(z

∫ Tα/z2

Sα/z2

Sufαdu, α ∈ {1, · · · , N} \ {xh}mh=1),

14

ΦN ′−m(z

∫ T ′β/z2

S′β/z2

Suf′βdu, β ∈ {1, · · · , N ′} \ {yh}mh=1) > (52)

By Theorem (2.1) the last scalar product in (3.23) tends to



and the same arguments as in the proof of Lemma (3.4) of [1] show that thet–integral term in (3.27) tends to

(g1−ε(h−1)|gε(h))−


dtqm · · · dtq1χ[Sxh ,Txh ](tqh)(fxh|gε(qh))

χ[S′yh,T ′yh

](tqh)(g1−ε(qh+1−1)|f ′yh) (54)

This proves our result.

§4. The limit of the non-negligible terms

In the previous section we have discussed the limit (1.19) for each fixedn and our main results are Theorems (3.2) and (3.4). The present section isdevoted to investigate:

1. the condition to exchange the limit z → 0 with the sum over n ∈ N;

2. the explicit form of the limit (1.19).

In the following, we shall use the notation

‖g‖2− := max

ε,σ∈{0,1}

∫ 0

−∞| < gε, Stgσ > |dt (55)

THEOREM(4.1) For each n,N,N ′ ∈ N, {fh}Nh=1, {f ′h}N′

h=1 ⊂ K,

∑ε∈{0,1}n

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn| < ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

15

(In(ε) + IIn(ε))ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > |

≤ n · 16n · C(N,N ′, {fh}Nh=1, {f ′h}N′

h=1) max0≤m≤n

( tn−m

(n−m)!· ‖g‖2m

−

[ maxG=g0,g1, F∈{fh}Nh=1∪{f

′h}N′h=1

∫ ∞−∞| < F, SuG > |du]2(n−m)

)(56)

where,

C(N,N ′, {fh}Nh=1, {f ′h}N′

h=1) := supz>0

max0≤m≤n

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

| < ΦN−m(z

∫ Tr/z2

Sr/z2

Sufrdu, r ∈ {1, · · · , N} \ {xh}mh=1),

ΦN ′−m(z

∫ T ′r/z2

S′r/z2

Suf′rdu, r ∈ {1, · · · , N ′} \ {yh}mh=1) > | (57)

PROOF By formula (3.3) and using the notation (4.3), we know thatthe left hand side of (4.2) is majorized by

∑ε∈{0,1}n

n−1∑m=0

∑2≤q1<···<qm≤n

∑1≤p1,···,pm≤n−1

ph<qh,h=1,···,m, |{ph}mh=1|=m

z2(n−m)

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

m∏h=1

| < Stphg1−ε(ph), Stqhgε(qh) > | · C(N,N ′, {fh}Nh=1, {f ′h}N′

h=1) (58)

this is the same, up to a constant, as the right hand side of (4.18) in [1],therefore the application of the same argument as in the proof of Lemma(4.3) of [1] leads to (4.2).

Combining together Theorem (4.1), Theorem (3.1), Theorem (3.2) andTheorem (3.4), one has the following

16

THEOREM(4.2) For each N,N ′ ∈ N, f1, · · · , fN , f ′1, · · · , f ′N ′ ∈ K,

{Sh, Th}Nh=1, {Sh, Th}Nh=1 ⊂ R, u, v ∈ H0, D ∈ B(H0), if

‖g‖2− <

1

16‖D‖(59)

the limit (1.19) exists and is equal to

< u⊗ΨN(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}), v⊗ΨN ′(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′} > +

+∞∑n=1

∑ε∈{0,1}n

n∑m=1

∑1=q1<q2<···<qm≤n

m∏r=1

qr+1−1∏h=qr+1

(g1−ε(h−1)|gε(h))−∫0≤tqm≤tqm−1≤···≤tq1≤t

dtqm · · · dtq1 < u,Dε(1) · · ·Dε(n)v >

∑1≤x1,···,xm≤N|{xh}mh=1|=m

m∏h=1


∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1


,T ′yh](tqh)(g1−ε(qh+1−1)|f ′yh)



PROOF By expanding the product

(−i)nV (t1) · · ·V (tn) (61)

to∑ε∈{0,1}n

Dε(1) · · ·Dε(n)⊗A+(St1gε(1))A(St1g1−ε(1)) · · ·A+(Stngε(n))A(Stng1−ε(n))

(62)one can write the scalar product in (1.19) in the form

∞∑n=0

∑ε∈{0,1}n

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn < ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

17

(In(ε)+IIn(ε))ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > · < u,Dε(1) · · ·Dε(n)v >

(63)Applying Theroem (4.1) to (4.9), one knows that if ‖g‖2

− < 1/16‖D‖, thelimit (1.22) is equal to

∞∑n=0

limz→0

∑ε∈{0,1}n

∫ t/z2

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

< u,Dε(1) · · ·Dε(n)v >< ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

(In(ε) + IIn(ε))ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (64)

By application of Theorem (3.2) and Theorem (3.3) we finish the proof.

§5 The quantum stochastic differential equation

From the Sections §2, §3 and §4 one has learnt

1. the limit space on which our limit processes lives;

2. the conditions allowing to take the limit in (1.19);

3. the explicit form of the limit (1.19).

Now we want to describe the quantum stochastic process arising in thelimit (1.19).

First of all notice that for each N,N ′ ∈ N, f1, · · · , fN , f ′1, · · · , f ′N ′ ∈ K,{Sh, Th}Nh=1, {Sh, Th}Nh=1 ⊂ R, u, v ∈ H0, D ∈ B(H0), the scalar product in(1.19) can be written as

< u, Fz(t;N,N′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) > (65)

and its limit i.e. (4.6) can be written in the form

< u, F (t;N,N ′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) > (66)

18

It is clear that both expressions are bounded.In the following we introduce the following notations: for σ ∈ {0, 1}

Dg(σ) :=∞∑n=0

(g1−σ|g1−σ)n−(gσ|gσ)n−(DσD1−σ)n (67)

D1(σ) := Dg(σ)Dσ D2(σ) := (g1−σ|g1−σ)−DσD1−σDg(σ) (68)

Our first and most important conclusion in this section is THEOREM(5.1)

For each N,N ′ ∈ N, f1, · · · , fN , f ′1, · · · , f ′N ′ ∈ K, {Sh, Th}Nh=1, {Sh, Th}Nh=1

⊂ R, u, v ∈ H0, D ∈ B(H0), under the conditions (1.8),(1.9) and (4.5), theexpressions (5.2) satisfy the system of differential equations

< u, F (t;N,N ′;

(f1, . . . , fNf ′1, . . . , f ′N ′

)) >=

=< u⊗ΨN(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}), v⊗ΨN ′(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′} > +

+

∫ t

0

ds∑

ε∈{0,1}

N∑i=1

N ′∑j=1

χ[Si,Ti](s) · χ[S′j ,T′j ]

(s)(fi|gε) · (g1−ε|f ′j) · (−1)i+j

<(

(gε|fi) · (f ′j|g1−ε)D+1 (ε)u+ (gε|fi) · (f ′j|gε)D+

2 (ε)u),

F (s;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > (69)

< u, F (0;N,N ′;

(f1, . . . , fNf ′1, . . . , f ′N ′

)) >=

< u⊗ΨN(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}), v⊗ΨN ′(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′} >

(70)and

< u, F (t; 0, 0; ∅) >=< u, v > (71)

REMARK This Theorem is the analogue of Theorem (5.10) of [1] and the

two proofs are also similar. We shall not repeat the details of the proof butonly give the main idea and outline the important steps.

PROOF By the change of variable

z2t1 ↪→ t1 (72)

19

in (1.22), one finds that (5.1) is equal to

< u⊗ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N), v⊗ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +

+ < u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N), z−2

∞∑n=1

∫ t

0

dt1(−i)V (t1/z2)

∫ t1/z2

0

dt2 · · ·∫ tn−1

0

dtn(−i)n−1V (t2) · · ·V (tn)ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) >

(73)Using the explicit form (1.6) of the interaction for V (t1/z

2) and with thechange of variables

m = n− 1, s1 = t2, · · · , sm = tn−1 (74)

then (5.7) becomes

< u⊗ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N), v⊗ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +

+z−2∑

ε∈{0,1}

∫ t

0

dt1 < D+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

1⊗ A+(St1/z2gε)A(St1/z2g1−ε)

Ut1/z2ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (75)

The first term of (5.9) tends, as z → 0, to

< u⊗ΨN(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}), v⊗ΨN ′(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′} >

(76)In the second term of (5.9), the action of the creator A+(St1/z2gε) on theΦN ′–number vector gives

N∑i=1

z

∫ Ti/z2

Si/z2

< St1/z2gε, Sufi > du(−1)iΦN−1(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N}\{i})

(77)

20

Therefore the second term of (5.9) is equal to

z−1

N∑i=1

∑ε∈{0,1}

∫ t

0

dt1

∫ Ti/z2

Si/z2

< St1/z2gε, Sufi > du · (−1)i

< D+ε u⊗ ΦN−1(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

1⊗ A(St1/z2g1−ε)Ut1/z2ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (78)

The expression 1 ⊗ A(St1/z2g1−ε)Ut1/z2 is handled with the same techniquesas in [1, · · ·, 6]. Namely: one expands Ut1/z2 using the iterated series andafter the change of variable z2 · t2 ↪→ t2, one finds

1⊗A(St1/z2g1−ε)+∞∑n=2

∑ε′∈{0,1}

1

z2·∫ t1

0

dt2Dε′⊗A(St1/z2g1−ε)A+(St2/z2gε′)A(St2/z2g1−ε′)

