On the problem of chaos conservation in quantum

physics

V.P.Maslov and O.Yu. Shvedov

Sub-faculty of Quantum Statistics and Field Theory,

Department of Physics, Moscow State University,

Vorobievy gory, Moscow 119899, Russia

December 12, 1995

Abstract

We develop a new method of constructing an asymptotic series in powers of N�1=2 as

N !1 for the function of N arguments which is a solution to the Cauchy problem for the

equation of a special type. Many-particle Schrodinger, Wigner and Liouville equations for

a system of a large number of particles are of this type, when the external potential is of

order O(1), while the coe�cient of the particle interaction potential is 1=N ; the potentials

can be arbitrary smooth bounded functions. We apply this method to equations for N -

particle states corresponding to the N -th tensor power of an abstract Hamiltonian algebra

of observables. In particular, we show for the case of multiparticle Schrodinger-like equations

that the property of N -particle wave function to be approximately equal at large N to the

product of one-particle wave functions does not conserve under time evolution, while the

same property for the correlation functions of the �nite order is known to conserve (such

hypothesis being the quantum analog of the chaos conservation hypothesis put forward by

M.Kac in 1956 was proved by the analysis of the BBGKY-like hierarchy of equations). In

order to �nd a leading asymptotics for the N -particle wave function, one should use not

only the solution to the well-known Hartree equation being derivable from the BBGKY

approach but also the solution to another (Riccati-type) equation presented in this paper.

We also consider another interesting case when one adds to the N -particle system under

consideration one more particle interacting with the system with the coe�cient of the

interaction potential of order O(1). It happens that in this case one should investigate not

a single Hartree-like equation but a set of such equations, and the chaos will not conserve

even for the correlation functions.

1

1 Introduction

The chaos conservation hypothesis is well-known in statistical physics. This hypothe-

sis put forward by M.Kac [1] in 1956 for the case of classical systems has the following

analog in the quantum case of the system of N bose-particles moving in �-dimensional

space. Consider the k-particle correlation functions [2]

Rtk;N (x1; :::; xk; y1; :::; yk) =Zdxk+1:::dxN

tN (x1; :::; xk; xk+1; :::; xN)

t�N (y1; :::; yk; xk+1; :::; xN) (1)

corresponding to N -particle wave functions tN(x1; :::; xN) which specify states of the

system and satisfy the N -particle Schrodinger equation (x1; :::; xN 2 R� are particlecoordinates, t 2 R is the time variable).The quantum analog of the chaos hypothesisis the following. Suppose that at the initial instant of time t = 0 the correlator (1)

factorizes as N !1; k = const as follows:Rtk;N (x1; :::; xk; y1; :::; yk)! 't(x1):::'t(xk)'t�(y1):::'t�(yk); (2)

where 't is one-particle wave function such thatRdxj't(x)j2 = 1. Then the property

(2) holds for arbitrary time t as well.The discussed hypothesis can be justi�ed for the case of the external potential

of order O(1) and the particle interaction potential of order O(1=N). The mathe-matical proof has been obtained in [3].The method of justi�cation of the property

(2) was based on the ideas of [2]: the N -particle equation for the density matrixtN (x1; :::; xN)

t�N(y1; :::; yN) was integrated over last N � k variables and the chain

of equations for Rtk;N was obtained in a way analogous to the method of derivationof the BBGKY hierarchy found almost simultaneously by Bogoliubov, Born, Green,Kirkwood and Yvon for the classical case. It was also found in [2,3] that the function

't obeys the Hartree equation being of widely use in physics for studying quantumsystems with a large number of particles.

The property (2) has the following physical meaning in terms of mean values of theobservables being operators acting in Hilbert space L2(R�N): Consider observablesAN with kernels of the special form

AN(x1; ::; xN; y1; :::; yN) =P0Xp=1

1

Npp!

X1�i1 6=::: 6=ip�N

A(p)(xi1; :::; xip; yi1; :::; yip)Yi6=il

�(xi � yi); (3)

where P0 2 N,A(p) are kernels of operators acting in L2(R�p).The property (2) impliesthat the mean values of the observables AN in the state

tN have the following limit

as N !1:(tN ; AN

tN ) �

Zdx1:::dxN

t�N (x1; :::; xN)(AN

tN)(x1; :::; xN)!

2

!P0Xp=1

1

p!

Z't(x1):::'

t(xk)'t�(y1):::'

t�(yk)�

�A(p)(y1; :::; yp;x1; :::; xp)dx1:::dxpdy1:::dyp:In this paper we consider a new formulation of a problem. Namely, we discuss if the

property (2) is valid when k also tends to in�nity, for example, k = N . We also study

whether the chaos property allows us to �nd limits as N ! 1 of mean values ofobservables of a more general form than (3).

To solve these problems, we construct an asymptotic formula as N !1 not forthe correlators (1) but for the full N -particle wave function obeying the initial condi-

tion of a product of one-particle wave functions (remind that a number of arguments

of the wave function also tends to in�nity as the parameter of the asymptotic expan-

sion tends to zero). We will see that the asymptotics will not factorize into a product

of one-particle wave functions. Therefore, the chaos hypothesis fails if k ! 1 in(1). Thus, when one makes an attempt to �nd mean values of general observables

uniformly bounded with respect to N , one can't use the property of factorizing of the

wave function, contrary to the case of the observables of the type (3).Consider the multiparticle Schrodinger equation

i�h@

@ttN(x1; :::; xN) =

24 NXi=1

� �h

2

2m�i + U(xi)

!+

1

N

X1�i

(7)

�i�h @@tvt(x) =

"� �h

2

2m�+W t(x)

#vt(x)

+'t�(x)ZdyV (x; y)('t�(y)ut(y) + vt(y)'t(y))

The system (7) can be formally obtained by the following procedure. One can write

the system consisting of the Hartree equation (5) and the equation conjugated to

it, consider the variation system for it and substitute the variations of ' and '� by

u and v that should not be conjugated. This variation system with independent

variations of ' and '� coincides with (7). The asymptotic formula for the N -particle

wave function is expressed through the solution to eq.(5) and through the operator

transforming the initial condition for the Cauchy problem for system (7) into the

solution to this Cauchy problem.

We can consider equations of a more general form than the multiparticle

Schrodinger equation (4). Namely, we can study equations for the functions N 2L2(X ), where X is an arbitrary measure space. Such equations are of the form

id

dttN = NAN

tN ;

where operator AN has a kernel of the type (3). Notice that the Schrodinger equationis a partial case of this equation. Asymptotic solutions to it that obey more general

initial conditions than a product of one-particle wave functions are constructed in sec-tion 5. These asymptotic formulas implying the results on the chaos non-conservationfor eq.(4) are proved in section 6. In section 7 we evaluate the corrections to theasymptotic formula.

We can consider not only Schrodinger-like equations but also sets of such equa-

tions. The generalization of the method which is applicable to such case is to bedeveloped in sections 8,9. The technique of these sections can be also used when oneconsiders the N -particle systems interacting with an additional particle, so that theevolution equation has the form:

i�h@

@ttN (y; x1; :::; xN) = [N

� �h

2

2M�y + U(y)

!+

NXi=1

V(xi; y)

+NXi=1

� �h

2

2m�i + U(xi)

!+

1

N

X1�i

be considered in sections 5-7 without modi�cation. But the coe�cients are N and 1,

so another approach is needed. It happens that the Hartree equation (5) should be

modi�ed, there will be no longer a single Hartree-like equation, there will be a set

of such equations, and the analog of the chaos hypothesis will then fail even for the

correlation functions.

We can consider the problem of chaos conservation not only for N -particle wave

functions obeying the multiparticle Schrodinger equation but also for other cases.

Namely, one can investigate N -particle density metrices obeying the N -particle

Wigner equation [13] or N -particle density functions (probability distributions) which

obey the multiparticle Liouville equation [11]. One can �nd [11,13] asymptotic for-

mulas for these cases. These asymptotics are also products of one particle densities

at the initial time moment, while there is no such factorization at time moment t. In

section 10 we will consider the equations for N -particle states corresponding to the

N -th tensor power of an abstract Hamiltonian algebra of observables. We generalize

a notion of a half-density (discussed in [11,13] for di�erent cases) by introducing the

notions of a half-density representation of an abstract Hamiltonian algebra and of an

abstract half-density. There are following examples of the latter notion:

(a) the square root of the N -particle probability distribution [11];(b) the square root of the N -particle density matrix [13];(c) the N -particle wave function.We show, that when our asymptotic method is applied to the N -particle half-

density equation, the average values of general bounded observables is unambigu-

ously de�ned by our approximation for the half-density. The results to be obtainedin section 10 imply, in particular, the asymptotic formulas found for the cases ofSchrodinger, Liouville, Wigner equations.

2 Violation of the chaos hypothesis for the N-particle

wave function

1.In the previous section we have seen that if the property (2) is satis�ed, onecan replace the wave function tN by the product of one-particle wave functions

't(x1):::'t(xN) in order to �nd a limit as N !1 of the mean values of the observ-

ables of the special form (3). Consider the problem if such replacement is valid for

�nding mean values of general observables AN uniformly bounded with respect to N ,

jjAN jj < C. What is necessary and su�cient condition for this replacement? Thefollowing lemma tells us that such condition isZ

dx1:::dxN jtN(x1; :::; xN)� ctN't(x1):::'t(xN )j2 !N!1 0 (9)

for some number ctN 2 C, jctN j !N!1 1:Lemma 1. Let �N and N be such elements of L

2(R�N) that (�N ;�N ) =

(N ;N ) = 1:

5

1.Let AN be operators acting in L2(R�N ) and uniformly bounded with respect to

N ,jjAN jj < C. Let for some set of numbers cN 2 C; jcN j !N!1 1

jjN � cN�N jj !N!1 0 (10)

Then the property (�N ; AN�N )!N!1 A implies that (N ; ANN)!N!1 A2.Let

(N ; ANN)� (�N ; AN�N )!N!1 0 (11)for arbitrary set of operators AN uniformly bounded with respect to N . Then the

property (10) is satis�ed for some number cN 2 C; jcN j !N!1 1Proof.

1. Denote XN = N � cN�N . We have

(N ; ANN)� (�N ; AN�N )

= cN (XN ; AN�N ) + c�N(�N ; ANXN ) + (XN ; ANXN )!N!1 0

because jjAN jj < C and jjXN jj !N!1 0.2. Consider the sequence XN of the form

XN = N � �N (�N ;N ):

Notice that (�N ;XN) = 0 and choose the following set of operators AN :

ANw = XN (XN ; w); w 2 L2(R�N)

that are uniformly bounded with respect to N : namely,

jjAN jj = jjXN jj2 � (jjN jj+ jj�N jj)2 = 4:

It follows from eq.(11) that

(N ; ANN )� (�N ; AN�N ) = j(XN ;N)j2 � j(XN ;�N )j2 =

= j(XN ;XN )j2 !N!1 0;so that the property (10) is satis�ed for cN = (�N ;N ): As 1 � jcN j2 = (N �cN�N ;N � cN�N )!N!1 0, one has jcN j !N!1 1. Lemma 1 is proved.

