数理解析研究所講究録
923 巻 1995 年 1-15

1TRACE FORMULAS IN BOSON FOCK SPACES AND APPLICATIONSASAO ARAI$’\backslash$Department of Mathematics, Hokkaido University, Sapporo 060, Japan
ABSTRACT. Trace formulas for the heat semi-groups of second quantization operators and theirperturbations in the abstract Boson Fock space are given in terms of path (functional) integral
representations. As applications, an inequality of Golden-Thompson type and a classical limit
are derived for the trace of the heat semi-group of a perturbed second quantization operator.
The abstract results are applied to a model of-type in quantum field theory.
$P(\emptyset)$

I. Introduction
In the previous work $[1,2]$ , we introduced (infinite dimensional) Dirac-type operators
acting in the abstract Boson-Fermion Fock space, which, in concrete realizations, describe
supercharges of some models of supersymmetric quantum field theory, and studied theirproperties (see also $[3]-[9]$ for further developments and related aspects). In particular, wederived a formula for their index in terms of a path (functional) integral. From a technicalview-point, however, the conditions assumed in $[1,2]$ to derive the index formula are not
optimal. It is desirable to formulate more optimal conditions in this respect. The present
paper concerns this problem.
In deriving the index formula, trace formulas for the heat semi-groups of second quantization operators and their perturbations in both the Boson and Fermion Fock spaces play
important roles. Hence, one of the basic tasks to refine the previous result on the index
formula should be to make more elaborate analysis on the trace formulas just mentioned.
A first step of work in this direction has been taken forward in [11], where some technically improved (possibly most general) results on trace formulas in the abstract Boson Fockspace were obtained. As applications of the trace formulas, one can derive an inequalityof Golden-Thompson type and a classical limit for the trace of the heat semi-group of a
perturbed second quantization operator [11]. We also apply the abstract results to a model
of $P(\phi)$ -type in QFT. This kind of results is well known in the case of Schr\"odinger operators
in finite dimensions (e.g., [24, \S 9, \S 10] and references therein), but, the corresponding results in the case of Schr\"odinger-type operators in infinite dimensions, including the case of
perturbed second quantization operators in the abstract Boson Fock space, seems to be lacking in the literature (see, however, [14], a pioneering work in the direction of mathematicaltheories of models in quantum field theory (QFT) at finite positive temperatures).
This work was supported by thecation, Japan.

$\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}_{- \mathrm{I}}\mathrm{n}$-Aid 06640188 for science research from the Minisitry of Edu-

2

In this paper, we summarize some results obtained in [11].
II. Preliminaries
In this section we review some fundamental facts in the abstract Boson Fock space.

2.1. Some definitions
be a real separable Hilbert space with normLetandbe the$(E,B,\mu)$Gaussian random process indexed by . We denote by
the underlying probability
space of the process, so that the Borel field is generated byand$\mathcal{H}$$\{\phi(f)|f\in \mathcal{H}\}$$||\cdot||_{\mathcal{H}}$$\mathcal{H}$$B$$\{\phi(f)|f\in \mathcal{H}\})$,$\int_{E}e^{i\emptyset}d(f)=\mu e-||f||_{\mathcal{H}}^{2}/2$$f\in \mathcal{H}$.

The complex Hilbert space $L^{2}(E, d\mu)$ is called the -space representation of the Boson Fock[ $21,$ \S I.3].
space overLet be a strictly positive self-adjoint operator acting in with domain $D(A)$ (i.e., there
exists a constant $c>0$ such that$f\in D(A))$ . Then, for each
, weon $D(A^{S/2})$ by
can define an inner product$Q$$\mathcal{H}$$A$$\mathcal{H}$$||Af||_{\mathcal{H}}^{2}\geq c||f||_{\mathcal{H}}^{2},$$s\in \mathrm{R}$$(\cdot, \cdot)_{s}$,$(f,g)_{s}=(A^{S}/2f, A^{s}/2)_{\mathcal{H}}g$$f,g\in D(A^{S}/2)$,

denotes the inner product of(note that, for $s<0,$ $D(A^{S/2})=\mathcal{H}$ ). For$s\geq 0,$ $D(A^{S/2})$ with the inner productbecomes a Hilbert space. We denote this. For $s<0$ , we denote by
Hilbert space by
the completion of in the norm.
dual space of
can be identified with
For allthrough the bilinear form$-s<f,g>_{s}=(f,g)_{\mathcal{H}}$ .
such that for all$-s<\cdot,$on
the ideal of the trace class operators on . Throughout the present
We denote bypaper, we assume the following:is in.
Assumption I. For somebe fixed. Then the embedding mapping ofLet
into
is Hilbert-Schmidt.
, we can takeHence, by a theorem ofwhere$\mathcal{H}$$(\cdot, \cdot)_{\mathcal{H}}$$(\cdot, \cdot)_{s}$$\mathcal{H}_{s}$.

$s\in \mathrm{R},$$>_{s}$$\mathcal{H}$$\mathcal{H}_{s}$$||\cdot||_{s}$$\mathrm{t}\dot{\mathrm{h}}\mathrm{e}$$\mathcal{H}_{-s}$$\mathcal{H}_{s}$$f\in \mathcal{H}_{-s}\cap \mathcal{H},g\in \mathcal{H}_{s}\mathrm{n}\mathcal{H},$$\mathcal{H}_{-s}\cross \mathcal{H}_{s}$$\mathcal{H}$$\mathcal{I}_{1}(\mathcal{H})$$\gamma_{0}>0,$$A^{-\gamma_{\mathrm{O}}}$$\mathcal{I}_{1}(\mathcal{H})$$\mathcal{H}$$\gamma>\gamma_{0}$$\mathcal{H}_{-\gamma}$$\mathrm{M}\mathrm{i}\mathrm{i}_{\mathrm{o}\mathrm{S}}- \mathrm{S}\mathrm{a}\mathrm{z}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{V}-\mathrm{G}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}$$E=\mathcal{H}_{-\gamma}$

and$\phi(f)=-\gamma<\emptyset,$ $f>\gamma$’$\phi\in E,$ $f\in \mathcal{H}_{\gamma}$.

