数理解析研究所講究録
923 巻 1995 年 87-123

87

Functional Integral Representation of a Model in
QED
Hokkaido Univ. Fumio HIROSHIMA
Abstract
This article presents functional integral representations for the heat semigroups with
the infinitesimal generators given by self-adjoint Hamiltonians describing an interactionof a non-relativistic charged particle and a quantized radiation field in the Coulomb
gauge without the dipole approximation. Special attention is paid to definition of
the “time-ordered Hilbert space-valued stochastic integrals associated with a familyof isometries from a Hilbert space t,o another one” and semigroup techniques. Someinequalities are derived, which are infinite degree versions of those known for finite
dimensional Schr\"odinger operators with classical vector potentials.

INTRODUCTION

1

The purpose of this paper is to construct a functional integral representa,tion for the heatselnigro\iota lp with the infinitesimal ellerator given by a Hamiltonian which describes an interaction of a non-relativistic chargecl particle in a scalar potential and a quantized radiation
field in the Coulomb gauge. Tlli b\n]oClel plays an important role for interpretations of some$\mathrm{g}$${ }$$1^{)}]_{\urcorner}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{C}\mathrm{a},11)\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{u}\mathrm{G}11\mathrm{a}_{\mathrm{t}}.\mathrm{f}\mathrm{o}1^{\cdot}\mathrm{e}\mathrm{X}\mathrm{a}\mathrm{l}\gamma 11)\mathrm{l}\mathrm{e}^{4},\mathrm{L}\mathrm{a}\mathrm{m}\mathrm{b}$$\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}^{:}:([1,32])$.

\ulcorner Fllere are many literatures wllic}\iota deal with models describing interactions of non-relativistic

example, the Pauli-Fierz model of non-relativistic
and a quantized field.
etc.. For, the Nelson lnode1([7,20]), andproblems [ the removal of an infrared cut-off ([7,10,11,20]), asympthis kind of models,
totic behaviors ([1.4,9,16]), resonance ([2.3]), scattering states ([3]), and dressed one electron
been discussed by many authors. These examples especially play an
ilnporfJant role as interaction lnodelIS of non-relativistic particles with quantized fields.
been studied extensively. In particular, wit, theWiener path integral nletllod
representa,tions for the heat semigroup generatedof stochastic integral. p.ath
by the Schr\"odinger Hamiltonian$1^{\urcorner},\mathrm{o}\mathrm{r}$$1^{3\mathrm{a}\Gamma \mathrm{t}\mathrm{i}\mathrm{c}1}\mathrm{e}\mathrm{s}$$1\supset \mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}([10,11])$$\mathrm{Q}\mathrm{F}_{\lrcorner}\mathrm{I})([1,2,3,6,16,23,32])$$\mathrm{t}1_{1}\mathrm{e}$$0$’$.\mathrm{s}’ \mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}([10,11])1\mathrm{l}\mathrm{a}\backslash \prime \mathrm{e}$$\mathrm{h}$$\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{s}$$\mathrm{T}1_{1}\mathrm{e}$$\mathrm{i}_{11\mathrm{t}\mathrm{r}\mathrm{a}}\mathrm{e}\mathrm{g}1$$\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}\mathrm{p}$(1. 1)$\mathrm{H}_{cl}=\underline{.\frac{1}{)}}\sum_{\mu=1}(-iD_{\mu}-A)\mu+d2V$

and a scalar potential were investigated. These are well known
has been studied extensively(FKI) formulas. The Hamiltonianas theby many authors ([5]), who used the path integral method.
wit,h a vector potential$A_{l^{l}}$$\Gamma^{}\mathrm{e}\mathrm{y}\mathrm{n}\mathrm{m}\mathrm{a}\mathrm{I}1- \mathrm{I}^{-}\backslash \mathrm{a}\mathrm{C}- \mathrm{I}\{\hat{o}$$V$$\mathrm{H}_{cl}$88

