Low density limit: Without rotating wave approximation

Vol. 37 (1996)

REPORTS

ON MATHEMATICAL

PtiMK

S

No. 2

LOW DENSITY LIMIT: WITHOUT ROTATING WAVE APPROXIMATION YUN-GANCJ Lu DipartimentoDi Matematica, UniversiG di Bari, Via E. Orabona 4, I-70 125 Bari Centro V. Volterra, Universita di Roma II, Italy (Received January 17, 1995) In the present paper we investigate the low density limit of a quantum “System + Reservoir” model in which the temperature is finite and the Hamiltonian of the system has a discrete spectrum. It is proved that the matrix elements of the time evolution operator, with a time resealing and some proper choice of collective vector, tends to matrix elements of a solution of a quantum stochastic differential equation driven by a quantum Poisson process.

1. Introduction The low density limit of quantum “System+Reservoir” (S+R) models have been investigated in [I, 2, 31 under the so called rotating wave approximation condition. We have proved that the time evolution of the model converges to a solution of a quantum stochastic differential equation driven by a quantum Poisson process. In the present paper, our physical model and basic assumptions are the same as those of [l, 2, 31 except that we do not assume the rotating wave approximation. Let be given a quantum S+R model, i.e. -the system Hilbert space l-lo; -a one-particle reservoir Hilbert space X1; -the system Hamiltonian Hs (which is a self-adjoint operator on ‘HO); -the one-particle reservoir Hamiltonian H1 (which is a self-adjoint operator on Bl).

In this paper we shall restrict ourselves to the case of a boson reservoir. The reservoir then is described by a boson Fock space over the Hilbert space X1 which will be denoted, as usual, by r(3-11). Moreover, the Hamiltonian of the reservoir is given by HR := dr(H1) ( second quantization of the one-particle Hamiltonian) and the total Hamitonian is given by a self-adjoint operator on the total Hilbert space X0 @ r(N1), which has the form Htot : = HS @ 1 + 1 @ HR + Hint =: Hfree + Hint. where Hint is the interaction Hamiltonian time evolution is given by the operator

61)

between the system and the reservoir. The

Ll771

178

Y.-G.LU e ztHfree ut *= .

Obviously it satisfies the differential

. e-it&t

(1.2)

equation

$Ut

= -iHi”,(t)Ut,

(1.3)

where (here and in the following) Hint(t) defined as eitHfr=C~i,te--itHrre. will be called the evolved interaction. Let us consider a gauge invariant quasi-free state v(Z) (z > 0 is the fugacity), i.e. for each f E ‘HI, @)(W(f))

= exp(-i(f,(l

+ ze-HP)(l

- ze-HP)-lf)),

(1.4)

where H is a self-adjoint operator and in the present article it is supposed to be commutative with the one-particle Hamiltonian. Up to the GNS-construction, one can write the quantity in (1.4) as

(@z , WU)@z). The interaction

Hamiltonian

will be assumed to have the following form:

Hint I= i(D @ A+(go)&l)

- D+ @ A+(gl)A(go)),

where D E B(7f1) and go, 91 E til. Therefore, St := ,@Hl, the evolved interaction

(1.5)

in terms of

D(t) := eifHsDe-itHs!

can be written in the form

Hi,,(t) := i(D(t) @ A+(&go)A(Stgl)

- D’(t) @ A+(Stg1)A(&go)).

(1.6)

Moreover, it will be assumed that there exists a subset K (which includes go,gl) of the one-particle Hilbert space HI such that J I(f> Stg)ldt rR

<

+cQ

VfYLl

E

x,

(1.7)

and for any f, g E K the series

I(.&St ev(-PnH)g) I dt ?I=1

R

has a positive convergence radius. Instead of the rotating wave approximation, in the present paper we shall assume the following condition, which practically means that the system Hamiltonian has a discrete spectrum (e.g. N-level case): D(t)

= g d=O

Dd . eit‘-’

(1.8a)

LOW DENSITY LIMIT

179

with wa = 0,

QO2 d # d’.

wd # Wd’,

(1.8b)

Our goal is to investigate the limit behaviour, as z + 0, of the time resealed operator Utlz. Practically, we shall study the limit of the expression like (u @ vZ(f),

&/z ‘u@ VW))?

where U, w E Ha and V,(f),Vi(f’) are two vectors (the so-called collective vectors) in the reservoir space and whose form will be determined according to the following principles: (i) K(O) = V:(O) is a vacuum vector; (ii) the limit limz_+a(Vt(f), .Vi(f’)) exists and has the form (V(f), .V’(f’)) on some (to be determined) limit space; (iii) {V(f): f E Kc) is a total family in the limit space. In order to understand the construction of the collective vectors, let us recall some well-known facts. Denote by 1-11the conjugate of El, i.e. /,: 3-11-

El.

W), then ‘FL”,is a Hilbert has

49))‘

L(Xf) := XL(f),

(1.9) (1.10)

:= (97 fh

space. It is well-known that up to a unitary isomorphism

r(xl) W)

=

r(xl)

(1.11)

@ r(xi)3

= @(Q+.f) @ @(Q-f) Gz = Q,@@,

one

Vf E XI,

(1.12) (1.13)

where @‘,@” are the vacuum of r(3-tr) and r(%!i), vector and the operators Q+, Q- are defined as:

respectively;

@(.) is a coherent

(1.14)

=

Moreover,

61

=: fiQ_.

up to the same unitary isomorphism, A(f) = A(Q+f)

(1.15) one has

@ 1+ 18 A+(Q-f).

Therefore A+(f)&)

= (A+(Q+f)

@ 1+ 1@ 4Q-f)) .

(A(Q+g) @ 1 + 18 A+(Q-d)

(1.16)

180

Y.-G.

LU

A+(Q+fMQ+d 8 1 + d$A+(Q+f) @ A+(Q-d+ + A(Q+g) 8 A(Q-0) + 21~ A(Q-W+(Q-d = A+(Q+f)A(Q+d 8 1 + d++(Q+f) 8 A+(Q-g)+ + A(Q+g) 8 A(Q-1)) + 4 @A+(Q-dA(Q-f) + (Q-f,Q-g)J. =

Throughout the paper, for simplicity, the dition is assumed: two test functions in the supports in the energy representation. Thus action of any function of H. This assumption the interaction Hamiltonian satisfy (go, Ste-‘jHgi) With the condition ((f?

(1.17)

following technical (unnecessary) coninteraction Hamiltonian have disjoint the disjointness is invariant under the means that two test functions gO,gl in vlt E LR.

= 0

(1.7), one can define, for each f. g, f’, g’ E K: and d = 0,l. g)l(f3

g’>>d

:=

s

ezdd”(.f,

&f’)(fk9’.

edBHg)

. . . ,

(1.18)

dt.

w It has been shown in [3, 71 that (1.18) defines a non-negative bilinear form. In the following the Hilbert space obtained from the set {(f, g); f, g E K}, by taking the (.].)d-quotient and completion, will be denoted by KS. As suggested by the discussion in [3], the collective creators are defined as A:(S,T;

{fd}, {gd}) := 2 d=O

fi

y’e’““‘a+(S,f,j)

C3A+(sSte-+dHgd)dt,

(1.19)

S/.:

are two families of test functions. From now on, we S L T and {.fd}r{gd} shall always take such a family {fd} in which there are only finite test functions different from zero. This finiteness guarantees that the collective creators are well defined. Moreover, we also define the collective annihilators A;(S: T; {fd}? {gd}) as the conjugate of the collective creators, i.e.

where

A,($ T; {fd}, {gd}) := 2 d=O

T/z ~6 s e?%l(S&)

(1.20)

@ A(tS&jHgd)dt.

S/:

In this notation, the main result in the present article can be briefly described as follows: the limit of II U@ @@I, ( rI AZ({Sj], {Tj]; {fl,j.d)r {f2.j.d))@ j=l

(1.21) exists for any U,U E ?-LO,a, N f N, {Sj), {T’}, {S$}, {T’} C will be called the test numbers) and for any test function

R

(in the following, they

{fl,j.d},

{f2,j,d},

{f{.j,d},

181

LOW DENSITY LIMIT

{f.,j,d}. Moreover, the limit (1.21) is described by a quantum stochastic process living on the Fock space over a Hilbert space determined by {Ic~}~=,, and the quantum stochastic process satisfies a quantum stochastic differential equation driven by a quantum Poisson process. 2. Preparations In this section, some preparations will be done for the following sections. First of all, we compute the limit scalar product of the collective test functions. LEMMA 2.1. For every N E N, S, T, S’, T’ E R and for all test functions {fi}$,

{ fd}yEO.

C K we have

=

(X[S,TJ, X(S,T~)L~IR)

c s

eiwdt(fd,

d=O

Wi)

dt.

(2.1)

R

Proof: We see that

=

5

5

d=Od’=O

i S

dZL “T””

dUi(fd,

(S’kl)/Z

where Sp := eitwdSt. By the Riemann-Lebesgue converges to x[S,T],

X[S’,T’])U(W)

The following result gives a description process lives.

fsj17 {Tjl

quantity

lemma, as z + 0, the above quantity

.I

’

(fd,

St.fA)eLwdt

dt.

(2.2)

oz

d=O

LEMMA

S,d:f~,)ei(Wd’-Wd)UIZ,

of the limit space on which our limit

2.2.For every n E N, E E (0, l}“, and for any choice of the test numbers C lR and the test functions { fj,d}, {gj,d}, the limit, as 2 + 0, of the n

(@@@L, n A:‘j’({s,),{Tj); j=l

{.fj,d}r

{gj.d})@

@ @t)

(2.3)

182

Y.-G. LU

exists and

is equal

to

(2.4) (Here and in the following, for a creation operator A, we use the notion

A+ and an annihilation

operator

(2.5) and A(.) (resp. A+(d)) in (2.4) is an annihilation Fock space r(L*(R+)

(resp. creation)

operator

on the

@ &:), d=O

and P is the vacuum vector of this space.) Proof First of all, notice that if R. is odd, it follows from the CCR that both (2.3) and (2.4) are equal to zero, so one needs to prove the lemma only in the case of n = 2N for some N E N. By the definition of the collective creation and annihilation operators, (2.3) is equal to Tt/t c

&2

dl,...,dnEN

5

-%/z

T,/z dtl . . .

s S,/z

dt, x

(2.6) By the CCR, the scalar product in (2.6) gives a sum of products of scalar products, i.e. has a form like C {product of TXscalar products with some frequency terms eitjwd3 }. Any scalar product contains one annihilator and one creator, Since, corresponding to every instant of time tj, we have two operators, it is possible that some terms in the sum have the form ...