For a probability measure on $(E, B)$ , we denote by: $\phi(f_{1})\cdots\phi(fn)$ : the Wick product$\phi(f_{n})(f_{j}\in \mathcal{H},j=1, \cdots , n)$ with respect to $\nu[21$ ,of the random variables
be the closed subspace (in $L^{2}(E,$) generated by\S I.l]. For each $n\geq 1$ , let, and set(the space of constant functions on $E$ ). Then one:
has the orthogonal decomposition$\nu$$\phi(f_{1}),$$\cdots,$$d\mu)$$\Gamma_{n}(\mathcal{H})$$\phi(f_{1})\cdots\phi(f_{n}):_{\mu}$$\Gamma_{0}(\mathcal{H})=\mathrm{C}$$L^{2}(E, d \mu)=\bigoplus_{n=0}\mathrm{r}_{n}(\mathcal{H})\infty$(see [21, \S I.l, Theorem I.6].).

3

As usual, we denote by

the second quantization of$d\Gamma(A)$$H_{0}=d\Gamma(A)$In the context of QFT,
Hamiltonian .

$H_{0}$[ $21,$ \S I.4] and set$A$.

describes the Hamiltonian of the free Boson field with one-particle$A$

2.2. Imaginary time Green functions
for all $\beta>0$ , since
is inUnder Assumption I,
with $C= \sup_{x>0}(X^{S}e-x)<\infty$ . Using the spectral property ofis in
one can easily prove that, for all $\beta>0,$$e^{-\beta A}$

for all $s>0$(see [17, \S VIII.10]),$e^{-\beta A}\leq\beta^{-s}CA^{-s}$$\mathcal{I}_{1}(\mathcal{H})$$e^{-\beta H_{\mathrm{O}}}$$d\Gamma(A)$$\mathcal{I}_{1}(L^{2}(E, d\mu))$$Z_{A}( \beta):=\mathrm{T}\mathrm{r}e-\beta H\mathrm{o}=\frac{1}{\det(1-e^{-\beta A})}$

with

,

where Tr denotes trace and $\det(1+\tau)$ with being a trace class operator is the determinantis called the “partitionof $1+T$ ([19, \S XIII.17], [23, Chapt.3]). In the context of QFT,
of
the
free
Hamiltonianwiththe“inverse
temperature”
function”.
The following estimates are well known (cf. [2], [18, \S X.7]): For all $n=1,2,$,$T$$Z_{A}(\beta)$$H_{0}$$\beta$$\cdots$$||\phi(f)^{n}\Psi||L2(E,d\mu)\leq C_{f,n}||(H0+1)n/2\Psi||_{L}2(E,d\mu),$$f\in \mathcal{H},$$\Psi\in D(H_{0^{/2}}n)$

,

(2.1)

is a constant depending on and . Hence, for all $t>0,$ $n=1,2,$, andis a bounded linear operator on $L^{2}(E, d\mu)$ , which implies that its
is also bounded. It follows thatadjoint
is bounded and its. For notational convenience, we denote the closure ofclosure is equal towithby the same symbol. Thus, for, we can define the “complex time Green function”, and $f_{j}\in \mathcal{H},j=1,$
at th . inverse temperature by
where

,

$f$

$C_{f^{n}}$$f\in \mathcal{H},$$n$$\cdots$$\phi(f)ne-tH\mathrm{o}$$(\phi(f)ne-tH\mathrm{O})^{*}$$e^{-tH_{0}}\emptyset(f)^{n}$$(\phi(f)ne-tH\mathrm{O})^{*}$$e^{-tH_{\mathrm{O}}}\emptyset(f)^{n}$$z_{1},$${\rm Re} z_{n}\leq\beta$$\cdots,$$0\leq{\rm Re} z_{1}\leq{\rm Re} z_{2}\leq\cdots\leq$$z_{n}\in \mathrm{C}$$\cdots,$ $n$$G_{n}(z_{1}, f_{1} ; z_{2}, f2;\ldots ; z_{n}, f_{n})$$\mathrm{a}$$\beta$$G_{n}(\mathcal{Z}_{1}, f_{1} ; \cdots ; z_{n}, f_{n})$$\mathrm{T}\mathrm{r}(e^{-z_{10}}\emptyset H(f1)e^{-(_{Z_{2}}}-z1)H\mathrm{o}\phi(f_{2})\cdots e^{-(z}n-n-1)zH\mathrm{o}\phi(fn)e-(\beta-zn)H\mathrm{o})$$=\overline{Z_{A}(\beta)}$ .
For any set
by$\{z_{1}, \cdots, z_{n}\}$with ${\rm Re} z_{j}\in[0,\beta],j=1,$$\cdots,$ $n$, we define$G_{n}(z_{1}, f_{1} ; \cdots ; z_{n}, f_{n}):=G_{n}(Z_{\sigma}(1), f_{\sigma(}1);\ldots$if;$z_{\sigma(n)},$$G_{n}(z_{1}, f_{1} ; \cdots ; z_{n}, f_{n})$$f\sigma(n))$,

denotes a permutation of $(1, 2, \cdots, n)$ .
The two-point function $G_{2}(z, f;w,g)$ can be explicitly computed $[14,2]$ :${\rm Re} z_{\sigma(1)}\leq{\rm Re} z_{\sigma(2)}\leq\cdots\leq{\rm Re} z_{\sigma(n)}$, where$\sigma$$G_{2}(z, f;w,g)=(f,$ $(1-e^{-\beta A})^{-1}(e^{-\beta}-z(w-z)A+e^{-\epsilon({\rm Re}}-w(z-w)A)A+\in({\rm Re}(w))(\mathcal{Z})))_{\mathcal{H}}g$ ,$f,g\in \mathcal{H},$${\rm Re} z,$ ${\rm Re} w\in[0,\beta]$

,

(2.2)

where $\epsilon(t)=1$ for $t\geq 0$ and $\epsilon(t)=-1$ for $t<0$ . Moreover, for all $n\geq 1$ ,$G_{2n-1}(_{Z}1, f_{1;}\cdots ; z_{2n-1}, f_{2}n-1)=0$,$G_{2n}( \mathcal{Z}_{1}, f_{1} ; \cdots ; z_{2n}, f_{2n})=\mathrm{P}\mathrm{a}\sum_{\mathrm{i}\mathrm{r}\mathrm{s}}G_{2}(Z_{i}, f1i_{1} ; z_{j1}, fj1)\cdots c2$($z_{i_{n}},$fin;$Z_{j_{n}},$$f_{j_{n}}$