2.Let U; V be smooth functions U : R� ! R; V : R� �R� ! R bounded with alltheir derivatives, V (x; y) = V (y; x): Denote by W12 (R

�) the following set:

W12 (R�) = ff : R� ! Cjf 2 W k2 (R�); k = 1;1g

Consider the initial condition '0 : R� ! C for the Hartree equation (5) such that

'0 2 W12 (R�);Zj'0(x)j2dx = 1: (12)

6

The following lemma is proved in [4,5].

Lemma 2. There exists a unique solution 't 2 W12 (R�) to the Cauchy problemfor eq.(5).

As the Schrodinger equation is the partial case of the Hartree equation, lemma 2

implies the following corollary.

Corollary. There exists a unique solution tN 2 W12 (R�N ) to eq.(4) which sat-is�es the initial condition

0N(x1; :::; xN) = '0(x1):::'

0(xN ): (13)

It occurs that the property (9) is not valid.

Theorem 1. Let V (x; y) 6= 0 for any x; y 2 R�: Then there is no such interval[t1; t2] that the property (9) is satis�ed for t 2 [t1; t2] for some number ctN 2 C:

This theorem is a corollary of the more general statement to be proved in section

6.

3. Let us consider a heuristic method to derive the result of theorem 1. Suppose

that for some function ctN the initial condition (13) evolve into the wave function

ctN't(x1):::'

t(xN) + ztN (x1; :::; xN); (14)

where jjzN jjL2 !N!1 0. One can then expect that the property of chaos conservationfor the full N -particle wave function is also valid when the initial condition for theHartree equation (5) is shifted by the quantity of order 1=

pN :

'0 ! '0 + �'0=pN:

In order to retain the property of the norm of the wave function to be of order O(1),let us choose the variation �'0 to be orthogonal to '0. As the Hartree equationcontains not only 't but also 't�, the function 't transforms as follows:

't ! 't + 1pN(At�'0 +Bt(�'0)�) +O(1=N); (15)

where At and Bt are some linear operators acting in L2(R�): Applying the transfor-

mation (15) to formula (14) and multiplying �'0 by eia, one obtains that the following

N -particle wave function

ctN ('t(x1) +

1pN(At�'0(x1)e

ia +Bt(�'0)�(x1)e�ia) + :::)

�:::('t(xN) + 1pN(At�'0(xN )e

ia +Bt(�'0)�(xN )e�ia) + :::) + ztN(x1; :::; xN) (16)

is also an asymptotic solution to eq.(4). At the initial time moment formula (16)

does not contain negative powers of eia, while at time moment t such powers arises.

7

For example,the contribution of the power e�ia is zero at initial time moment and is

equal to

atN't(x1):::'

t(xN)

+ct0N

1pN

NXi=1

't(x1)::'t(xi�1)B

t(�'0)�(xi)'t(xi+1)::'

t(xN ) (17)

at time moment t for some numbers atN ; ct0N 2 C (the term with atN arises from

the coe�cient ctN that may be a-dependent). Because of the linearity of eq.(4), the

N -particle wave function (17) is also expected to be asymptotic solution to eq.(4).

Notice that the function Bt(�'0)� can be decomposed into two parts: one of them

being proportional to 't and another being orthogonal to 't:

Bt(�'0)� = bt't +Bt0(�'0)�

The contribution of the term bt't to eq.(17) can be involved to the �rst term of eq.(17),

while another term can't be treated in this way. As the norm of its contribution is of

order O(1), we are faced with the di�culty: the N -particle wave function being equal

to zero at the initial time moment evolves into non-zero wave function. The only

possible way to resolve the di�culty is to adopt the violation of the chaos hypothesis(9) when Bt

0 6= 0. The theorem 1 is heuristically justi�ed.One can also expect that investigation of the operators At; Bt being obtainable

from the variation system (7) can lead us to the correct asymptotic formula for thewave function. This is to be done in the following sections.

3 Multiparticle canonical operator and asymptotic

formula for the N-particle wave function as N !

1

1. In the previous section we have seen that the wave function (17) may play animportant role in constructing the N -particle wave function asymptotics as N !1:We can also notice that such wave function satis�es the chaos property (2) and doesnot satisfy the propery (9). Therefore, the wave function (17) gives us an example of

the state which can be replaced by the product 't(x1):::'t(xN ) in order to �nd limits

of mean values of observables of the special form (3) but not of the general form.Let us give a generalization of the example (17). Introduce a notion of a multipar-

ticle canonical operator being a partial case of the canonical operator corresponding

to Lagrangian manifold with complex germ in Fock space [6,7].

Let us introduce the following notations. By F we denote the space of sets(g0; g1(x1); g2(x1; x2); :::) of functions gn : R

�n ! C which are symmetric with re-spect to xi 2 R�, belong to the space L2(R�n) and satisfy the condition

1Xn=0

Zdx1:::dxnjgn(x1; :::; xn)j2

By gn we denote the n-th component of g 2 F . De�ne an inner product in F as

(g(1); g(2)) =1Xn=0

Zdx1:::dxng

(1)�n (x1; :::; xn)g

(2)n (x1; :::; xn):

By F', ' 2 L2(R�), we denote the subspace of F which consists of all the elementsg 2 F such that Z

dx1:::dxn'�(x1)gn(x1; :::; xn) = 0; n = 1;1: (19)

Consider the following element of L2(R�N ):

(KN' g)(x1; :::; xN) =NXp=0

pp!pNp

X1�i1

Corollary 3. Let f; g be such non-zero elements of F' that for some cN 2 C

jjKN' f � cNKN' gjj !N!1 0:

Then for some c 2 C f = cg.2. It occurs that the approximate solution to eq.(4) which satis�es the initial

condition (13) should be expressed not only through the solution 't to the Hartree

equation but also through the solution to another equation. This is a Riccati-type

equation:

i�h@

@tRt(x; y) = V (x; y)'t(x)'t(y) +

� �h

2

2m�x � �h

2

2m�y +W

t(x) +W t(y)

!Rt(x; y)

+Zdy

0't(y)V (y; y

0)'t�(y

0)Rt(x; y

0) +

Zdx

0't(x)V (x; x

0)'t�(x

0)Rt(y; x

0)

+Zdx

0dy

0Rt(x; x

0)Rt(y; y

0)V (x

0; y

0)'t�(x

0)'t�(y

0); (23)

where W t has the form (6).

By W' we denote the space of complex functions R(x; y) : R� � R� ! C such

that(i) R 2 W12 (R2�); R(x; y) = R(y; x);(ii) Z

R(x; y)'�(y)dy = �'(x); (24)

(iii) the operator M in L2(R�) with the kernel

M(x; y) = R(x; y) + '(x)'(y) (25)

satisfy the property jjM jj < 1:Lemma 4. Let R0 2 W'0: Then there exists a solution Rt 2 W't to the Cauchy

problem for eq.(23) with the initial condition R0.In order to prove this lemma, we will express the function Rt through the evolution

operator transforming the initial condition for variation system (7) into the solution

to this system.

Let us �rst construct this operator and study its properties. Denote by L; Y t andZ t the following operators in L2(R�):

L = � �h2m

�; (Z tw)(x) = �h�1ZdyV (x; y)'t(x)'t(y)w(y);

(26)

(Y tw)(x) = �h�1[W t(x)w(x) +ZdyV (x; y)'t(x)'t�(y)w(y)]:

10

The solution to the Cauchy problem for the system (7) is then as follows: ut

vt

!=

At Bt

Bt� At�

! u0

v0

!; (27)

where At; Bt are operators in L2(R�) that satisfy the following equation

id

dt

At Bt

Bt� At�

!=

" L 0

0 �L!+

Y t Z t

�Z t� �Y t�!#

At Bt

Bt� At�

!(28)

and the initial condition

A0 = E;B0 = 0:

We are going to show that the solution to eq.(28) is the following: At Bt

Bt� At�

!=

e�iLt 0

0 eiLt

!1Xn=0

Atn B

tn

Bt�n At�n

!; (29)

where operators Atn; Btn are determined from the following recursive relations:

At0 Bt0

Bt�0 At�0

!=

E 00 E

!

Atn B

tn

Bt�n At�n

!= �i

Z t0d�

Y �i Z

�i

�Z��i �Y ��i

! A�n�1 B

�n�1

B��n�1 A��n�1

!; (30)

Y ti = eiLtY te�iLt; Z ti = e

iLtZ te�iLt:

Notice that the series (29) is well-de�ned, since the functions U(x) and V (x; y), as

well as the operators Y ti ; Zti are bounded. The estimation

jjAtnjj �cntn

n!; jjBtnjj �

cntn

n!

for some constant c can be justi�ed by induction. Thus, the series (29) converges.

The following property is satis�ed for operators (29).Lemma 5.The kernels of operators Bt and AtR0 belong to the space W12 (R

2�).Proof. Introduce notations:

jjSjj = supjj�jj=1

jjS�jj; jjSjj2 =pTrS+S;

jjSjj(m) = jj(��+ 1)mS(��+ 1)�mjj;jjSjj(m)2 = jj(��+ 1)mS(��+ 1)mjj2

for each operator S in L2(R�): Let us show by induction that

jjAtnjj(m) �cnmt

n

n!; jjBtnjj(m)2 �

cnmtn

n!(31)

11

for some constants cm. Inequalities (31) are correct if n = 0: Suppose them to be

correct as n < k and check eq.(31) as n = k. It follows from eq.(30) that

jjAtkjj(m) �Z t0d� [jjY �i jj(m)jjA�k�1jj(m) + jjZ�i jj(m)2 jjB�k�1jj(m)2 ];

jjBtkjj(m)2 �Z t0d� [jjY �i jj(m)jjB�k�1jj(m)2 + jjZ�i jj(m)2 jjA�k�1jj(m)]:

For cm > max[jjY �i jj(m); jjZ�i jj(m)2 ], eq.(31) is justi�ed.It follows from eq.(31) that

1Xn=0

jjAtnjj(m)

1.The operator At� +Bt�R0 is boundedly invertable.

2. The kernel of the operator

Rt = (Bt +AtR0)(At� +Bt�R0)�1

belongs to W't.

3. The following relation is satis�ed:

jjiRt+�t �Rt�t

� (Z t + (L+ Y t)Rt +Rt(Y t� + L) +RtZ t�Rt)jj2 !�t!0 0 (33)

Proof.

1. To prove the �rst statement, notice that eq.(30) implies that the matrix (29)

is boundedly invertable:

At Bt

Bt� At�

!�1=

1Xn=0

At�n B

t�n

Bt��n At��n

! eiLt 0

0 e�iLt

!

where At�n; Bt�n obey the following recursive relations: At�n B

t�n

Bt��n At��n

!= i

Z t0d�

A��n+1 B

��n+1

B���n+1 A���n+1

! Y �i Z

�i

�Z��i �Y ��i

!

which imply that At Bt

Bt� At�

!�1=

At+ �BtT�Bt+ AtT

!(34)

This means that the matrix (29) is a matrix of a canonical transformation [8]. Thisimplies [8] that the operator At is boundedly invertable. As

E � ((At)�1Bt)((At)�1Bt)� = (At)�1(At)��1 > 0;

the operator (At�)�1Bt� has the norm lesser than 1. Therefore, the operator

(At� +Bt�R0)�1 = (E + (At�)�1Bt�R0)�1(At�)�1

is bounded, since jjR0jj = 1 and jj(At�)�1Bt�jj < 1: The �rst statement of lemma 7 isproved.