∫ t2/z2

0

dt3 · · ·∫ tn−1

0

dtn(−i)n−2V (t3) · · ·V (tn) (79)

MoreoverA(St1/z2g1−ε)A

+(St2/z2gε′)A(St2/z2g1−ε′) =

=< St1/z2g1−ε, St2/z2gε′ > A(St2/z2g1−ε′)+A+(St2/z2gε′)A(St2/z2g1−ε′)A(St1/z2g1−ε)

(80)and by (1.9) the scalar product is not equal to zero only when ε′ = 1 − ε.Thus (5.12) can be rewritten as

z−1

N∑i=1

∑ε∈{0,1}

∫ t

0

dt1

∫ Ti/z2

Si/z2


< D+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

1⊗ A(St1/z2g1−ε)ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +

21

+z−1

N∑i=1

∑ε∈{0,1}

∫ t

0

dt1

∫ Ti/z2

Si/z2


< D+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

( 1

z2·∫ t1

0

dt2 < St1/z2g1−ε, St2/z2g1−ε > A(St2/z2g1−ε′)Ut2/z2+

+1

z2·∫ t1

0

dt2A+(St2/z2gε′)A(St2/z2g1−ε′)A(St1/z2g1−ε)∫ t2/z2

0

dt3 · · ·∫ tn−1

0

dtn(−i)n−2V (t3) · · ·V (tn))

ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (81)

Now let see the third term of (5.15) and try to move the annihilationoperator A(St1/z2g1−ε) to the right hand side of product V (t3) · · ·V (tn) suchthat we can let the annihilator act on the ΦN ′–number vector. In order todo this, from the formulas (1.13) and (5.14) we know that the annihilatorA(St1/z2g1−ε) can appear in two ways:

1. it is used to produce a scalar product with a creator A+(Stjg1−ε)

< St1/z2g1−ε, Stjg1−ε > (82)

where j = 3, 4, · · · , n;

2. the annihilation operator is simply exchanged with the product V (t3) · · ·V (tn).

In the case 1), since j ≥ 3, one obtains a term of type II, therefore its limitis zero: all the terms of this type are collected in o(1) below. Thus (5.16) isequal to

z−1

N∑i=1

∑ε∈{0,1}

∫ t

0

dt1

∫ Ti/z2

Si/z2


< D+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

22

1⊗ A(St1/z2g1−ε)ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +

+z−1

N∑i=1

∑ε∈{0,1}

∫ t

0

dt1

∫ Ti/z2

Si/z2


< D+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),( 1

z2·∫ t1

0

dt2 < St1/z2g1−ε), St2/z2g1−ε > A(St2/z2g1−ε′)Ut1/z2+

+1

z2·∫ t1

0

dt2A+(St2/z2gε′)A(St2/z2g1−ε′)∫ t2/z2

0

dt3 · · ·∫ tn−1

0

dtn(−i)n−2V (t3) · · ·V (tn)A(St1/z2g1−ε))

ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +o(1) (83)

By letting the annihilator A(St1/z2g1−ε) act on the number vector ΦN ′(z∫ T ′k/z2

S′k/z2 Suf

′kdu, k = 1, · · · , N ′) one obtains

N ′∑j=1

z(−1)j∫ T ′j/z

2

S′j/z2

< St1/z2g1−ε, Suf′j > duΦN ′−1(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k ∈ {1, · · · , N ′}\{j})

(84)Finally recall that as z → 0 one has∫ Ti/z

2

Si/z2

< St1/z2gσ, Sufi > du −→ χ[Si,Ti](t1)(gσ|fi) ; σ = ε, 1− ε (85)

and

z−2

∫ tn−1

0

dtn < Stn−1/z2gσ, Stn/z2gσ >−→ (gσ|gσ)− ; σ = ε, 1− ε, n ∈ N

(86)From the above, in the notations (5.1), (5.2), we deduce:

< u, Fz(t;N,N′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) >=

23


+N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0

dt1χ[Si,Ti](t1)(fi|gε) · χ[S′j ,T′j ]

(t1)(g1−ε|f ′j)(−1)i+j

< D+ε u, F (s;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > +

+z−1

N∑i=1

∑ε∈{0,1}

∫ t

0

dt1

∫ Ti/z2

Si/z2


< D+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

1

z2·∫ t1

0

dt2 < St1/z2g1−ε, St2/z2g1−ε > A(St2/z2g1−ε′)Ut2/z2+

ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +o(1) (87)

Notice that in (5.20) the last term is similar to (5.12), therefore byrepeating the discussion from (5.12) to (5.20), we have

< u, Fz(t;N,N′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) >=


+N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0

dt1χ[Si,Ti](t1)(fi|gε) · χ[S′j ,T′j ]

(t1)(g1−ε|f ′j) · (−1)i+j

< D+ε u, F (t1;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > +

+N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0

dt1χ[Si,Ti](t1)(fi|gε)·

·(g1−ε|g1−ε)− · χ[Sj ,Tj ](t1)(gε|f ′j) · (−1)i+j

< D+1−εD

+ε uF (t1;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > +

24

+z−1

N∑i=1

∑ε∈{0,1}

∫ t

0


·z−2

∫ t1

0

dt2 < St1/z2g1−ε, St2/z2g1−ε > ·z−2

∫ t2

0

dt3 < St2/z2gε, St3/z2gε > ·

< D+ε D

+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

1⊗ A(St3/z2gε)Ut3/z2ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +o(1) (88)

Iterating n times the above procedure one finds that the scalar product

< u, Fz(t;N,N′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) >

is expressed as a sum of several terms. Denoting by Tk the sum of all theterms obtained in the k-th step with the exception of the first and the lastsummands, one has

Tn = Tn−1 +N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0


·(g1−ε|g1−ε)−(gε|gε)− · · · (gεn|g1−εn)− · χ[Sj ,Tj ](t1)(gεn|f ′j) · (−1)i+j

< D+1−εn · · ·D

+1−εD

+ε uF (t1;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) >

(89)where

εn :=

{1− ε, if n is oddε, if n is even

(90)

Moreover the last summand in the n-th step is

z−1

N∑i=1

∑ε∈{0,1}

∫ t

0


·z−2

∫ t1

0

dt2 < St1/z2g1−ε, St2/z2g1−ε > ·z−2

∫ t2

0

dt3 < St2/z2gε, St3/z2gε > ·

25

· · · · · ·z−2

∫ tn−1

0

dtn < Stn−1/z2gεn , Stn/z2gεn > ·

< D+1−εn · · ·D

+ε D

+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

1⊗ A(Stn/z2gεn)Utn/z2ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > +o(1) (91)

Notice that the term (5.24) differs from the corresponding one in the(n-1)-st step in that the operator A(Stn−1/z2gεn−1) has been replaced by

z−2

∫ tn−1

0

dtn < Stn−1/z2gεn , Stn/z2gεn > ·A(Stn/z2gεn)

and with operator D+1−εn−1

· · ·D+ε D

+1−εD

+ε repalced by D+

1−εn · · ·D+ε D

+1−εD

+ε .

Therefore it follows, from the induction and formula (5.19b) that

< u, Fz(t;N,N′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) >=


+∞∑n=1

N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0


·(g1−ε|g1−ε)−(gε|gε)−(g1−ε|g1−ε)− · χ[Sj ,Tj ](t1)(gε|f ′j) · (−1)i+j

< D+1−εD

+ε D

+1−εD

+ε uF (t1;N−1, N ′−1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > + o(1)

(92)Finally by rewriting the right hand side of (5.25) as the sum of two terms

corresponding to n odd or even and letting z tend to zero, we obtain (5.5a).It is easy to check (5.5b) and (5.5c).

Now let us introduce some notations on the Fock space Γ(L2(R⊗(K, (·|·)).For each ξ ∈ B(L2(R)), T ∈ B(K), denote N(ξ ⊗ T ) the number operator,characterized by the property

< ΨN(ηr ⊗ fr, r = 1, · · · , N), N(ξ ⊗ T )ΨN ′(η′r ⊗ f ′r, r = 1, · · · , N ′) >

26

=N∑j=1

N ′∑k=1

(−1)j+k < ηj, ξη′k > · < fj, T f

′k >

< ΨN−1(ηr⊗fr, r ∈ {1, · · · , N}\{j}), N(ξ⊗T )ΨN ′−1(η′r⊗f ′r, r ∈ {1, · · · , N ′}\{k}) >(93)

For each f, g ∈ K, s ≥ 0, define the number process Ns(f, g) by N(χ[0,s]⊗|f >< g|). Consider the quantum stochastic differential equation

U(t) = 1 +

∫ t

0

∑σ∈{0,1}

(D1(σ)⊗dNs(σ, 1−σ) +D2(σ)⊗dNs(σ, σ)

)U(s) (94)

where,Ns(σ, ε) := Ns(gσ, gε) (95)

THEOREM(5.3) The quantum stochastic differential equation (5.27)has a unique and unitary solution.

PROOF The exsistence and uniquness of the solution of q.s.d.e. (5.27)follows form the fact that D1(σ), D2(σ) are the bounded operators. Theproof of unitarity is the same as the one of Theorem (6.3) of [1]. Now our

last assertion can be stated and proved as following:

THEOREM(5.4) Under the conditions (1.8), (1.9) and (4.5), the limit(1.19) is of form

< u⊗ΨN(χ[Sr,Tr]⊗ fr, r = 1, · · · , N), U(t)v⊗ΨN ′(χ[S′r,T′r]⊗ f

′r, r = 1, · · · , N ′)

(96)and where U(t) is the solution of the quantum stochastic differential equation(5.27).