),
(2.3)

4

where

... ,

means the sum over all $(2n)!/2^{n}n!$ ways of choosing distinct pairs$i_{n}<j_{n}$ .
from $\{1, \cdots, 2n\}$ with $i_{1}<i_{2}<\cdots<i_{n}$ ; $i_{1}<j_{1},$$n$$\sum_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}$$\{i_{n},j_{n}\}$$\{i_{1},j_{1}\}$,$\cdots,$.S.) of (2.2) is defined for all
Remark : Obviously the right hand side (with symmetricity in $\{z, f\}$ and $\{w,g\}$ , provided that
Hence, if, thencan be extended to a function.
in time varibles $z_{j},j=1,$
on$z,$ $w\in \mathrm{C}$$\mathrm{R}.\mathrm{H}$$f,g\in \mathrm{n}_{\alpha>0}D(e^{\alpha})A$$f_{j}\in \mathrm{n}_{\alpha>0}D(e)\alpha A,j=1,$ $\cdots,$ $n$$\mathrm{C}^{n}$$G_{n}(z_{1}, f_{1;}\cdots ; z_{n}, f_{n})$$\cdots,$ $n$

In what follows, we are concerned with functions
with $t_{j}\in[0,\beta]$ ,called the imaginary time Green fucntions (ITGF’s), and their generalizations. By (2.2) wehave$G_{n}(t_{1}, f_{1} ; \cdots ; t_{n}, f_{n})$$G_{2}(t, f;s,g)=(f,$ $(1-e^{-\beta A})^{-}1(e^{(}-(\beta-|t-s|)A+e-|t-S|A)g)_{\mathcal{H}}$$f,g\in \mathcal{H},$ $t,$ $S\in[0,\beta]$

,

.

(2.4)

2.3. Path Integral Representations of the ITGF’s
It was proven in $[1,2]$ that the ITGF’s introduced in the last subsection can be represented
in terms of path (functional) integrals. In this subsection we review this aspect.
We first recall a fundamental result. Let $\beta>0$ and$E_{\beta}=C([0,\beta];E)$, we denotebe the space of $E$ -valued continuous functions on. For each
at $t\in[0, \beta]$ by $\Phi_{t}\in E$ . Let $F$ be the Borel field on
the value ofgenerated by. The following theorem is a key to deriving the path integral
representations of the ITG .
$[0,\beta]$$\Phi\in E_{\beta}$$\Phi$$E_{\beta}$$\Phi_{t}(f),$ $f\in \mathcal{H}_{\gamma},t\in[0,\beta]$$\mathrm{F}’ \mathrm{s}$Theorem 2.1 $[1,2]$ . There exists a $pr$ obability measureon$t\in[0, \beta]\}$ is a family ofjoin
Gaussian andom variables on$(E_{\beta}, \mathcal{F})$$l^{\text{ノ}}\beta$$r$$tl\mathrm{y}$$\mathcal{H}_{\gamma},$such that $\{\Phi_{t}(f)|f\in$with covariance$(E_{\beta}, F, \nu_{\beta})$$\int_{E_{\beta}}\Phi_{t}(f)\Phi_{s}(g)d\nu\beta(\Phi)=(f,$ $(1-e^{-\beta})A-1(e-(\beta-|t-s|)A+e-|t-s|A)g)\mathcal{H}$$s,t\in[0,\beta],$ $f,g\in \mathcal{H}_{\gamma}$

’

.

(2.5)

is an abstract form of a measure introduced in [14] toRemark: (1) The measuredescribe finite positive temperature states of Boson field models. The measurewith$\beta=+\infty$ (“zero-temperature state”) is discussed in [12].
,(2) It follows from (2.4) that, for all$\nu_{\beta}$$\nu_{\beta}$$f\in \mathcal{H}_{\gamma}$$\int_{E_{\beta}}|\Phi_{0}(f)-\Phi_{\beta(}f)|^{2}d\nu_{\beta}(\Phi)=0$,, implies that. . Hence, if we
which, together with the separability of, thendenote by $L([0,\beta], E)$ the space of continuous loops of $E$ with parameter space. Thus can be regarded as a probability measure on the
we haveloop space $L([0, \beta], E)$ .
(3) The random variable
can be extended to allas. We denote it by the same symbol.
an element in$\mathcal{H}_{\gamma}$$\Phi_{0}=\Phi_{\beta},$$\mathrm{a}.\mathrm{e}$$\Phi$$[0,\beta]$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\nu_{\beta}\subset L([0,\beta], E)$$\nu_{\beta}$$\Phiarrow\Phi_{t}(f)(t\in[0, \beta], f\in \mathcal{H}_{\gamma})$$\mathrm{n}_{p<\infty}L^{p}(E\beta, \nu_{\beta})$

By Theorem 2.1 and $(2.3)-(2.5)$ , we obtain the following:$f\in \mathcal{H}$5Theorem 2.2. Let $f_{j}\in \mathcal{H},t_{j}\in[0,\beta],j=1,$ $\cdots,n$ . Then
$G_{n}(t_{1}, f_{1} ; \cdots ; t_{n}, f_{n})=\int_{E_{\beta}}\Phi_{t_{1}}(f_{1})\cdots\Phi t_{n}(f_{n})d_{\mathcal{U}}\beta(\Phi)$.

We can also derive more general trace formulas. For this purpose, we introduce a classof measurable functions on $(E,B)$ .
be a self-adjoint operator in $L^{2}(E, d\mu)$ such that, for all $t>0$ ,$\exp(-tH)$ is in. We say that a measurable function $F$ on $(E, B)$ is in the set
for all $t>0$ .
if $e^{-tH}|F|e^{-t}H$ is in

Definition 2.3. Let$H$$\mathcal{I}_{1}(L^{2}(E, d\mu))$$\mathcal{I}_{H}$$\mathcal{I}_{1}(L^{2}(E, d\mu))$

Theorem 2.4 [1]. Let$F_{1},$$\cdots,$

and $0<t_{1}<t_{2}\cdots<t_{n}<\beta$ . Then

$F_{n}\in \mathcal{I}_{H_{\mathrm{O}}}$$\frac{\mathrm{T}\mathrm{r}(e^{-}F_{1}t_{1}H_{\mathrm{O}}-e-t_{1})H_{\mathrm{O}}F(t_{22}\ldots-(\mathrm{t}_{n}-tn-1)H_{\mathrm{O}}F_{n}e-(\beta-\mathrm{r}_{n})H\mathrm{o})e}{Z_{A}(\beta)}$$– \int_{E\rho}F_{1}(\Phi_{t_{1}})\cdots F_{n}(\Phi \mathrm{r}n)d\mathcal{U}\beta(\Phi)$.