2. Let us check the property Rt 2 W12 (R2�): It follows from the Cauchy-Schwarz-Bunyakovskii inequality that

jj�MRt�Kjj22 = Tr(�KRt��2MRt�K) � jj�2MRtjj2jj�2KRtjj2The �rst statement of lemma 7 and lemma 5 imply that

jj�2KRtjj2 � jj�2K(Bt +AtR0)jj2jj(At�+Bt�R0)�1jj

Therefore, Rt 2 W12 (R2�):Let us check that Rt't� = �'t. As the set of functions

ut = i't; vt = �i't� (35)

is a solution to eq.(7), one has

't� = At�'0� �Bt�'0 = (At� +Bt�R0)'0�:

Therefore,

Rt't� = (Bt +AtR0)'0� = Bt'0� �At'0 = �'0:Eq.(24) is proved.

Let us prove that the operator with the kernel (25) has the norm lesser than 1. It

is su�cient to prove that (�; (E �Rt+Rt)�) � 0, (�; (E �Rt+Rt)�) = 0 if and only if� = �'t�: We have

(�; (E �Rt+Rt)�) = ((At� +Bt�R0)�1�; (E �R0+R0)(At� +Bt�R0)�1�):

This quantity is non-negative and equals to zero if and only if (At�+Bt�R0)�1� = �'0�,

i.e. � = �'t�. The second statement is proved.3. The proof of statement 3 is by straightforward substitution. Lemma 7 is proved.Remark. As the quantity �h(Z t+ (L+ Y t)Rt+Rt(Y t�+L) +RtZ t�Rt) coincides

with the right-hand side of eq.(23), lemma 4 is a corollary of lemma 7.3. Let us give an asymptotic formula for the N -particle wave function being a

solution to eq.(4). Let ' 2 W12 (R�); R 2 W': Denote by �R the following elementof F':

�R;2n(x1; :::; x2n) =1

2nn!q(2n)!

X1

Consider also the solution 't to eq.(5) which is equal to '0 at initial time moment

and the solution Rt to the Cauchy problem for eq.(23).Consider the functions:

St =Z t0dt

"i

Zdx't�(x)

d

dt't(x)�H0('t�; 't)

#; (38)

where

H0('�; ') =

1

�h

Zdx'�(x)

� �h

2

2m�+ U(x)

!'(x)+

1

2�h

ZdxdyV (x; y)j'(x)j2j'(y)j2;

ct = exp

�� i2�h

Z t0d�ZdxdyV (x; y)'��(x)'��(y)R� (x; y)

�: (39)

Theorem 2. The following formula is satis�ed;

jjtN � cteiNSt

KN't�Rtjj !N!1 0:

This theorem is a corollary of the more general statement to be proved in section6. It follows from theorem 2 and lemma 1 that one can use the N -particle wave

function

cteiNSt

(KN't�Rt)(x1; :::; xN) = cteiNS

t[N=2]Xl=0

1

(2N)ll!

� X1�i1 6=::: 6=i2l�N

M t(xi1; xi2):::Mt(xi2l�1 ; xi2l)

Yi6=is

't(xi) (40)

(M t(x; y) = Rt(x; y) + 't(x)'t(y))

instead of the exact wave function tN in order to �nd limits as N ! 1 of meanvalues of the observables uniformly bounded with respect to N . We see that not theform of the product of one-particle wave functions but the more complicated form(40) of the N -particle wave function conserves under time evolution. The product(13) is a partial case of the wave function (40) which is realized when

Rt(x; y) = �'t(x)'t(y): (41)

Therefore, one can make use of theorem 2 for �nding an approximate solution to the

Cauchy problem for eq.(4) with the initial condition (13). As the solutioin to eq.(23)does not, in general, have the form (41), the asymptotic solution to this problem

has the form (40) with M t 6= 0. This implies that eq.(9) is not satis�ed because ofcorollary 3 from lemma 3. Thus, theorem 1 is a corollary of theorem 2.

15

4 Heuristic derivation of the asymptotic formula

In this section we consider a heuristic method to derive eq.(40). This method is

analogous to the procedure of section 2 which is based on shifting the solution to

the Hartree equation and allows us to conclude that the product of one-particle wave

functions is not an asymptotic solution to eq.(4).

As the chaos property (2) for correlation functions is satis�ed for the N -particle

wave function (20) for arbitrary g 2 F', as well as for the function tN [3], it isreasonable to look for the asymptotic expression for tN in the following form:

tN = eiNStKN'tg

t; St 2 R; 't 2 L2(R�); gt 2 F': (42)

It occurs that the heuristic method to be developed allows us to �nd the function

St up to an additive quantity that does not depend on the solution to the Hartree

equation, while vector gt 2 F't is de�ned up to a multiplier ct 2 C; jctj = 1:1. The function St can be found by the following technique. Consider the small

shift of the function ' by the quantity �=N of order 1=N in eq.(20). The function �

can be decomposed into two parts:

� = '(';�) + �0; (43)

where �0is orthogonal to '. It follows from eq.(20) that the contribution of shifting

by �0to formula (20) is small. We will see that the coe�cient of �

0chould be of order

1=pN for making such shifting of ' by �

0appreciable. The contribution of the �rst

term of eq.(43) to eq.(20) is as follows: the p-th term of eq.(20) is multiplied by

(1 +1

N(';�))N�p:

As p = O(1), all these quantities are approximately equal to exp(';�). Because ofthe remark after lemma 3, norms of terms of order p = O(N) in eq.(20) are small.Therefore, the following approximate formula takes place:

KN'+ 1

N�g ' e(';�)KN' g (44)

This formula can be also proved rigorously [9].Let us make use of eq.(44) for �nding St. Let us shift the initial condition for

eq.(5) '0 ! '0 + 1N�0; so that the solution to eq.(5) will be transformed as

't ! 't + 1N�t:

The factor St depending on the initial condition '0 will be changed as St('0) !St('0 + 1

N�0): It follows from eq.(44) that

eiNSt('0+ 1

N�0)e('

t;�t)�('0;�0) ' eiNSt('0):

16

Therefore, the variations of St and '0; 't are related as follows:

�St = i[('t; �'t)� ('0; �'0)]: (45)

Eq.(45) determines the function St up to a constant that depends on t but does not

depend on '0. Let us check that eq.(38) really satis�es eq.(45). One has:

�St =Z t0dtZdx[i(�'t)�(x)

d

dt't(x)� i d

dt(�'t)�(x)�'t(x)� (�'t)�(x) �H0

�'�t(x)�

��'t(x) �H0�'t(x)

] +Z t0dt

Zdxi

d

dt['t�(x)�'t(x)]:

The �rst term in this formula vanishes because of the Hartree equation, while the

second term is equal to eq.(45). Thus, the function St (38) can be found by the

developed technique up to a function depending only on t but not on the solution to

eq.(5).

2. Let us derive conditions on the function gt 2 F't:We are going to consider thevariations of 't which are of order 1=

pN and to obtain formula analogous to eq.(44)

for the N -particle wave function

KN'+ 1p

N�(g +O(1=

pN)); (46)

where g 2 F': Notice that the vectors g + O(1=pN) belong to di�erent subspaces

F'+ 1pN� of the space F and, therefore, should be di�erent in general.

First of all, consider the expression (46) in the case of vacuum vector g = �(0),i.e. g0 = 1; g� = 0; � � 1: The wave function (46) is then equal to the product ofone-particle wave functions

('(x1) +1pN�(x1))::('(xN) +

1pN�(xN)) (47)

which has been considered in section 2. Examine the case (';�) = 0: It follows fromeq.(20) that formula (47) can be written as

KN' C�;

where C� is the following element of F':

C�;n(x1; :::; xN) =1pn!�(x1):::�(xn): (48)

When (';�) 6= 0, one has

KN'+ 1p

N��(0) = (1 +

1pN(';�))NKN' C�0 ; (49)

17

where �0is expressed from eq.(43).

Consider now the following case of vector g in formula (46): g + O(1=pN) =

CfN ; fN = f +O(1=pN): As

('+1pN�; fN) = 0; (50)

one has

KN'+ 1p

N�CfN = K

N'+ 1p

N(�+fN )

�(0) = (1 +1pN(';�+ fN ))

NKN' C�0+fN�'(';fN): (51)

It follows from eq.(50) that

('; fN) = �(�; fN)=pN = O(1=

pN );

(1 +1pN(';�+ fN))

N ' (1 + 1pN(';�))Ne�(�;f):

Since ('; f) = 0, one has

KN'+ 1p

N�(Cf +O(1=

pN)) = constKN' (e

�(�0;f)Cf+�0 +O(1=

pN)); (52)

where constant in the right-hand side of eq.(52) does not depend on f . As the vectors(48) make up a full system of the vectors in the space F' [8], eq.(52) can be used forconstructing an approximation for eq.(46).

It is convenient to introduce creation and annihilation operators [8] in Fock spaceF : These operators are the following:

(a+(x)g)n(x1; :::; xn) =1pn

nXi=1

�(x� xi)gn�1(x1; :::; xi�1; xi+1; :::; xn);

(a�(x)g)n�1(x1; :::; xn�1) =pngn(x; x1; :::; xn�1):

Notice that the operators

a+[�] =Za+(x)�(x)dx; a�[�] =

Za�(x)��(x)dx;

transform the subspace F' into F' if and only if ('; �) = 0: Consider the operator

V';� = exp(Zdx(a+' (x)�(x)� a�' (x)��(x)); (53)

where

a�' (x) = a�(x)� '(x)a�[']; a+'(x) = a+(x)� '�(x)a+[']:

The canonical commutation relations

[a�(x); a�(y)] = 0; [a�(x); a+(y)] = �(x� y)

18

and the representation of Cf through the creation operators

Cf = exp(a+[f ])�(0)

imply that the operator (53) transforms the vector (48) as follows:

V';�Cf = exp(�(�0; f)� 12(�

0; �

0))Cf+�0 : (54)

The factor e�(�0;�0)=2 can be involved into the constant in eq.(52). Therefore, one

has

KN'+ 1p

N�(g +O(1=

pN)) = constKN' (V';�g +O(1=

pN)) (55)

for the case g = Cf . Since the system of vectors (48) is full [8], eq.(55) is correct for

arbitrary g.