PROOF Clearly (5.29) can be written in the form:

< u,G(t;N,N ′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) > (97)

Using the QSDE (5.27) it is easy to show that (5.30) satisfies the system ofdifferential equations (5.5a,b,c).

Since D1(σ), D2(σ) are the bounded operators, one knows that the dif-ferential equation has a unique solution. This allows to identify (5.29) with(5.2) and therefore, by Theorem (5.1), with the limit (1.19). This completesthe proof.

27

2 REFERENCES

[1]. L.Accardi, Y.G.Lu : The Number Process as Low Density Limit ofHamiltonian Models. Commu. Math. Phys. Vol. 637 (1991), pp.1–31.

[2]. L.Accardi, Y.G.Lu : Low Density Limit: The Finite TemperatureCase, to appear in The proceedings of Nagoya conference. [3]. Y.G.Lu: Low

Density Limit: Without Rotating Wave Approximation, accepted by Rep.Math. Phy. [4]. L.Accardi, Y.G.Lu : The Low Density Limit: Langevin

Equation of Boson Fock Case, to appear in Ideas and methods in Mathema-tics and Physics [5]. L.Accardi, Y.G.Lu : The low density limit for quantum

systems. Jour. Phys. A: Math. and Gene. 24 (1991). [6]. L.Accardi, R.

Alicki, A, Frigerio, Y.G.Lu: An invitation to the weak coupling and low den-sity limits, Quantum Probability and Applications V, World Scientific (1991).[7]. R.Dumcke: The low density limit for an N-level systems interacting with

a free Boson and Fermi gas. Comm. Math. Phys. 97 (1985), 331–359.

[8]. E.B. Davies: Markovian master equation. Comm. Math. Phys. 39(1974), 91-110.

[8a]. E.B. Davies: Markovian master equation III. Ann. Inst.H.Poicare,B 11 (1975), 265-273.

[8b]. E.B. Davies: Markovian master equation II. Math. Ann. 219(1976), 147-158. [9] L.Accardi, Y.G.Lu: The Weak Coupling Limit Without

Rotating Wave Approximation. Ann. Inst. H.Poincare, Phys. Th. Vol. 54,No 4 (1991). [10] L.Accardi, A.Frigerio, Y.G.Lu: The weak coupling limit as

a quantum functional central limit. Comm. Math. Phys, 131 (1990), 537–570. [11] L.Accardi, A.Frigerio, Y.G.Lu: The Weak Coupling Limit (II): The

Langevin equation and finite temperature case, submitted to PublicationsRIMS. [12]. P.F.Palmer: Dr. Phil. Thesis. Oxford 1977.

28

[13] L.Accardi, A.Frigerio, Y.G.Lu: The Weak Coupling Limit For Fer-mions. Journ. Math. Phys. 32 (6), June 1991, pp. 1567–1581. [14]

R.Alicki, A.Frigerio: Quantum Poisson Process and Linear Quantum Bol-tzmann Equation, preprint. [15] A.Frigerio, H.Maassen: Quantum Poisson

Process and Dilations of Dynamical Semigroup, Prob. Th. Rel. Fields 83(1989), 489–508.

[5] Accardi, L., Frigerio, A. and Lu, Y.G.: Quantum Langevin equa-tion in the weak coupling limit . to appear in: Quantum Probability andApplications V.

[7] Accardi, L. , Lu, Y.G.: Squeezing Noises as Weak Coupling Limit ofHamiltonian Systems to appear in: Reports in Mathematical Physics. [8]

Accardi, L. , Lu, Y.G.: The Weak Coupling Limit For Non-linear Interac-tions, submitted to Ascona. [9] Accardi, L. , Lu, Y.G.: The low density limit:

Boson Fock case, submitted to Comm. Math. Phys.

[10] Accardi, L. , Lu, Y.G.: The low density limit: Langevin equation forBoson Fock case, invited contribution to the Memorail Volume for RaphaelHoeg-Krohn “Ideas and methods in Mathematics and Physics.

[11] Accardi, L. , Lu, Y.G.: The low density limit of the Boson models,to appear in Quantum Probability and Applications V. [12] Accardi, L. , Lu,

Y.G.: The low density limit: finite temperature case, to appear. [13] Accardi,

L. , Lu, Y.G.: The low density limit and the quantum Poisson process,invited talk at the 5-th Vilnius Conference on Probability and Mathematicsstatistics, to appear. [14] Accardi, L. , Lu, Y.G.: The low density limit: the

quasi-free case, without rotating wave approximation, to appear.

[15] Accardi, L. , Lu, Y.G.: The low density limit for Fermion Fock case,to appear.

[16] Accardi, L. , Lu, Y.G.: The collective coherent vectors for the weakcoupling limit of quadratic interactions. In preparation.

29

[17]. Accardi, L., Bach, A.: Quantum central limit theorems for stronglymixing random variables. Z.W.V.G. 68 (1985), pp. 393-402.

[18]. Accardi,L., Bach, A.: The harmonic oscillator as quantum centrallimit theorem of Bernoulli processes, to appear in Prob. Th. Rel. Fields.

[19]. Accardi, L., Lu, Y.G.: Quantum central limit theorems for weaklydependent maps, to appear.

[20]. Davies, E.B.: Markovian master equation. Comm. Math. Phys. 39(1974), 91-110.

[21]. Davies, E.B.: Markovian master equation III. Ann. Inst.H.Poicare,B 11 (1975), 265-273.

[22]. Davies,E.B.: Markovian master equation II. Math. Ann. 219 (1976),147-158.

[23]. Davies,E.B. : Quantum Theory of Open Systems, Academic Press,London and New York (1976). [23a] Dell’Antonio, G.F.: Large time, small

coupling behavior of a quantum particle in a random field. Ann. Ist. HenriPoincare 35 (1983), 339-384

[24]. Dumcke, R.: Convergence of multi-time correlation functions in theweak and singular coupling limit. J. Math. Phys. 24 (1983), 311-315.

[25]. Dumcke, R.: The low density limit for an N-level system interactigwith a free Bose or Fermi Gas. Comm. Math. Phys. 97 (1985), 331-359.

[26]. Dumcke, R.: Makovian limits of multi-time correlation functions foropen quantum system. in: Quantum Probability and Applications. SpringerLNM N. 1055 (1984), 113-118.

[27] Frigerio, A.: Quantum Poisson processes: physical motivations andapplications, in: Quantum Probability and Applications III. Springer LNMN. 1303 (1988), 107-127.

[28]. Palmer, P.F.: The singular coupling and weak coupling limits. J.Math. Phys. 18 (1977), 527-529. [28a]. Palmer, P.F.: D. Phil. Thesis.

30

Oxford 1977.

[29]. Pule, J.V.: The Bloch equations. Comm. Math. Phys. 38 (1974),241-256.

Let consider the quantum Resevoir+System model in Fermion field, H1

be resevoir Hilbert space, H0 be system Hibert space. Let be given a resevoirHarmiltonian HR, system Harmiltonian HS. As usual we suppose that HS

has finite spectrum andHR = dΓ(∆) (98)

with the self-adjoint operator ∆ on the one partical space H1 (e.g. H1 isL2(R3) and ∆ is Laplacian). The total Hilbert space should be

H0 ⊗ Γ(H1) (99)

and the total Hamiltonian should be

HTO := HS ⊗ 1 + 1⊗HR + V (100)

with the interactor V . In this paper, as in [1], we consider the V as the form

V := i(D ⊗ A+(g0)A(g1)−D+ ⊗ A+(g1)A(g0)) (101)

DenoteSt := e−it∆ t ∈ R (102)

a one parameter unitary group and W (H1) the CAR-algebra on H1, i.e. thealgebra generated by the set

{A(g), g ∈ H1}

where, A is the Fermion annihilator.Let be given a gauge invariant quasi free state ϕQz (Q := Qz ≤ 1) which

is discribed by

ϕQz(A+(f)A(g)) =< f,

1−Qz

2g > ∀f, g ∈ H1 (103)

For low density limit, we consider the case

Qz :=1− z2e−βH/2

1 + z2e−βH/2(104)

31

with the self-adjoint operator H, where β−1 is the temperature and z2 thefagulity.

Denote (HQ, πQ,ΦQ) the GNS-triple of (W (H1), ϕQ), for simplicity, weshall denote πQ(A(f)) as A(f), πQ(A+(f)) as A+(f) in the following.