On the other hand, E.Nelson
the “generalized path space” ill confield models from markoff fields (so called thenection with the construction offunctional integral method). In $[1 l]$ . the authors introduced a natural elllbedding of therelativistic Boson Fock space in . space dimensions into a constant time subspace in the ,in $d+1$ dimensions, by which,space over the “generalized path
Feynman-KacNelson (FKN) formula relating thetheory to the Euclidean
waswas studied nlore generally and abstracted
obtained. Th.e‘generalized path
The classlcal path integral alld the functional integral methods have beell applied ,inultaneously to interaction models ofparticles alld quantized fields. In [4],weak coupling limits for Hamiltonialls describing a quantum syst
of finiteof nonrelativistic particles interacting with a lnassive or massless bose field was studied, where theFKN formula and the Wiener path integrals were applied. And in $[12,13]$ , analyzing the
QED by usin. the functional integrals and stochastic
Pauli-Fiertz model ofproblem is tofunctional integral representations for
integrals was suggested. Our
the Pauli-Fiertz mo .
The Harniltonian,of the model which we consider is defined as an operator
of
in
the
tensortwo Hilbert spaces
product
andbyacting$([^{\underline{)}[.\underline{\rangle}\mathit{2}}‘])\mathrm{i}_{11}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{d}_{\mathrm{U}\mathrm{C}}\mathrm{e}\mathrm{d}$$\mathrm{q}_{11\mathrm{a}\mathrm{I}1}\mathrm{t}_{\mathrm{U}}\mathrm{m}$$d$$I^{2}$’$\mathrm{t}1_{1}\mathrm{e}^{1}$$\mathrm{s}_{1^{)\mathrm{a}\mathrm{C}\mathrm{e}}’}$$P(\phi)_{2}$$1^{\cdot}e1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{i}_{\mathrm{S}\mathrm{t}}\mathrm{i}_{\mathrm{C}}P(\phi)_{1+1}$”$\}_{)}\mathrm{y}[19].\cdot$$\llcorner^{\backslash },\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$$‘\backslash _{1-}$$11\mathrm{o}\mathrm{n}-_{1\mathrm{e}1}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}_{\mathrm{S}\mathrm{t}}\mathrm{i}_{\mathrm{C}}$$11\mathrm{U}11\mathrm{t}\}_{)\mathrm{e}}\mathrm{r}$$e\mathrm{m}$$\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}.\mathrm{i}_{\mathrm{S}\mathrm{t}}\mathrm{i}_{\mathrm{C}}$$\mathrm{g}$$\mathrm{m}\mathrm{a}\ln$$\mathrm{g}_{\mathrm{l}\mathrm{V}\mathrm{e}}$$\mathrm{d}\mathrm{e}\mathrm{l}$$\mathrm{H}_{\rho,B}+V\otimes I_{\backslash }$$\mathcal{M}_{B}$$L^{2}(\mathrm{F}_{\mathrm{c}}^{d}))$$\mathcal{F}^{\cdot}(\mathcal{W})$$\mathrm{H}_{\rho,B}=‘\frac{1}{2}\sum_{\mu=1}^{d}(-iD_{\mu}\otimes I-A_{\mu}(\rho(\cdot)))^{2}+I\cap-\sim\neg d\mathrm{r}_{B(}\tilde{\omega}_{B})$Here$\mathcal{F}(\mathcal{W})$denotes the Boson Fock space over.

(1. 2)$A_{\mu}(p(\cdot))$$\mathcal{W}=\frac{L^{2}(\mathbb{R}^{dd})\oplus\ldots\oplus L^{2}(\mathrm{F}’)}{d-1},$

the-th direction time-zero radiation field with an ultraviolet cut-off function in the Coulomb
is the free Hamiltonian of the quantized radiation field andgauge,
is a scalarpotential (see section 3). Comparing (1.1) and (1.2), functional integral representations forseem to rely on the FKN and the FKI formulas heavily. Actually, as it will become
clear later, these formulas are fundamental in this article.
In [1,2,3,6,16], instead ofdefined by taking the dipole ap. the Hamiltonian
proximation
was studied. This approximation implies replacing $p(x)$ in
with
$\mu$