3. Let us �nd vector gt entering to eq.(42). Make use of eq.(55) for �nding operator

W t transforming g0 into gt, gt =W tg0: Suppose that variation of this operator is alsoof orde N�1=2 as the solution to eq.(5) is shifted by the quantity O(1=

pN ). The

N -particle wave functions

KN't+ 1pN�tW tg +O(1=

pN)

andconst(KN'tW tV'0;�0g +O(1=

pN))

are then asymptotic solutions to eq.(4) if 't is a solution to eq.(5) and the set ut =�t; vt = �t� is a solution to the system (7). As the asymptotic solutions coincide atinitial time moment because of eq.(55), the same property should be satis�ed at timemoment t. Making use of eq.(55), one �nds that

V't;�tW t = constW tV'0;�0: (56)

It is eq.(56) that allows us to �nd operatorW t up to a multiplicative factor. It followsfrom eq.(27) that eq.(56) is satis�ed if and only ifZ

dx[a+'t(x)(Atu0 +Btv0)(x)� a�'t(x)(Bt�u0 +At�v0)(x)]W t =

W tZdx[a+'0(x)u

0(x)� a�'0(x)v0(x)]: (57)

Eq.(57) is a straightforward corollary of eq.(56) when u0 = �0; v0 = �0�: In order toprove eq.(56) at arbitrary u0; v0, one should write eq.(56) for u0 = �0eia; v0 = �0�e�ia

and integrate it over a with the weight e�ia:

Eq.(57) shows us that the operator W t corresponds to a linear canonical transfor-mation of the creation and annihilation operators [8]. It has been shown in [8] that

W t is de�ned up to a multiplicative factor. In order to �nd the vectorW t�R0 and to

19

show a role of the Riccati-type equation (23), it is convenient to introduce a notion

of complex germ analogous to [10,6].

4. LetR 2 W'. Consider the following subspace GR of the space L2(R�)�L2(R�) :GR = f(v; u)ju = Rvg:

De�nition 2. A subspace GR will be referred to as a complex germ correspondingto R 2 W':

The vector �R satis�es the following interesting property.

Lemma 8. 1. Let g 2 F' be such vector thatZdx[a+' (x)u(x)� a�' (x)v(x)]g = 0 (58)

for any (v; u) 2 GR. Then g = c�R.2. The vector g = �R 2 F' satis�es eq.(58).To prove this lemma, one can make use of the de�nition of creation and annihi-

lation operators, rewrite eq.(58) in terms of components gn of g and �nd them by

induction to be equal to expression (36) up to a multiplier.Lemma 7 implies the following statement.Lemma 9.

GRt = f(Bt�u0 +At�v0; Atu0 +Btv0)j(v0; u0) 2 GR0gCorollary.

W t�R0 = ct�Rt (59)for some constant ct 2 C.

Namely, consider the vectorW t�R0 : It follows from eq.(57) that this vector satis�esthe condition (58) for (vt; ut) 2 GRt, and, therefore, is equal to c�Rt.

Therefore, theorem 2 is heuristically justi�ed. Notice that the developed methodcan be applied also to the simpler case of the complex germ approximation for quan-

tum mechanics as �h! 0, see appendix A for more details.Notice also that eq.(57) allows us to construct another asymptotic solutions to

eq.(4) with the help of the complex germ creation operators:

�'+u;v =Zdx[a+' (x)v

�(x)� a�' (x)u�(x)]: (60)

It follows from eq.(57) that

W t�'0+u01;v01

:::�'0+

u0k;v0k

�R0 = ct�'

t+

ut1;vt1

:::�'t+

utk;vtk

�Rt: (61)

As any element of the space F' can be presented as a linear (maybe, in�nite) combi-nation of vectors (61), eq.(61) allows us to reconstruct the operator W t and to �ndasymptotic solutions

cteiNSt

KN'tW tg

20

to eq.(4).

The approach developed here can be used for constructing asymptotic solutions

to equations of a more general form than (4). Let us formulate the corresponding

theorem.

5 General case : formulation of the theorem

Let X be a measure space, HN be a self-adjoint operator in L2(XN) of the form

HN = NHN0 +H

N1 + :::+N

1�kHNk ; (62)

where

(HNl )(x1; :::; xN) =P0Xp=1

1

Npp!

X1�i1 6=::: 6=ip�N

Zdyi1:::dyipH

(p)l (xi1; :::; xip; yi1; :::; yip)

�(x1; :::; xi1�1; yi1; xi1+1; :::; xip�1; yip; xip+1; :::; xN); (63)where H

(p)l � H(p)l is a kernel of the operator acting in L2(X p) which is symmetric

separately over xi1; :::; xip and over yi1; :::; yip and

H(p)�l (x1; :::; xp; y1; :::; yp) = H

(p)l (y1; :::; yp;x1; :::; xp):

Analogously to section 3, denote by F the space of sets (g0; g1(x1); g2(x1; x2); :::) offunctions gn : X n ! C which are symmetric with respect to xi 2 X , belong to thespaces L2(X n) and satisfy eq.(18). By F' � F we denote such subspace of F thatconsists of all the elements of F which satisfy eq.(19). The multiparticle canonicaloperator of the form (20) is denoted by KN' : F' ! L2(XN). Denote by �R 2 F'eq.(36).

Let tN be a solution to the Cauchy problem

id

dttN = HN

tN ;

(64)

0N = KN'0�

'0+

u01;v01

:::�'0+

u0k;v0k

�R0;

where �+ has the form (60).

Denote

Hl('�; ') =

P0Xp=1

1

p!

Zdx1:::dxpdy1:::dypH

(p)l (x1; :::; xp; y1; :::; yp)

�'�(x1):::'�(xp)'(y1):::'(yp); ' 2 L2(X ); (65)

21

Hl('�; ') � Hl('�; '): Let 't be a solution to the Cauchy problem for the following

equation:

id

dt't(x) =

�H0('t�; 't)

�'�(x); (66)

such thatRdxj'0(x)j2 = 1, Rt satisfy the equation:

id

dtRt(x; y) =

�2H0

�'�(x)�'�(y)+Zdx

0 �2H0

�'�(x)�'(x0)Rt(x

0; y)

+Zdy

0Rt(x; y

0)

�2H0

�'(y0)�'�(y)

+Zdx

0dy

0Rt(x; x

0)

�2H0

�'(x0)�'(y

0)Rt(y

0; y); (67)

where arguments 't�; 't of the function H0 are omitted, uti; v

ti satisfy the following

system:

id

dtuti(x) =

Zdy

�2H0

�'�(x)�'(y)uti(y) +

�2H0

�'�(x)�'�(y)vti(y)

!;

(68)

�i ddtvti(x) =

Zdy

�2H0

�'(x)�'(y)uti(y) +

�2H0

�'(x)�'�(y)vti(y)

!:

By St we denote expression (38), while

ct = exp

�iZ t0d�

"1

2

Zdxdy

�2H0

�'(x)�'(y)Rt(x; y) +H1

#!; (69)

where arguments 't�; 't of the functions H0;H1 are also omitted.Let the following functions of x1; :::; xm; z1; :::; zs�2j�i:Z

dxm+1:::dxpdy1:::dyp't�(xm+1):::'

t�(xp)H(p)l (x1; :::; xp; y1; :::; yp)

�'t(ys+1):::'t(yp)Rt(y1; y2):::Rt(y2j�1; y2j)Rt(y2j+1; z1):::Rt(ys�i; zs�2j�i)��tJ1(ys�i+1):::�tJi(ys) (70)

belong to L2(Xm�s�2j�i), where J1; :::; Ji 2 1; :::; k, �J = v�J �Ru�J .Theorem 3. The following relation takes place:

jjtN � cteiNSt

KN't�'t+

ut1;vt1

:::�'t+

utk;vtk

�Rtjj !N!1 0:

Remarks.

1. For some special choice of H(p), eq.(4) is a partial case of eq.(64), the Hartree

equation (5) is a partial case of eq.(66), eq.(23) is the analog of eq.(67), eq.(68) is avariation system (7). Therefore, theorems 1 and 2 are corollaries of theorem 3.

2. An asymptotic formula being approximately equal to the wave function tNaccurate to O(N�L=2) for arbitrary L > 0 is to be presented in section 7.

22

3. The requirement for the functions (70) to belong to L2 means that the N -

particle wave function (61) belongs to the domain of an operator HN . This require-

ment can be easily checked for the case of bounded operators HNl , while for general

case one should prove this independently. For the case of conditions of theorem 2, the

square integrability of eq.(70) is a corollary of the properties of the solutions to the

Hartree equation (5) and to Riccati equation (23); some other cases are presented in

[11-13].

4. We have used di�erent notations, H(p)l and H

(p)l , for the same quantity. This

has been done in order to simplify formulas to appear in section 8. In that sectoion

we will denote by H(p)l an operator in l

2, while its eigenvalue will be denoted as H(p)l .

The same remark is correct for the notations Hl and Hl.

6 Proof of the theorem

1. Let us consider the more convenient representation for the operator HN that allows

us to �nd a commutation rule between HN and multiparticle canonical operator KN' .

Consider the space L2(XN) as a subspace FN of the Fock space F of the formFN = fg 2 Fjg� = 0; � 6= Ng

Consider the operator

HNl =P0Xp=1

1

Npp!

Zdx1:::dxpdy1:::dypH

(p)l (x1; :::; xp; y1; :::; yp)

�a+(x1):::a+(xp)a�(y1):::a�(yp): (71)Lemma 10. The operator (71) transforms FN into FN and coincide on the

subspace FN with the operator (63).The proof is by making use of the de�nition of creation and annihilation operators.As the operator HN is expressed through the operators a

�, it is su�cient to �ndtheir commutation rules with the multiparticle canonical operator.

Lemma 11. The following relations are satis�ed:

a+' (x)a�[']pN

KN' = KN' a

+' (x);

(72)

a�' (x)KN' =

a�[']pNKN' a

�' (x):

The proof is by making use of the following expression for the element KN' g 2 FN �F :

KN' g =NXp=0

1pp!

Zdx1:::dxpgp(x1; :::; xp)

23

�a+'(x1)a

�[']pN

:::a+'(xp)a

�[']pN

a+[']NpN !

�(0) (73)

and of the commutation relations between operators a�(x).

Consider now the operator ddtKN' .

Lemma 12. The following relation takes place;

d

dtKN' = K

N' (

d

dt+ a+[']a�' [

d'

dt] + (';

d

dt')(N � n̂)

�(1 � n̂N)pNa�' [

d'

dt] +

pNa+' [

d'

dt]); (74)

where an index t on ' is omitted,

a+' [�] = a+[�]� a+['](';�); a�' [�] = a�[�]� a�['](�;'); n̂ =

Zdxa+' (x)a

�'(x):

Proof. Consider the N -particle wave function

d

dtKN' g =

d

dt

NXp=0

sN !

Npp!

1

(N � p)!

�Zdx1:::dxpgp(x1; :::; xp)a

+(x1):::a+(xp)a

+[']N�p�(0); g 2 F': (75)When one takes a derivative with respect to t, there will be two terms: one of themcontains d

dtgp, another contains

ddta+[']. Consider the �rst term. The function dg

dtcan

be decomposed into two parts:

d

dtgp(x1; :::; xp) = �

pXi=1

'(xi)Zd'�(yi)

dtgp(x1; :::; xi�1; yi; xi+1; :::; xp)dyi

+d0

dtgp(x1; :::; xp):

The second part being equal to

d0

dtg = (

d

dt+ a+[']a�' [

d

dt'])g

belongs to F', since the property g 2 F' (19) conserves under time evolution, andcontribute to eq.(75) as KN'

d0

dtg. The �rst part gives rise to the following contribution

to the expression (75):

�NXp=0

sN !

Npp!

1

(N � p)!pp

Z(a�' [

d

dt']g)p�1(x1; :::; xp�1)

�a+(x1):::a+(xp�1)dx1:::dxp�1(a+['])N�p+1�(0)

24

which can be presented as

� 1pNKN' (N � n̂)a�'

"d'

dt

#:

Consider now the terms containing ddta+[']. They contribute to eq.(75) as follows:

NXp=0

sN !