In the analogy with Boson case, we shall firstly investigate the low densitylimit for Fock case and moreover, suppose that

e−itHS D eitHS = D (105)

In the point of view of our work, the difference between the condition (1.8)and N-levels case is only to use Reimann-Lebesgue Lemma repeatedly (cf.[3]), it will be our follow-up works. Consider the evolution of the model

U(t) := eitHFRe−itHTO (106)

withHFR = HTO − V (107)

then U(t) satisfies the differential equation

d

dtU(t) = −iV (t)U(t) (108)

So,

U(t) =∞∑n=0

(−i)n∫ t

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtnV (t1) · · ·V (tn) (109)

As usual, we suppose that there exists a subspace K ⊂ H1 satisfying∫R| < f, Stg > |dt <∞ ∀f, g ∈ K (110)

so, one obtains the Hilbert space (K, (·|·)) with the scalar product (·|·) definedby

(f |g) :=

∫R< f, Stg > dt ∀f, g ∈ K (111)

Our conclusions about low density limit are(i) the low density limit

limz→0

< u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k = 1, · · · , N),

32

U(t/z2)v ⊗ ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (112)

(ii) the low density limit (1.15) is equal to

< u⊗ΨN(χ[Sk,Tk]⊗fk, k = 1, · · · , N), U(t)v⊗ΨN ′(χ[S′k,T′k]⊗f ′k, k = 1, · · · , N ′) >

(113)where, U(t) is the solution of a quantum stochastic differential equation andfor each n ∈ N, g1, · · · , gn ∈ H1

Φn(fk, k ∈ {1, · · · , n}) := A+(f1) · · ·A+(fn)Φ (114)

In this paper, we shall consider mainly the Fock case, i.e. Qz = 1. anddenote

D0 := D D1 := −D+ (115)

t1z2 ↪→ t1, · · · , tnz2 ↪→ tn (116)

the left hand side of (3.19) becomes to

n∑m=1

∑1=q1<q2<···<qm≤n

z−2(n−m)

∫ t

0

dt1

∫ t1

0

dt2 · · ·∫ tn−1

0

dtn

m∏r=1

qr+1−1∏h=qr+1

< Sth−1/z2g1−ε(h−1), Sth/z2gε(h) >

∑1≤x1,···,xm≤N|{xh}mh=1|=m

m∏h=1

(−1)(xh,N)

∫ Txh/z2

Sxh/z2

< Sufxh , Stqh/z2gε(qh) > du

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(yh,N′)

∫ T ′yh/z2

S′yh/z2

< Stqh/z2gε(qh), Svf′xh> dv

< ΦN−m(z

∫ Tα/z2

Sα/z2

Sufαdu, α ∈ {1, · · · , N} \ {xh}mh=1),

ΦN ′−m(z

∫ T ′β/z2

S′β/z2

Suf′βdu, β ∈ {1, · · · , N ′} \ {yh}mh=1) > (117)

33

since m(m− 1) is even. Now change the variables as following:

tqh = sqh , h = 1, · · · ,m (118)

tqh+1 − tqh = sqh+1, tqh+2 − tqh+1 − tqh = sqh+2, · · · ,

tqh+1−1 − tqh+1−2 − · · · − tqh+1 − tqh = sqh+1−1, h = 1, · · · ,m (119)

then (3.21) becomes to

n∑m=1

∑1=q1<q2<···<qm≤n

z−2(n−m)

∫ t

0

dsq1

∫ 0

−sq1dsq1+1

∫ 0

−sq1+1−sq1dsq1+2 · · ·

∫ 0

−sq2−2−···−sq1+1−sq1dsq2−1

∫ sq2−1+···+sq1+1+sq1

0

dsq2 · · ·∫ sqh−1

0

dsqh

∫ 0

−sqh

dsqh+1

∫ 0

−sqh+1−sqh

dsqh+2 · · ·∫ 0

−sqh+1−2−···−sqh+1−sqh

dsqh+1−1

∫ sqh+1−1+···+sqh+1+sqh

0

dsqh+1· · ·

∫ sqm−1+sqm−1+···+sqm−1+1

0

dsqm

∫ 0

−sqmdsqm+1

∫ 0

−sqm+1−sqmdsqm+2 · · ·

∫ 0

−sn−1−···−sqm+1−sqmdsn

m∏r=1

qr+1−1∏h=qr+1

< g1−ε(h−1), Ssh/z2gε(h) >

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(xh,N)

∫ Txh/z2

Sxh/z2

< Sufxh , Ssqh/z2gε(qh) > du

m∏h=1

(−1)(yh,N′)

∫ T ′yh/z2

S′yh/z2

< S(sqh+1−1+sqh+1−2+···+sqh )/z2g1−ε(qh+1−1), Svf′yh> dv

< ΦN−m(z

∫ Tα/z2

Sα/z2

Sufαdu, α ∈ {1, · · · , N} \ {xh}mh=1),

ΦN ′−m(z

∫ T ′β/z2

S′β/z2

Suf′βdu, β ∈ {1, · · · , N ′} \ {yh}mh=1) > (120)

34

Change the variables sα/z2 = tα, α ∈ {1, · · · , n} \ {qh}mh=1, sqh = tqh , h =

1, · · · ,m, (3.23) becomes to

n∑m=1

∑1=q1<q2<···<qm≤n

∫ t

0

dtq1

∫ 0

−tq1/z2

dtq1+1

∫ 0

−tq1+1−tq1/z2

dtq1+2 · · ·

∫ 0

−tq2−2−···−tq1+1−tq1/z2

dtq2−1

∫ z2(tq2−1+···+tq1+1)+tq1

0

dtq2 · · ·∫ tqh−1

0

dtqh

∫ 0

−tqh/z2

dtqh+1

∫ 0

−tqh+1−tqh/z2

dtqh+2 · · ·

∫ 0

−tqh+1−2−···−tqh+1−tqh/z2

dtqh+1−1

∫ z2(tqh+1−1+···+tqh+1)+tqh

0

dtqh+1· · ·

∫ tqm−1+z2(tqm−1+···+tqm−1+1)

0

dtqm

∫ 0

−tqm/z2

dtqm+1

∫ 0

−tqm+1−tqm/z2

dtqm+2 · · ·∫ 0

−tn−1−···−tqm+1−tqm/z2

dsn

m∏r=1

qr+1−1∏h=qr+1

< g1−ε(h−1), Sthgε(h) >

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(xh,N)

∫ (Txh−tqh )/z2

(Sxh−tqh )/z2

< Sufxh , gε(qh) > du

m∏h=1

(−1)(yh,N′)

∫ (T ′yh−tqh )/z2−tqh+1−1−tqh+1−2−···−tqh+1

(S′yh−tqh )/z2−tqh+1−1−tqh+1−2−···−tqh+1

< g1−ε(qh+1−1), Svf′yh> dv

< ΦN−m(z

∫ Tα/z2

Sα/z2

Sufαdu, α ∈ {1, · · · , N} \ {xh}mh=1),

ΦN ′−m(z

∫ T ′β/z2

S′β/z2

Suf′βdu, β ∈ {1, · · · , N ′} \ {yh}mh=1) > (121)

Let z → 0 one finishes the proof.

35

3 REFERENCES

[1] Accardi, L., Frigerio, A. and Lu, Y.G.: On the weak coupling limit pro-blem in: Quantum Probability and Applications IV Springer LNM N. 1396(1987), 20-58. [2] Accardi, L. , Frigerio, A. and Lu, Y.G.: The weak coupling

limit as a quantum functional central limit, to appear in Comm. Math. Phys(1990). [3] Accardi, L. , Frigerio, A. and Lu, Y.G.: The Weak Coupling Limit

(II): The Langevin equation and finite temperature case, submitted to Pu-blications RIMS Oct. 1989 [4] Accardi, L. , Frigerio, A. and Lu, Y.G.: The

Weak Coupling Limit For Fermions, submitted to Journ. Math. Phys. [5]

Accardi, L., Frigerio, A. and Lu, Y.G.: Quantum Langevin equation in theweak coupling limit . to appear in: Quantum Probability and ApplicationsV. [6] Accardi, L. , Lu, Y.G.: The Weak Coupling Limit Without Rotating

Wave Approximation, to appear in Ann. Inst. H.Poicare. [7] Accardi, L. ,

Lu, Y.G.: Squeezing Noises as Weak Coupling Limit of Hamiltonian Systemsto appear in: Reports in Mathematical Physics. [8] Accardi, L. , Lu, Y.G.:

The Weak Coupling Limit For Non-linear Interactions, submitted to Ascona.[9] Accardi, L. , Lu, Y.G.: The low density limit: Boson Fock case, submitted

to Comm. Math. Phys.

[10] Accardi, L. , Lu, Y.G.: The low density limit: Langevin equation forBoson Fock case, invited contribution to the Memorail Volume for RaphaelHoeg-Krohn “Ideas and methods in Mathematics and Physics.

[11] Accardi, L. , Lu, Y.G.: The low density limit of the Boson models,to appear in Quantum Probability and Applications V. [12] Accardi, L. , Lu,

Y.G.: The low density limit: finite temperature case, to appear. [13] Accardi,

L. , Lu, Y.G.: The low density limit and the quantum Poisson process,invited talk at the 5-th Vilnius Conference on Probability and Mathematics

36

statistics, to appear. [14] Accardi, L. , Lu, Y.G.: The low density limit: the

quasi-free case, without rotating wave approximation, to appear.

[15] Accardi, L. , Lu, Y.G.: The low density limit for Fermion Fock case,to appear.

[16] Accardi, L. , Lu, Y.G.: The collective coherent vectors for the weakcoupling limit of quadratic interactions. In preparation.

[17]. Accardi, L., Bach, A.: Quantum central limit theorems for stronglymixing random variables. Z.W.V.G. 68 (1985), pp. 393-402.

[18]. Accardi,L., Bach, A.: The harmonic oscillator as quantum centrallimit theorem of Bernoulli processes, to appear in Prob. Th. Rel. Fields.

[19]. Accardi, L., Lu, Y.G.: Quantum central limit theorems for weaklydependent maps, to appear.

[20]. Davies, E.B.: Markovian master equation. Comm. Math. Phys. 39(1974), 91-110.

[21]. Davies, E.B.: Markovian master equation III. Ann. Inst.H.Poicare,B 11 (1975), 265-273.

[22]. Davies,E.B.: Markovian master equation II. Math. Ann. 219 (1976),147-158.