(&,fj,d,r

&fk.dlc)

S-e (&,e-3PHgm,d,,,5

Stje-3pHgj,d,)

.+.

(2.7)

with Ic # m. As in [2, 3, 71, such a term will be called the cross term and it was shown in [2, 3, 71 that the cross terms contribution to the limit of (2.6) is equal to zero. In other words, in order to consider the limit of (2.6), one has only to investigate the terms with the following property: if an annihilator A(&$ fj,d,) is used to produce a scalar product with a creator A+(&, fk,d,,), then the annihilator A(~S~~e-ifl~g~,d,) is used to produce a scalar product with A+(~&~e-~P~gk,d~). Moreover, it is obvious, by the CCR, that the scalar product in (2.6) is not zero only if n

&(j) j=l

=

in =:N,

(2.8a)

LOW DENSITY LIMIT

183

and for any 1 5 nz. 5 n, [{k;

k I WE(~)

where 1x1 denotes cardinality of annihilation operators and for any the time variables tl, ..*, t m, there For every E E (0.1)” satisfying {TI....,

= O}l 2 I{k; k I m,E(k)

= l}[,

(2.8b)

the set X. That is, we have N creation and N 1 5 m 5 n, among the operators corresponding to are more annihilators. (2.Q let us denote TN} := {j; E(j) = 0)

(2.9a)

with the order ~1 < .. . < TN, and {ICI.. . ., kjv} := {j; Obviously,

and {kl.. .., kN}

{rr , . . .,T-N}

I

= 1).

(2.9b)

are determined

uniquely

by a given E E

(0, 1)“. With the above arguments term of order o(l), as Tl/Z

and notation,

we are able to rewrite (2.6), up to a

Tl%/z

,N

(2.10) The multidimensional I? z s

dtTJ

S,,lr

equal

to

TkO(J)/z

TT312 j=l

integral in (2.10) is

tk+j dt%,, eXp(i(Wd,0C3,

s

wdrJ tr3 )>x

skb(3)/z ’

(‘tvj

fr,‘dp)

1 Stkocjj

fkc(,),d~oc,j

)(Stko(j)gk”(~,,d~~(j)

(2.11)

Y StP3 e-pHgr,,d,,).

By a change of variables (2.12) and the Riemann-Lebesgue

lemma, the limit of (2.11), as z + 0, is

equal

to

(2.13)

184

Y.-G. LU

By the CCR, the sum of the right-hand equal to

sides of (2.13) for all rll, . . . , d, E iV is

dP3=0

j=l

Taking the sum over all ~7 E S,v, such that k-,(j) > ~j. j = 1, . . . , N, we finish the proof. 0 3. The uniform estimate and negligible terms In order to consider the limit of (1.21) operator U, into a series: U,,, =

as usual (see [l, 2, 31) we expand the

E ydtl j:dt2. n=O 0

(3.1)

0

The first problem to investigate is the possibility of exchanging the limit lim++a . . with the sum Cr.o (i.e. the uniform estimate). Let us study, for every n, N, N’ E N, the scalar product N

( rI UC3

A,+(b%h

{Tj);

{fl,j,d)~

{f2,j.d})@

‘8 @L,

j=l

Tzdtl jdt2...‘T1 dt,(-i)n.Hi”t(tl). 0

0

. . Hint(t,)x

0

By the definition of the evolved Hamiltonian Hint(t) and formula (1.17), it is easy to verify that the absolute value of (3.2) is dominated by

c

II4 * Ibll *11w

1p(

&tl

jl(It2...

0

0

‘7’

fi A,+Wjh {Tj);

{fl,j,d)r

{.f2,j.d))@

j=l

EE{o,~l,~2}-,~E{o,l}-

dt,A(E(‘)+(‘))(tl,

0 N’

go, gl). . . A(‘(+@))(tn:

go, gl)x

@ @L,

185

LOW DENSITY LIMIT

where A(‘,‘)(t,go,gl)

:=

A(‘90)(t, go7 gl) := A+(&Q+go)

A(-‘*“)(t,go, 91) := A(&Q+d

(3.4a)

@ 1.

A+(StQ+go)A(&Q+gl)

8 A+(&Q-d,

(3.4b)

@ A(&Q--So).

(34c) (3.4d)

Ac2*‘)(t, go, 91) := 1 8 A+(LStQ-g1)A(LStQ-go).

(3.4e)

A(-2.0)(t. go. 91) := (LQ-go, LQ-gI)t. A@,‘) := (AhO))*,

(3.4f)

and m(e) := ;I{$

E(j) = &l}] + j{j: F(j) = f2}/.

(3.5)

Due to CCR and the definitions of the collective creation operators At and of the evolved Hamiltonian Hint(t), for any E E {O.fl, f2)” and E E (0, l}“, the absolute value term in (3.3) is less than or equal to

where F. G E {Q&SE. _fc,j,dv~~~~f~,~,~}; s := l$tl{Sj,

s;, : fE,j,d # 0, f:,jr,d’

T := ~II{T,,T;,

:

.fQ,d

#

0:

.f&,d’

# 0 for c = 1, 2).

(3.7a)

#

(3.7b)

0

for

6 =

1, 2).

Thanks to the discussion in [2, 71, we have the following result. THEOREM

3.1. For each 71,N. N’ E RI, we have

AL+t{Sj)t {Tj}i {fj.d}r

{gj.d})@ ‘8 @t:

j=l

n A:t{S;). {T’); {f’i

x 21~8

4. {&})@ 8 @‘)I

j=l

2 C . n5. 16”

1lD11” . o~m~~~(llglls + ~z(l))~~f-, -

where o,( 1) --+ 0 for zz-+ 0; C is a constant determined

(3.8)

by S. T, N, N’ and the test

186

Y.-G.

LU

(3.9) Moreover, if (3.10) then the limit (1.21)

is equal to the sum over all n of the limit of (3.2).

An easy consequence

of Theorem

3.1 is:

THEOREM 3.2. The limit of (3.2) for z + 0 is equal to the limit of N (

U@

l-I A:({SjI,

{q);

{fl,j,d),

{f2,j,d})@

@ @L?

j=l

y&I

](It2...

0

0

‘7’

dt&l:)W;&)

.~1 I$&)x

0

X ’ @ fiAZ(lS:I>

IT;);

(f;,j,dI,

I.fi,j,dI)@

(3.11)

@ @t)>

. . where, up to a urutaly Isomo~h~~cl 9 f&(t)

= i(A+(StQ+so)A(StQ+s~) + A(StQ+gl)

8 A(&&-go))

@ 1 + fi(A+(&Q+go)

~3A+(&Q-gl)

+ ~18 A+(LStQ-gI)A(LStQ-go)

+

- CL).

(3.11)’

Remark: The difference between H&(t) and the original evolved interaction miltonian Hint(t) is the t independent term z(LQ_gg, LQ_gl)! and its conjugate.

Ha-

Proof: We must prove that the limit of

I,m(E)( fi AZ({Sj), {Tj);

{fl,j,d),

{f2,j,d))@

@@L,

j=l

y&,]dk2... ‘j;ldt,A@(‘)*‘(‘))(tl,

go, gl). . . A(+),‘(“))(tn,

for z -+ 0 is zero if there exists a Ic E { 1, . . . ,n} (3.12) is less than or equal to z

&Q-so,Q-dl 0

+m(E)-l(

fi A:({Sji, j=l

go, gl) x

such that c(k) = -2. In this case,

{T3); {.fl,j,d)r

{f2,j,d))@

@ @L>

LOW DENSITY LIMIT

tlz Sdtl

Jdtz...

0

0

x &(*)+(‘))(tl,

187

t~2dtk_lfj;1dt~+~...t~1dt~ 0

go, gl)

. . .

x 0

A(E(*l~.‘(“l))(tb-l,

go, gl) x

x A(E(“+l).‘(“+l))(tk+~, go, gl). . . A(E(n).E(“))(t,, go, gl) x

A:W;l.

x fi

V;b

kf;,j,J~

U;,j,dH@

@ @L)i.

(3.13)

j=l

Now let us define

0(l) = E(l),

/3(l) = E(l), . . *. a(lc - 1) = &(k - l), /3(/C- 1) = E(lG- l), . *. , a(?2 - 1) = e(n).

o(k) = &(IC+ 1) ,0(/C) = C(k + l),

B(n - 1) = c(n),

(3.14)

then 0 E (0, fl, f2}~z--1, In order to complete

m(a) = m(e) - 1,

our proof, due to Theorem !i_moz ~,(Q-go&-sMs

0 E {o,l}n-l.

(3.15)

3.1 it is sufficient to show that (3.16)

= 0.

0

However, this is obvious because of our basic assumption on the test functions go! ,gl. Thanks to Theorem

3.2, we are lead to study the following quantity:

where E E (0. fl, 2)” and

By the definition equal to

of the collective

annihilation

and creation

operators,

(3.17) is

188

Y.-G.LU

Applying the CCR and forgetting the frequency terms (since they are less than or equal to one), we see that the second scalar product in (3.18) is dominated by a quantity like a sum of

i.e. a sum of products of scalar products, where each scalar product comes from one commutator [A, A+]. Thus, (3.18) is controlled by a sum of the following type of terms:

T; ,I2

l-:/z X

s

dvl...

s

S;/Z

dv~v ydtl

~dt,...‘r’d,,,~~(S~~f,S,,g)~.

0

S:,,/z

0

0

(3.20)

a.0

Remark: In expression (3.20), each product like (3.19) corresponds to a possibility of using all operators to produce scalar products and the “sum” is a sum for all such possibilities. The test functions in (3.19), i.e. f, g, can be equal to Q&go, Qkgt , .fi,j,d,

.f2,j,d,

or

e-+PH.f;

j 3 dr

?

epbpHfi.j

d’

Now let us analyse carefully the above expression. how the operators in the product

Practically, we shall examine

_@)~‘(‘))(tt, go, gl) . . . A(E(n),E(n))(tn, go, gl)

(3.21)

are used to produce the scalar products. For simplicity we shall omit the test functions and the frequency terms, and denote the operator corresponding to the time variable t, (where p = 1, . . . , n) by A+(t,)A(t,)@l, A+(tp)@AA+(tp), A(tp and 1 @IA+(t,)A(t,) respectively. Similarly, the operator A+(S,F) @ A+(LSue-+oHG) will be denoted by A+(u)@A+(,u)(=A+(,~)@l~l@A~(u)) for any UE {u~,...,u,v,zII,..., UN’} and the test functions F, G. First of all, let us consider the operators of the type A+(t,)A(t,) @ 1. Applying the arguments developed in [l, 2, 71, we have the following results:

LOW DENSITY LIMIT

189

LEMMA 3.3. If the annihilation operator A(&) (or, respectively, the creation operator A+(&,)) is used to produce a scalar product with a creation operator A+(&,) (resp. an annihilation operator A(&)) and (q -pi > 1, then the limit of (3.18), for z + 0, is equal to zero. LEMMA 3.4. If -the annihilation operator A(tp) is used to produce the scalar product with a creation operator A+ (vk); -the creation operator A+(&,) is used to produce the scalar product with an annihilation operator A(uj); -the operator 18 A+ (Q) is used to produce the scalar product with an operator 1 8 A(u) and u # uj (such term is still a cross term), then the corresponding limit of (3.18) is equal to zero.