Npp!

1

(N � p)!Zgp(x1; :::; xp)a

+(x1):::a+(xp)dx1:::dxp

�(N � p)a+[ ddt'](a+['])N�p�(0):

According to the de�nition of a�' , this expression is equal to

KN'

(';

d

dt')(N � n̂) +

pNa+' [

d

dt']

!:

Combining all the terms, we obtain eq.(74). Lemma 12 is proved.

Corollary.

(id

dt�HN )eiNStKN't = eiNS

t

KN't

id

dt+ ia+['t]a�'t[

d

dt't]�H0N

!; (76)

where

H0N = Nd

dtSt � iN('t; d

dt't)(1 � n̂

N)� i

pNa+'t[

d

dt't] + i

pN(1� n̂

N)a�'t[

d

dt't]

+kXl=0

N1�lP0Xp=1

pXm;s=0

p!

m!(p�m)!s!(p� s)!1

Nm+s2

Zdx1:::dxpdy1:::dyp

�'t�(xm+1):::'t�(xp)H(p)l (x1; :::; xp; y1; :::yp)'t(ys+1):::'t(yp)a+'t(x1):::a+'t(xm)�(1� n̂=N)(1 � (n̂ + 1)=N):::(1� (n̂+ p�m� 1)=N)a�'t(y1):::a�'t(ys): (77)

Remark. The corollary implies that the operator H0N is a product of N by a poly-nomial in N�1=2:

H0N = NH00 +N

1=2H01 +H

02 + :::+N

1�K=2H0K (78)

for some K. The coe�cients H00;H

01;H

02 can be presented as follows:

H0

0 = H0('t�; 't) +

d

dtSt � i('t; d

dt't); (79)

H01 =

Zdx

"a+'t(x)

�H0

�'�(x)� i d

dt't(x)) + a�'t(x)(

�H0

�'(x)� i d

dt't�(x)

!#; (80)

25

H02 = n̂

Zdx't�(x)

id

dt't(x)� �H0

�'�(x)

!+H1('

t�; 't)

�12

Zdxdy't(x)'t(y)

�2H0

�'(x)�'(y)+Zdxdy[

1

2a+'t(x)

�2H0

�'�(x)�'�(y)a+'t(y)

+a+'t(x)�2H0

�'�(x)�'(y)a�'t(y) +

1

2a�'t(x)

�2H0

�'(x)�'(y)a�'t(y)]; (81)

where the arguments 't�; 't of the functional H0 are omitted. One can notice that

eqs.(38),(66) provide the nulli�cation of H00;H

01.

2. Let us prove that

jj(i ddt�HN )eiNStKN'tgtjj !N!1 0; (82)

where

gt = ct�'t+

ut1;vt1

:::�'t+

utk;vtk

�Rt:

Lemma 13.

jjH 0kgtjj

�t; �t are some complex functions. This system for ut0; vt

0, as well as eq.(84), is

satis�ed when (ut; vt) satis�es eq.(68).

2. It is su�cient then to prove lemma for the case k = 0. From eq.(83) one has:

(id

dt+ ia+[']a�' [

d

dt'])ct�Rt = i

dct

dt�Rt +

i

2ctZdxdya+'t(x)

dM t

dt(x; y)a�'t(y)�Rt;

H02c

t�Rt = ct(H1 +

1

2

Zdxdy(M t(x; y)� 't(x)'t(y)) �

2H0

�'(x)�'(y))

+ct

2

Zdxdya+'t(x)a

+'t(y)[

�2H0

�'�(x)�'�(y)+Zdz

�2H0

�'�(x)�'(z)M t(z; y)

+ZdzM t(z; x)

�2H0

�'(z)�'�(y)+Zdzdz

0M t(z; x)M t(z

0; y)

�2H0

�'(z)�'(z0)]�Rt:

Lemma 14 is then proved as a corollary of eqs.(67),(69).

Eq.(82) implies the statement of the theorem,the proof is analogous to [14]. Con-

sider the quantity eiNSt

KN'tgt �tN being equal to

eiNSt

KN'tgt �tN =

Z t0d�e�iHN (t��)(i

d

d��HN )(eiNS�KN'� g� ��N )

The following estimation takes place

jjeiNStKN'tgt �tN jj �Z t0d� jj(i d

d��HN )eiNS�KN'� g� jj !N!1 0:

Theorem 3, as well as theorem 1,2, is proved.

7 Corrections to the asymptotic formula

Let us construct now the N -particle wave function that approximates the solution tothe Cauchy problem for eq.(64) accurate to O(N�L=2) for arbitrary L > 0. It happensthat such wave function has the form:

�tN;L = eiNStKN't(g

t0 +N

�1=2gt1 + :::+N�(L�1)=2gtL�1);

where vectors gtm 2 F't are of the form

gtm =n0(m)Xn=0

1pn!

Zdx1:::dxng

tm;n(x1; :::; xn)a

+'t(x1):::a

+'t(xn)�Rt;

n0(m) is a �nite quantity, functions gm;n(x1; :::; xn) are to be de�ned by induction.

Let gtm;n be de�ned for m < l and all functionsZdxq+1:::dxpdy1:::dyp'

t�(xq+1):::'t�(xp)H

(p)r (x1; :::; xp; y1; :::; yp)

27

�'t(ys+1):::'t(yp)Rt(y1; y2):::Rt(y2j�1; y2j)Rt(y2j+1; z1):::Rt(ys�n; zs�2j�n)�gtm;n(ys�n+1; :::; ys) (85)

belong to L2(Xm�s�2j�n) for m < l. De�ne the functions �tl;n(x1; :::; xn) being sym-metric with respect to xi and obeying the requirement

Rdx1'

t�(x1)�tl;n(x1; :::; xn) = 0

by the following relation:

Xn

1pn!

Zdx1:::dxn�

tl;n(x1; :::; xn)a

+'t(x1)::a

+'t(xn)�Rt =

= H03g

tl�1 + :::+H

0Kg

tl�K ; (86)

where all gtn�i = 0 by de�nition as n < i. Note that de�nition (86) is correct and

�tl;n = 0 for su�ciently large n, the sum in the left-hand side of eq.(86) is then �nite.

When l = 0, let �t0;n = 0 by de�nition.

Let gtl;n be a solution to the Cauchy problem for the following equation:

i(d

dt� d ln c

t

dt)gtl;n(x1; :::; xn) =

nXk=1

Zdyk[

�2H0

�'�(xk)�'(yk)+

ZdzkR

t(xk; zk)�2H0

�'(zk)�'(yk)]gtl;n(x1; :::; xk�1; yk; xk+1; :::; xn) +

(m+ 1)(m+ 2)

2

�Zdy1dy2

�2H0

�'(y1)�'(y2)gtl;n+2(y1; y2; x1; :::; xn) + �

tl;n(x1; :::; xn); (87)

where ct has the form (69). Let tN be a solution to eq.(64) that satis�es the initialcondition

tN = KN'0(g

00 +N

�1=2g01 + :::+N�(L�1)=2g0L�1):

Theorem 4. Under the conditions of theorem 3

jjtN ��tN;Ljj = O(N�L=2):Remarks.

1. The requirement for the functions (79) to be square integrable is analogous tothe same requirement for the function (68) and means that the asympyotic solution�tN;L belongs to the domain of an operator HN .

2.For the case of eq.(4), the solvability of eq.(81) and the square integrability of

eq.(79) can be proved in a way analogous to the proof of solvability of eqs.(7) and(23). When HN is a bounded operator, such statements can be also proved.

Proof. It is su�cient to show that

jj(i ddt�HN )�tN;Ljj = O(N�L=2);

i.e.

(id

dt+ ia+['t]a�'t[

d

dt']�H 02)gtm = H

03g

tm�1 + :::+H

0Kg

tm�K : (88)

The proof of this relation is straightforwrd. Theorem 4 is proved.

28

8 Some aspects of problems with operator-valued

symbols

We have seen that the discussed method of constructing asymptotic solutions can

be applied to equations of the form (64). However, one can be interested in the

problem of generalization of the considered approach to the case of the set of such

equations. Investigation of it is very important when one considers the quantum

mechanical system consisting of two subsystems (one of them is the examined system

of N bose-particles, another subsystem interacts with the �rst one), some examples

are to be discussed in section 9. Notice that some of the results to be obtained in this

section and in section 9 can be also derived by making use of the technique analogous

to the derivation of the Ehrenfest theorem in ordinary quantum mechanics [20], see

appendix B for more details. Let us now consider the speci�cation of the form of the

set of equations which is to be approximately solved.

Let X be a measure space. Denote by l2 L2(XN) the Hilbert space of sets ofcomplex functions I(x1; :::; xN); I = 1;1; x1; :::; xN 2 X such that I 2 L2(XN)and

P1I=1

Rdx1:::dxN jI(x1; :::; xN)j2

where g 2 l2F , while gI is the element of F of the form (g0;I; g1;I(x1); g2;I(x1; x2); :::)at �xed I.

We are going to �nd approximate solutions to eq.(64) for the case of the operator-

valued function H(p)l and

tN 2 l2L2(X ) by the technique analogous to the method

discussed in sections 5-7. These asymptotic solutions are to be looked for in the

following form:

tN = eiNStKN't(g

t0 +N

�1=2gt1 + :::); (89)

where gti 2 l2F't. We will formulate the theorem in section 9, while in this sectionwe are to �nd gti heuristically. One should substitute the expression (89) to eq.(64)

and make use of the commutation rule between operators KN't and (id=dt �HN ).Lemma 14. The following relation is satis�ed:

(id

dt�HN )eiNStKN't = eiNS

t

KN't(iDt �H0N ); (90)

where H0N has the form (77) and

Dt = d=dt + a+['t]a�'t[

d

dt't]:

The proof of this lemma for the case of the operator-valued function H(p)l is anal-

ogous to the proof of eq.(76) for the case of sections 5-7.By Hl('

�; ') we denote the operator in l2 of the form (65). The operator H0Nin l2 F' is then written in a form (78), where operators H 00;H

01;H

02 have the form

(79),(80),(81). For the simplicity, the operators in l2 of the form like �E, where � isa number, are denoted by �.

One can notice that one should choose gt0;gt1; ::: in such a way that

(�NH 00�N1=2H01+ iDt�H

02�N�1=2H

03� :::)(gt0+N�1=2gt1+ :::) = O(N�L=2) (91)

when the asymptotics accurate to O(N�L=2) is looked for.

An interesting feature of the operator-valued case is that the operators H00;H

01

cannot be set to zero by varying 't, since the operator H0('�; ') is not, in general,

equal to H0('�; '). Therefore, in order to provide satisfaction of the relations like

eq.(82), one cannot choose gt to be independent on N , gt must be choosen as gt =gt0 +N

�1=2gt1 +N�1gt2:

Let us consider the recursive relations for gti, which are derivable from eq.(91).First of all, consider the term of order O(N) in eq.(91) which has the form H

00g

t0 = 0.

It follows from eq.(79) that H00 can be presented as H0('

t�; 't)��t for some number�t. Therefore, gt0 should be chosen as g

t0 = � gt0, i.e.

gt0;n;I(x1; :::; xn) = �Igt0;n(x1; :::; xn);

where � 2 l2 is the eigenvector of the operator H0('t�; 't) acting in l2.