[23]. Davies,E.B. : Quantum Theory of Open Systems, Academic Press,London and New York (1976). [23a] Dell’Antonio, G.F.: Large time, small

coupling behavior of a quantum particle in a random field. Ann. Ist. HenriPoincare 35 (1983), 339-384

[24]. Dumcke, R.: Convergence of multi-time correlation functions in theweak and singular coupling limit. J. Math. Phys. 24 (1983), 311-315.

[25]. Dumcke, R.: The low density limit for an N-level system interactigwith a free Bose or Fermi Gas. Comm. Math. Phys. 97 (1985), 331-359.

37

[26]. Dumcke, R.: Makovian limits of multi-time correlation functions foropen quantum system. in: Quantum Probability and Applications. SpringerLNM N. 1055 (1984), 113-118.

[27] Frigerio, A.: Quantum Poisson processes: physical motivations andapplications, in: Quantum Probability and Applications III. Springer LNMN. 1303 (1988), 107-127.

[28]. Palmer, P.F.: The singular coupling and weak coupling limits. J.Math. Phys. 18 (1977), 527-529. [28a]. Palmer, P.F.: D. Phil. Thesis.

Oxford 1977.

[29]. Pule, J.V.: The Bloch equations. Comm. Math. Phys. 38 (1974),241-256.

[30]. van Hove, L. : Quantum mechanical perturbations giving rise to atransport equation. Physica 21 (1955), 517-540. [31] Hudson, R.L., Partha-

sarathy, K.R.: Quantum Ito’ s formula and stochastic evolutions. Comm.Math. Phys. 93 (1984), 301-323. [32] Friedrichs, K.O.(1948): On the per-

turbation of continuous spectra. Commun. Pure Appl. Math. 1, 361-406.[33] Grad, H.: Principles of the Kinetic Theory of Gases. In: Handbuch der

Physik, Vol. 12, edited by S. Flugge. Berlin: Springer 1958. [34] Martin

P., Emch G.G.: A rigorous model sustaining van Hove’s phenomenon. Helv.Phys. Acta. 48 (1975), 59-78. [35] Spohn, H.(1980): Kinetic equations from

Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 53, 569-615.[36] Gorini, V., Kossakowski, A. and Sudarshan, E. C. G.: Completely po-

sitive dynamical semigroups of n-level systems. J. Mathematical Phys. 17(1976), 821. [37] Lindblad, G.: On the generators of quantum dynamical

semigroups. Comm. Math. Phys. 48 (1976), 119-130. [38] Accardi, L.,

Frigerio, A., Lewis, J.: Quantum stochastic processes Publications of theResearch institute for Mathematical Sciences Kyoto University, 18 (1982),

38

97-133. [39] Alicki, R., Frigerio, A.: Quantum Poisson noise and linear quan-

tum Boltzmann equation. Preprint. [40] Frigerio, A., Maassen, H.: Quantum

poisson processes and dilations of dynamical semigroups. Probab. Th. Rel.Fields 83, 489-508 (1989). [41a] Gardiner, C.W., Collet M.J.: Input and out-

put in damped quantum system: quantum stochastic differencial equationand master equation. Phys. Rev. A 31 (1985), 3761-3774. [41b] Barchielli,

A.: Input and output channels in quantum systems and quantum stocha-stic differential equations. LNM N 1303 (1988), Springer. [41c] Barchielli,

A.: Measurement theory and stochastic differential equations in quantummechanics. Physical review A 34 (1986), 1642-1649. [42] Haken, H.: Laser

theory. Springer 1984 [43a] Lax, M.: Quantum noise theory of noise sources.

Phys. Rev. 145 (1966), 110-129. [43b] Lax, M.: Description of quantum

laser fluctuations. Phys, Rev. 172 (1968), 350-361. [44] von Waldenfels, W.:

Ito solution of the linear quantum stochastic differential equation describinglight emission and absorption . in: Quantum Probability and ApplicationsII , Accardi,L. , W.von Waldenfels (eds.) ; Springer LNM 1055 (1984). [45]

Petz, D.: An invitation to the C∗–algebra of the canonical commutation re-lation. Leuven Notes in Math. and Theor. Phys., eds. Fannes M., VerbeureA. , Leuven University Press 1990. [46] Kummerer, B.: Markov Kilations

on W ∗–Algebras. J. Func. Anal. 63 (1985), 139–177. [47] Kummerer, B.,

Schroder, W.: A new construction of unitary dilations: singular couplingto white noise . in: Quantum Probability and applications II . Accardi L.,von Waldenfels W. (eds.) Springer LNM N 1136 (1985). [48] Maassen, H.:

The construction of continuous dilations by solving quantum stochastic dif-ferential equations. Preprint. [49a] Applebaum, D., Frigerio, A.: Stationary

dilations of W-dynamical systems constructed via quantum stochastic diffe-rential equations. preprint. [49b] Applebaum, D.: Stochastic dilations of the

39

Bloch equations in boson and fermion noise. J.Phys. A: Math. Gen. 19(1986), 937–959

[50] Kummerer, B., Maassen, H.: The essentially commutative dilationsof dynamical semigroups on Mn. Commu. Math. Phys. 109 (1987), 1-22.[51] Cartwright, N.: How the laws of Physics lie. Clarendon Press (1988). [52]

Agarwal, G.S.: Quantum Optics. Springer 1974 [53] Barchielli, A.: Quantum

stochastic differential equations: An Application to the electron shelvingeffect. Preprint. [54] Robinson, P.: Quantum Diffusions on the Rotation

Algebras and the Quantum Hall effect. Preprint.

+z−1

N∑i=1

∑ε∈{0,1}

∫ t

0


· < D+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

{1⊗ A(St1/z2g1−ε), Ut1/z2}ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (122)

where, the symbol {·, ·} means anti-commutator. In the last term of (5.16),by the formula (), we have

{1⊗ A(St1/z2g1−ε), Ut1/z2} =∞∑n=1

∫ t1/z2

0

dt2 · · ·∫ tn−1

0

dtn

(−i)n−1{1⊗ A(St1/z2g1−ε), V (t2) · · ·V (tn)} =

=∞∑n=1

∫ t1/z2

0

dt2 · · ·∫ tn−1

0

dtn(−i)n−1[1⊗A(St1/z2g1−ε), V (t2)]V (t3) · · ·V (tn)+

+∞∑n=1

n∑p=3

∫ t1/z2

0

dt2 · · ·∫ tn−1

0

dtn(−i)n−1V (t2) · · ·V (tp−1)[1⊗A(St1/z2g1−ε), V (tp)]V (tp+1) · · ·V (tn)

(123)

40

The second term of (5.17) gives always, at least one of the scalar product ofthe form

< SαG,SβG′ >, β − α > 1 (124)

where G,G′ ∈ {g0, g1}. Therefore in the last term of the right hand side of(5.16), the scalar product is a sum of two terms and one of them tends, sinceit is the type II term in the sence of (), to zero as z → 0. Thus it is valid torewrite (5.16) to

< u, Fz(t;N,N′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) >=

= o(1)+ < ΨN(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}),ΨN ′(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′} > −

−N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0

dt1χ[Si,Ti](t1)(fi|gε)·χ[S′j ,T′j ]

(t1)(g1−ε|f ′j)∫ t1/z2

0

dt2 · · ·∫ tn−1

0

dtn

< D+ε u, F (s;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) >

+z−1

∞∑n=2

N∑i=1

∑ε∈{0,1}

∫ t

0


· < D+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

(−i)n−1{1⊗ A(St1/z2g1−ε), V (t2)}V (t3) · · ·V (tn)

ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (125)

In the third term of the right hand side of (5.18) the anti-commutator isequal to, by using the condition (),

i · {1⊗ A(St1/z2g1−ε), D1−ε ⊗ A+(St2g1−ε)}A(St2gε) =

= i· < St1/z2g1−ε, St2g1−ε > D1−ε ⊗ A(St2gε) (126)

Hence with changing of variable, in the third term of the right hand side of(5.18): z2t2 ↪→ t2, one finds that that term is equal to

41

z−1

N∑i=1

∑ε∈{0,1}

∫ t

0


·z−2

∫ t1

0

dt2 < St1/z2g1−ε, St2/z2g1−ε >< D+1−εD

+ε u⊗ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N}\{i}),

1⊗ A(St2/z2gε)Ut2/z2ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k = 1, · · · , N ′) > (127)

we have

< u, Fz(t;N,N′;

(f1, . . . , fNf ′1, . . . , f

′N ′

)) >=

= o(1)+ < ΨN(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}),ΨN ′(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′} > −

−N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0

dt1χ[Si,Ti](t1)(fi|gε)·χ[S′j ,T′j ]

(t1)(g1−ε|f ′j)∫ t1/z2

0

dt2 · · ·∫ tn−1

0

dtn

< D+ε u, F (s;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > +

+N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0


·(g1−ε|g1−ε)− · χ[Sj ,Tj ](t1)(gε|f ′j)·

< D+1−εD

+ε uF (t1;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > +

+N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0


(g1−ε|g1−ε)− · (gε|gε)− · χ[S′j ,T′j ]

(t1)(g1−ε|f ′j)·

< D+ε D

+1−εD

+ε u, F (t1;N − 1, N ′ − 1;

(f1, . . . , fi, . . . , fNf ′1, . . . , f ′j, . . . , f ′N ′

)) > +

+z−1

N∑i=1

N ′∑j=1

∑ε∈{0,1}

∫ t

0


42

(g1−ε|g1−ε)− · (gε|gε)− · χ[S′j ,T′j ]

(t1)(g1−ε|f ′j)·

< D+ε D

+1−εD

+ε u⊗ ΦN(z

∫ Tk/z2

Sk/z2

Sufkdu, k ∈ {1, · · · , N} \ {i}),

{1⊗A(St3/z2gε)Ut3/z2}ΦN ′(z

∫ T ′k/z2

S′k/z2

Suf′kdu, k ∈ {1, · · · , N ′} \ {j}) > (128)

Now notice that, if add the condition

< g0, Stg1 >= 0, ∀t ∈ R (129)

then the low desity limit (4.7) exists still and its form can be changed toother form.