As the second step let us consider the operators 81 . 1 @ A+(Q).

like A+(t,) @ A+(&,)( = A+ (t,) @

LEMMA 3.5. if the operator A+(&,) @I1 is used to produce the scalar product with an operator A(&), p - q > 1, then the limit of (3.18), for s -+ 0, is equal to zero. Remark: In Lemma 3.5, we have only considered the operator A+(t,)@l. One has not a similar conclusion for the operator 18 A+($,), and this point will be explained carefully in the following section.

LEMMA 3.6. If -the

operator A+(&,) @ 1 is used to produce

the scalar product with an operator

A(W) @ 1; -the operator 18 A+(t,) is used to produce the scalar product with an operator 1 cs A(q); -lc # j (such term is still a cross term), then the limit of (3.18), for z -+ 0, is equal to zero.

The discussion of an operator like A(tp)@A(tp) is analogous to that of A+ @A+, and we shall omit the details and go over to investigate the 1 @ A+(t,)A(t,)-type operators. Since in this case there are “too many” small quantities z ( there is a z corresponding to each operator), the discussion in [2] provides that: LEMMA 3.7. If there is a p E (1,. . . , n} such that E(P) = 2, then the limit of (3.18),

for z --+ 0, is equal to zero.

4. The low density limit Having investigated the uniform estimate and verified the negligible terms, in the present section we shall study the relevant terms. The discussion in the preceeding section asserts that the investigation of the limit of (3.2) can be reduced (if we forget the system part) to studying the limit of the

190

Y.-G. LU

sum, for all E E (0, &l}“, E E (0, l}“, of N p(4 (n

AZ+({SjI,

{Tj);

{.fl,j,dI~

{f2.j,dI)@

@@IT

j=l

tl j- dtl j- dt2... t/z

0

‘7’ dt,,A(‘(1)3’(1))(tl. go> gl)

. . .

A(E(“)+))(t,,

go, gl)

x

0

0

In order to discuss clearly our problem, let us first of all introduce some and terminology. For any (I 2 p, we say that t, is connected with t, if there -an annihilator with the time variable t, being used to produce a scalar with a creation operator with the time variable t,+i, -an annihilator with the time variable 2;,+1 being used to produce a scalar with a creation operator with the time variable t,+z,

notation are product product

-an annihilator with the time variable t,_i being used to produce a scalar product with a creation operator with the time variable t,. Remark: It is obvious that if 4 > p and t, is connected A@(j).‘(j)) must be of the A+A @ l-type for any p < j < q.

with t,, the operators

Now let us consider the operator A (E(l)+(l))(tl, go, gt ). It is clear that corresponding to every product like (3.19) there exsits a unique pr 5 rz such that tl is connected with t,, and is not connected with t,, +i (if the oper,ator is of the form A+ 8 A+, pl must be equal to 1). The same argument asserts that there is a unique p2 E {pi + 1,. . . , n}, such that tpl+i is connected with t,, and not connected with tpz+l. Repeating the procedure we divide the set { 1:. . . . n} into {po(:= 0) + 1,.

. . ,Pl}

u {Vl

+

1,.

.3p2}

u . . . u {pm-1

+

1,.

. . ,p,(

=

n)},

(4.2)

where m h. = O,... length of reponding

5 n. is also determined uniquely by E and E. In the following, for each ,nz - 1, the set {tPh+l.. . . , tP,,+l} we shall call a chain; ph+l - (ph + 1) the the chain; the operators with the time variables {t,,+l, . . . , tph+l} the coroperators and {tp, +I. . . . . tph+,}~c;l a composition of chains. For simplicity, operators we shall also call a chain. {t,,+lt~ . . * tph+l } 7 together with the corresponding There are the following types of chains: (i) A+A @ l-chain. If each corresponding operator is of the form AfA @ 1, then in the product like (3.19) the product of operators A+(tl)h+l)A(tYh+l)~l...A+(tph+l)A(tph+l)~l

plays a role like A+($+,+, )AtL,+l)

@ Wtph+Mtph+zg’)

... (StPh+l-lg)StPh+lg’)~

(4.3)

LOW

DENSITY

LIMIT

191

where g,g’ E {Q+go, Q+gl}. Such term, up to a constant, is still like A+A @G 1. (ii) A+ 8 A+-chain. If the product of the corresponding operators is of the form A+&,,+l)A&+l)

8 1 . ..A+(tph+l-l)A(tPh+l-l)

then it will play a role, in the product

@ lA+G,,,+,) @ A+&+,).

(3.19), like

A+ttph+1)~A+(tPh+l)(StPh+lg~StPh+2g’)...(St,,h+l--~,StPh+lg’).

(4.4)

Such term, up to a constant, is still like A+ 8 A+. (iii) A @ A-chain. If the product of the corresponding operators is of the form A(tP,,+l) @ A(tPh+l)A+(tPh+2)A(tPh+2) 8 1 .. . A+@,,+,bW,,,+,) then it will play a role, in the product A&,+1) ~

(3.19), like

A(t,,+l)(St,,+l~.Stp,+,g’) ...

Such term, up to a constant, is still like A 18 A. (iv) Constant chain. If the product of the corresponding operators A(t,,+l)

@ 1.

@ A(tph+l)A+(tp,+2)A(tPh+2)

(4.5)

is of the form

@ 1.. .

. ..A+(t.,+,-l)A(t,,+,-l)~

then it will play a role, in the product

(StPh+,-lg,StPh+lg’).

lA+(t,,,+,)~A+(t,,+,).

(3.19), like

1~ A(t,,+l)A+(t,,+,)(StPh+~g~

Stph+d)

... Pt~h+l-,,v.

Stph+J).

(4.6)

Moreover, the arguments on the cross- and non-cross-negligible terms (see Section 3 and [2, 71) make sure that, up to an o(l), the contribution of the operator l@A(tp,+l)A+(t,,+,) is just the scalar product (SI,,+,_le-$‘Hg. StPh+, e-jBHg’). Therefore, instead of (4.6), in practice we have (St Ph+l~.~tp~+Z~‘)~~~~~tP~+L_,9’~tP~+,~’~~~tp,,+,_-l~~~‘~H~~.~t,,L+l~~~9~~‘).

(4.7)

Thus, for every fixed E and E, the integral t,,-1

s

dt,

0

0

A(E(‘)s’(l))(tl, go. g,) . . A(‘(“)+))(t,,

.yo, gl)

0

in (4.1) becomes a sum of the following quantities:

t/z j- dtl

j? dt2.. .

0

0

‘j;’dt,

composition

of chains(tl.

. . , t,{).

(4.8)

0

where, in order to enhance the time variables, we have used the notation composition of chains (tl.. . . : t,,).

192

Y.-G. LU

In (4.8), as explained before, up to an o(l), in any fixed composition of chains, any operator that remained will not be used to produce a scalar product with another operator which remained in other chains. That is, all remaining annihilation operators must be used to produce scalar products with creation operators in the product

(4.9) j=l

and all remaining creation operators annihilation operators in the product iv

must be used to produce

scalar products with

hj,d)~

(4.10)

j=l

Moreover, up to an o(l), -if in a certain A+A @ l-chain the annihilation operator A(t,,+l) is used to pro,. _. duce a scalar product with a creation operator A+(erk) @ 1 and the creation operator is used to produce a scalar product with an annihilation operator A(q)@l, A+(k+l) then the annihilation operator 1 @ A(uj) must be used to produce a scalar product with the creation operator 1 RIA+(vk); -if in a certain ABA-chain (resp. A+ @A+-chain) the annihilation (resp. creation) operator A(&,, +I) @ 1 (resp. A+(t ph+l)~ 1) is used to produce a scalar product with a creation (resp. annihilation) operator A+(wk)@l (resp. A(vk)@l), then the annihilation (resp. creation) operator 18 A(tph+l) (resp. 1 CGA+(&,+,)) must be used to produce a scalar product with the creation (resp. annihilation) operator 1 @ A+(Q) (resp. I@ A(Q)). In order to understand the role played by the products of operators with the time variables $,,+I,. . . , tph+l in the A+A@ 1, A+ @A+ and A @ A-chains, we must change variables in (4.1) as follows: “j

:=

ztj.

j = 1,2 (..., 72.

(4.11)

Thus, (4.1) becomes N

pw-n

AT({Sj),

{q);

{fl,j,d),{f2,j,d})@

@ @L,

j=l 3*-l

Sl

i

0

da

s

ds2.

0

..

s

du,A(‘(‘),‘(‘))(sl/z,

go, gl) . ’ . A(E(+(n))(s,/~,

go, gl) x

0

x fi':({SiI> j=l

f','I;(.fi,,,dI> (f$,j,d)P@

@L),

(4.12)

and respectively, in all chains, any time variable tj is replaced by a new time variable sj/z. Now we are going to explain the role played by the chains for any jixed composition of chains.

LOW

DENSITY

193

LIMIT

Since we are considering the non-zero contributions, the role of the frequency terms must be understood. In order to do this, let us recall that for a time variable tk, if we have an A+A 61 l-type operator, the corresponding term in the evolved interaction must be

De(b) @A+(St,Q+g,)A(St,Q+gl-,)

@1

c-s

=

c

D,

&=O Thus, the complete

k

&

e’(-l)t%tbA+

( S tb Q +9,)A(S,, Q+sI-e) 8 1.

form of (4.3) is

2 _ e :l:,‘+,(-l)‘(j),,,t,A+(S

t vh+‘Q+ge(ph+l))A(StPhilQ+gl-F(P,,+l)) @ 1 x x (St,,+,Q+gl-E(Y,,Sl). St,,+,Q+g<(p,+q) ... . . . (St,,+,-1Q+gl-t(P,,+,-1)’

Therefore,

(4.13)

*

we have to investigate quantities

Stp,,+,Q+g+h+J

(4.14)

like

First of all, we consider the case of Ed + 1) = 0 and ph+i --2)h - 1 (i.e. the length of the chain) being even. Thus, due to our basic assumption on the test functions go$gl, (4.15) becomes equal to

x exp ’ . . -

Wd pht1-l

%htl-l - 2

i Wd,,+l -SP,, +1 z U

+ wd,,t,

SPhtt - Z

-

wd Ph +2

iWdI& + b,d’Pi

-2ph +2 z .