$\rho$$V$$d\Gamma_{B}(\tilde{\omega}_{B})$$\epsilon^{-t\mathrm{H}_{\rho,B}}$$\mathrm{H}_{\rho,B}^{D}$$\mathrm{H}_{\rho,B}$$\mathrm{f}\mathrm{o}1^{\cdot}\mathrm{H}_{\rho,B}$$p(0)$$\mathrm{H}_{\rho,B}$;$\mathrm{H}_{\rho,B}^{D}=‘\frac{1}{2}\sum_{\mu=1}^{d}(-iD_{\mu’}\overline{d}I-I\otimes A_{\mu}(p(\mathrm{o})))^{2}+I\otimes d\Gamma_{B}(\tilde{\omega}_{B})$However, for the original Hamiltonian.

there are few mathematicallyresultscome from the coupling term of the.
Thestrategy to achieve our goal will be certain semigroup idea anding the “time-ordered Hilbert space-valued stochastic integrals associated with a family of”. As in the method used in [14,21,22,26]. we construct a unitary operator fromto the tensor product $M$ ofand the -space over generalized
as an .perator acting inpath space. Wee defineby the unitary transform of
to some
Supposlng some regularity conditions for ultraviolet cut-off functions , we shall show that the contraction semigroup generated by a self-adjoint extension(see below) can be constructed on. Applying the FKN, the FKI formulas and theof,time ordered stochastic integral. the functional integral representation for, is obtained.
$\mathrm{H}_{\rho,B}$,$1^{\cdot}\mathrm{i}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{S}$$\mathrm{d}(\mathrm{e}\mathrm{r}\backslash ^{\gamma}\mathrm{a}\mathrm{t}\mathrm{i}\iota^{\mathrm{v}}\prime \mathrm{e}D_{\mu}.\mathrm{a}\mathrm{n}\mathrm{d}A_{\mu}7_{\rho}).\mathrm{I}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{h}\mathrm{t}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{m}_{\mathrm{h}\mathrm{f}}\mathrm{a}\mathrm{t}\mathrm{h}e\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}1\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{C}\mathrm{u}\rho(_{X}))\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{I}\mathrm{a}\mathrm{m}\mathrm{i}1\mathrm{t}_{0}\mathrm{n}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{H}1,\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}B$$\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{l}\cdot \mathrm{o}\mathrm{d}\mathrm{u}\mathrm{C}-$$\mathrm{m}\mathrm{a}\ln$$\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{n}\urcorner \mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s}$$,kt_{B}\equiv L^{2}(1\mathrm{R}^{d})\otimes F(\mathcal{W})$$L^{2}(\mathrm{F}_{\wedge}^{d}’ \mathrm{I}$$.\mathrm{H}_{\rho}$$1^{\cdot}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}$$L^{2}$$\mathcal{M}$$0$$\mathrm{H}_{\rho,B}$$\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{n}.$$p’ \mathrm{s}$$\mathrm{H}_{\rho,0}$$\mathcal{M}$$\langle F^{1},$$F^{\mathrm{I}},$$c’\in \mathcal{M}$$e^{-t}\mathrm{H}_{\rho}G^{\prime\rangle_{\mathcal{M}}}$89