2.4. A circle $ac$ tion
Let $\beta>0$ . Since $\coth x>0$ for all $x>0$ , we can define, via the functional calculus, aself-adjoint operator$B( \beta):=(\coth\frac{\beta A}{2})^{1/2}$

on$\mathcal{H}$,, which is strictly positive and bounded with
$1\leq B(\beta)\leq\sqrt{\coth\frac{\beta\lambda_{1}}{2}}$where$\lambda_{1}>0$

is the lowest eigenvalue of

$A$

,

.

Lemma 2.5. The operator $B(\beta)-1$ is in$\mathcal{I}_{1}(\mathcal{H})$.

By Lemma 2.5, $B(\beta)-1$ is Hilbert-Schmidt. It follows from Shale’s theorem ([20],on $(E,B)$ mutually[21, p.41, Theorem I.23] that there exists a probability measureabsolutely continuous to such that$)$$\mu_{B(\beta)}$$\mu$$\int_{E}e^{i\phi}d\mu_{B(\beta})(\phi)(f)=\int_{E}e^{i\emptyset(B(\beta}d\mu)f)(\phi)=e^{-}||B(\beta)f||^{2}\mathcal{H}/2$

andwith $G_{\beta}\in L^{p}(E, d\mu)$ for some $p>1$ and$d\mu_{B(\beta)}=G_{\beta}d\mu$$q>1$ .

,

$f\in \mathcal{H}_{\gamma}$,$G_{\beta}^{-1}\in L^{q}(E, d\mu B(\beta))$for some, we can extend
as a function of
By Remark (2) after Theorem 2.1, for, there exists awith period . It follows that, for eachto a periodic function onsuch thatfrom $L^{2}(E, d\mu_{B(\beta)})$ into
unique linear isometry$\mathrm{R}$$t\in \mathrm{R}$$\beta$$L^{2}(E_{\beta}, d\nu_{\beta})$$J_{\mathrm{t}}$$J_{t}1=1$$J_{t}$:$t$$\Phi_{t}$$\mathrm{a}.\mathrm{e}.\Phi\in E_{\beta}$,$\emptyset(f1)\cdots\emptyset(fn):=\mu B(\beta):\Phi \mathrm{r}(f1)\cdots\Phi_{t}(f_{n}):_{\nu_{\beta}}$,$n\geq 1,$ $f_{j\gamma}\in \mathcal{H},j=1,$ $\cdots,n$.

6

It is easy to show that$tarrow J_{t}$is strongly continuous and$J_{t+\beta}=J_{t}$,$t\in \mathrm{R}$.

Moreover, in the same way as in a standard case (e.g., [21, p.34, Theorem I.17]), we can showtois positivity preserving and extends uniquely to a contraction fromthatfor each $p\in[1, \infty]$ . By a standard limiting argument, we can show that, for all$F\in L^{2}(E, d\mu_{B(\beta)})$ and,$L^{\mathrm{p}}(E, d\mu B(\beta))$$J_{t}$$L^{p}(E_{\beta}, d\nu_{\beta})$$t\in \mathrm{R}$$(J_{l}F)(\Phi)=F(\Phi_{t})$.

In particular, the mapping $tarrow F(\Phi_{t})$ is continuous in$L^{2}(E_{\beta}, d\nu_{\beta})$.

III. Perturbation ofby the multiplication operator defined by a realWe now consider a perturbation of$(E,B)$. For generality, we take the perturbation in theonvalued measurable function
is bounded from below andsense of quadratic forms. We first consider the case wherethen the case where is not necessarily bounded from below.
$H_{0}$$H_{0}$

$V$

$V$

$V$

3.1. The case where is bounded from below
In this subsection, we assume the following:
$V$

(V.1)

$V$

is bounded from below and $D(H_{0^{/2}}^{1})\cap D(|V|^{1}/2)$ is dense in $L^{2}(E, d\mu)$ .

Under this condition, we have the self-adjoint operator$H_{V}:=H_{0}\dotplus V$

the self-adjoint
and(we denote bydetermined by the quadratic form sum ofand $B$ if
operator determined by the quadratic form sum $qA,B$ of self-adjoint operators
is bounded from below and closed, see, e.g., [17, \S VIII.6] .
$H_{0}$$B\dotplus C$

$V$

$A$

$)$

$q_{A,B}$sa tisfies $(V.l)$ . Then, for all $\beta>0,$ $\exp(-\beta H_{V})$ is in
Theorem 3.1. Suppose that
and, for ffi $t_{i}\in[0,\beta],$ $0\leq t_{1}<t_{2}<\ldots<t_{n}\leq\beta$ , and$F_{n}\in L^{\infty}(E, d\mu)$ ,$\cdots,n-1),$$V$$F_{j}\in \mathcal{I}_{H_{V}}(j=$$\mathcal{I}_{1}(L^{2}(E, d\mu))$$2,$$F_{1},$$\frac{\mathrm{T}\mathrm{r}(e-t_{1}H_{V}F_{1}e-(t_{2}-l_{1})H_{V}F_{2}\cdots e-(t\mathfrak{n}-t-1)nH_{V}Fne-(\beta-t_{\mathfrak{n}})H\mathrm{O})}{Z_{A}(\beta)}$.