+

. . .

(4.15’)

194

Y.-G. LU

LEMMA 4.1. &pression

(4.15’) for z - 0 converges to

03

c

6Wd

-%ht2

Phfl

-+WdPht3-udPh+4 +...twd

+wdl

p,,tl

,Wd* ’ ’

d,d’,d,h+l,...,dPhtl=O

SPhtl

‘Ph-1

s dS~hyds,,,l s jdt,,+z

“’

jdtphi3

‘..

0

0

0

db(SaflAd~gO)(gl~

-cc

sbf~,k’,d’j(Sb+t,h+l

-_oo x

’ v; +

jdtpht4...

-m

-m

jdt,,,, x

-cc

~g1rst~h+~g1~~~o~S~,,t,~o)(~1,St,h+,g~)~~~(g x

’ x TdaT

-m

d tPh+2” ’ bhtl d’ WdPh+z’.. wd,,+,

+...+tpht2e-BHf;,lc,,d,,

Saf2,k,d) x

(4.16)

x[s,~)(sP,+l)x[sr,~')(sPh+l)~

>

where vD1 0

d

+

d’

1

&%+2 ... tm+1 ’ ’ wd,,+,

WdPh+2*

is defined as e-iwdae-iwdpht2tpht2

e

. ..e

Proof

.,t4(t,,+2+t,ht3+t,ht4).

--iuld

i~d,,t3(b,,t2+tp,,t3)

Zwdphtl(‘~~+z+...+t,,,,)ei~d,(b+t~htl

.. +-,.+tPht2)

With the change of variable: +2

_SPh z

_

sPh+l z

=

(4.17a)

tph+2,

(4.15’) becomes cc

c

y’du

4d’,d,h+~,...,d,,t,

=0 s/z

~id.~.~s~-idsph s’/z

0

~dsp,+l

x

0

0

X

s

dtph+~ez(Yd~~t~~~d~~+~)s~~t1~2p~iur~ph+~t~ht~(~+gl,

stpht,~+gl)

x

-%htll~

%ht’+“b,t2

s

Ph+l-l

8Phtl

S S ...(S. s~~~,,,,Q~g~~~‘““.h x .,t2+,,,,,,zLlo; X(SSph+3/zQ+gll Ss,h~4,zQ+g~je-iWd~h+4sPhf4/z.. . X

s

dsph+3.”

dSPh+l

.**

. . * (S, Phtl_l/ZQ+goI SSPhtl,ZQ+go)eiWdPh+L8ph+l’Zx

195

LOW DENSI’IY LIMIT

With the change of variable:

sph+l+ tPh+2

%‘h+3

z

2

=

(4.17b)

tph+3.

(4.15’) becomes C-3

c

d,d’,d ph+l.....dph+l

0

0

s/z

=0 s/z 0

X

S -%$+1/f 0

X

dtPh+2e-i-'dph+2tph+2(Q+gl,

St,h+2Q+gl)ei(Wdph+~-*idph+~)~ph+1/~

dtPh+3eauld~~+3tph+3(Q+g0)~tph+3Q+gO)ei~~ph+a(tph+Z+Bph+~/~)

S

x

x

-SP,,tl/Z-t,,+2

* Ph+l-l

sph+l+~t,h+2+zt,ht” X

dSph+4... S

S

0

0

Repeating

the procedure

SPh+l

dSPh+l S ... 0

. ..(SSpht3~iQ+gllSSPh+4.iQ+gl)e-iWdph+4JPht4”

...

***(S, Ph+l-I/ZQ+go,SBPhtl,ZQ+yn)e'wdPh+l"Pht"'

x

we find finally that (4.15’) is equal to

m

c

y’du

d.d’,d ph+lr....dph+l

=0.5/z

T=du..

. ‘T-Ids,, 0

S/r

ydsph+l

x

0

0 X

S

dtPh+2e-iWdph+~tPh+2(Q+g,r Stt,ht2Q+gl)ei(WdPh+,-~dpht2)~Ph+ll:

x

-&1/2 0 X

S -%tllZ-bht2

dtPh+3eZwd~h+3tph+3(Q+g0, St,h+3Q+gO)eiW~,h+~ft~~+z+~~h+~/~)

n

n

S

dtp,,+d,..

S

dtPh+lx

Sph+l+Ltph+2+...+ZtPh+l-l X

S

. ..e

0 X

x

(Q+g~,St,,t4Q+gl)e-iWdPhtrOl+3+t~~+~ts~~t~/~)~..

-iwd

Ph

t4tPh t4

x

196

Y.-G.LU (Q+go, . ..e iYtdm,+ltPh+l

Q+gO)ei~d,~+lftPh+l-~+“‘+t~h+~+d~~+~/z)

StPhil

x (Sufl,k,d.Ssph+~/=Q+Yoj(StPh+l+...+tph+2+~pB+1/~Q+gl,

Svfl,k’,d’)eiwd’u-i~du

-$flH

x (S,,e

x

I

x (4.20)

f2.k’.d’? sue -L’Hf2,k.d). ’

Now, with the change of variables: sPh+l

u--=a

,

2

21-

G+&&+z+.. z

(4.21)

b>

=

. + tP,,+l

(4.20) becomes 00

(T’-~p~+l)/~--ft~~+2+...+t~~+~)

CT-+,,+I)/z

c

da

s

d,d’&,+~,...&,.+,

=o(s-s,,+l)/z

%/& -1

db...

s

s

SPh dsph

0

(S’-Sph+l)lZ-(tph+2+...+tPh+l)

Sdsp,+lx

0

0

S

X

dtph+2e-iWdPh+2tph+2(Q+gl,

St,h+lQ+gl)ei(Wd,h+l--Cldph+2)sp~+i/f

x

-SPh+‘l’

0

X

S

dtPh+3eiWdph+3tPh+3(Q+g0,

St~h+3Q+go)ei~dph+.i(tPhtZ+sphtl/') x

-%htllZ-t,,+2 0

X

0

S

%t* x

S

dt ph+4”’

-%tllZ-tph+2-tpht3

-Sph+l/Z-t,,+2-...-tP,,fl_I

Sph+l+itph+a+...+zt

PI+1 --1

S

X

ph+4htf4 x ...e--i&d

0 X (Q+gl,

St,h+aQ+g~)e-iwd~~+~(t~~+~tt~hC~tJ~~+~/2).

. ..e lWdph+‘tpil+l(Q+go,Stp(,+l

Q+gO)ei'cldph+l(tPh+l-l+...+tPh+2+Pph+l/i) x

X (Saeiwda fl.k,d. xe x

By putting together

S

SbeiWd’bf;.k’,d’) X

Q+go)(Q+gI, /Ze

i”)dfh,+l

+“‘+tph+2+bp,,+l/f)

-4PHfZi.k’,d’? Sne-3PHf2,k.d). (4.2;)

factors, (4.22) becomes

(T’-sph+1)lZ-_(tph+2+...+tPh+l)

(T-Sw,+l)/Z d,d’.d Ph+lr...,dPh+l=O(~-~ph+l)/~

-zWdSph+l

(Sb+, Ph+l +...+t,,+ze

the frequency

..

S

da

db..-‘y-Ids,,

(S'-s,h+l)lZ-(t,h+2+...+tPh+l) 0 X

S -%h+‘l~

dt,,

+2e

0

~d~p*+~

x

0

-iwd

Ph+ZtPh+2(Q+gl, StPh+ZQ+gl)

x

LOW DENSITY LIMIT

197

0

s

X

dtPh+3eiridph+3tpht3(Q+go,

-%h+llZ--tph+2 0

s

X

StPh+,Q+gO) x

0

s

dtPh+4..-

dbh,l x

-Sph+llZ-t,h+2-...-t,h+,_l

-%h+ll~-bh+2-f,h+3

Sph+l+Ztph+2+...+Ztph+l_,

s

X

. ..e -itid ph+4tPh+‘i X

0

X (Q+gl yStph+4Q+glj . . . eiwdph+l tph+l (Q+go, St,,+, Q+go) x x (Saeiwda f1,k.d. Q+go)(Q+n, e-3PHf;,k!.dr,

X (Sb+t Ph+, +...+tPh+z Xe

xe

i(W Ph+l

+...+w,j

+Wd P,,+3-udP,,+4

-Ld,,,t2

Sbei”d’bfi.k’,d’)X

Ph+l

Sae-3”Hf2.k.d)

-dd+wdr)sph+~/i

e

iwn

X

ph+:3tPh+~ x

ph+4(tPh+3+tPh+2).. . eiwd,h+l(tPh+l-l+“‘+tph+2)eilr’d’(t~h+l+’ +tph+2).

-iwd

Applying the Riemann-Lebesgue

(4.23) 0

lemma we complete the proof.

Similarily, we can consider the other cases. 4.2.Exprmion

LEMMA

(i) in the being odd:

case

of

E(P~

(4.15) converges, fir z + 0, to 1) = 0 and ph+l - ph - 1 (i.e. the length of the chain)

+

02

d.d’,d

c

phtl,....dphtl =o

sWd Phtl -Wd,,,+2+WdPh+3-wdP,,t4+...+wd

‘Ph-1 ” ’ s x

0

.‘.

1 dtPh+Z j

dtph+g

-m

-m

0

* * .

jdt,,w

j -m

--o:

dtPh+l

~g~~~t,,,+~g~~~go~~tPh+3gO~~glt~tPh+4gl~~~~(g0,~tPh+l_1g0)(gl,StP~+lgl)

x

x

x

,Wd

‘P,,+l

*Ph

d%, s dSp/,+l s

0

+wd’

Ph+l-l -%‘h+l

v,: -o

d d’

&,,+2 “,d,h+,l

... . . .

7 -m (Sb+t

da

7 --oo

&%fi.k,dr -/3H

P,,+l

+“+t,,t~e

tph+, ‘,jdPh+l

1

0 d -

d’

‘P,,+2 . . . wdph+z

I f2,k’,d’r

x

Sbf;,k’,dJ)

‘%f2,k,d)

X

X

(4.24a) XIS,T)(Sph+l)XIS’.T’)(Sph+l)r

where @

!lO)(go,

x

***

tph+l wdph+l

198

Y.-G. LU

is defined as e -iwda

e -zWdph+2t,h+2ezWd,h+3 iWd

. ..e

(ii) in the case of E(P~ + 1) = 1 and ph+l - ph - 1 (i.e. the length being even:

of the

chain)

02 c drd’&,tl....rdp,,+l

6_ (Wd

‘P,,-1 ...

s

p,,t3 -wdp,,+4t..‘tWd Phtl )+‘dd!,Wd’ ’ ’

+Wd

Phtl-"dPht2

=O %h

jdt,,.z

s %,,+I +f’-.