The outline of the present paper is as follows. In Section II, following the standard,end stochastic integrals to Hilbert space valued one
cedure, we
stochastic integral
and define “time-ordered Hilberl space-valued
chastic integral associated with a familyof isometries from a, Hilbert space to another one ” (Theorem 2.5). In Section III, we$d-1$ . Two Hilbert spaces
introduceandare. vectors(lefined for]. and we construct a unitary operator fromto
. The Hilbert space $L^{2}(Q_{-1}, d\mu_{-}1)$ is the -space overthe underlying measure space for he Gaussian random
cess indexed by. Moreover,
into a constant time subspace in $L^{2}(Q_{-}2, d\mu-2)$ ,using a natural embedding of, we derive a simple
and the Markoff property foron
operat])$L^{2}(Q_{-2},d\mu_{-}2)$ is the. The Hilbert spaceextension of the FKN formula$\mathrm{e}\mathrm{x}\dagger$$\mathrm{p}\mathrm{r}o$$\mathrm{s}\mathrm{t}o$$e^{r}.\uparrow\cdot=1\ldots.,$$\mathrm{p}_{01\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{z}}.\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$[\overline{\mathcal{H}}_{-1}]$$\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{l}\cdot \mathrm{S}$$[\overline{\mathcal{H}}_{-2}]$$\mathcal{M}=$$\mathcal{M}_{B}$$\mathrm{g}_{\mathrm{l}\mathrm{V}}\mathrm{e}\mathrm{n}\mathrm{p}\mathrm{o}$$L^{2}$$L^{2}(\mathrm{P}^{d})\otimes L2(Q_{-1}, d\mu_{-}1)(\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{n}1.3.[)$$[\overline{\mathcal{H}}_{-1}]$$\mathrm{t}$$\mathrm{p}\mathrm{r}o$$L^{2}(Q_{-1\cdot f^{\ell_{-}}}d1)$$\mathrm{s}\mathrm{o}\mathrm{m}\langle \mathrm{Y}$$o\mathrm{r}\mathrm{s}$$\mathrm{r}o.|\backslash .\mathrm{C}\mathrm{e}(,\mathrm{i}\mathrm{o}\mathrm{n}$$L^{2}.(Q_{-2}, d\mu_{-}2)$$L^{2}-$$(\mathrm{p}\mathrm{r}(|)\mathrm{o}\mathrm{S}\mathrm{l}\mathrm{t}\mathrm{l}\mathrm{o}\mathrm{n}3.4)$$\mathcal{H}_{-2}]$.

3.4));,$\mathrm{H}_{\rho}$$\equiv$$\mathrm{H}_{\rho.0}+I\mathfrak{O}\mathrm{H}_{0}$$\mathrm{H}_{\rho.\mathrm{U}}$$\equiv$$\frac{1}{\iota \mathit{2}}\sum_{\mu=1}^{f}(_{-}iD_{\mu}\otimes \mathrm{J}I-\phi\rho \mathcal{F},\mu)\prime 2$