30

Let H0('t�; 't) be eigenvalue of the operator H0('

t�; 't) and smooth function of

t, �('t�; 't) be projector on the corresponding eigenspace, so that

(H0('t�; 't)�H0('t�; 't))�('t�; 't) = 0: (92)

The term of order O(N) vanishes then, if St has the form (38).

Suppose that this eigenspace is one-dimensional. The method under consideration

can be also applied in analogous way to the case of �nite and 't-independent dimen-

sionality of the eigenspace. The case of terms intersection, ehen this dimensionality

depends on ', requires the more careful treatment.

Assume that H0 is an isolated point of the spectrum of the operator H0 in l2, so

that there exists a unique operator R such that

R� = 0; (H0 �H0)R(1 ��) = 1��:

We will denote this operator R as

(H0 �H0)�1(1��) = R;

the arguments 't�; 't of the operator � and functional H0 are omitted.To each operator A in l2 with the matrix AIJ we assign the operator in l

2 F'that transforms the vector with components gn;I(x1; :::; xn) into the vectorP1

J=1AIJgn;J (x1; :::; xn): This operator in l2 F' will be also denoted by the same

symbol,A.

Let us consider other terms of eq.(91). It is convenient to present gti as

gti = gtki + g

t?i ;

where

gtki = �g

ti;g

t?i = (1 ��)gti:

It follows from eq.(38) that H00g

tki = 0: Therefore, eq.(91) can be written as

(H0 �H0)gt?m +H01g

tm�1 + (H

02 � iDt)gtm�2 + :::+H

0Kg

tm�K = 0: (93)

The vector gt?m is determined in a unique fashion from this relation if and only if

�(H0

1gtm�1 + (H

0

2 � iDt)gtm�2 + :::+H0

Kgtm�K) = 0; (94)

since �(H0 �H0) = 0 (because H0 is a self-adjoint operator).When m = 1, eq.(94) implies that �H

01g

t0 = 0. It follows from eq.(80) that one

should require that

�

id

dt't(x)� �H0

�'�(x)

!� = 0;

31

i.e.

�

id

dt't(x)� �H0

�'�(x)

!� = 0: (95)

This equation is the analog of Hartree equation for the operator-valued case.

Furthermore, one can �nd gt?m�1 from eq.(93), substitute it to eq.(94), make use

of formula (95) implying that �H01g

tkm�1 = 0 and obtain the equation for g

tm�2. In

order to �nd its solution, one can �rst �nd gt?m�2 from eq.(93) and obtain the following

equation for gtkm�2 :

��H 01(H0 �H0)�1(1��)(H01(g

tkm�2 + g

t?m�2) + (H

02 �Dt)gtm�3 +H

03g

tm�3 + :::)

+�((H02 � iDt)(gtkm�2 + gt?m�2) + :::) = 0; (96)

where gt?m�2 are found from eq.(93). Therefore, one can look for the quantities gt?m

and gtkm by induction. Let gt?0 ;g

tk0 ; :::;g

t?m�3;g

tkm�3 be already found. Then one should

�nd gt?m�2 from eq.(93) and gtkm�2 from eq.(96). The equation for g

tkm�2 has the form

like�(iDt �H 02 +H

01(H0 �H0)�1(1 ��)H

01)g

tkm�2 = �

tkm�2 (97)

for some right-hand side �tkm�2 such that (1��)�tkm�2 = 0. Eq.(97) is analogous then

to eq.(88).Let us simplify our main equations, (95) and (97). As �(H0�H0) = (H0�H0)� =

0, one has

0 =�

�'�(x)(�(H0 �H0)�) = �

�(H0 �H0)�'�(x)

!�:

Therefore, eq.(95) takes the form of eq.(66), while

H01 =

Zdx

"a+(x)

�(H0 �H0)�'�(x)

!+ a�(x)

�(H0 �H0)�'(x)

!#

In order to simplify eq.(97), denote by � t 2 l2, jj� tjj = 1, the eigenvector ofH0('t�; 't).One has

gtkm�2 = �

t gtm�2; �tkm�2 = � t �tm�2: (98)Making use of the relations like

��(H0 �H0)�'(x)

= � ���'(x)

(H0 �H0);

2��

�'(y)

�(H0 �H0)�'(x)

� + ��2(H0 �H0)�'(x)�'(y)

� = 0;

��(H0�H0)�'(y)

(H0 �H0)�1(1��)�(H0 �H0)�'(x)

�

32

= � ���'(y)

(1��)�(H0�H0)�'(x)

� = � ���'(y)

�(H0 �H0)�'(x)

�

and commutation relations between operators a�(x), one obtains that eq.(97) takes

the following form:

[iDt + t �

Zdxdy[

1

2a+'t(x)

�2H0

�'�(x)�'�(y)a+'t(y)

+a+'t(x)�2H0

�'�(x)�'(y)a�'t(y) +

1

2a�'t(x)

�2H0

�'(x)�'(y)a�'t(y)]g

tm�2 = �

tm�2; (99)

where t is a number of the form:

t = i

� t;

d� t

dt

!+ (� t;

1

2

Zdxdy['(x)

�2H0

�'(x)�'(y)'(y)� '(x)�

2(H0 �H0)�'(x)�'�(y)

'�(y)]

�H1)� t)�Zdx

�� t

�'(x);�2(H0 �H0)�'�(x)

� t!: (100)

Notice that eq.(92) has been already solved in sections 6,7. Therefore, the functionsgtm are found.

Note that the leading asymptotics has the following form:

tN = ~cteiNS

t

KN't(�t �'t+

ut1;vt1

:::�'t+

utk;vtk

�Rt) +O(1=pN): (101)

It is remarkable that the functions 't; utk; vtk; R

t obeys the equations coinciding with

the equations obtained in sections 5-7 for the case of a single Schrodinger-like equation,not for a set. The only di�erence with the considered case is that the quantity (100)arises in the left-hand side of eq.(99), so that the phase factor ~ct di�ers from thefactor ct obtained in sections 5-7.This con�rms the heuristic arguments of section 4that predict the form of the asymptotics by making use of the equation for 't only.

Consider the terms of the additional factor t in more details. The �rst term wasstudied in ref.[14] for the case of semiclassical approximation (the Maslov canonicaloperator with real phase) for quantum mechanics. When one considers adiabaticperturbation theory, this term known as Berry phase [18] also arises.

Consider the last term of eq.(100). One can formally rewrite it as

1

2

Zdx(� t;

�2(H0 �H0)�'(x)�'�(x)

� t): (102)

We will study some examples for which the form (100) is correct, while the expression(102) contains divergences to be eliminated. Therefore, we will use eq.(100).

Discuss now the problem of chaos conservation for the operator-valued case. One

can study di�erent eigenvalues and eigenvectors of the operatorH0 and obtain asymp-

totic solutions being superpositions of the formulas like eq.(101):

1XJ=1

~ctJeiNStKN't

J(� tJ �'

t+

ut1;J

;vt1;J

:::�'t+

utkJ ;J

;vtkJ ;J

�RtJ) +O(1=

pN): (103)

33

One can consider the quantities analogous to the k-particle correlators (1)

Rtk;N (x1; :::; xk; y1; :::; yk) =1XJ=1

Zdxk+1:::dxN

tN;J(x1; :::; xk; xk+1; :::; xN)

t�N;J(y1; :::; yk; xk+1; :::; xN); (104)

where tN 2 l2 L2(XN). The correlation functions (104), as well as the correlators(1), allow us to predict the limits as N !1 of mean values of the observables of thespecial form (3). We can notice that for the element (103) of the space l2 L2(XN)such correlators (104) have limits

Rtk;N (x1; :::; xk; y1; :::; yk)!N!11XJ=1

�J'tJ(x1):::'

tJ(xk)'

t�J (y1):::'

t�J (yk);

where �J are some numbers. For example, one can choose the functions 'tI to be

coinciding at the initial time moment. Since the functions 'tI obey di�erent Hartree-

like equations corresponding to diferent eigenvalues of H0, they will not, in general,

coincide at time moment t. Therefore, the chaos property (2) being satis�ed at t = 0does not hold at arbitrary time moment. Therefore, even for the correlation functions,the chaos conservation hypothesis fails for the operator-valued case.

9 Operator-valued case: the theorem and some ex-

amples

1. Let H0('�; ') be non-degenerate eigenvalue of the operator H0('

�; '), 't be asolution to eq.(66), St have a form (38). Let gn satisfy recursive relations (93) andthe norm of the vectors H

0mgn;Dtgn be �nite. Consider the solution to the equation

id

dttN = HN

tN ;

tN 2 l2 L2(XN)

that satis�es the initial condition

0N = KN'0(g

00 +N

�1=2g01 + :::+N�L=2g0L): (105)

Theorem 5. The following relation is satis�ed:

jjtN � eiNst

KN't(gt0 +N

�1=2gt1 + :::+N�L=2gtL)jj = O(N�

L+12 ):

Remarks.1. The initial condition (105) for the Cauchy problem is not arbitrary. In partic-

ular, g00 should be an eigenvector of the operator H0. Moreover, the vectors g0?m can

be expressed through g0m�1; :::;g00 with the help of eq.(93). When the initial condition

34

does not satisfy these properties, one can present it as a superposition of the permiss-

able initial conditions corresponding to di�erent eigenvalues of H0. Therefore, one

should make use of the solutions to di�erent Hartree-like equations (66). The chaos

property (2) will not conserve then under time evolution.

2. When the initial condition (105) is allowable, one can solve the Cauchy problem

for the recursive relations by induction with the help of the technique analogous to

the previous section: one can �rst express gt?m through gtm�1; :::;g

t0, then one should

�nd the solution to the Cauchy problem for eq.(97) by using the substitution (98)

and reducing eq.(97) to eq.(99). One can notice that if the initial condition for gtiis expressed as a result of action of a polynomial in creation operators a+' (x) to the

vector �R, then the function �tm�2 is also expressed in such a way:

�tl =Xn

1pn!

Zdx1:::dxn�

tl;n(x1; :::; xn)a

+'t(x1):::a

+'t(xn)�Rt:

The vector function gtm�2 has then the form

gtl = exp(iZ t0�tdt)

Xn

1pn!

Zdx1:::dxng

tl;n(x1; :::; xn)a

+'t(x1):::a

+'t(xn)�Rt;

where l = m� 2,�t = t � 1

2

Zdxdy

�2H0

�'(x)�'(y)M t(x; y); (106)

t has the form (100), while the set of functions gtl;n(x1; :::; xn) is a solution to theCauchy problem for eq.(87).

3. The proof of theorem 5 is analogous to the proofs of theorems 3,4. Nevertheless,one should take into account that the function

KN't(gt0 +N

�1=2gt1 + :::+N�L=2gtL):

approximately satis�es eq.(64) accurate to O(N�L�12 ), not to O(N�

L+12 ). Therefore,

one should substitute to eq.(64) the asymptotic formula with two additional terms.