4,

0

0 x

(90,

jdt,,ts

StPh+?SO)h

pia~

tph+2

a.. ***

tp,+

StPh+490).

. . (a,

gl)(gO,

e-PHfl

PI&+1+...+tPh+z X[S,T)(%

wd,,,+,

x

stphtlgl)

x

Sbf;,&d’)

x

‘%f2,k.d)

x

-c-c

x {Sb+t

WdPhfZ

jdt,,,, --oo

db&.fl,k,dr

-cc

Do: 1 d + d’

dtPht4...

-m

St,,+3gl)(gO> x

x

j

-m

--oo

0

2,k’,d’,

tl)X(S’,T’)(~ph

(4.24b)

t1h

where Do:

1 +

d d’

&,tz

.+.

Wd,,tz

tpht,

“.

wdpht,

>

is defined as e -iwdaeiwdp,+2tP,t2e-iWd,,+,(t,,+2+t,,+3)~iwd~~t~(t~~t2tt~~t~tt~~+4)... . ..e -iWdphtl(tph+Zt”.ttPhtl)eiWd,(b+tPh+l

(iii)

in the case

of E(P~ + 1)

t...+t&+++

= 1 and ph+l - ph - 1 (i.e. the length

of the chain)

being odd: m %i

c

d,d’,d ph+l....rdph+l

0

Phtl-"dPh+s

twd

p~+~-~dp~+4+~~~+~dp~+,_-l

--Yd,h+l)+Wd’.“d

. ’ .

=o

0

x (903 St,,+2go)h

0

-co

-cu

-0.2

Stpht3a)(g07

StPh+4gO) x

x

* . . (91, StPL+l_lS1)(gO1

Id”7 --m (Sb+t

-cxJ

d%%fl,k,dr~l)(~l,

q,h+lgo)

x

Sbf;,k’,d’)

x

-cc Ph+* ++tJ+.tze

-/3H

I f2,k’,d’,

Saf2,k,d)

x

199

LOW DENSITY LIMIT

x

vi -1

d

tph+2 . .- tp,,+l WdPhfZ . * * ‘ddPh+l >

d’

(4.24~)

X[S,T)(sph+l)X[S’.T’)(Sph+l),

where is

d’

WdPh+

***

wd,,+,

>

defined as e --Iwdaeiwdph+2tph+2e-iWdph+l(tPh+2+tph+3)eiWdph+4(tPhfZ+tPh+3+tPh+4) -iwd

.,.e

. . .

Ph+l--l(tPh+l+...+tPh+l--I)eiWdph+l(tph+z+...+tph+l)eiWdl(b+tph+,

+...+tP,,+2)

Now we are going to investigate the role played by the A+ ~3A+-chains. For the time variable tk, if we have the operator A+(tk)@A+(tk), the corresponding term in the evolved interaction must be equal to De(&) @ A+(St,Q+d =

00 c

8 A+WQ-a-4

dk =o

Thus, the complete

D, IC8 ei(-‘)‘WdktkA’(StEQ+gE) ~3A+(&,Q_gl_,). ’

(4.25)

form of (4.4) is

x (St,,+lQ+gl-c(p,,+v StPh+zQ+gE(p,,+2)).e. . . . (St,h+l-‘Q+gl-e(Ph+l-l):

Similarly, as in the consideration stigated should be

Stph+lQ+iqp,,+J.

of the A+A 8 l-chains, the quantity to be inve8Ph+l-l

sPh

s

s

dSph+l. . .

%+1 dSPh+l

s 0

0

0

. . . (S~fl,k.drss,h+,/tQ+ge(p~+l))(Ss,,,,/,Q-g~-,(ph+,),

'

”

Sue-3PHf2,&d) x

,’ Q + g1_Ep,,+l)’ ( eiC,Ph::“(-1”(~)~d19t’Z-iiJd~(S %,,+I

X

(4.26)

Q+ge(ph+2)).

SBph+2,

. .

z

(4.27)

..’ (SsPh+l--l/zQ+gl--e(ph+l-l)'SSPh+l/tQ+g~(ph+l)).

Expression (4.27) converges, for z --_)0, to (i) in the case of &%+I) = 0 and ph+l - ph - 1 (i.e. the length of the chain) being even: LEMMA

4.3.

%,,-I ...

s

0

‘Ph 4s

%h+l

j&h+1

J

0

0

...

Tda

--Di,

jdtl.h+2

--oo

jdt,h+s

--DC)

j&,+4...

--03

jdt,,,,

--cm

x

200

Y.-G. LU

x (glJtPh+2gl)(go) %h+3so)(sl~ stph+,gl).. . (go, St,,+,go) ’ (S~fl.~.d~gO)(StPh+l +...+tph+2glr 0 d &,,+2 ‘.. tphtl

v; +

x

wd

Pht2

Sae-PHf2,k,d) x (4.28a)

X[S.T)(h+l)r

wd

‘*’

x

where

0 d

&x,+2 . . . wdPh+z

tphtl wdph+l

‘.’

is defined as ... e -kia e -iWdp,+2tp,+2e i~dph+3(tPh+2+tPht3)

. ..e

-iwd

. .e iWdph+l(tph+2+“‘+tPhtl)

,,t4(t,,t2+t,ht3+t,ht~~.

(ii) in the case of ~(%+I) = 0 and being odd:

ph+l

-

ph

1 (i.e. the length of the chain)

-

00 -

):

6-wdph+l +Wdpht” +-~dpht,_l

+Wdph+l,‘dd

’

.

.

d+Gh+l ,...>d,,+, =O SPh -1

.”

*Ph

s dS~h

ydajdtp,,+2

Sds~h+~s]tleea

0

0

-m

0

jdt,,+j

-cc

x ~gO~~t~~+2gO~~gl~~tP~t3g1~~gO~~tP~t4gO~ ’

&+2

x

-m

. ..(gl.StPh+l_lgl)(gO.StPh+lgO)X

...

wd Ph+2

jdt,,,,

-‘x

(Safi,lc,d,gl)(StPhtl+...+tPhfZgl,

0 d -

v;

x

$dtphh4...

--ocI

tph+l wd

‘..

&epPHf2,k,d)

x[s,T)(s~,, +I),

Ph+l

X

(4.28b)

where 0

d

tp,,+2

“.

tphtl

wd Ph+2

...

wd ph+l

is dejined as e -i~daeiwdpht2tPhtZe-iWdpht~(tPht2ttPh+3) . ..e

(iii)

in

the

of

case

being even:

c s

‘-‘d,

-iWd

&h+l)

ph+l

iWdpht4(t,ht2tt,ht3ttPh+4)

.ht~-,(t~~+z”“+t~~+~-~)eiwd~ht~

=

1

and

+wdpht2-Wd,h+3

ph+l

-

...

(tph+2+...+tph+l+tPh+l)

>

1 (i.e. the length of the chain)

ph -

+“‘+Wd,h+l-l

-Wdphtl

.wd

‘. *

d,dp,,+l,...>d,,+, =O =Ph

‘Ph-1 ...

0

dSPh

‘Phtl

Sd%h+l

s

0

0

...

Tda

--DC1

jd&,+,

--03

x (gOJtPh+2g0)(g1~

jdt,,+s

-cc q‘t,glHgo>

jdtph+v

-m

StPh+4gO)...

jdtpi+Ix

-c-z

(91, St,,+,gl)X

LOW DENSITY

&emJHf2,k,d) x

X bWi,k,dr9i)(StPh+l

1 d

+...+tph+2903

$x+2 wd

201

LIMIT

tph+l

*a*

Pht2 ***

wd

Ph+l

>

X[S.T)(Sph

+I)?

(4.28~)

where 1

d

&x+2 WdPhi-2

...

tp,,+l

..’

wd Ph+l

is defined as e -iwdaeiwd,ht2t,h+2e-i”d,h+3(t,,+z+t,,+3). .

.,

ii*‘d,h+4(t,,+z+t,,t3+t,,+4).

..e

. . e-~wd,ht,(tph+2+...+tPh+l),

(iv) in the case of E(P~+I) = 1 and ph+l - ph - 1 (i.e. the length of the chain) being odd: cc

c

d.d ~p,+l....rd~h+l

0

=o

&d,Phtl -ddPht2 +...+Ldd

0

0

-m

plL+1--1

.Wd . . *

-iLldphtl

-co

--3o

-cm

-cm

x ~90~~tPh+Z90~~91~~tPh+39l~~~~~90!~tPh+l-,9O)(9l.StPh+lgl)X x (Safi.k,d~~l~(~tPhtl+...+tPh+~gO~

x

VA -1 d

tph+2 -.. ph+Z .*.

wd

&,,+I wd

phtl

Sae-‘H_f2,k.d)

X[S.T)&,+lh

X

(4.28d)

where @I is

(

1 _

d

tp,,+2 wd Ph+2

a+. *..

tph+l wd pII+1

defined as

Proof We only give the proof for case (i) because the proofs for the other cases are similar. In this case, it follows from our basic assumption on the test functions that E(JI~+ 1) must be equal zero. That is, one must investigate the limit of

y’d,. ..‘I-‘&,,&&,,tl...sp*j:-ldSph+,s~l .,. d.d,,+l.....d,,+,=os/z

0

0

0

0

202

Y.-G.LU x eiC,P1::+l(-l)‘(j)WdISJIZ--ill)du~S s,,+l/zQ+w

Ssph+z/zQ+gd.

.a.(& ,h+l_l/zQ+go> SaPh+l/zQ+go).

-.

(4.29)

With the change of variables like (4.17) and (4.21) expression (4.29) becomes

=o

d,dw,+l.-&,+l

0

0

(s-%Jh+l)/z 0

0 X

S -sPh+l

%,,+dQ+glr

St,,+zQ+gl)

S -%+llZ--tPh+2

/z

dtp,,+3(Q+go, StPh+zQ+go) 0

0 X

x

&,,+a +. .

S -%htllZ-bht2-tph+3

S -Sph+l/+-t,ht2-...-t.h+l-1

dbh,l x

sphtl+ztph+2+...+ztph+l-l X

. . .

