An inverse problem in quantum field theory and canonical correlation ...

J. Math. Soc. Japan Vol. 51, No. 2, 1999

An inverse problem in quantum field theory and canonical correlation functions An application of a solvable model called the rotating wave approximation. By Masao Hirokawa (Received June 9, 1995) (Revised June 12, 1997)

Abstract. In this paper, we treat a quantum harmonic oscillator in thermal equilibrium with any systems in certain classes of bosons with infinitely many degrees of freedom. We describe the following results: (i) when a canonical correlation function is given, we so reconstruct a Hamiltonian by the rotating wave approximation from it that the Hamiltonian restores it. Namely, we solve an inverse problem in the quantum field theory at finite temperature in a finite volume. (ii) Taking an infinite volume limit for the result in (i), we consider long-time behavior of the canonical correlation function in the finite volume limit.

1. Introduction In this paper, we shall treat long-time behavior of the canonical correlation function in an infinite volume limit. For that purpose, we shall apply Arai’s results [5] concerning long-time behavior of two-point functions to a class of canonical correlation functions of position and momentum operators in the infinite volume limit. In [5], Arai argued long-time behavior of two-point functions of position operators for some models of a quantum harmonic oscillator interacting with bosons. We consider a quantum harmonic oscillator in thermal equilibrium with any system in certain classes of bosons with infinitely many degrees of freedom in a finite volume V > 0. Our models describe photons in a laser interacting with oscillation caused by a heat bath, which can be observed when the laser passes in the heat bath as well as photons in a laser interacting with oscillation caused by phonons on the surface of a material, which can be observed when we irradiate the weak laser on the surface. When a two-point function (or canonical correlation function) R V ðt1 ; t2 Þ of the position operator or momentum operator of the harmonic oscillator is given by an observation at a temperature, we take the infinite volume limit, V ! y, for R V ðt1 ; t2 Þ, V and get Ry b ðt1 ; t2 Þ 1 limV !y R ðt1 ; t2 Þ at the inverse temperature, b, under suitable conditions. And we argue long-time behavior of Ry b ðt1 ; t2 Þ. For a positive parameter V > 0, we set G V 1 2pZ=V :

1991 Mathematics Subject Classification. Primary 82B10; Secondary 46N50, 47D40, 47N50, 47N55, 82C22. Key Words and Phrases. Inverse problem, quantum field theory, finite temperature, infinite volume limit, rotating wave approximation.

338

ð1:1Þ

M. Hirokawa

GV

  2pZ 2pn 1 k k ¼ ; n ¼ 0; G1; G2;


: ¼ V V

def

The Hilbert space of state vectors of our system in the finite volume case is taken to be the symmetric Fock space F over C l l 2 ðG V Þ. We denote by a, a  the annihilation and creation operators of the quantum harmonic oscillator, respectively, and by bk , bk ðk A G V Þ those of bosons, acting in F. We assume that there exists a self-adjoint operator H on F, called a Hamiltonian, which governs the time development of the system, such that eÿtH is trace class for all t 2 ð0; bŠ and ðb 1 def V dl trðeÿð bÿlÞH e iHt1 BeÿiHt1 eÿlH e iHt2 BeÿiHt2 Þ; R ðt1 ; t2 Þ ¼ b trðeÿbH Þ 0 pffiffiffi  Þ= 2 or the momentum opwhere B denotes either the position operator q ¼ ða þ a pffiffiffi  erator p ¼ iða ÿ aÞ= 2 of the quantum harmonic oscillator and tr denotes trace on F (we use units where p (the Planck constant divided by 2p) ¼ 1). However, we do not specify the concrete form of H. The Hamiltonian H may be a complicated function of a, a  , bk and bk ðk A G V Þ. To apply Arai’s result [5, Theorem 1.3] to our case, we first solve the following inverse problem: In terms of R V ðt1 ; t2 Þ only, determine positive frequencies x 0 , xk (k A G V ) of the quantum harmonic oscillator and scalar bosons, respectively, and coupling constants yk A C ðk A G V Þ, appearing in the Hamiltonian of the rotating wave approximation (RWA), X X def ð1:2Þ xk bk bk þ ð yk a  bk þ yk bk aÞ; HRWA ðx; yÞ ¼ x 0 a  a þ k A GV

def

x ¼ x 0 l ðxk Þk A G V ;

k A GV

def

y ¼ 0 l ð yk Þ k A G V ;