Firstly, for each m ≤ n ∈ N, 1 = q1 < q2 < · · · < qm ≤ n we call thatε ∈ {0, 1}n is good if

ε(qh + 1) = 1− ε(qh), ε(qh + 2) = 1− ε(qh + 1) = ε(qh), · · · ,

ε(qh+1 − 1) =

{1− ε(qh), if qh+1 − qh ∈ N0

ε(qh), if qh+1 − qh ∈ N1(130)

for h = 1, · · · ,m with N0 := {2, 4, · · ·}, N1 := {1, 3, · · ·}.

LEMMA(4.3) For each m ≤ n ∈ N, 1 = q1 < q2 < · · · < qm ≤ n, withthe condition (4.11), one has

m∏r=1

qr+1−1∏h=qr+1

(g1−ε(h−1)|gε(h))− =

∏m

h=1

((g1−ε(qh)|g1−ε(qh))− · (gε(qh)|gε(qh))−

)[qh+1−qh

2]−χN0

(qh+1−qh)

·

·(g1−ε(qh)|g1−ε(qh))χN0

(qh+1−qh)− , if ε is good,

0, if ε is not good(131)

PROOF For each r = 1, · · · ,m, and ε is not good, there exists a r =1, · · · ,m and qr + 1 ≤ h ≤ qr+1 − 1 such that

1− ε(h− 1) 6= ε(h)

43

So, by (4.11)(g1−ε(h−1)|gε(h))− = 0 (132)

On the other hand, for ε is good, by (4.12)

qr+1−1∏h=qr+1

(g1−ε(h−1)|gε(h))− = (g1−ε(qr)|g1−ε(qr))− · (gε(qr)|gε(qr))− · · · · ·

((g1−ε(qr)|g1−ε(qr))−χN0(qr+1 − qr) + (gε(qr)|gε(qr))−χN1(qr+1 − qr)

)(133)

Cleraly the right hand side of (4.15) is the product of qr+1 − qr − 1 scalarproducts. So, if qr+1 − qr ∈ N0, one gets a product of [ qh+1−qh

2] − 1 fac-

tors of the form (g1−ε(qr)|g1−ε(qr))−· (gε(qr)|gε(qr))− and one factor of the form(g1−ε(qr)|g1−ε(qr))−, i.e.

((g1−ε(qr)|g1−ε(qr))− · (gε(qr)|gε(qr))−

)[qr+1−qr

2]−χN0

(qr+1−qr)

·(g1−ε(qr)|g1−ε(qr))χN0

(qr+1−qr)− (134)

Simlarly, if qr+1 − qr ∈ N1, one gets a product of [ qh+1−qh2

] factors of theform (g1−ε(qr)|g1−ε(qr))−· (gε(qr)|gε(qr))−, (4.16) is obtained again. This givesthe proof of (4.13).

Now we investigate that with the change ε ↪→ ε′ what is the new form ofDε(1) · · · Dε(n).

LEMMA(4.4) For each m ≤ n ∈ N, 1 = q1 < q2 < · · · < qm ≤ n,ε ∈ {0, 1}n is good

Dε(1) · · ·Dε(n) =

q2−q1︷︸︸︷Dε′(1)D1−ε′(1)Dε′(1)(Dε′(1)χN1(q2 − q1) +D1−ε′(1)χN0(q2 − q1)) ·

q3−q2︷︸︸︷Dε′(2)D1−ε′(2)Dε′(2)(Dε′(2)χN1(q3 − q2) +D1−ε′(2)χN0(q3 − q2)) ·

· · ·qm−qm−1︷︸︸︷

Dε′(m−1)D1−ε′(m−1)Dε′(m−1)(Dε′(m−1)χN1(qm − qm−1) +D1−ε′(m−1)χN0(qm − qm−1))

44

qm+1−qm︷︸︸︷Dε′(m)D1−ε′(m)Dε′(m)(Dε′(m)χN1(qm+1 − qm) +D1−ε′(m)χN0(qm+1 − qm))

(135)

PROOF The proof is almost the same ones as Lemma (4.4).

LEMMA(4.5) For each m ≤ n ∈ N, 1 = q1 < q2 < · · · < qm ≤ n :=qm+1, ε ∈ {0, 1}n is good,

m∏h=1

(tqh)(fh|gε(qh))(g1−ε(qh+1−1)|f ′h)

=m∏h=1

(fh|gε(qh))(

(g1−ε(qh)|f ′h)χN1(qh+1−qh)+(gε(qh)|f ′h)χN0(qh+1−qh))

(136)

PROOF (4.18) is the directely results of (4.12).

Combine tocheger Theorem (4.2), Lemma (4.3), Lemma (4.4) and Lemma(4.5), one has the following

THEOREM (4.6) Under the condition (4.11) and (4.6), the low densitylimit (4.7) exists and is equal to

< u⊗ΨN−m(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}), v⊗ΨN ′−m(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′}) > +

+∞∑n=1

n∑m=1

∑1=q1<q2<···<qm≤n

∑ε∈{0,1}m

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

m∏h=1

((g1−ε(h)|g1−ε(h))− · (gε(h)|gε(h))−

)[qh+1−qh

2]−χN0

(qh+1−qh)

·(g1−ε(h)|g1−ε(h))χN0

(qh+1−qh)−∑

1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(xh,N)χ[Sxh ,Txh ](th)(−1)(yh,N′)χ[S′yh

,T ′yh](th)(fxh |gε(h))

45

((g1−ε(h)|f ′yh)χN0(qh+1 − qh) + (gε(h)|f ′yh)χN1(qh+1 − qh)

)< u,

q2−q1︷︸︸︷Dε(1)D1−ε(1)Dε(1)(Dε(1)χN1(q2 − q1) +D1−ε(1)χN0(q2 − q1)) ·

q3−q2︷︸︸︷Dε(2)D1−ε(2)Dε(2)(Dε(2)χN1(q3 − q2) +D1−ε(2)χN0(q3 − q2)) ·

· · ·qm−qm−1︷︸︸︷

Dε(m−1)D1−ε(m−1)Dε(m−1)(Dε(m−1)χN1(qm − qm−1) +D1−ε(m−1)χN0(qm − qm−1))

qm+1−qm︷︸︸︷Dε(m)D1−ε(m)Dε(m)(Dε(m)χN1(qm+1 − qm) +D1−ε(m)χN0(qm+1 − qm)) v >



PROOF (4.8) is equal to


+∞∑n=1

n∑m=1

∑1=q1<q2<···<qm≤n

( ∑ε∈{0,1}n is good

+

∑ε∈{0,1}n is not good

) m∏r=1

qr+1−1∏h=qr+1

(g1−ε(h−1)|gε(h))−


dtqm · · · dtq1 < u,Dε(1) · · ·Dε(n)v >

∑1≤x1,···,xm≤N|{xh}mh=1|=m

m∏h=1


∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1


,T ′yh](tqh)(g1−ε(qh+1−1)|f ′yh) (138)

46

By Lemma (4.3) and condition (4.11), the terms corresponding to the sum∑ε∈{0,1}n is not good is zero and ∑

ε∈{0,1}n is good

=∑

ε∈{0,1}m(139)

So, change the variablestqh ↪→ th (140)

we finish the proof.

COROLLARY (4.7) Under the condition (4.11) and (4.6), the low densitylimit (4.7) exists and is equal to

< ΨN−m(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}),ΨN ′−m(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′}) > +

+∞∑n=1

n∑m=1

∑1≤p1,···,pm,

∑mh=1 ph=n

∑ε∈{0,1}m

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

m∏h=1

((g1−ε(h)|g1−ε(h))− · (gε(h)|gε(h))−

)[ph2

]−χN0(ph)

· (g1−ε(h)|g1−ε(h))χN0

(ph)−∑

1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1


,T ′yh](th)(fxh|gε(h))(

(g1−ε(h)|f ′yh)χN1(ph) + (gε(h)|f ′yh)χN0(ph))

< u,

p1︷︸︸︷Dε(1)D1−ε(1)Dε(1)(Dε(1)χN0(p1) +D1−ε(1)χN0(p1)) ·


· · ·pm−1︷︸︸︷

Dε(m−1)D1−ε(m−1)Dε(m−1)(Dε(m−1)χN0(pm−1) +D1−ε(m−1)χN0(pm−1))

47

pm︷︸︸︷Dε(m)D1−ε(m)Dε(m)(Dε(m)χN0(pm) +D1−ε(m)χN1(pm)) v >



PROOF Put

p1 := q2−q1, p2 := q3−q2, · · · pm−1 := qm−qm−1, pm := qm+1−qm (142)

then, p1, · · · , pm ≥ 1 and∑m

h=1 ph = n and (4.19) becomes to


+∞∑n=1

n∑m=1

∑1≤p1,···,pm,

∑mh=1 ph=n

∑ε∈{0,1}m

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

m∏h=1

((g1−ε(h)|g1−ε(h))− · (gε(h)|gε(h))−

)[ph2

]−χN0(ph)

· (g1−ε(h)|g1−ε(h))χN0

(ph)−∑

1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1


,T ′yh](th)(fxh|gε(h))(

(g1−ε(h)|f ′yh)χN1(ph) + (gε(h)|f ′yh)χN0(ph))

< u,



· · ·pm−1︷︸︸︷

Dε(m−1)D1−ε(m−1)Dε(m−1)(Dε(m−1)χN0(pm−1) +D1−ε(m−1)χN1(pm−1))

pm︷︸︸︷Dε(m)D1−ε(m)Dε(m)(Dε(m)χN0(pm) +D1−ε(m)χN1(pm)) v >

48


ΨN ′−m(χ[S′α,T′α] ⊗ f ′α, α ∈ {1, · · · , N ′} \ {yh}mh=1) >

This is the same as (4.23).