$= \int_{E_{\beta}}F_{1}(\Phi_{t_{1}})\cdots Fn(\Phi t_{n})e-\int^{t}\mathrm{o}(\Phi s)ddVsn\nu_{\beta()}\Phi$Remark. (1) This theorem is a refinement of [1, Appendix
Theorem 3.5 below).
(2) Under condition (V.1), we have$||H_{0}^{1/2}\Psi||\leq||(H_{V}+c)^{1/}2\Psi||$

where

$c$

,

$\mathrm{D}$, Proposition D.3] (see$\Psi\in D(H^{1/})0\cap D(|\mathrm{v}|1/2)2$,

is a constant such that $V+c\geq 0$ . It follows from this estimate and (2.1) that.
for all$\phi(f)\in \mathcal{I}_{H_{V}}$$f\in \mathcal{H}$7

Theorem 3.1 can be proven by applying the Trotter product formula [16] and limit theorems on trace class operators $[13,23]$ and quadratic forms $[15,22]$ .

3.2. The case where is not necessarily bounded from below
of self-adjoint operators in $L^{2}(E, d\mu)$ :In this case, we introduce a class$V$$\mathfrak{S}_{V}$is in
if the following conditions$H$
is bounded from below ; (ii) there exists a sequence(i) and (ii) are satisfied: (i)of real-valued measurable fucntions on $(E, B)$ satisfying (V.1) such that $V_{n}\geq V$ for all ,. as $narrow\infty$ and, for all $t>0$ ,Definition 3.2. We say that a self-adjoint operator$H$$\mathfrak{S}_{V}$$\{V_{n}\}$$n$$V_{n}arrow V\mathrm{a}.\mathrm{e}$$e^{-tH_{V_{n}}}arrow e^{-tH}$

weakly as $narrow\infty$ .

Remark. (1) It is easy to see that the weak convergence condition on $\exp(-tH_{V_{n}})$ in.
Definition 3.2 implies in fact that, for all $t>0,$ $\exp(-tH_{Vn})arrow^{\mathit{8}}\exp(-tH)$ as
satisfies (V.1). Then, inwith $\beta=+\infty$ . Assume thatbe the measure(2) Letthe same way as in the proof of Theorem 3.1, we can show that, for all $t>0$ ,$narrow\infty$$V$$\nu_{\infty}$$\nu_{\beta}$$( \Psi_{1}, e-tHV\Psi_{2})=\int_{E_{\infty}}\Psi_{1}(\Phi_{0})^{*}\Psi 2(\Phi_{t})e^{-\int(}\mathrm{o}^{V}s)\ell d\Phi ds\infty\nu(\Phi)$

which is a standard Feynman-Kac-Nelson (FKN) formul.
can show that, in this case,$a$,$\Psi_{j}\in L^{2}(E, d\mu),j=1,2$$[12,21]$,. Using this formula, we$\mathfrak{S}_{V}=\{Hv\}$

The following proposition shows that, for wide classes of $V,$

is not empty.

$\mathfrak{S}_{V}$Proposition 3.3. (i) Suppose that$V,$$e^{-V} \in\bigcap_{0<p}<\infty L^{p}(E, d\mu)$.

Then $H=H_{0}+V$ is essentially self-adjoint on $C^{\infty}(H_{0})\cap D(V)$ and bounded from below.
of $H$ ). Moreothe closby(we den(ii) Suppose that, for a constant $\alpha>1,$ $D(H_{0^{/2}}^{1})\cap D(|V|^{\alpha/2})$ is dense in $L^{2}(E, d\mu)$ and$ot\mathrm{e}$$\overline{H}$$\mathrm{u}re$$\mathrm{r}^{\gamma}er,$$\mathfrak{S}_{V}=\{\overline{H}\}$$||H_{0} \Psi 1/2||^{2}+\int_{E}V(\phi)|\Psi(\emptyset)|2d\mu(\phi)\geq-c||\Psi||^{2}$

with a constant $c>0$ . Then$\mathfrak{S}_{V}\neq\emptyset$,$\Psi\in D(H_{0}^{1})/2(\cap D|V|\alpha/2)$,.

Remark. In Proposition 3.3, we do not need Assumption I for ; as for part (i) (resp.
becomes a
is strictly positive (then, it is sufficient to assume that
hypercontractive semi-group [18, Theorem X.61] (resp. nonnegative).
$A$

$A$

$(\mathrm{i}\mathrm{i}))$$e^{-tH_{\mathrm{O}}}(t>0)$$)$

In what follows, we assume that
Theorem 3.4. Let(i) If$H\in \mathfrak{S}_{V}$$\mathfrak{S}_{V}\neq\emptyset$. We state the main result of this subsection.

. Then the following (i) and (ii) hold:$\int_{E_{t}}e^{-}df_{0}^{t}V(\Phi_{s})d_{\mathit{8}}(\Phi\nu t)<\infty$

,

(3.1)

8

for $t>0$ , then $\exp(-tH)$ is in$\mathcal{I}_{1}(L^{2}(E, d\mu))$and$\frac{\mathrm{T}\mathrm{r}e^{-tH}}{Z_{A}(t)}\leq\int_{E_{\ell}}e^{-\int_{0^{V}}S}d\nu t((\Phi_{S})d\Phi)t$.

(ii) Let $\beta>0$ be fixed and $0<\delta<\beta$ . Suppose that, for all$\exp(-tH)$ is in
andThen, for all$t\in[\delta,\beta],$$(\mathit{3}.\mathit{1})$is satisfied.

$\mathcal{I}_{1}(L^{2}(E, d\mu))$$t\in[\delta,\beta],$$\frac{\mathrm{T}\mathrm{r}e^{-\gamma}e^{-}H(\beta-l)H_{\mathrm{O}}}{Z_{A}(\beta)}=\int_{E_{\beta}}e^{-\int_{0}^{t}V}sd\nu\beta(\Phi)dS(\Phi)$.

Theorem 3.4 can be generalized as follows.

Theorem 3.5. Let$\exp(-tH)$ is in$H\in \mathfrak{S}v$$\mathcal{I}_{1}(L^{2}(E, d\mu))$$F_{j}\in \mathcal{I}_{H}(j=2, \cdots,n-1),$ $F_{1},$and (3.1) holds for all $t\in(0,\beta]$ . Then, for all $t\in(0,\beta]$ ,and, for all $t_{j}\in[0,\beta],$ $0\leq t_{1}<t_{2}<\cdots<t_{n}\leq\beta$ , and$F_{n}\in L^{\infty}(E, d\mu)$ ,$\frac{\mathrm{T}\mathrm{r}(e^{-}F_{1}t_{1}H(t_{2^{-}}t1)H-(t-ntn-1)HFne^{-}(\beta-l_{n})H\mathrm{o})e^{-}F_{2}\cdots e}{Z_{A}(\beta)}$.