(where c means the complex conjugate of c A C) such that the Hamiltonian HRWA ðx; yÞ recovers R V ðt1 ; t2 Þ in the following sense: fenergy levels of HRWA ðx; yÞgnf0g ðwhere ‘energy level’ is the notion in the quantum theory of many-particle systems ½32; x2-1-aŠÞ ¼ fspectra ðenergy levelsÞ of the one particle Hamiltonian hRWA of HRWA ðx; yÞ more than its lowest oneg ðwhere this ‘energy level’ is the notion in the quantum mechanics ½33; x1-3ŠÞ   ðy itz V dte R ð0; tÞ ¼ positive poles of the meromorphic function 0

with hRWA a self-adjoint operator on C l l 2 ðG V Þ such that HRWA ¼ dGðhRWA Þ, the second quantization of hRWA (see §2.2), and ð1:3Þ

R V ðt1 ; t2 Þ ¼ a representation in terms of W V ðt1 ; t2 Þ;

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339

with def

W V ðt1 ; t2 Þ ¼ ðW0 ; e iHRWA ðx; yÞt1 BeÿiHRWA ðx;yÞt1 e iHRWA ðx; yÞt2 BeÿiHRWA ðx; yÞt2 W0 ÞF ; the two-point vacuum expectation value (two-point function) of B, with the Hamiltonian HRWA ðx; yÞ, where W0 is the Fock vacuum in F (in fact, it is the vacuum (ground state) of HRWA ðx; yÞ), and ð ; ÞF is the inner product of F. There are some negative criticisms in a physical sense against RWA [13, §V.D], claiming that a model, called the independent-oscillator model, is more suitable and useful in physics than RWA model [12, 13, 22]. But, in this paper, we use RWA model, since we find that it is suitable for our purpose of investigating the long-time behavior of R V ðt1 ; t2 Þ in the infinite volume limit. The reason why we employ RWA is nothing but easy to argue the infinite volume limit for the Hamiltonian of RWA in a mathematically rigorous way, and mathematical theories on RWA model are established in [4, 5, 19]. If the inverse problem stated above is solved, then we can investigate the long-time behavior of the infinite volume limit of R V ðt1 ; t2 Þ through the infinite volume limit W y ðt1 ; t2 Þ of W V ðt1 ; t2 Þ with representation (1.3). On the other hand, the longtime behavior of the latter function is investigated in detail by Arai [5]. An answer to the inverse problem above is given by Theorem 3.1 in this paper. By V representation (1.3), we can consider the infinite volume limit Ry b ðt1 ; t2 Þ of R ðt1 ; t2 Þ, through the right hand side of (1.3). Then, we have a representation of Ry b ðt1 ; t2 Þ by y using W ðt1 ; t2 Þ at (4.50) in §4.2. And, applying Arai’s results in [5] to the reprey sentation, we consider the long-time behavior of Ry b ðtÞ 1 Rb ð0; tÞ for B in Theorem 3.2 of this paper. The present paper is organized as follows. In §2 we review some basic facts on the Liouville space and RWA. In §3 we state our main results. In §4 we give proofs for the main results. In the last section, as two appendices, we review Mori’s memory kernel equation, and find an example. The author would like to express his thanks to Professor H. Ezawa of Gakushuin University for valuable suggestion and comments, and he wishes to express his gratitude to Dr. M. Ban, Dr. S. Ogawa, and Dr. T. Miyake of Advanced Research Laboratory, Hitachi Ltd., for helpful suggestions on the experimental side. He also acknowledges helpful discussions on the inverse problem with participants in the international workshop of ‘‘Constructive Results in Field Theory, Statistical Mechanics and Condensed Matter Physics,’’ satellite colloquium of ICMP-Paris, at Ecole Polytechnique on 27th of July ’94. He is grateful to the referee for his many useful and instructive advice. Research on §4.2 is supported by the Grant-in-Aid No.09740092 for Encouragement of Young Scientists and No. 09640014 for Scientific Research (C). 2. Preliminaries 2.1. Liouville space def We give a complex Hilbert space l 2 ðG V Þ by l 2 ðG V Þ ¼ fðck Þk A G V j ck A C; k A G V ; P 2 0 2 k A G V jck j < yg. For each f A C l l ðG V Þ, we denote f by f l ð fk Þk A G V , where