S

(Q+gl,

Stpht4Q+gl) ...(Q+go, StPh+l Q+go) x

0

x &fi,k,d,

Q+go)(StPhtl +...+t,,,+2/ZQ-gl,

x

Sae-3PH.f2,k,d)

Xe -iwdae-2wd,,+2t~htz e ‘Wd,h+3 (t Pht2+tPhtdX -iwd Pht4(tPh+2+tph+3+tpht4)., . eiwdph+l(tpht2+“‘+tphtl +tPh+l)X xe i(w Phfl -wdPht2tad Ph+3-udph+4+...+wdph+l-wd)%tl/~, (4.30) xe

Applying the Riemann-Lebesgue

0

lemma, we obtain the thesis.

The analysis of the A @ A-chain is the same as before similar.

and the result is also

Now we shall investigate the constant chain. It is clear from our basic assumptions that the length of any non-trivial constant chain is odd. Therefore, one must study expressions like *Ph-1

00

c

d Phtl-dw,+l

... =o

s 0

+/I

dsPh sda,,,l...si~-‘ds,ht~s~l... 0

. ..e X

0

iwd phfl-2SPh+f-

(S,Phtl/zQ+go,

0

. ..e ~~~dp~+~yPh+l/2eiwd,h~~s~ht~/z.. 2/z e-Zud,htl-lS~htl- l~zeiwd~~+lsQhtl~z

SSph+2/=Q+go)~SSphtP/=Q+gl(S=ph+3/=Q+gl). . ..(S. _-a/zQ+go> Ssph+l_2/zQ+go)

. X

.. x

x ~S~ph~1-/~Q+g~~S~phf1~/~Q~g~~~S~pht~-~/=Q+~o~Ss~htL/zQ+~o~~ x (SSgh+l/LQ-g~l

SSph+l/zQ-gd

(4.314

LOW DENSITY

203

LIMIT

and

...sJ-lds,h &s,,+l...‘r~-Lda,,,.s~l ...

30

dP,,+iY.“?

c d

P,,+l

=o

0

0

0

0 l@d ph+l~Ph~~l~e-i~dph+2sph+2/~

.._e . ..e

-

. . .

iW,jph+~-zs~~+~-2~ieiwdph+l-1sph+l-1/ze-i~dph~,~ph~,~~ x

x (SSph+l/ZQ+gl.SSph+2/ZQ+gl)(SBph+2/fQ+go'

Ssph+3/zQ+go) a- e

. *. (S, ph+,-~~~Q+glrS~Ph+l-~/~Q+gljx

x (83~h+~--Z/~~+g~~S~Ph+I-,/~Q+gO~(S~ph+l_~/~Q+g1~SSPh+ll~Q+gljx /a&-got x (ssPh+l

It follows easily from a similar argument

(4.31b)

Ssph++Q-go).

as the above that

LEMMA 4.4. Expressions (4.31a,b) converge,

for z +

0, to

m d ph+l

c v..,dPh+l

s-Wdph+l

+Wdphfp

+...--wd

Ph+l

+iJd

-I

Ph+l

,o

* * *

=O

yds,,,l

. ..‘yj.‘ds,h

0

j

dtPh+2 j

dtPh+3 j) dt,,+z,.

.. j

dtph+ls~+‘.

--oc

0

..

0

---~.90.~~+,90~~9~~~,,+:,91~~~~~0.S~~~+~_~90)X x (91,s

Ph+l-l9i~~9o~~tP~+l9o~~9i~~tph+Z+...+tPh+l~~BHgl)~ 0

x ‘0;

4%+2 wd

(-

*‘*

Phi2

‘I’

tph+l

(4.32a)

wd Phtl >

and m 6Wd

c

Ph+l

-WdPh+2

+‘.‘+Wd

-Wd

Ph+l-'

d P&+1.*... d Ph+l =o

-‘*~-ldsPh ~ds,,,r j 0

0

-cc

Here

,o*..

dtph+2 j dtphg

x I$

respectively.

Ph+l

-cc

1

-

j

--m

&++2 wd

Phf2

dtph+‘, . . . j

-cc

...

.-*

dt,,+lsT’.

0

tphfl wd

Ph+l

, >

..

(4.32b)

Y.-G. LU

204

are defined as iwd

e

ph+2tph+ae-iWdph+J(tPh+a+tPh+3)...eZ~dph+l_-2(tph+2+...+tPh+l-2).,.

. ..e

_-iuJd ph+2+...+tPh+l-l)ei% Ph+l(tPh+?+...+tPh+l) ph+l --I ct

and e

-iwd

-iuJd (tPh+2+".+tPh+l2).. ~,,+~-2 ph+2tPh+2eZu'dph+3(t,h+2+tPh+3).. .e icdd lJe-iwd ph+l-l(tPh+2+...+tPh+lPh+l(tPht2+...+tPhtl) . ..e

respectively. Summing up, any A+A 8 l-chain acts like a number operator; any A+ 8 A+-&& like a creation operator; any A @ A-chain like an annihilation operator, and any constant chain like a constant. Moreover, the above arguments guarantee the following theorem. TI-IEoREM 4.5. With the assumption 5. The limit

quantum

stochastic

(3.10), the low density limit (1.21) exists.

process

In the present section, we shall put together the results obtained in the preceeding sections to describe a limit quantum stochastic process which is the LDL of our original evolution operator U,,, . We return to (3.11). As argued in Section 3, instead of (3.11), one should confine to consider, for all E E (0. &l}” and E E (0, l}V1, the sum of

Following the arguments in [l, 2, 3, 71, in order to understand the limit stochastic process, one should look for the possible contributions of the operators in (5.1) with respect to the time variable tl. According to assumption (1.8a), our basic object to be investigated, i.e. (1.21), up to o(1) is equal to

LOW

x &(2)(t2)

8 A

DENSITY

(42)342))(t2,

X fi

205

LIMIT

go, g1) . . . D+)(tn)

A:((Si),

(T’);

(fi,j,d)r

@ A(E(n)+(n))(&,

ifl,j,d))u

, go, 91) x

@ @ @ @‘)I

(5.2)

j=l

where D l.d

if E = 0, if f = 1.

Dd :=

-0;

(5.3)

As the first step, we consider the case of ~(1) = 0. In this case we have, starting from ti, A+A @ l-chains (corresponding to the case of the last operator in the chain being A+A@l-type) or A+@A+-chains (corresponding to the case of the last operator in the chain being A+ 8 A+-type). Let us study the case of the A+A ~3 l-type operator contributing to our limit in A+A 8 l-chains. As argued in Sections 3 and 4, taking into account all possible lengths of chains, one is able to rewrite (5.2), up to an o(l), as N UC!9 AZ({Sj), {Tj); ( rI

{fl,j,d)>

{f2,j,d))@

@@L,

j=l

+E n=*

k=l

CE{O.fll~.

c

z”‘@)j-

c(l)=...=.(k)=0

x A+(S,,/,Q+g,(1))A(S,,/.Q,gl-~(l)).

XD 4l),d1

dsl

0

rE{O,l)”

. . A+(S,,/,Q+g,(k))AtS,,/,Q+gl-,(k)) s1/3 tt+1 x s dtk+, / dtk+2...

@

A(E(“+l)~‘(“+l))(t~+l,gO,

gl).

. .

0

DE&n)

@ lx ‘Tldt,,x 0

@ A(+++))(t,.

go, gl)x

I

As

argued in Section 2, the first term of (5.4) goes,

(u @ *, fiA(X[sj,Tj) j=l

‘8 &fi.j.dT d=O

f2,j.d)) $A’

x

0

. . DEckj,dk@ ei ~~&1)L(3)wd~8~~ix

0 x D++q(tk+l)

j!d.s2.. .'I1d.sk 0

for z + 0, to

(x~.s;.TJ @ G(f;.j,dj d=O

f;.j.d))’

@ ‘).

206

Y.-G.

LU

Exchanging the order of the sums for k and n, the second term of (5.4) becomes CC k=l

z+) lE{O,l}”

j dsI a ds2... 0

"j;'dsr

0

x

0

X

2

(u

@ fi

dl....3dk=0

A,+Wjh

D 4lMl X A+

tS,,,2Q+g,(I))A(S,,/.Q+gl-~(l)). ce X

c

@

h,d,

. . .&(q,ci~

UwH@

@@,,

@e

i C”= (-lywLJdj”j/z 3 1

. . A+(S,,/,Q+g,(~))A(S,,/,Q+gl_,(k)) Sl/Z h+1

c {k+l,..., n}-+{O,+l}, 2: {k+l,....n}~{o,l}

z+‘)

0

X fi

A,+W;L

{T;h

fj;ldt,,x 0

0

A(E’(“+l),“(“+l))(t~+~, go, gl). . . D+)(tn) I

X

8 lx

d&+1 j- dtk+2-..

s

n=k E: X D~~(lc+l)(~lz+l)

W;

j=l

8 A(E’(n)yE’(n))(t,, go, gl)x

{f;,j,dh

t.f;,,jdd)w

‘8 @ @ @$

j=l

which in fact is equal to .z+)

LX k=l &{O,l}k

j dsl j! ds:!. . ‘r’dskx 0

0

0

Moreover, up to an o(l), the product (5.5) is equal to

of the creation

and annihilation

A+(S,,/,Q+g,(1))A(S,,/,Q+gl-,(k)) 8 lx Q+s(z,) . . . ($_+Q+gl-e(w), x C%,/zQ+g1-,(1), Ss,/z

operators

%/zQ+ge(d

in

(5.6)

The creation and annihilation operators in (5.6) act on the collective vectors according to the principles explained in the proceeding sections. Thus from the above discussion it follows LEMMA

is equal to

5.1. With the assumption (3.10),

the limit of (5.4), for z + 0, exsits and

LOW DENSITY LIMIT

t

s

2

c

dsl

0

l
2 7 7dbx

XIS,,T,)(SI)X,S:,,T~,)(SI)da k=l-‘33

d,d’=O

x

0 -

x v; (

x

d d’

t2 w,&

... ‘. .

(901 ~t,!Jo~ka,

jdh

d

x 2);

jdtv

--x

jdtzkx --3o

-cc

Ddl(-D;&’

Stdlo)

%91)(90?

t2

m

t2k wdzk >

(Safl,~,d,Sl)(Stz,,+...+tZgl,

1

--3o

[dl,,;k_o_jd,, -

+ fLZk(l,l;l,-)+w&wd

207

Ddw

*. (a*

St,,_,g1)(go,

Sbfi.,f~d’)(Sbe-‘Hf~,.T’.d’,

...

(-%zk)

+

%.YO)X &.f2.x.d)

x

t2k

(

x

1

x

x Do: ( +6

st,9kIo.

St,so)h.