4. If the dimensionality of the eigenspace is a constant D more than 1, one shouldconsider the orthonormal basis � t! ; :::; �

tD in this eigenspace and present g

tkm�2 as

gtkm�2 =

DXI=1

� tI gt(I)m�2:

The equation for gt(I)m�2 has then also the form (99), but

t should be considered as a

matrix D �D of the form

tMN = i

� tM ;

d� tNdt

!+ (� tM ;

1

2

Zdxdy['(x)

�2H0

�'(x)�'(y)'(y)� '(x)�

2(H0 �H0)�'(x)�'�(y)

'�(y)]

35

�H1)� tN )�Zdx

�� tM�'(x)

;�(H0 �H0)�'�(x)

� tN

!:

2. We have considered the case of �nding asymptotic solutions to the in�nite set

of equations for functions I(x1; :::; xN); I = 1;1. The same method is applicablewhen one considers the set of d equations for d functions I(x1; :::; xN); I = 1; d,i.e

one studies the equation for the element of Rd L2(XN):Example 1. Let us consider the simple example. Consider the set of two

Schrodinger-like equations

i�h@

@t

tN;1(x1; :::; xN)

tN;2(x1; :::; xN)

!=

NXi=1

�h

B11(xi) B12(xi)

B21(xi) B22(xi)

! tN;1(x1; :::; xN)

tN;2(x1; :::; xN)

!

+

24 NXi=1

� �h

2

2m�i + U(xi)

!+

1

N

X1�i

One �nds the following asymptotics for the solution to eq.(107):

tN;I = eiNSt

++iR t0dt�t

+KN't+(�+I �

't++

ut1;+

;vt1;+

:::�'t++

utk+;+

;vtk+ ;+

�Rt+)+

eiNSt�+i

R t0dt�t�KN't�

(��I �'t�+

ut1;�;v

t1;�:::�

't�+

utk�;�

;vtk�;�

�Rt�) +O(1=pN);

where St� has the form (38), �t� has the form (106), '

t� obey eq. (108), (ui;�; vi;�)

obey the variation system (68), Rt� obey eq.(67),

��1 = aH12; ��2 = a(

H22 �H112

� �);

a�2 = 2�(H22 �H11

2� �):

Let us now consider the problem of divergences in eq. (102). When one formally

calculates the quantity (�; �2H0

�'�(x)�'(x)�); one obtains:

�(0)[1

2(B11(x) +B22(x))

� 12�

�(B11(x)�B22(x))H11 �H22

2+H21B12(x) +H12B21(x)

�]:

It happens that this in�nite quantity is equal to the divergent part of (�; �2H0

�'�(x)�'(x)�):

Therefore, the divergences can be eliminated in eq.(102), but they arise in calculation.In order to avoid arising of in�nite quantities, we have used eq.(100) instead of eq.(102).

Example 2. The developed approach can be also applied to the more interestingcase of eq.(8). To construct asymptotic solutions to eq.(8),one should �rst considerthe operator

H0 = H00 +

1

�h

� �h

2

2M�y + U(y) +

ZdzV(z; y)j'(z)j2

!

depending on ' and �nd the eigenvalues H(I)0 ('

�; ') and eigenfunctions �I ['�; '](y) of

this operator. Then one should make use of the solutions to the Hartree-like equations

(66), as well as to eqs.(67),(68). The asymptotic solution has then the form (103).The argumentation on the chaos non-conservation for the correlation functions which

has been presented in the end of section 8 is also valid for this example.

37

10 The problem of chaos conservation and

asymptotic formulas for abstract Hamiltonian

algebras

1. We have constructed asymptotic solutions to equations of the type (64) and to sets

of such equations. It was shown that the L2(XN) norm of the di�erence between exactand approximate solutions tended to zero as N !1. Since tN was interpretted asa multiparticle wave function, lemma 1 told us that its approximation eiNS

t

KN'tgt

constructed in sections 5,6 could be used instead of tN for �nding limits as N !1of mean values of general observables uniformly bounded with respect to N .

On the other hand, there are some other interesting cases. For example, one can

consider the case of classical statistical mechanics when the algebra of observables is

presented not as an operator algebra but as algebra of real-valued functions on the

phase space. The case of quantum statistical mechanics when states are speci�ed not

by wave functions but by density matrices can be also studied [13]. Examination of

these cases requires one to formulate analogs of lemma 1.In this section we are going to generalize lemma 1 for the case of abstract Hamilto-

nian algebras of observables (see, for example, [3,19]) which involves all the cases con-sidered earlier. Our results will imply, for example, the conclusion of refs.[11,13] thatasymptotic solutions to Liouville and Wigner equations which are found by the de-

veloped technique should be interpretted not as approximate densities but as approx-imations for N -particle half-densities which are equal to the square roots of densityfunctions (matrices) and also obey Liouville (Wigner) equations. We will introducethe notion of an abstract half-density which generalizes the notions of refs.[11.13].We will also consider the equation for it and �nd its asymptotic solutions.

These asymptotics are to be expressed through the solutions to eqs.(66),(68). Ifwe consider 't and 't� to be independent, the set of equation (66) and equationconjugated to it will form a Hamiltonian system playing an important role in con-structing tunnel asymptotics [13]. It happens that for the case under considerationin this section such Hamiltonian systems can be simpli�ed, so that the number ofequations can be cut in half.

2. De�nition 3. [3,19] A Hamiltonian algebra A is a set of complex linear spaceA and mappings � : A � A ! A, � : A � A ! A, j : A ! A denoted also as�(A;B) � AB, �(A;B) � fA;Bg, j(A) � A+; A;B 2 A, if the following axiomshold:

A1). for any A;B;C 2 A; �; � 2 Ca) A(�B + �C) = �AB + �AC; fA;�B + �Cg = �fA;Bg+ �fA;Cg;b) (A+)+ = A; (�A+ �B)+ = ��A+ + ��B+;c) (AB)+ = B+A+; fA;Bg+ = fA+; B+g;d) fA;BCg = fA;BgC +BfA;Cg;e) A(BC) = (AB)C;

38

f) fA;Bg = �fB;Ag; ffA;Bg; Cg+ ffB;Cg; Ag+ ffC;Ag; Bg = 0;A2). there exists such element I 2 A that AI = IA = A; fA; Ig = 0 for any

A 2 A;A3). for any A;B 2 A and some �h 2 R AB �BA = i�hfA;Bg.Remark. The case �h = 0 is also allowable, so that one cannot write fA;Bg =

1i�h(AB �BA).By L we denote the complex linear space of all linear functionals � : A ! C.

Introduce also the notation

L+ = f� 2 Lj�(I) = 1; �(A+) = (�(A))�; �(A+A) > 08A 2 Lg:

Suppose that L+ 6= ;.By AF � A we denote the set of such elements A 2 A that sup�2L+ j�(A)j

are usually called non-negative. The property �(A+A) � 0 for elements of L+ meansthat average values of non-negative observables are also non-negative. The functional

p(A) has the following physical meaning: it is the largest possible average value of the

observable A. The role of the norm jj�jj is the following: if the quantity jj�1 � �2jj issmall, the di�erence between average values �1(A)� �2(A) of any abservable A suchthat p(A) = 1 is also small.

Let us give now some examples of Hamiltonian algebras.

Example 1. Denote by Acn the algebra of smooth functions A(p1; q1; :::; pn; qn)on R2�n. De�ne the algebra operations as follows:

(AB)(X) = A(X)B(X); A+(X) = A�(X);X = (p1; q1; :::; pn; qn);

(109)

fA;Bg(p1; q1; :::; pn; qn) =nXi=1

@A

@qi

@B

@pi� @A@pi

@B

@qi

!(p1; q1; :::; pn; qn):

It is easy to see that the axiom A1 is satis�ed. The element I(X) � 1 playes the roleof I, so that the axiom A2 is also checked. The axiom A3 is satis�ed when �h = 0.Therefore, the algebra Acn is Hamiltonian. Note that it is the algebra of observablesfor classical satistical mechancs.

Lemma 17. The functional p(A) is the following in the case of example 1: p(A) =supX2R2�njA(X)j:

Proof. Let jA(X)j < C. Then 2C � A(X)ei� � A+(X)e�i� = B+B for someobservable B. Therefore, for any � 2 L+ one has 0 < �(CI) � Re(ei��(A)),i.e.j�(A)j < C. Thus, p(A) � supX2R2�n jA(X)j: Consider now the element � 2 L+ ofthe form �X0(A) = A(X0). We see that p(A) � j�X0(A)j = jA(X0)j. We prove lemma17.

Let us give the examples of elements of LF .A) To any function � from L1(R2�n) one can assign te element � 2 LF of the form

�(A) =ZdX�(X)A(X);X 2 R2�n: (110)

One has: jj�jj = R dXj�(X)j.B) The following element of LF : �X0(A) = A(X0) can be also formally written

in the form (110), but �(X) is a generalized function �(X) = �(X � X0). One hasjj�jj = 1.

Note that the function � playes the role of a probability distribution.

Example 2. Let Aqn be algebra of bounded operators in L2(Rn). Let

�(A;B) = AB;�(A;B) =1

i�h(AB �BA); j(A) = A+; I = E:

Analogously to the previous example, one can check axioms A1-A3. The functional

p(A) is equal to the ordinary operator norm:

p(A) = jjAjj = supjj'jj=1

(';A'):

40

Checking this property is analogous to the proof of lemma 17: one can present op-

erators cE �Aei� �A+e�i� as B+B. Note also that elements of LF play the role ofdensity matrices.

3. Let us now generalize the notion of half-density to the case of an abstract

Hamiltonian algebra.

De�nition 4. A half-density representation of a Hamiltonian algebra A is a setof a Hilbert space H and mappings � : AF �H ! H; � : AF �H ! H; denoted alsoas

�(A;') � A' � �A';�(A;') � fA;'g � �A';A 2 AF ; ' 2 H;if the following axioms hold:

H1). the mappings �A and �A are linear operators in H which are de�ned on acommon domain D � H;

H2).�I = E;�I = 0;

H3). the following relations are satis�ed for any A;B 2 AF , ';� 2 D:a) ��A+�B = ��A + ��B;��A+�B = ��A + ��B;

b) fA;B'g = fA;Bg'+BfA;'g;c) �AB = �A�B;�fA;Bg = �A�B ��B�A;d) (';�A�) = (�A+';�):(';�A�) = �(�A+';�):Elements of H are called half-densities.To any element of ' 2 D one can assign the following element of LF :

�'(A) = (';�A'): (111)

It is important that the norm jj�'1 � �'2 jj is small when jj'1 � '2jj is small.Lemma 18. The following relation is satis�ed:

jj�'1 � �'2 jj � jj'1 � '2jj(jj'1jj+ jj'2jj) (112)

Proof. One has

jj�'1 � �'2jj = supp(A)=1

j('1;�A'1)� ('2;�A'2)j (113)

Notice that �' 2 L+ when jj'jj = 1. therefore, j�'(�A)j = j(';�A')j � 1 in thiscase. Thus, jj�Ajj � 1 when p(A) = 1. Eq. (113) implies then eq.(112). Lemma 18 isproved.

Let � 2 LF . Denote by fH; �g the following functionalfH; �g(A) = �(fA;Hg):

De�nition 5. Let H 2 A;H = H+. An abstract Liouville equation is thefollowing equation for �t 2 LF ; t 2 R:

d�t

dt= fH; �tg: (114)

41

An abstract half-density equation for 't 2 H isd't

dt= fH;'tg: (115)

Lemma 19. Let 't obey eq.(115). Then �'t obey eq.(114).