340

M. Hirokawa

f 0 A C and ð fk Þk A G V A l 2 ðG V Þ. An inner product ð ; ÞCll 2 of C l l 2 ðG V Þ is given by P def ð f ; gÞCll 2 ¼ f 0 g 0 þ k A G V fk gk ð f ; g A C l l 2 ðG V ÞÞ. We denote the symmetric Fock space over C l l 2 ðG V Þ by def

y

F ¼ 0 Sn ðC l l 2 ðG V ÞÞ n ;

ð2:1Þ

n¼0 n

where Sn ðC l l 2 ðG V ÞÞ is the n-fold symmetric tensor product of C l l 2 ðG V Þ for each def n A N and S0 ðC l l 2 ðG V ÞÞ 0 ¼ C (see [29, p. 53, Example 2]). We denote by að f Þ ð f A C l l 2 ðG V ÞÞ the (smeared) annihilation operator on F and set a ¼ að1 l 0Þ, bk ¼ að0 l ek Þ ðk A G V Þ with ek A l 2 ðG V Þ such that the k 0 -th component of ek is given by ðek Þk 0 ¼ dkk 0 (the Kronecker delta) for k; k 0 A G V . The operators a and a  (the adjoint of a) physically denote the annihilation and creation operators of the quantum harmonic oscillator, respectively, and bk , bk ðk A G V Þ those of free bosons. We consider a quantum harmonic oscillator in thermal equilibrium at an inverse temperature b > 0 with a system of bosons with infinitely many degrees of freedom in a finite volume. We take the Hilbert space of state vectors of the system to be F. Let H be a strictly positive self-adjoint operator on F satisfying the condition ðHÞ eÿtH is a trace class operator on F for every t A ð0; bŠ: The operator H physically denotes the Hamiltonian of the quantum system under consideration. Condition ðHÞ implies that the spectrum of H consists of only eidef genvalues, each having a finite multiplicity. We set N  ¼ f0; 1; 2; . . .g. We denote the eigenvalues of H by ln ðn A N  Þ with order 0 < l0 U l1 U


U ln U lnþ1 U . . . ; ln % y as n % y, counting multiplicities. We can take a complete orthonormal system fjn j n A N  g of F in such a way that each jn is a normalized eigenvector of H with eigenvalue ln : Hjn ¼ ln jn . For the Hamiltonian H, we can construct a space Xc ðHÞ consisting of suitable operators on F [18, 19, 20], called a Liouville space. We denote the linear hull of fjn jn A N  g by D, i.e., ð2:2Þ

def

D ¼ L:h: ½fjn j n A N  gŠ:

From here on, we denote the linear hull of a set S by L:h: ½SŠ. Obviously D is dense in F. Further, we denote by BðD; FÞ the space of bounded linear operators from D to F. Every element A in BðD; FÞ has a unique extension to an element in BðFÞ, the space of bounded linear operators on F. We denote the extension of A by Aÿ , and A  d D (the restriction of A  to D) by Aþ . We first define a class TðHÞ of linear operators on F. For a linear operator A on F, we denote its domain by DðAÞ. We say that a linear operator A on F is in the class TðHÞ if the following conditions (T.1) and (T.2) are satisfied: (T.1) DðAÞ ¼ D and DðA  Þ I D. When DðAÞ I D, we regard A as A d D and denote A d D by A only.

An inverse problem in quantum field theory

341

(T.2) For all t in ð0; bŠ, eÿtH A and AeÿtH are in BðD; FÞ with ðeÿtH AÞÿ and ðAeÿtH Þÿ being Hilbert-Schmidt operators on F. It is easy to see that TðHÞ is a complex vector space with the natural operations of addition and scalar multiplication. We also remark that TðHÞ may contain not only bounded operators but also unbounded ones. We can introduce an inner product in TðHÞ, called the Bogoliubov (Kubo-Mori) scalar product, by ðb 1 def ð2:3Þ dl trððeÿð bÿlÞH A  Þÿ ðeÿlH BÞÿ Þ hA; BiH ¼ bZðbÞ 0 for A; B A TðHÞ with def