unitary transform ofisMoreover it is shown that)
restricted to some domain(Theorem 3.1). Inon. we construct the contraction -semigroupis a self-adjoint extension of the formally definedsuch that the infinitesimal generator(Lenlmas 4.6,4.7 and 4.8). We give a rigorous definition ofHamiltonian
in terms. Applying the Trotter product formula $([18])$ , theof the form sum
alld
FKN formula, a functional inte.gral representation
integral. andfor the heat semlgroup generated
an extended self-adjoint HalniltoIllan ofis a suitable scalar potential. Moreover, they areare derived in Theorem 4. 10, where
extended for a more general class of potentials in Theorem 4.12. In Section V, we derivesome inequalities which are known
the classical case as a diamagnetic inequality ([5,31])
inequality
and an abstract
the functional integr.al representation.
Section VI, we
comparlng our model with the classlcal one ([31]) and‘,,calar field theory ([26]).
It is a pleasure to thank Prof. A.Arai for raising a problem which led toconsiderationof the functional integral representation of a model in QED.
$1\}_{1\xi^{\mathrm{Y}}}$$\mathrm{H}_{\rho.B}$$\mathrm{H}_{\rho}$$C_{0}$$\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{t}_{)}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{I}\mathrm{V}$$G_{\rho}(t)$$\mathcal{M}$$\overline{\mathrm{H}}_{\rho.0}$$\mathrm{H}_{\rho,0}$$\mathrm{H}_{\rho}$$\dotplus \mathrm{o}\mathrm{f}\overline{\mathrm{H}}_{\rho,0}$.hastic$\mathit{1}\prime 3\mathrm{H}_{0}$$\mathrm{t}1_{1}e$$\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}- \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}_{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{S}\mathrm{t}\mathrm{o}\mathrm{C}$$\mathrm{H}_{\rho}+I\Theta V$$\mathrm{b}.\backslash ^{r}$$V$$\mathrm{i}_{1}\iota$$\mathrm{I}\backslash ^{\mathit{7}}\mathrm{a}\mathrm{t}_{0}.’$\ln$$([\mathrm{t}.5,\underline{\prime}7.29]).\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}$\mathrm{s}$$\mathrm{g}_{1}\mathrm{v}\mathrm{e}\mathrm{t}‘,\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{l}\backslash \prime \mathrm{s}.$$\mathrm{n}\mathrm{l}\mathrm{y}$2TIME ORDERED STOCHASTIC INTEGRALIn this section we extend the standard stochastic integral to a Hilbert space-valued one andthe “time-ordered Hilbert-valued stochastic integral associated with a family$\mathrm{i}11\mathrm{t}_{c}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}$$\mathrm{s}^{\backslash }.\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$)”. (A general reference
For a Hilbert spaceover C. we denote the inner product and the associated norm by, respectively. The inner product is linear in. and antilinear
and
. The
$<*,$
domain of an operator is denoted by $D(A)$ . The notation
denotes the space ofto the Hilbert space . For $n=1,2,$ , we denote bystrongly continuous functions
the subspace of -times strongly differentiable functions inand defineof$1\mathrm{S}[31]$$\mathrm{i}_{\mathrm{S}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{f}_{j}\mathrm{r}}\mathrm{i}\mathrm{e}\mathrm{S}$$‘ \mathrm{Y}$$\cdot>\chi$$\mathrm{i}\mathrm{n}*$$||\cdot||_{\mathrm{Y}}.$$A$$C(\mathrm{F}_{-}^{d}’;\mathcal{X})$$\mathrm{f}\mathrm{r}\mathrm{o}\ln \mathbb{R}^{d}$$\mathcal{X}$$\ldots$$n$$C_{\mathrm{c}}^{\mathrm{v}n}(\mathbb{R};\mathcal{X}d)$$C_{1y}^{n}(\mathrm{F}’(l;‘ \mathrm{t}’)$$=$$C(\mathbb{R}^{d:};‘ V)$$\{f\in C^{\prime n}(\mathbb{R}^{d}\mathrm{i}\mathcal{X})|_{|k|}\sup_{\leq n,x\in \mathrm{l}\mathrm{R}d}||\partial^{k}f(X)||_{\mathrm{Y}}.\cdot<\infty\}$,90$H^{n}(\mathbb{R}^{d};X)$$=$$\{f\in(^{\urcorner n}(\mathrm{F}’;\mathcal{X}d)|||\partial^{k}f(\cdot)||\lambda^{\text{ノ}}\in L^{2}(\mathrm{F}^{d}\rangle),$$|k|\leq n\}$,is a multi-index, $|k|=k_{1}+k_{2}+\ldots+k_{d}$ , and the derivative
is taken in the strong topology in . We fix probabilils tic notations. Let $(\Omega.Db)$
be a probability space for -dimensional Brownian motion
andbethe Wiener measure on
defined by $d\mu=dx\otimes Db$ . Let denote the expectation value
with respect to $(\Omega, Db)$ . Following [2 , XIII.16], we use the following identification;$\partial^{k}=$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{l}\cdot \mathrm{e}k=(k_{1}, k_{2}, \ldots, k_{d})$$\partial_{1}^{k_{1}}\partial_{2}^{k\underline{\circ}}\ldots\partial dk_{d}$$\mathcal{X}$$d$$b(t)=(b_{\mu}(t))_{1\leq\leq d,t\geq 0}\mu$$d\mu$$\mathrm{p}_{\mathrm{c}}^{d}\cross\Omega$$F_{\lrcorner}$$l$$L^{2}(M, ( \int,n)\mathrm{G}\mathcal{X}\cong\int_{M}^{\oplus}i\mathrm{t}_{C}’ l\uparrow)l$Let$\mathcal{H}$.