(Safl,r,d,90)(St21r--1+...+t2gl, 0

d

t2

**.

+

d’

wdz

‘. .

t2k-1

1

d

t2

...

d’

wdz

*

**

Dd&@z)Dd,

&SO)(Sl>

(Safi,~,d~~i)(St~~--l+~~~+t~~O~

+

t2k_1 WdZlc--l >

St&I*).

.

(elLuii~

Sbf:.2’,d’)(Sbe-13Hf;.T’,d’,

WdZb_-l >

WZk--l(l,l;l,+)+w~~,w~~90~

x

x v;

bl!

~W2k--1(l,l;0,+)+Wd’,Wd

h3~1)kJO~

x

Snf2,r.d)

x

‘+Dcizn-_2)Ddw

klo)

.

+

. . (91. ~t2k-,,91)

Sbfi,z/.d’)(Sbe-3Hf~,s’,d“

Snf2.s.d)

x x

,>Ix

(-L)d:)Ddz(-Dd+3)“‘DdZ~--2(-Dd:k_,

where

(5.8a)

wk(l,

1;0,

+)

:=

wdl

-

Wdz + wd3 -

wd4 +

. . . + Wd,, ,

wk(l,

1;0,

-)

:=

wdl

-

wdz

wd4 +

. . ’ + ‘ddkml

1 : 1, +)

:=

-(wdl

-

WdZ + Wd3 -

Wdq +

. .

1; 1, -)

:=

-(wdl

-

wdz

wd4

. . . + WdL_,

wk(1, Wk(lr

+ wd3 -

+ Wdg -

+

-

Wdk,

(5.8b) (5.k)

+ Wdb), -

‘dd&).

(5.8d)

Y.-G. LU

208 and hereinafter, for

r = 1, the operator

product of integrals Jf,

dt2 ‘. ’ J:,

S,,.+...+,,

is defined

dt,.hz(tz) ’ ’ ’ h,(t,)

as identity, and

the

is equal to one.

Now let us study the case in which ~(1) = 0 and the operator A+(tl)A(tl) @ 1 contributes to our limit in the A+ 8 A+-chains. In this case, the last operator of the chain must be an A+ 8 A+-type operator. Moreover, our discussion in the following will include the case of the operator with time variable tl being of the A+ @ A+-type (by looking at it as an A+ 8 A+-chain with zeros A+(tl)A(tl) 8 l-type operator), i.e.

E(1) = 1. As argued in Sections 3 and 4, taking into acount all possible lengths of chains, one is able to rewrite the second term of (5.4) as

k=l

6

dl,...&=o

0

0

A+(S,,/,Q+g,(l))A(S,,/,Q+gl_,(l)). *. . . . A+(S,,_,,,Q+S,(k-I))A(s~~-~/=Q+gl-E(k-l)) @1 X tn-1 Sl/Z tlL+1 63A+(&.,,Q-&-c(k)) s &+I s d&+2 * *s s dtn X x

x A+(S,,/,Q+s&

0 X DE(k+l)(tk+l)

@ A

(E(k+l).E(k+l))(tk+l,

X fiA:((Si].

0

. . D+)(tn)

go, gl).

(‘J’,‘); (f;,j,d)r

0

@ A(+++))(tn, if&,d)b

go, 91) x

@@@@L).

(5.9)

j=l

Similarly as in the case of the A+A 8 l-chains, we have the following 5.2. With the assumption (3.10),

LEMMA

the limit of (5.8) for z + 0 twits and

is equal to j 0

dsl F z=l

F

x~s,,T,~(sI) c

7 da[

k=l

d=O

x 1 ~wzk(l,o;o,-),wd(go~ x DA + ~Wd1.0;1,-bd

-cc

2

St,go)(g1,

0 -

d

j

dt:! j

dt3 j

-m

-m

dl,...,d2k=0-m

t2 ‘d&

St,g1)(go,

’ *. . ‘.

d&e

+.

i

dt2k

St,go) . . . (91, st,,_,gl)(go,

St,,go)x

t2k Wda

>

(-Dd:

)h

’ ’ * (-D&_,

(91~~tz91~(90,~t~~0)(91r~t,~~)~~~(go,St,,_,90)(g*,StZbgl) x

x

--co

(S=fl,s,drgO)(Stzb+...+tzgo,

S,e-PHf2,,,d)

)Ddzlc

LOW DENSITY LIMIT

2

+

j

209

dt2 j

dl,....d2~-~=0-m

x

{

6W*k_1(1.0;0.+).Wdkw

ml)kIo.

dt3 j

--3c St,go)kQ.

dt4.‘.

%Yl)

. .. (go.

X (Safi.~.d,g0)(St?k_-l+...+t2gl.

x

d

VA 0 +

t2

...

‘d&

. . .

t2k_1 Wd2k--1 >

(90.

+ &JZk-l(l,O;l,+),Wd

x

x DA

d

1

St,go)h.

Ddl(-%2)Dd, S*,s1)(90.

j

d&-l

X

Strs_,.qo)

x

--x

--?o

Sae-“Hf2.r.d)

x

. ‘. (-Dd+Lle_-2)&b_-l

+

&,YO)

...

(91. &,,_,Yl)

x

&e-!‘Hf2,s.d)

(Snfl,s.d~gl)(Stzk-l+...+t~gO~

x

t2 ... Wdn “’

+

X

Fyo(u@n

~~({~~}~{~~}~{fl,~,d}~{f~.~,d})~~~~~

l
n

A:({S$},

{T,‘};

{fi,j,d},

{f;,j,d}b’

@ @ @ @l)q

(5.10)

l
where ‘&(l,O;0,+):=W&

-W&

+W& -W&

+“.+‘&j,.

(5.11a)

“&(l, 0;O. -) := --wdl + w,& + ‘. ’ - ‘d&_-l + f-d&.

(5.11b)

w&O;

(511c)

1, +) := --wdl + w& - W& + .‘. + wd,_, -w&.

‘&+(l.0; 1, -) := wdl - WdZ+ ” ’ + ‘ddk_-l - W&.

(5.11d)

As the second step, we consider the case of ~(1) = -1, i.e. corresponding to the time variable tl, in which we have an A@ A-type operator. In this case, starting from tl, we have the A 8 A-chains (corresponding to the case of the last operator in the chain being A+A 8 l-type), or the constant chains (corresponding to the case of the last operator in the chains being A+ @ A+-type). Since the totality of the A @ A-chains is the adjoint of the totality of the A+ 18 A+-chains, the discussion of the A @ A-chains is analogous to the discussion of the A+ ~3 A+-chains. Now we are going to deal with the constant chains, i.e. we consider the chains begining with an A 8 A-type operator and ending with an A+ 63A+-type operator. It is clear that what one should treat is a quantity like

210

Y.-G.

x

1

LU

q+)(--~+(s2/4)~

. . ~(S2k-2/~)(--D+(S2lc-l/~))~(S21,/~)

A(&,/zQ+go)

@

8

8 AWs,/zQ-91)

x

x A+(S,,,,Q+go)A(S,,/,Q+gl)A+(S,,/,Q+gl)A(S~~/~Q+go).

..

s~~_,/~Q+go)A(S,,,_,/=Q+gl)A+(S~~k_~/~Q+g1)A(S~~k_1/ZQ+go)

-.A+(S

x A+C%,,/zQ+go) +

(-D+(sl/4qs2/4f.

x

@ A+Wsz+Q-91)

+

. (-D+(s2k-2/~>)o(s2k-l/Z)(-D+(SfE/Z))

Ws,/zQ+d

8

8

8 +%,/zQ-go)

x

x A+(S,,/,Q+gl)A(S,,/,Q+go)A+(S,,/,Q+go)A(S~~/~Q+gl).

..

. ..A+(S.,,_,,,Q+S~)A(S,,,_,/,Q,S~)A+(S,,,_,/,Q+~~)A(S,,,_,/,Q,~~) x A+b%,,/zQ+d X us,/, fiAZ((SiJ,

(‘jll;

x @ A+W,,,/zQ-go)

(.fi,j.dI> I.f;,j,dIb

@ @ 8 @L)*

x (5.1’)

j=l A simple application of the results from the preceeding ments from the present section gives the following LEMMA 5.3. With the assumption

jds$ 2 k=l

0

(3.10), the limit of (5.12) exsits and is equal to

jdt2-

dl,...,dzk

j dt2k { ~w&o;o,-),0(90,Stz90)(91,St3gl) **. -cc

=O -m

* ** (90, %,_,90)(91,

x vi

1

-

t2

w&

*'.

...

sections and of the argu-

St2k_-lgl)(gO) S,,,go)(gl,

t2k WdZk (-Dd:)Ddz > ’

Fyo

(u

Stz+,,,+tzke-PHgI)

* * * (-D+d2k-Z)Ddz&D+d2k)}

@ fiA:({sjJ,

iTj);

(fi,j,d),

x

X

{f2.j,d})@

8

@p,,

j=l iv’ us,/= nAZ({s;>> j=l

{‘;I;

{f[,j,dIl

{fi,j,d})v

@@@@c)>

(5.13)

where w2k(o,o;o, -) :=

-‘ddl

+ ‘d&

+

.‘.

-

wdzk_,

+

‘d&,

W2k(O,O;1, -) := w& - w& + ... + “&2E_-1- W&..

(5.14a) (5.14b)

211

LOW DENSITY LIMIT

In order to describe the limit process, we must find a clear way to rewrite the above limits. Let us introduce some notation. For any k 2 1, let Tg0,s1.2~_1be a linear operator (corresponding to the SWZk_,(lrl;~,+)+lJd,,Wd-term) defined by cc

(@

f2.d

(fl,dr

~g”,sl.2k-l d’=O

d=O

i

$dt3 j dt4.. . j dt2k_l 7 da 7 db x

dt2

d,d’=Odl,...,dzk-l=O-x

-m

-cc

(91, st,91)~90~

X Lzk(l,l;O,+)+w,p,wd

st,~oH5?l~zI)~~~

x (Snfl.d,gO)(St~E--l+...+tzgl~

x q

Similarly,

we

bwZk--l(l,l;l,+)+wd,.wd

can -term).

-term) s IJ2E(111;0.-)+W&Wd

iLt,:-:&

x Snf2.d)

s6fi,dl)(sbe-“Hf~,d,,

0

d

t2

+

d’

wd2

t2k_1

..’

. . ’

(5.15)

.