ZðbÞ ¼ trðeÿbH Þ

ð2:4Þ

(see [19, Lemma 3.2]). We denote by Xc ðHÞ the completion of TðHÞ in the norm k
kH of TðHÞ induced by the inner product h
;
iH . The Hilbert space Xc ðHÞ has a structure of partial algebra ([19, Proposition 3.14]). We also note here that an element in Xc ðHÞ is not always an operator acting in F. It is noteworthy that Naudts et al. attempted to argue in general about linear response theory on the Hilbert space which is constructed by a completion of a von Neumann algebra with KMS-state [27]. Similarly we can formulate Mori’s theory in statistical physics in terms of Xc ðHÞ. We define an operator L acting in Xc ðHÞ, called the Liouville operator in physics, by ð2:5Þ

def

DðLÞ ¼ fA A TðHÞ j HA; AH d D A TðHÞg; def

ð2:6Þ

LA ¼ ½H; AŠ ¼ HA ÿ AH;

A A DðLÞ:

It is shown that L is essentially self-adjoint [19, Lemma 3.8]. We denote the closure of L by the same symbol. The spectral properties of L can exactly be known as is shown below. Let Fm; n : D ! D be an operator defined by def

ð2:7Þ ð2:8Þ

DðFm; n Þ ¼ D; def

1=2 Fm; n x ¼ b 1=2 ZðbÞ 1=2 Wm; n ðjn ; xÞF jm ;

x A D; m; n A N  ;

where ð2:9Þ Note that

def

Wm; n ¼

8 > < > :

ln ÿ lm eÿblm ÿ eÿbln

if lm 0 ln ,

bÿ1 e blm

if lm ¼ ln .

ð2:10Þ

Wm; n > 0;

ð2:11Þ

Wm; n ¼ Wn; m ;

m; n A N  ; m; n A N  :

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M. Hirokawa

It can be shown that fFm; n gm; n A N  is a complete orthonormal system of Xc ðHÞ with ð2:12Þ

LFm; n ¼ ðlm ÿ ln ÞFm; n ;

ð2:13Þ

Fþ m; n ¼ Fn; m ;

m; n A N  ;

m; n A N  ;

(see [20, Proposition 3.9]). Moreover, we can prove ð2:14Þ ð2:15Þ

sðLÞ ¼ flm ÿ ln j m; n A N  g; sess ðLÞ ¼ fe A sðLÞ j e is an eigenvalue of L of infinite multiplicityg;

where sðSÞ (resp. sess ðSÞ) denotes the (resp. essential) spectrum of an operator S. Remark. One can easily show that the set of possible limit points of sðLÞ is f0g. The referee remarked this fact to the author. The author is thankful to the referee for it. We note here that 0 is an eigenvalue of L of infinite multiplicity. For every A A Xc ðHÞ and t A R, we define def

ð2:16Þ

AðtÞ ¼ e iLt A;

ð2:17Þ

RA ðt1 ; t2 Þ ¼ hAðt1 Þ; Aðt2 ÞiH ;

def

t1 ; t2 A R:

Remark. The vector AðtÞ can be regarded as a generalization of the Heisenberg operator for A with the Hamiltonian H [19, Proposition 3.13]. With an interpretation that A is an observable in the quantum system under consideration, the function RA ðt1 ; t2 Þ physically means a two-point correlation function of the Heisenberg operator AðtÞ of A at inverse temperature b. Putting ð2:18Þ

RA ðtÞ ¼ RA ð0; tÞ;

we have ð2:19Þ

RA ðt1 ; t2 Þ ¼ RA ðt2 ÿ t1 Þ:

Thus, as for the correlation function RA ðt1 ; t2 Þ, we need only to consider RA ðtÞ. By (2.12) and (2.14), we have ð2:20Þ

RA ðtÞ ¼

y X

Am; n e itðlm ÿln Þ

m; n¼0

with ð2:21Þ

Am; n ¼ jhFm; n ; AiH j 2 V 0:

We can enumerate elements of the set ð2:22Þ

def

fep gy p¼ÿy ¼ sðLÞ V ð0; yÞ

with order 0 <


< eÿðpþ1Þ < eÿp <


< eÿ1 < e0 < e1 < e2 <


< ep < epþ1 <


;

An inverse problem in quantum field theory

343

ep ! y as p ! y. It is easy to see that sðLÞ is reflection symmetric with respect to the origin, so that we have def

fÿep gy p¼ÿy ¼ sðLÞ V ðÿy; 0Þ:

ð2:23Þ Introducing

X

¼ AðGÞ p

ð2:24Þ

Am; n ;

p A Z;

m; n; lm ÿln ¼Gep

X

RA0 ¼

ð2:25Þ

Am; n ;

m; n; lm ÿln ¼0

we can write 0 þ RA ðtÞ ¼ Rÿ A ðtÞ þ RA ðtÞ þ RA ;