be a Hilbert space over .
$\mathbb{C}$

Lemma 2.1 Let$f\in C_{b}^{1}(\mathbb{R}^{d};\mathcal{H})$and define$\mathrm{J}_{n}^{\mu}(f, b)=\sum_{k=1}^{2^{n}}f(b(\frac{k-1}{9^{n},\cup}t))\{b_{\mu}(\frac{k}{2^{n}}t)-b_{\mu}(\frac{k-1}{\underline{9}^{n}}t)\},$$t\geq 0,$ $\mu=1,$ $\ldots,$ $d$.

Then the strong limit$s-1 \mathrm{i}\mathrm{n}_{\infty}1\mathrm{J}^{\mu}narrow n(f, b)\equiv\int_{0}^{t}.f(b(s))db_{\mu}$$ex\cdot ists$in$L^{2}(\Omega;\mathcal{H})$. Moreover,$fo\uparrow$. any$\mathit{9}\in$.

$C_{b(;}^{1}\mathrm{R}^{d}\mathcal{H}$),(2. 1)$\langle\int_{0}^{t}f(b(s))db_{\mu},$ $\int^{t}\mathrm{o}(gb(s))db_{l^{\text{ノ}}\rangle_{L}}2(\Omega:\mathcal{H})=\delta_{\mu\nu}E(\int_{0}^{t}(f(b(S)), g(b(_{S}))\rangle \mathcal{H}sd),$

where$\delta_{\mu\iota \text{ノ}}$is Kroneker’s delta.

Proof: In the same way as in tllc

is a Cauchy sequence in
can see that$L^{2}(\Omega_{\}\mathcal{H})$of of, one can see that
. lIence the strong limit ofexists in$\mathrm{p}\mathrm{r}o$$[31,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}1l.2]$$\{\mathrm{J}_{n}^{\mu}(f, b)\}_{n}\geq 1$$L^{2}(\Omega;\mathcal{H})$$\mathrm{J}_{n}^{\mu}(.f, b)$

. One

.

$\langle \mathrm{J}_{n}^{\ell\iota}(.f, b).\mathrm{J}_{n}^{\nu}(g, b)\rangle_{L(;\mathcal{H})}2\Omega=E(.\sum_{k=1}^{2^{11}}\frac{t}{\underline{9}^{n}}\langle f(b(\frac{k-1}{2^{n}}t)),g(b(\frac{k-1}{2^{n}}t))\rangle_{\mathcal{H}})\delta_{\mu\iota \text{ノ}}$Since$\langle f(b(S)).g(b(S\mathrm{I}\mathrm{I}\rangle_{\mathcal{H}}$is contillltous in$\lim_{narrow\infty}\sum_{k=1}^{2^{\eta}}\frac{t}{2^{n}}\langle.f(b(\frac{k-]}{2^{n}}t)),$$s\mathrm{a}.\mathrm{s}.b\in\Omega$, we have$g(b(_{\frac{k-1}{2^{n}}t}\mathrm{I})\rangle_{\mathcal{H}}$$=$$\int_{0}^{t}ds$$\langle.f(b(s)),g(b(s))\rangle_{\mathcal{H}}$$a.s.b\in\Omega$,.