Wdah_-l

X

the (corresponding to Tgl .go ,2k - 1 operators be a linear operator (corresponding to the Let Tgo ,g,, ,2k defined by define

m

(@(fl,d>

f2.&

Tgo,go.2k

&r;.,) d’=O

d=O

f;.,,)

c

c

=

j- dt2

d,d’=Odl....,dz,,=O-cc

f --3o

sdt2k x

dt3 j- dt4... -cc

(91:~~~~l)(9o.~,,~O)(~l.St,91)~~~(~0~Sf~~_~~O)(~~~~2k!~,~

X LJzIc(l.l;O,-)+Wd’,Wd

X(S,fl,d,gO)(StZli+...+tZgO,

x

x &f2,d)

s,fi,d’)(Sbe-BHf~.d’,

;D;

0

-

d d'

t2 ‘dd2

..

’

. . .

t2k wdzk

.

X

(5.16)

>

Analogously we can define T gl,gl,2k (corresponding to the 6~~(1,1;1,-)+~~,.~~-term). It is clear that the operators introduced above are well defined linear bounded operators on the Hilbert space @,“=,K$. Now we introduce some notation to describe the limit of the A+ @IA+-chains. For be an element (corresponding to the Sw2koO;.,_J,wd-term) in any k > 1, let ng,,g,,,k the Hilbert space @,“=, PC:, defined by cc (@

(fl,d>

f2,dh

)

ng,,g,,2k

d=O

=

2

9

d=Odl,....dz,,=O X6 w?k(l.o;l,-),wh~

x

(Safi.s,d,SO)(StZk+...+tZgOt

Sae-PHf2,Z,d)Di

7

da j

dt2 i

dt3 i

-m

-cc

-cc

-cc

%g1)(so7

kJo)(g1,

1

_

d

%571)~

dt4.. . J dtZk x . . bo:~*liw~

t2

...

t2k

Wd2

“’

Wdzb

.

x

(5.17)

212 and

Y.-G.LU let

flg1.go,2ic--l

be

the

element

(corresponding

to the rj~*‘2k_,(l,0;l,+).Wd-term) in the

Hilbert space $rzO X2, defined by

0

d

t2

...

t2k-1

WC12 .

.

(5.18)

W&k--1>

In a similar way we define IIg1,s1.21i and ng0,9,,~k_1. Finally, let us denote by PYo,y0.2kand P,, ,91,~k the quantities 0

dt2 . . . dl,...,dzk=O-m

dt2kS,,,(o.o;o,-).o(90,

s

St,go)h.

St,sl)

..

-m

x

“‘(g0~StZk-_2g0)(g1~St~k_~g1)(go~St,kgo)(g1rSta+.,.+tzke-9Hg1)

0 -

t2

”

w&

’

’ -.

t2k

(5.19a)

WdZk

and 0

dt2 dl,...,d2k

=0 -m

s

dt2kS~~k(0,0;1,-).0(g1,

%d(gO.

&go)

”

*

--cc

“’

(SlJ

St~k_~~l)(gO~

St~~._IgO)(gl~

s&gl)(gO,

1 -

Sta+,,,+tzke-‘3HgO)

t2

...

wdz

. . .

t2k

.

x

(5.19b)

Wdzk >

respectively. Now we are ready to state our main result. In order to do this, we use some common notation. For any Hilbert space X, J: E ‘H and T E B(‘K), by A,(a), AZ(x) and N,(T) we denote the annihilation, creation and number operators A(x,~,~) @ x), A+(x[Q) @ x) and N(x[o,~J @ T), respectively. For the number operator, the function x[o+) means the operator defined as (x[o,.3,f)(~) = x[o,.s)(W(~), THEOREM 5.4. With the assumptions esists and is equal to

v’f E L2(R+ ): t E R+.

(1.7) and (3.10),

the low density

limit (1.21)

213

LOW DENSITY LIMIT

d=O

where {U(t)},,0

is a qunatum stochastic process on the Hilbert space ‘l-to@T( L2(R)@

@YE;“=, I$). It satisfies the following quantum stochastic differential equation driven by the quantum B.M. and quantum number processes:

k-l +

( n

D~,d~j_~D1-~,d~,

. &_I)

8 dA,+(17,c,,,_c,w)

+

j=l

Moreover, the stochastic process U(t) is a unitaiy operator.

{U(t)} ~0

is unitary, i.e. for every t 2 0 the operator

Proof: The proof can be completed by application of the preceeding arguments and the discussion in [l, 2, 31. What must be made clear are the following points: (i) The operator of the system part for each A,(17gC,,E,2k) is the conjugate of the operator of the system part for Af(n,,_E ,91_C,2k); (ii) For the ds-term we have only P,E,,C,2k; (iii) The Ito formula. Now let us explain the first point (i). Without loss of generality, we shall deal with the question for the pair A,(Lr90,9,,2k), A,+(n,,,,,,,k). The creation operator A:(IT,, ,gl ,2k) comes from an A+ 8 A+-chain like A+(st,Q+sl)A(S,,Q+go)A+(St,Q+go)A(St,Q+gl).. . ..A+(st.,_,Q+g~)A(S,,,_,Q+go)A+(st,,Q+go)

. @ A+(&,,LQ-91).

(5.22a)

214

Y.-G.

and the annihilation 4%Q+d

operator

A,(17 go,ga,2k) comes from an A ~3A-chain like

@ A(S,,LQ_go)A+(St,Q+g1)A(St2Q+go)... . . . A+(S,,,_,Q+go)A(S,,,-~

Therefore, the operator (5.22a) is equal to (--D+(tl))W*) and the operator qh>(-~‘(t2)).

LU

Q+gM+@t,,

Q+gM(&,,Q+go).

of the system part corresponding

* . . (--~+(~2k-1))~(~2/4

=

(--1)ICD+(h)w2)

of the system part corresponding . . q~2lcM)(-~+t~2k))

=

(522b)

to the A+ @ A+-chain * ’ * ~+(t24qt2k>

(5.23a) to the A @ A-chain (5.22b) is

(4wl)~+(t2)~

. . q~2/4)~+(~2k),

(5.23b) which is nothing but the conjugate of (5.23a). In order to understand the second point (ii), we should explain why dAs(17,~1,,~1,2k-1)dAs+(17y.*.SLZ.2n)

= 0

(5.24a)

for any ej,~j E {O,l} (j = 1,2) and k,n E N. As argued in [3], putting together two limiting operators is equivalent to connect the two corresponding chains (the procedure of connecting two chains can not be realized for a fixed composition of chains, i.e. such procedure changes one composition of chains into another). Thus a quantity like the left-hand side of (5.24a) comes, in the limit, from the operation of putting together an A @JA-chain (with an odd length) and an A+ @ A+-chain (with an even length). Putting them together means that the operators A @ 1 and 1 @ A in the A 69 A-chain are used to produce scalar products with the operators A+ 8 1 and 1 @A+ in the A+ @A+-chain, respectively. If the test function gE corresponds to the operator A @ 1, then the test function corresponding to the operator 1 @ A is still gE since the length of the chain is odd. Similarly, if the test function gE corresponds to the operator A+ @ 1, then the test function corresponding to the operator 1 18A+ must be gl+ since the length of the chain is even. Thus, it follows from our basic assumption on the test functions go,gi that putting together such two chains gives only a trivial (zero) contribution to our limit since the product (gE, gE) . (gE:gt_) = 0 for any E, E E (0.1). This means that (5.24a) must be true. By the same reason one has (5.24b) for any Ej, Ej E (0,1) (j = 1,2) and k,n E N. Now, let us explain the Ito formula. Notice that if in our limit two chains are put together, one finds a new chain in another composition of chains. Let us consider several cases: -Putting together an A+A @al-chain with another A+A @Il-chain. It means that the annihilation operator in the first A+A ~3 l-chain is used to produce a scalar product with the creation operator in the next A+A @Il-chain. Thus the new chain (a chain in a new composition of chains) is still an A+A 8 l-chain. The limit effect of this phenomenon is nothing but, up to a constant, dNdN = dN.

LOW DENSITY LIMIT

215

-Putting together an A*A @ l-chain with an A+ ~3 A+-chain. It means that the annihilation operator in the A+A @ l-chain is used to produce a scalar product with the operator A+ @I1 in the A+ @ A+-chain. Thus the new chain (a chain in a new composition of chains) is still an A+ @ A+-chain. The limit effect of this phenomenon is nothing but, up to a constant, dNdA+ = dA+. -One can not put together an A+A 8 l-chain with an A @ A-chain. The limit effect of this phenomenon means that dNdA = 0. -One can not put together an A+A 8 l-chain with a constant chain. The limit effect of this phenomenon means that dNdt = 0. -An A+ @ A+-chain can nor be put together with any type of chains. The limit effect of this phenomenon means that dA+dN = dA+dA+ = dA+dA = dA+dt = 0. -Putting together an A 8 A-chain with an A+A ~3 l-chain. It means that the operator A 8 1 in the A 8 A-chain is used to produce a scalar product with the operator A+ @ 1 in the A+A 8 l-chain. Thus the new chain (a chain in a new composition of chains) is still an ABA-chain. The limit effect of this phenomenon is nothing but, up to a constant, dAdN = dA. -Putting together an A @ A-chain with an A+ @ A+-chain. It means that the operators A@1 and l@A in the ABA-chain are used to produce scalar products with the operator A+@1 and l@A+ in the A+@A+ chain, respectively. Thus the new chain (a chain in a new composition of chains) is a constant chain. The limit effect of this phenomenon is nothing but, up to a constant, dAdA+ = dt. -An A @ A-chain can be put together with neither an A @ A-chain nor with a constant chain. The limit effect of this phenomenon means that dAdA = dAdt = 0. -A constant chain can not be put together with any type of chains. The limit effect of this phenomenon means that dtdN = dtdA+ = dtdA = dtdt = 0. Moreover, the unitarity of U(t) is a direct consequence of the general result from quantum stochastic calculus (see [5, 61). Thus we complete the thesis. 0 REFERENCES [l] [2] [3] [4] [5] [6] [7] [8]

L. Accardi and Y.-G. Lu: Commun. Math. Phys. 141 (1991), 9. L. Accardi and Y.-G. Lu: Nagoya Math. 126 (1992) 25. L. Accardi and Y.-G. Lu: J. Phys. A: Math. Gen. 24 (1991) 3483. A. Frigerio: Lecture Notes in Math. 1303 (1988), 107. A. Frigerio and H. Maassen: Prob. Th. Rel. Fie. 83 (1989) 489. R. L. Hudson and K. R. Parthasarathy: Commurr. Mark Phys. 93 (1984) 301. Y.-G. Lu: 1. Math. Phys. 33 (9) (1992), 3060. P. F. Palmer: The rigorous theory of infinite quantum mechanical system. Master equations and the dynamics of open systems. D. Phil. Thesis. Oxford University, 1976.