ð2:26Þ with

RG A ðtÞ

¼

y X

Gitep AðGÞ : p e

p¼ÿy

Here we set the following condition: ðGÞ (A.0) There exists p0 A N such that Aÿp ¼ 0 for each p with ÿp < ÿ p0 . Let ð2:27Þ

Gþ V 1 fk A G V j k V 0g;

Gÿ V 1 fk A G V j k U 0g:

þ Under assumption (A.0), we reorder fe p gy p¼ÿ p0 as ek 1 ep for k A G V and p A fÿ p0 ; ÿp0 þ 1; ÿp0 þ 2;


g with k ¼ 2pð p þ p0 Þ=V. We note here that for k A G ÿ V, þ ek ¼ eÿjkj ¼ ÿejkj ðÿk ¼ jkj A G V Þ by the reflective symmetry above. Thus, under (A.0), we have X ðGÞ ðtÞ ¼ ð2:28Þ Ak eGitek : RG A k A Gþ V

~ by We define a self-adjoint operator L ~ def L ¼ PLP;

ð2:29Þ

where P is the orthogonal projection onto L:h: ½fFm; m jm A N  g U fFm; n jm; n A N  with lm ÿ ln ¼ 0; Gek ; k A G þ V gŠ: Then, under (A.0), we have ð2:30Þ

~

AðtÞ ¼ e iLt A;

t A R:

~ Thus, as far as the operator A satisfying (A.0) is concerned, it is su‰cient to consider L instead of L. Since we are concerned with only A satisfying (A.0) in what follows, henceforth ð2:31Þ

~ by L under ðA:0Þ: we denote L

344

M. Hirokawa

For a bounded measurable function f on ½0; yÞ, we can define the Fourier-Laplace transform ½ f ŠðzÞ ðIz > 0Þ by ðy def ð2:32Þ dt e itz f ðtÞ; Iz > 0: ½ f ŠðzÞ ¼ 0

Note that ð2:33Þ

RA ð0Þ ¼

y X

2 Am; n ¼ kAkH < y;

m; n¼0

and hence 2 jRA ðtÞj; jRG A ðtÞj U kAkH ;

ð2:34Þ

t A R:

By this fact, ½RA Š and ½RG A Š exist. For all M; N A N and t > 0, we have X M X N 2 ÿt I z itðlm ÿln Þ itz Am; n e e U kAkH e : m¼0 n¼0

Hence, by the Lebesgue dominated convergence theorem, under (A.0) we obtain þ ½RA ŠðzÞ ¼ ½Rÿ A ŠðzÞ þ ½RA ŠðzÞ þ

ð2:35Þ

iRA0 z

with ½RG A ŠðzÞ

ð2:36Þ

X AðGÞ k ¼i ; þ z G ek

Iz > 0:

k A GV

The function on the right hand side of (2.35) is obviously meromorphic on C with possible poles f0; Gek j k A G þ V g. Thus, ½RA ŠðzÞ can be extended uniquely to a meromorphic function on C. We denote the extension of ½RA ŠðzÞ by the same symbol. It follows from (2.35) and (2.36) that ð2:37Þ

ðGÞ

Ak

1 ðz G ek Þ½RA ŠðzÞ; z!Hek i

¼ lim

k A Gþ V;

1 RA0 ¼ lim z½RA ŠðzÞ: z!0 i

ð2:38Þ

Lemma 2.1. Assume (A.0). Then the function RA ðtÞ is twice continuously di¤erentiable if and only if  X 1 lim ðz ÿ ek Þ½RA ŠðzÞ ek2 < y z!e k i þ k A GV

and X

k A Gþ V

 1 lim ðz þ ek Þ½RA ŠðzÞ ek2 < y: z!ÿek i

An inverse problem in quantum field theory

345

Proof. This lemma follows from the following well-known lemma and (2.37). r Ð

Lemma 2.2. Let m be a finite measure on a measure space X. Let CðtÞ ¼ X e dmðxÞ, t A R. Then CðtÞ is twice continuously di¤erentiable if and only if ð x 2 dmðxÞ < y: itx

X

Here we note the following lemma: Lemma 2.3. Assume (A.0). If A A DðLÞ, then RA ðtÞ is twice continuously di¤erentiable. Proof. Let A A DðLÞ. Then we have 0