$= \int_{E\rho}F_{1}(\Phi_{t_{1}})\cdots p_{n}(\Phi_{t_{n}})e-\int_{0^{t}}nV(\Phi_{S})dsd\nu_{\beta}(\Phi)$

Remark. Theorem 3.5 may be the most general form for trace formulas w.r.t. the heatsemi-group generated by a second quantization operator perturbed by a multiplication operator.
IV. A Golden-Thompson Inequality
By applying Theorem 3.4, we can establish an inequality of Golden-Thompson type for.
withthe partition function Tr$e^{-\beta H}$

Theorem 4.1. Let $\beta>0$ and$H\in \mathfrak{S}_{V}$$H\in \mathfrak{S}_{V}$. Assume that$V$$\int_{E}e^{-\beta V}d\mu B(\beta)<\infty$

Then$e^{-\beta H}$

is in$\mathcal{I}_{1}(L^{2}(E, d\mu))$ansatisfies

.

(4.1)

$d$

.

$\frac{\mathrm{T}\mathrm{r}e^{-\beta H}}{Z_{A}(\beta)}\leq\int_{E}e^{-\beta V}d\mu B(\beta)$

Moreover, if$\int_{E}|V|d\mu B(\beta)<\infty$in addition to (4.1),$t\Lambda en$$e^{-\beta\int_{E}Vd\mu_{B(}} \beta)\leq\frac{\mathrm{T}\mathrm{r}e^{-\beta H}}{Z_{A}(\beta)}$.

As a Corollary of Theorem 4.1, we can obtain some information about properties of a.
limiting operator given as a limit of operators$H^{(N)}\in \mathfrak{S}_{V_{N}}$9Corollary 4.2. Let $V_{N}(N=1,2, \cdots)$ be a real-valued meas urable$fu\mathrm{n}$ction on $(E, B)$ with.

$C_{\beta}:= \sup_{N\geq 1}\int_{E}e^{-\beta V_{N}}d\mu B(\beta)<\infty$. Suppose that there exists a self-adjoint opera $torHbo$ un $ded$ from beLet
low such th at $\exp(-\beta H(N))arrow\exp(-\beta H)$ weakly as $Narrow\infty$ . Then, $\exp(-\beta H)$ is inand$H^{(N)}\in \mathfrak{S}_{V_{N}}$$\mathcal{I}_{1}(L^{2}(E, d\mu))$$\frac{\mathrm{T}\mathrm{r}e^{-\beta H}}{Z_{A}(\beta)}\leq C_{\beta}$

V. Classical Limit, we defineFor$\lambda>0$$V_{\lambda}$.

by$\phi\in E$$V_{\lambda}(\emptyset)=V(^{\sqrt{\lambda}\phi)},$.

be a parameter, which physically means the Planck constant devided by , andLet. We are interested in the limiting behavior of the scaled partition function
let, which, in concrete realizations, corresponds to the classical limit of theasTr.
quantum system whose Hamiltonian is given by$2\pi$$\hslash>0$$H_{\hslash}\in \mathfrak{S}_{V_{\hslash}/\hslash}$$e^{-\beta\hslash H_{\hslash}}$$\hslasharrow 0$$\hslash H_{\hslash}$5.1. A simpler caseWe first consider the case where
(V.2)

$V$

obeys the following condition:is bounded from below and there exists a polynomial $P(x,y)$ of two real variables
with positive coefficients such that$V$$|V(\phi)-V(\phi;)|\leq||\emptyset-\phi’||EP(||\emptyset||E, ||\phi’||E)$

Note that (V.2) implies thatthe fact that$V$

is continuous on $E$ and$|| \phi||_{E}^{2}=\sum_{=n0}^{\infty}\frac{|\phi(e_{n})|2}{\lambda_{n}^{\gamma}}$

,

$V$

,

$\phi,$$\phi’\in E$.

is polynomially bounded. Using$\phi\in E$,

one can show that$\int_{E}||\phi||_{E}^{p}d\mu(\phi)<\infty$

for all $0<p<\infty$ . Hence, under condition (V.2), it follows that, for allis dense in $L^{2}(E, d\mu)$ ; In particular, (V.1) is satisfied with replaced byHence we have$\lambda>0,$ $D(H_{0})\cap D(V_{\lambda})$$V$$H_{\hslash}=H_{V_{\hslash}/\hslash}=H_{0} \dotplus\frac{1}{\hslash}V\hslash$$V_{\lambda}/\lambda$,$\lambda>0$.

.

For any constant $c>0$ , the operator $cA^{-1/2}$ is a continuous bijection from
Hence it extends to a continuous bijection from $E$ to itself. We set$V^{(A)}(\phi)=V(^{\sqrt{2}}A-1/2\emptyset)$

for all$\phi\in E$.

$\mathcal{H}_{\gamma}$

to itself.

10

Theorem 5.1. Suppose that$V$sa tisfies $(V.2)$ and$H_{\hslash}$be as above. Then)$\lim_{\hslasharrow 0}\frac{\mathrm{T}\mathrm{r}e^{-\beta\hslash}H_{\hslash}}{\mathrm{T}\mathrm{r}e-\beta\hslash H\mathrm{o}}=\int_{E}e^{-\beta V}d\langle\beta A$

$\mu$

.

(5.1)

The method of proof of this theorem is similar to that of [10].

5.2. A more general caseWe next consider the case where obeys a more general condition than (V.2). To describe
it, we introduce two bounded operators:$V$$C(\beta;\epsilon)$$:=\sqrt{\frac{\epsilon}{\beta}}B(\epsilon)$,$C(\beta)=\sqrt{\frac{2}{\beta}}A^{-1/2}$$\epsilon>0$,.