For each m ≤ n ∈ N, 1 ≤ p1, p2, · · · , pm, there are k ≤ m and 1 ≤ i1 <· · · < ik ≤ m such that

{pih}kh=1 ⊂ N0

{pα, α ∈ {1, · · · ,m} \ {ih}kh=1}

}⊂ N1 (143)

So, (4.23) is equal to


+∞∑n=1

n∑m=1

∑ε∈{0,1}m

m∑k=0

∑1≤i1<···<ik≤m∑

1≤p1,···,pm, {pih}kh=1⊂N0,

∑mh=1 ph=n

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

k∏h=1

((g1−ε(ih)|g1−ε(ih))− · (gε(ih)|gε(ih))−

) pih2−1

· (g1−ε(ih)|g1−ε(ih))−

∏α∈{1,···,m}\{ih}kh=1

((g1−ε(α)|g1−ε(α))− · (gε(α)|gε(α))−

) pα−12

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

k∏h=1

(fxih |gε(ih))(gε(ih)|f ′yih )∏

α∈{1,···,m}\{ih}kh=1

(fxα|gε(α))(g1−ε(α)|f ′yα)

m∏h=1


,T ′yh](th)

< u, (Dε(1)D1−ε(1))p1−1

2 Dε(1) · · · (Dε(i1)D1−ε(i1))pi12 · · ·

(Dε(ik)D1−ε(ik))pik2 · · · (Dε(m)D1−ε(m))

pm−12 Dε(m)

49



Change the order of sum∑∞

n=1

∑nm=1, one gets the sum

∑∞m=1

∑∞n=m. Mo-

reover, in (4.26), {pih}kh=1 ⊂ N0, so n =∑m

h=1 ph ≥ m + k. Replace n −mwith n′, (4.26) becomes to


+∞∑m=1

∑ε∈{0,1}m

m∑k=0

∞∑n′=k

∑1≤i1<···<ik≤m∑

1≤p1,···,pm, {pih}kh=1⊂N0,

∑mh=1 ph=n′+m

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

k∏h=1


) pih2−1

· (g1−ε(ih)|g1−ε(ih))−

∏α∈{1,···,m}\{ih}kh=1

((g1−ε(α)|g1−ε(α))− · (gε(α)|gε(α))−

) pα−12

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

k∏h=1


α∈{1,···,m}\{ih}kh=1


m∏h=1


,T ′yh](th)

< u, (Dε(1)D1−ε(1))p1−1

2 Dε(1) · · · (Dε(i1)D1−ε(i1))pi12 · · ·


pm−12 Dε(m)



50

Notice that

∞⋃n=k

{(p1, · · · , pm) ∈ Nm : {pih}mh=1 ⊂ N0,m∑h=1

ph = n+m} =

= {(p1, · · · , pm) ∈ Nm : {pih}mh=1 ⊂ N0} (146)

so, (4.27) is equal to


+∞∑m=1

∑ε∈{0,1}m

m∑k=0

∑1≤i1<···<ik≤m

∑1≤p1,···,pm<∞, {pih}

kh=1⊂N0

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

k∏h=1


) pih2−1

· (g1−ε(ih)|g1−ε(ih))−

∏α∈{1,···,m}\{ih}kh=1

((g1−ε(α)|g1−ε(α))− · (gε(α)|gε(α))−

) pα−12

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

k∏h=1


α∈{1,···,m}\{ih}kh=1


m∏h=1


,T ′yh](th)

< u, (Dε(1)D1−ε(1))p1−1

2 Dε(1) · · · (Dε(i1)D1−ε(i1))pi12 · · ·


pm−12 Dε(m)



Put

pih2− 1 =: dih ,

pα − 1

2=: dα, ∀α ∈ {1, · · · ,m} \ {ih}kh=1 (148)

51

then dh ∈ {0} ∪N and (4.29) becomes to

< ΨN−m(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}),ΨN ′−m(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′}) > +

+∞∑m=1

∑ε∈{0,1}m

m∑k=0

∑1≤i1<···<ik≤m

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

k∏h=1


α∈{1,···,m}\{ih}kh=1


m∏h=1


,T ′yh](th)

< u,∞∑d1=0

((g1−ε(1)|g1−ε(1))− · (gε(1)|gε(1))−

)d1

(Dε(1)D1−ε(1))d1Dε(1) · · ·

(g1−ε(i1)|g1−ε(i1))−

∞∑di1=0

((g1−ε(i1)|g1−ε(i1))−·(gε(i1)|gε(i1))−

)di1(Dε(i1)D1−ε(i1))

di1+1 · · ·

(g1−ε(ik)|g1−ε(ik))−

∞∑dik=0

((g1−ε(ik)|g1−ε(ik))−·(gε(ik)|gε(ik))−

)dik(Dε(ik)D1−ε(ik))

dik+1 · · ·

∞∑dm=1

((g1−ε(m)|g1−ε(m))− · (gε(m)|gε(m))−

)dm(Dε(m)D1−ε(m))

dmDε(m)v >



Denote, for σ ∈ {0, 1}

D1(σ) := Dg(σ)Dσ D2(σ) := (g1−σ|g1−σ)−DσD1−σDg(σ) (150)

and

Dg(σ) :=∞∑n=0

(g1−σ|g1−σ)n−(gσ|gσ)n−(DσD1−σ)n (151)

52

we have the following

THEOREM(4.8) Under the condition (4.11) and (4.6), the low densitylimit (4.7) exists and is equal to

< u⊗ΨN−m(χ[Sα,Tα]⊗fα, α ∈ {1, · · · , N}),⊗ΨN ′−m(χ[S′α,T′α]⊗f ′α, α ∈ {1, · · · , N ′}) > +

+∞∑m=1

∑ε∈{0,1}m

m∑k=0

∑1≤i1<···<ik≤m

∫ t

0

dt1 · · ·∫ tm−1

0

dtm

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

k∏h=1


α∈{1,···,m}\{ih}kh=1


m∏h=1


,T ′yh](th)

< u,D1(ε(1)) · · ·D2(ε(i1)) · · ·D2(ε(ik)) · · ·D1(ε(m))v >



53

4 §5 The Quantum Number Stochastic Dif-

ferential Equation

In the section, we investigate the relation between the low density limit (4.7)and the solution of some Quantum Stochastic Defferential Equation with thecondition (4.11).

Let consider the Fock space Γ(L2(R ⊗ (K, (·|·)), for rach ξ ∈ L2(R),T ∈ B(K), denote N(ξ ⊗ T ) the number operator by

< ΨN(ηr ⊗ fr, r = 1, · · · , N), N(ξ ⊗ T )ΨN ′(η′r ⊗ f ′r, r = 1, · · · , N ′)

=N∑j=1

N ′∑k=1

(−1)j+k < ηj, ξη′k > · < fj, T f

′k >

< ΨN−1(ηr⊗fr, r ∈ {1, · · · , N}\{j}), N(ξ⊗T )ΨN ′−1(η′r⊗f ′r, r ∈ {1, · · · , N ′}\{k}) >(153)

For each f, g ∈ K, s ≥ 0, difine the number process Ns(f, g) by N(χ[0,s]⊗|f >< g|). Let consider the quantum stochastic differential equation

U(t) = 1+

∫ t

0

∑σ∈{0,1}

(D1(σ)⊗dNs(σ, 1−σ)+D2(σ)⊗dNs(σ, σ)

)U(s) (154)

where,Ns(σ, ε) := Ns(gσ, gε) (155)

SinceD1(σ), D2(σ) are the bounded operators, (5.2) has a unique solutionand the solution can be obtained by iterative, i.e. put

U0(t) = 1 (156)

Un+1(t) = 1 +

∫ t

0

∑σ∈{0,1}

(D1(σ)⊗ dNs(σ, 1− σ) +D2(σ)⊗ dNs(σ, σ)

)Un(s)

(157)then, U(t) =

∑∞n=0 Un(t). Clearly,

Un(t) =

∫ t

0

∑σ1∈{0,1}

(D1(σ1)⊗dNt1(σ1, 1−σ1)+D2(σ1)⊗dNt1(σ1, σ1)

)Un−1(t1)

54

=

∫ t

0

∑σ1∈{0,1}

(D1(σ1)⊗ dNt1(σ1, 1− σ1) +D2(σ1)⊗ dNt1(σ1, σ1)

)∫ t1

0

∑σ2∈{0,1}

(D1(σ2)⊗ dNt2(σ2, 1− σ2) +D2(σ2)⊗ dNt2(σ2, σ2)

)Un−1(t2)

= · · · · · · =∫ t

0

∫ t1

0

· · ·∫ tn−1

0

∑σ∈{0,1}n(

D1(σ(1))⊗ dNt1(σ(1), 1− σ(1)) +D2(σ(1))⊗ dNt1(σ(1), σ(1)))· · ·(

D1(σ(n))⊗ dNtn(σ(n), 1− σ(n)) +D2(σ(n))⊗ dNtn(σ(n), σ(n)))

=

∫ t

0

∫ t1

0

· · ·∫ tn−1

0

∑σ∈{0,1}n

n∑k=0

∑1≤i1<···<ik≤n

D1(σ(1)) · · ·D2(σ(i1)) · · ·D2(σ(ik)) · · ·D2(σ(n))⊗

⊗dNt1(σ(1), 1− σ(1)) · · · dNti1(σ(i1), σ(i1)) · · ·

dNtik(σ(ik), σ(ik)) · · · dNtn(σ(n), σ(n)) (158)

First of all, one has

THEOREM(5.1) The solution of (5.2) is unitary.