Proof. Let A 2 A. One hasd

dt�'t(A) = ('

t;�A�H't) + (�H't;�A't) =

= ('t; [�A;�H ]'t) = ('t;�fA;Hg't) = �'t(fA;Hg):Therefore, eq.(114) is satis�ed. Lemma 19 is proved.

Let us give examples of half-density representations.

Example 1. Consider the Hamiltonian algebra Acn. De�ne oprations AB ��AB; fA;Bg � �AB, where A 2 Acn; B 2 L2(R2�n) by eq. (109). Let D = S(R2�n).The axioms of the half-density representation are satis�ed. We can also notice that

the element �' 2 LF is equal to

�'(X) = j'(X)j2:

for example, if ' is real, � is presented as a square of the function '. Because of thisreason, the function ' has been called half-density function in ref.[11].

Example 2. Consider the Hamiltonian algebra Aqn. Let H be Hilbert spaceL2(X n). Determine operators �A;�A as

�A' = A';LA' =1

i�hA': (116)

It is not hard to check axioms H1-H3.The half-densities play the role of the wave

functions in this example. The density matrix �' is proportional to the projectionoperator on one-dimensional subspace.

Example 3. Consider the same Hamiltonian algebra Aqn. Choose the Hilbertspace H as the space of Hilbert-Schmidt operators on L2(X n) with the inner product(';�) = Tr'+�: Let �A;�A be the following:

�A' = A';LA' =1

i�h[A;']: (117)

The density matrix �' is �' = ''+; the matrix ' has been called half-density [13]

for reasons analogous to example 1.

We can notice that di�erent equations (Schrodinger and Wigner) are treated fromthe same point of view: they are abstract half-density equations for the cases ofexamples 2 and 3 correspondingly. Contrary to the half-density representations, the

Hamiltonian algebras are identical for examples 2 and 3.

42

4. For the simplicity, consider the case when elements A of the Hamiltonian

algebra A are presented as functions AY on a measure space Y, while the algebraoperations are as follows:

(AB)X =ZdY dZdXY ZAYBZ; fA;BgX =

ZdY dZfXY ZAYBZ;

(118)

(A+)X = A�X;X; Y; Z 2 Y; A;B 2 A;

f and d are (generalized) functions Y�Y�Y ! C. To simplify the notations, denotethe integrals like (118) as dXY ZAYBZ,fXY ZAYBZ, i.e. we integrate over repeated

indices.

Elements of LF are presented as (generalized) functions on Y, the quantity �(A)can be formally written as �(A) = �XAX.

A tensor product AN = A� :::�A is then presented as an algebra of functionsAY1:::YN : Y � :::� Y = YN ! C, Y1; :::; YN 2 Y. The algebra operations are de�nedas:

fA;BgX1:::XN =NXp=1

dX1Y1Z1:::dXp�1Yp�1Zp�1fXpYpZp

�dXp+1Zp+1Yp+1 :::dXNZNYNAY1:::YNBZ1:::ZN ;(AB)X1:::XN = dX1Y1Z1 :::dXNYNZNAY1:::YNBZ1:::ZN ; (A

+)X1:::XN = A�X1:::XN

;

A;B 2 AN ;Xi; Yi; Zi 2 Y:It is not hard to check axioms of a Hamiltonian algebra for AN . Note that the axiomA3 is important for such check.

Let be a polynomial functional : L ! R of the form

[�] =K0Xk=1

1

k!�X1:::�XkW

(k)X1:::Xk

(119)

for some observables W (k) 2 Ak.De�nition 6.[3] An �-uniformization of the functional (119) is a set of observables

H(N) 2 AN :

H(N)X1:::XN

=K0Xk=1

�k�1

k!

X1�j1 6=::: 6=jk�N

W(k)Xj1 :::Xjk

Yp6=ji

IXp: (120)

Consider now tensor products of half-density representations. Without loss ofgenerality, one can assume that the Hilbert space H is presented as L2(X ) for somemeasure space X . Elements of H will be denoted as 'x; x 2 X ; integrals like

Rdx'�x�x

43

will be denoted as '�x�x. Let (L2(X );�;�) be a half-density representation of the

Hamiltonian algebra A. The operations �;� are(A')x = bXxyAX'y; fA;'gx = cXxyAX'y:

for some (generalized) functions b; c : Y � X � X ! C. Consider the Hilbert spaceHN = H� :::�H = L2(XN ) and the following operations:

(A')x1:::xN = bX1x1y1 :::bXNxNyNAX1:::XN'y1:::yN ;

(121)

fA;'gx1:::xN =NXp=1

bX1x1y1 :::bXp�1xp�1yp�1cXpxpyp�Xp+1xp+1yp+1 :::�XNxNyN

�AX1:::XN'y1:::yN ;where �Xxy = bXxy � i�hcXxy;Xi 2 Y; xi; yi 2 X . One can check the axioms ofde�nition 4 for the operations (121). This half-density representation is called a

tensor product of N half-density representations (L2(X );�;�):Let N be �xed, � = 1=N , H have the form (120). Then the abstract half-density

equaton (115) takes the form

id

dt'N;tx1:::xN =

K0Xk=1

1

Nk�1

X1�l1

+1

2

Zdp1dp2dq1dq2V (q1; q2)�(p1; q1)�(p2; q2)

both for the classical and quantum cases. One should only take into account in

quantum case that �(p; q) is a symbol of the operator � 2 L in L2(R�) which isde�ned as

�(p; q) =1

(2��h)�

Zdy�K(x; y)e

i�hp(y�x);

where �K is a kernel of the operator �.

3.For the case of the Hamiltonian (120), eq.(114) has been considered in [3]. It

was shown that the property of the correlation functions

R(N;k;t)X1:::Xk = �N;tX1:::XN

IXk+1:::IXN (123)

to be approximately equal to the products of one-particle correlators

R(N;k;t)X1:::Xk !N!1 �tX1 :::�tXk (124)conserves under time evolution, while �X obeys the abstract Vlasov equation

d�tXdt

= fY XZ@

@�Z(�t)�tY : (125)

This justi�es the chaos conservation hypothesis.

5. One can notice that eq.(122) for the half-density is of the type (64), if HN =NHN0 ,H

N0 has the form (63) and

H0('�; ') =

K0Xk=1

1

k!

kXq=1

W(k)S1:::Sk

bS1x1y1 :::bSq�1xq�1yq�1icSqxqyq�Sq+1xq+1yq+1 ::�SNxNyN

�'�x1:::'�xk'y1 :::'yk: (126)Therefore, the technique developed in sections 5-7 can be applied to eq.(122). Onecan construct such function 'N;tas 2 L2(XN) that

('N;tas � 'N;t; 'N;tas � 'N;t)!N!1 0; (127)where 'N;t is a solution to the Cauchy problem for eq.(122). Lemma 18 tells usthat estimation (127) means that one can use approximate half-densities in order to�nd limits as N ! 1 of average values of the observables uniformly bounded withrespect to N . In particular, one can con�rm the chaos conservation hypothesis for

the correlation functions of �nite orders and deny it for the N -particle densities in a

way analogous to section 3.It is interesting that eq.(124) is the analog of the Ehrenfest theorem. Namely, one

can introduce creation and annihilation operators a� and apply lemma 10. Eq.(120)

takes the following form:

i

N

d

dt'N;t = H0(a

+=pN; a�=

pN)'N;t; (128)

45

where 't is such element of F that only N -th component of it di�ers from zero, while

H0(a+=pN; a�=

pN) =

K0Xk=1

1

Nkk!

kXq=1

W(k)S1:::Sk

�bS1x1y1 :::bSq�1xq�1yq�1icSqxqyq�Sq+1xq+1yq+1 ::�SNxNyN a+x1:::a+xka�y1:::a�yk: (129)The correlation functions R(N;k;t)X1:::Xk can be presented through the average values of thefunction of operators a�=

pN :

R(N;k;t)X1:::Xk =(N � k)!N !

('N;t; bX1x1y1 :::bXkxkyka+x1:::a+xka

�y1:::a�yk'

N;t):

As the commutator between operators a�=pN tends to zero as N !1, one obtains

eq.(125) from Heisenberg equations. Thus, one con�rms the chaos conservation for

correlation functions.

Let us present the theorem for the N -particle density which implies the results of

[11,13] for multiparticle Liouville and Wigner equations. Let H have the form (120),

� = 1=N . Consider the solution to eq.(114) that satis�es the initial condition

�N;0X1:::XN

= bX1x1y1:::bXNxNyN (KN'0�R0)

�x1:::xN

(KN'0�R0)y1:::yN

Let 't be a solution to the Cauchy problem for eq.(66), Rt be such a solution toeq.(67) that obeys the initial condition R0. Denote

ct = exp

� i2

Z t0d�

�2H0

�'x�'yR�xy

!;

�N;t;asX1:::XN = jctj2bX1x1y1 :::bXNxNyN (KN't�Rt)�x1:::xN (KN't�Rt)y1:::yNTheorem 6. The following relation takes place:

jj�N;t;asN � �N;tN jj !N!1 0:

This theorem is a corollary of theorem 3, lemma 18 and reality of St (eq.(38)).Consider now eq.(66) for our case in more details. Note that it can be treated as

a Hamiltonian system if one introduces the following Poisson brackets:

f'x; 'yg = f'�x; '�yg = 0; f'x; '�yg = �i:

One can present the functional H0 as

H0 =1

�h[[�]� [�� �h�]]; �h 6= 0;

(130)

46

H0 =@

@�X�x; �h = 0;

where

�X = icXxy'�x'

�y; �X = bXxy'

�x'

�y: (131)

The Poisson brackets between �; � are the following:

f�X ; �Y g = �hfZXY �Z; f�X ; �Y g = fZXY �Z; f�X; �Y g = fZXY �Z:

Therefore, the Hamiltonian system in terms of �; � is divided in quantum case into

two independent parts: as f�X; �Y � �h�Y g = 0, the equations for � and � � �h� areindependent. In classical case the Hamiltnian system consists of two equations: the

equation for � coincides with eq.(122), another equation for � is linear.

11 Conclusions

We have developed a new asymptotic method that allows us to �nd approximationsfor functions of a large number N of arguments as N !1; as well as the correctionsto the leading order of the asymptotic formula. We have seen that this technique isapplicable to eq.(64) being of a general form, as well as to the set (�nite or in�nite) ofsuch equations. Multiparticle Schrodinger, Liouville and Wigner equations are partialcases of eq.(64).

We have noticed that for the case of Schrodinger equation our approximate wave

function can be used instead of the exact wave function for �nding limits of mean val-ues of general observables uniformly bounded with respect to N . We have consideredthe case of a general Hamiltonian algebra, justi�ed the chaos conservation hypothesisfor the correlators and denied it for the N -particle densities. It is interesting that forthe operator-valued case the chaos does not conserve even for the correlators.

Two methods have been used for constructing such asymptotics. One of them isheuristic and allows us to construct the asymptotic formula up to a multiplicativefactor cteiNC

t

, where constant C t does not depend on the solution to the Hartree-like