By the functional calculus, we can show that
$C(\beta;\epsilon)arrow^{\mathit{8}}C(\beta)$

defines a norm equivalent to. Sinceas$(E,B)$
such thaton
measure$\epsilonarrow+0$$||C(\beta;\epsilon)f||_{\mathcal{H}}$$||f||_{\mathcal{H}},$$\mathrm{t}\mathrm{h}\mathrm{e}\dot{\mathrm{r}}\mathrm{e}$exists a probability$\mu c(\beta;\epsilon)$$\int_{E}e^{i\emptyset}d(f)\mu C(\beta;\epsilon)(\emptyset)=e-||C(\beta;\epsilon)f||_{\mathcal{H}}^{2}/2=\int_{E}e^{i\sqrt{\epsilon/\beta}\emptyset}d(f)\mu B(\epsilon)$

Similarly there exists a probability measure$\mu c_{(\beta)}$,$f\in \mathcal{H}_{\gamma}$.

on $(E,B)$ such that$\int_{E}e^{i\emptyset(f}d)\mu c(\beta)(\phi)=e^{-}||c_{(\beta)}f||^{2}\mathcal{H}/2=\int_{E}e^{i\emptyset(C}d(\beta)f)\mu$,$f\in \mathcal{H}_{\gamma}$.

It follows that, for all $F\in L^{1}(E, d\mu c(\beta;\epsilon))$ ,$\int_{E}F(\sqrt{\epsilon/\beta}\phi)d\mu B(\Xi)(\emptyset)=\int_{E}F(\emptyset)d\mu_{C}(\beta;\epsilon)$and, for all $G\in L^{1}(E, d\mu c(\beta))$ ,$\int_{E}G(c(\beta)\emptyset)d\mu(\emptyset)=\int_{E}G(\emptyset)d\mu c(\beta)$

We now consider the case where(V.3) There exists a sequence
following properties:$V$.

satisfies the following condition:$\{V_{N}\}_{N}$of functions on $E$ obeying condition (V.2) with the11(i) For all $p>0$ ,$F_{1}(p):=0< \epsilon\leq\geq\sup_{\epsilon_{\mathrm{O}},N1}\int_{E}e^{-pV_{N}}d\mu c_{(}\beta;\epsilon)<\infty$$F_{2}(p):= \sup_{N\geq 1}\int_{E}e^{-pV_{N}}d\mu c(\beta)<\infty$,.

(ii) There exists some $q\in(1, \infty)$ such that$\lim||V_{N}-V||_{L}q(E,d\mu_{C(\in}\beta,))=0$$Narrow\infty$

uniformly inand$\epsilon\in(0, \epsilon_{0}]$$\lim||V_{N}-V||_{L}q(E,d\mu_{C(\beta\rangle})=0$.

$Narrow\infty$

We can prove the following theorem.

. Then, for ffi $t>0$ andsatisfies
Theorem 5.2. Suppose th at$\exp(-tH_{\hslash})$ is inand (5.1) holds.
$V$$(V.\mathit{3})$$\hslash\in(0, \epsilon_{0}/\beta]$,$\mathcal{I}_{1}(L^{2}(E, d\mu))$

VI. Application to a Model in QFT
In this section we apply the results in the preceding sections to a QFT model of $P(\phi)$ -type. Let
on a finite volume in the -dimensional space$\mathrm{R}^{d}(d\geq 1)(\mathrm{e}.\mathrm{g}.,[21,14,10])$$d$$\Lambda=[-l_{1}/2,l_{1}/2]\cross\cdots\cross[-l_{d}/2,l_{d}/2]$

be a rectangle in $\mathrm{R}^{d}(l_{j}>0,j=1, \cdots, d)$ and set$\Lambda^{*}=\{p--(p1, \cdots,p_{d})=(\frac{2\pi}{\ell_{1}}n_{1},$ $\cdots,$$\cdots,n_{d}\in \mathrm{Z}\}$$\frac{2\pi}{l_{d}}n_{d})|n_{1},$.

the space of real distributions on A (regarded as a -torus). For
We denote by, we define its ourier transform by$d$$D_{\mathrm{r}\mathrm{e}\mathrm{a}1}’(\Lambda)$$\hat{\phi}$$\dot{\mathrm{F}}$$\phi\in D_{\mathrm{r}\mathrm{e}\mathrm{a}1(\Lambda)}’$$\hat{\phi}(p)=\emptyset(f^{*}p)$where$f_{p}(x)= \frac{1}{\sqrt{|\Lambda|}}e^{ipx}$

Let

$a$

be a real-valued function on$\Lambda^{*}$$(| \Lambda|:=\prod_{j=1}dl_{j})$such that$a(p)\geq C(p+m_{0}^{2}2)^{d/}2$,$p\in\Lambda^{*}$

,

.

12

where $C>0$ and $m_{0}>0$ are constants. Then the set$\mathcal{H}(\Lambda)=\{\phi\in D_{\mathrm{r}}’1(\mathrm{e}\mathrm{a}\sum_{*}\frac{|\hat{\emptyset}(p)|^{2}}{a(p)}\Lambda)|_{p\in \mathrm{A}}<\infty\}$

becomes a real Hilb$e\mathrm{r}\mathrm{t}$space with the inner product.

$( \phi,\psi)_{\mathcal{H}}(\Lambda):=\frac{1}{2}\sum_{p\in\Lambda^{*}}\frac{\hat{\phi}(p)^{*\hat{\psi}}(p)}{a(p)}$In$\mathcal{H}(\Lambda)$, we define an operator$D(A(\Lambda))$$A(\Lambda)$by$:= \{\phi\in \mathcal{H}(\Lambda)|_{p\in\Lambda}\sum_{*}\frac{|a(p)\hat{\phi}(p)|2}{a(p)}<\infty\}$$(\overline{A(\Lambda)\phi})(p):=a(p)\hat{\phi}(p)$The operator$A(\Lambda)$,$\phi\in D(A(\Lambda))$.

is self-adjoint and satisfies$A(\Lambda)\geq^{cm}0d$

It is easy to see that the spectrum of$A(\Lambda)$.

is equal to$\{a(p)|p\in\Lambda^{*}\}$$\sum_{p\in\Lambda^{*}}\frac{1}{a(p)^{\gamma}}\leq\frac{1}{C^{\gamma}}\sum_{p\in\Lambda^{*}}\frac{1}{(p^{2}+m^{2}0)^{d\gamma}/2}<\infty$. If$\gamma>1$, then.