PROOF The proof is the same as ones of Theorem (6.3) of [1].

Now, we want to express the quantity

dNt1(σ(1), 1− σ(1)) · · · dNti1(σ(i1), σ(i1)) · · ·

dNtik(σ(ik), σ(ik)) · · · dNtn(σ(n), 1− σ(n)) (159)

LEMMA (5.2). For each n,N,N ′ ∈ N, T1, · · · , Tn ∈ B(H), H is a

Hilbert space, {fh}Nh=1, {gh}N′

h=1 ∈ H, then,

< ΦN(f1, · · · , fN), N(T1) · · ·N(Tn)ΦN ′(g1, · · · , gN ′) >

55

=n∑

m=1

∑(S1,···,Sm)∈Pn,m

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

m∏h=1

(−1)(xh,N)+(yh,N′) < fxh , TShgyh >< ΦN(f1, · · · , fx1 , · · · , fxm , · · · , fN),

ΦN ′(g1, · · · , gy1 , · · · , gym , · · · , gN ′) > (160)

where, Pn,m is the set of the ordered partitions of {1, · · · , n} and for each(S1, · · · , Sm) ∈ Pn,m, Sj ⊂ {1, · · · , n} and Si ∩ Sj = ∅ for i 6= j,

⋃mh=1 Sh =

{1, · · · , n}. The order is defined by the following

Si < Sj ⇐⇒ min{s : s ∈ Si} < min{s : s ∈ Sj} (161)

This Lemma is same as Lemma (2.2) of [8], so the proof is omitted.

LEMMA (5.3). For each n,N,N ′ ∈ N, T1, · · · , Tn ∈ B(H), H is a

Hilbert space, {fh}Nh=1, {gh}N′

h=1 ∈ H, {ξh}Nh=1, {ηh}N′

h=1 ∈ L2(R), then,

< ΦN(ξ1⊗f1, · · · , ξN⊗fN), dNt1(T1) · · · dNtn(Tn)ΦN ′(η1⊗g1, · · · , ηN ′⊗gN ′) >

=∑

1≤x1,···,xn≤N|{xh}nh=1|=n

∑1≤y1,···,yn≤N|{yh}nh=1|=n

n∏h=1

(−1)(xh,N)+(yh,N′) < fxh , Thgyh > ξxh(th)ηyh(th)dth

< ΦN(ξ1⊗f1, · · · , fx1 , · · · , fxn , · · · , ξN⊗fN),ΦN ′(η1⊗g1, · · · , gy1 , · · · , gyn , · · · , ηN ′⊗gN ′) >(162)

PROOF By Lemma (5.1), one has

< ΦN(ξ1⊗f1, · · · , ξN⊗fN), dNt1(T1) · · · dNtn(Tn)ΦN ′(η1⊗g1, · · · , ηN ′⊗gN ′) >

=n∑

m=1

∑(S1,···,Sm)∈Pn,m

∑1≤x1,···,xm≤N|{xh}mh=1|=m

∑1≤y1,···,ym≤N|{yh}mh=1|=m

56

m∏h=1

(−1)(xh,N)+(yh,N′) < fxh , Thgyh > ξxh(th)ηyh(th)dtSh

< ΦN(ξ1 ⊗ f1, · · · , fx1 , · · · , fxm , · · · , ξN ⊗ fN),

ΦN ′(η1 ⊗ g1, · · · , gy1 , · · · , gym , · · · , ηN ′ ⊗ gN ′) > (163)

Notice that for h = 1, · · · ,m if |Sh| ≥ 2, one has dtSh = 0. This gives theproof.

Applying (5.10) to (5.6), we know that for each n ∈ N, k = 1, · · · , n,1 ≤ i1 < · · · < ik ≤ n, N,N ′ ∈ N, {fh}Nh=1, {f ′h}N

′

h=1 ⊂ K, {Sh, Th}Nh=1,{S ′h, T ′h}N

′

h=1 ⊂ R, g0, g1 ∈ K,

< ΦN(χ[Sh,Th]⊗fh, h = 1, · · · , N), dNt1(σ1, 1−σ1) · · · dNti1(σi1 , σi1) · · · dNtik

(σik , σik) · · ·

dNtn(σn, 1− σn)ΦN ′(χ[S′h,T′h] ⊗ f ′h, h = 1, · · · , N ′) >

=∑

1≤x1,···,xn≤N|{xh}nh=1|=n

∑1≤y1,···,yn≤N|{yh}nh=1|=n

k∏h=1

(−1)(xih ,N)+(yih ,N′)(fxh|gσ(ih))(gσ(ih)|f ′yh)χ[Sh,Th](th)χ[S′h,T

′h](th)dth∏

α∈{1,···,n}\{ih}kh=1

(−1)(xα,N)+(yα,N ′)(fxα|gσ(α))(1−gσ(α)|f ′yα)χ[Sα,Tα](tα)χ[S′α,T′α](tα)dtα

(164)

Comdine together (5.6), (5.12) and Theorem (4.8), we obtain the following

THEOREM(5.4) Under the condition (4.11) and (4.6), the low densitylimit (4.7) exists and is equal to

< ΨN(χ[Sr,Tr] ⊗ fr, r = 1, · · · , N), U(t)ΨN ′(χ[S′r,T′r] ⊗ f

′r, r = 1, · · · , N ′) >

(165)where, U(t) is the solution of (5.2).

we don’t care what is exact ϑ. This dues to that

57

1. in our low density limit the type II terms will contribute only zero,therefore ϑ = 1 or −1 does not influence our limit;

2. in our discussion the function of the type II terms is only influence ouruniform estimate. Since our uniform estimate is done on the module–absolutely uniform estimate, the exact form of ϑ does not play anyfunction.

PROOF The normal ordered form of product (3.2) should be the follo-wing form

n−1∑m=0

Cn,m(A+ · · ·)n−m(A · · ·)n−m (166)

where, (A · · ·)n−m is a product of n−m annihilators; (A+ · · ·)n−m is a productof n−m creators and their time-indices order are considered as the nuturalone, i.e. increasing; Cn,m is a product of a power of (−1) and some scalarproducts which come from taking the product (3.2) to the normal orderedform. For each m = 0, 1, · · · , n− 1, there exist m pairs operator being usedto produce scalar products. Clearly, each “pair includes one of creator andone of annihilator. Lebal the m creator with q1, · · · , qm and without loss thegenerality, suppose that q1 < · · · < qm. Notice that A+(St1gε(1)) has beenthe ordered form, so q1 ≥ 2 and naturally, qm ≤ n. Label the m annihilatorswith p1, · · · , pm, then pm ≤ n− 1 and ph < qh, h = 1, · · · ,m, |{ph}mh=1| = m.So we get the following

n−1∑m=0

Cn,m(A+ · · ·)n−m(A · · ·)n−m =n−1∑m=0

∑2≤q1<···<qm≤n

∑1≤p1,···,pm≤n−1

ph<qh,h=1,···,m, |{ph}mh=1|=m

ϑ

m∏h=1

< Stphg1−ε(ph), Stqhgε(qh) >∏α∈{1,···,n}\{qh}mh=1


α∈{1,···,n}\{ph}mh=1

A(Stαgε(α)) (167)

where, ϑ = ±1 is some power of (−1) which comes from the CAR-relation.Now we split the right hand side of (3.6) to two parts according to ph = qh−1

58

for any h = 1, · · · ,m or otherwise, i.e. the right hand side of (3.5) can bewritten to

n−1∑m=0

∑2≤q1<···<qm≤n

ϑ

m∏h=1

< Stqh−1g1−ε(qh−1), Stqhgε(qh) >

∏α∈{1,···,n}\{qh}mh=1


α∈{1,···,n}\{qh−1}mh=1

A(Stαg1−ε(α))

+n−1∑m=0

∑2≤q1<···<qm≤n

′∑(p1,q1,···,pm,qm)

ϑ

m∏h=1

< Stphg1−ε(ph), Stqhgε(qh) >

∏α∈{1,···,n}\{qh}mh=1


α∈{1,···,n}\{ph}mh=1

A(Stαg1−ε(α)) (168)

Comparing (3.7) with (3.4), we find that in order to prove the theorem oneneeds only to know what is ϑ in the first term of (3.7).

and by Theorem (2.1), the factor with form, in (3.12), of

|< ΦN−m(z

∫ Tα/z2

Sα/z2

Sufαdu, α ∈ {1, · · · , N} \ {αh}mh=1),

ΦN ′−m(z

∫ T ′β/z2

S′β/z2

Suduf′β, β ∈ {1, · · · , N ′} \ {βh}mh=1) > |·

59

The Fermion number processes as a functional central limit of quantum Hamiltonian models

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