.
is inHence, for all $\gamma>1,$
the
case
where
the Hilbert space
and the self-adjoint
In what follows, we consider, respectively. We remark
andoperator in the abstract theory are realized asthat the case $a(p)=(p^{2}+m_{0}^{2})^{1}/2$ (independently of ) gives the standard framework for a(hence, for $d\geq 2$ , the present model differs
neutral scalar QFT on the space-time A
one).
from the stand. For $N=1,2,$ , we define
We fix a constant $\gamma>1$ and set$A(\Lambda)^{-\gamma}$$\mathcal{I}_{1}(\mathcal{H}(\Lambda))$$\mathcal{H}$$A$$\mathcal{H}(\Lambda)$$A(\Lambda)$$d$$\cross \mathrm{R}$$a\mathrm{r}\mathrm{d}$$E=\mathcal{H}(\Lambda)_{-\gamma}$$\phi_{N}(_{X})=\sum^{N}p\emptyset(fp)*f_{p}(X)$

where$\sum_{p}^{N}=\sum_{|p_{1}|\leq}2\pi N/\ell_{1},\cdots,|pd|\leq 2\pi N/\ell d$$\cdots$,$x\in\Lambda,$. Note that$\int_{E}||\emptyset N-\emptyset||2d\mu-\gammaarrow 0$

as $Narrow\infty$ .

$\phi\in E$,

13

Let $g\in L^{q}(\Lambda),g\geq 0(1<q\leq 2)$ . Then we can show that, for all $p\geq 1$ and $j=1,2,$$\lim_{Narrow\infty}\int_{\mathrm{A}}$:$\phi_{N}(x)^{j}:_{\mu}g(x)dx^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}\int_{\Lambda}$:$X\in \mathrm{R}$

$P$

,

(6.1)

$\emptyset(X)^{j}:_{\mu}g(X)dX$

exists in $L^{p}(E, d\mu)$ [ $10$ , Appendix].
Let be a polynomial of the form $P(X)= \sum_{j=}2n1jCx^{j},$$2n-1$ , and set$\cdots$, with $c_{2n}>0,$$c_{j}\in \mathrm{R},j=$$1,$ $\cdots,$$V_{N}( \phi)=\int_{\Lambda}$$V( \phi)=\int_{\Lambda}$: $P(\phi_{N}(X)):_{\mu}g(x)dX$ ,(6.2): $P(\phi(x)):_{\mu}g(x)dX$ .

(6.3)

Then, by (6.1), we have for all $p\geq 1$$||V_{N}-V||_{L^{p}(}E,d\mu)arrow 0$

way as in the case of the standard
as $Narrow\infty$ . Moreover, in the$N\geq
1$ ,
$t>0$
and
for
all
that,
can show$\mathrm{s}a\mathrm{m}\mathrm{e}$$e^{-tV_{N}},$$e^{-\mathrm{r}V}\in L1(E, d\mu)$$P(\phi)_{2}$model [21], we.

(cf. also [10, \S III].) Hence, applying a general theorem [18, p.261, Theorem X.58], we seethat$H(V_{N}):=H_{0}+V_{N}$and$H:=H0+V$
are essentially self-adjoint on $C^{\infty}(H_{0})\cap D(V_{N})$ and $C^{\infty}(H_{0})\cap D(V)$ , respectively, andconverges to in norm-resolvent sense as $Narrow\infty$ .
bounded from below. Moreover,(resp. ) desribes a Hamiltonian with (resp. without) momentum
The operator
cutoff.
. Hence wegiven by (6.3) satisfies the assumption of Proposition
The potential
have the following fact.
$\overline{H}$$\overline{H(V_{N})}$$\overline{H(V_{N})}$$\overline{H}$$V$

Lemma 6.1. Let$V$$3.3(\mathrm{i})$be as in (6.3). Then$\mathfrak{S}_{V}=\{\overline{H}\}$.

6.1. Bounds for the partition function of
We now apply Theorem 4.1 to obtain bounds for the partition function Tr$\overline{H}$

Theorem 6.2. For all $\beta>0,$$e^{-\beta\overline{H}}$is in$\mathcal{I}_{1}(L^{2}(E, d\mu))$$e^{-\beta\overline{H}}$of$\overline{H}$.

and$e^{-\beta\int_{B}Vd} \mu_{B(}\beta)\leq\frac{\mathrm{T}\mathrm{r}e^{-\beta\overline{H}}}{Z_{A(\Lambda)}(\beta)}\leq\int_{E}e^{-\beta V}d\mu B(\beta)$.

6.2. Classical limit
As for classical limit of the present model, we first consider the case of the cutoff Hamil.
tonian$\overline{H(V_{N})}$

14

Lemma 6.3. Let$V_{N}$

be given by (6.2). Then, for all $N\geq 1,$$V_{N}$

sa tisfies $(V.2)$ .

By Lemma 6.3, we can apply Theorem 5.1 to obtain the following result. Let$H(V_{N})$ with.
replaced by$V_{N}$$H_{N,\hslash}$be$(V_{N})_{\hslash}/\hslash$

Theorem 6.4. For all $\beta>0$ and $N\geq 1$ ,$\lim_{\hslasharrow 0}\frac{\mathrm{T}\mathrm{r}e^{-}\beta\hslash HN,\hslash}{\mathrm{T}\mathrm{r}e^{-\beta}\hslash H0}=\int_{E}e^{-\beta V_{Nd}^{\rho_{A}(\mathrm{A}}}\mu\rangle$

Finally we consider the classical limit for Tr
Theorem 6.5. Let$H_{\hslash}$be$\overline{H}$

with

$V$

$e^{-\beta\overline{H}}$replaced by.

$V_{\hslash}/\hslash$$\sum_{p\in\Lambda^{*}}\frac{1}{a(p)}<\infty$.

. Suppose that.

Then, for all $\beta>0$ ,.

$\lim_{\hslasharrow 0}\frac{\mathrm{T}\mathrm{r}e^{-\beta\hslash}H_{\hslash}}{\mathrm{T}\mathrm{r}e-\beta\hslash H\mathrm{o}}.=\int_{E}e^{-\beta V^{\beta A(}}d\mu \mathrm{A})$REFERENCES
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$P(\emptyset)_{2}$