Moreover,$| \sum_{k=1}^{2^{n}}\frac{t}{2^{n}}\langle f(b(‘\frac{k\cdot-1}{2)l}t)),g(b(\frac{k-1}{2^{n}}t))\rangle_{\mathcal{H}}|\leq c_{0}c_{0}’t$

where

and
convergence theorem yields (2.1).
$c_{0}= \sup_{x\in \mathrm{R}^{d}}||.f(X)||_{\mathcal{H}}$We call$\int_{0}^{t}f(b(s))db_{\mu}$the “$\mathcal{H}$$c_{0}’= \sup_{x\in \mathrm{R}^{d}}||g(.’\iota\cdot)||_{\mathcal{H}}$,. Hence the Lebesgue-valued stochastic integral for$f$”$\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\coprod$

91

Relllark 2.2 (1) As in$\int_{0}^{t}f(b(s))db_{\mu}$from. Lemma 2.1 suggests that one can extend the definition of
to arbitrary functions such that$[\mathit{3}\mathit{1},p\mathit{1}\mathit{5}_{\sim}^{)}\mathrm{t}’]$$C_{b}^{1}(\mathrm{P}^{d};\mathcal{H})$$f$$|| \int_{0}^{t}.f(b(\iota \mathrm{s})\mathrm{I}^{d}b\mu||_{L^{2}(\Omega\kappa)}^{2};$(2) In an obvious way, we can$(t_{3}, t_{4}]=\phi$

and

$=$

$E( \int_{0}^{t}||f(b(s))||2\mathcal{H})ds$$=$$\int_{()}^{t}(\int_{\mathrm{J}\mathrm{R}^{d}}d_{X}(2\pi s)^{-}\frac{d}{2}||.\mathrm{f}(X)||_{\mathcal{H}}2-\in\frac{x^{2}}{2^{\sigma}})ds<\infty$$e.\iota\cdot \mathit{1}endI_{0}^{t}f(b(S))db_{\mu}$

to $J_{t}Sf(b(S))db_{\mu}$ .

.

Then for$[t_{1}, t_{2})\cap$$f,g\in C^{1},b(\mathrm{F}_{-}’;\mathcal{H}d)$$\langle\int_{t_{1}}^{t_{2}}f(b(s))db_{\mu},$ $\int^{t}t_{3}(g(bS)4)db\nu\rangle_{L(}2\Omega,\mathcal{H})=0$(3) From (2.1) and (2.2) it follows that

Lemma 2.3. Let$f\in\dot{c}_{b(}^{2}\mathbb{R}^{d};\mathcal{H}$$\int_{0}^{t}f(b(s))db_{\mu}$.

(2. 2)is strongly continuous in) and define for $t\geq 0,$ $\mu=1,$$d$$\ldots,$$t$in$L^{2}(\Omega;\mathcal{H})$,$b_{\mu}( \frac{k}{2^{n}}t)-b_{\mu}(\frac{k-1}{2^{n}}t)\}$$\mathrm{S}_{n}^{\mu}(f, b)=\sum_{k=1}^{2^{n}}\frac{1}{2}\{f(b(\frac{k}{2^{n}}t))+f(b(\frac{k-1}{\mathit{2}^{n}}‘ t))\}\{$.

.

$I^{1}he?l$(2. 3)$s- \lim_{arrow n\infty}\mathrm{S}_{n}^{\mu}(f, b)=\int_{0}^{t}f(b(s))db\mu+‘\frac{1}{\mathit{2}}\int_{0}^{t}(\partial_{\mu}.f\mathrm{I}(b(s))ds$in$L^{2}(\Omega;\mathcal{H}),$$(\partial_{\mu}f)(b(\cdot))$$P\uparrow\cdot oof.\cdot$$u)he^{l}’ \epsilon\int_{0}^{f}(\partial f\mu.)(b(S))dS$on$\mathrm{F}_{\mathrm{c}}^{d}’$We divide$\mathrm{S}_{n}^{\mu}(f, b)$$=$.
$\mathrm{S}_{n}^{\mu}(f, b)$is th Bochner integral of$\rho$$L^{2}(\Omega;\mathcal{H})$-valued function

in two parts as follows$\sum_{k=1}^{2^{n}}f(b(\frac{k^{\wedge-}1}{2^{n}}t))\{b_{\mu}(\frac{k}{\underline{9}^{n}}t)-b_{\mu}(\frac{k-1}{2^{n}}t)\}$$+ \sum_{k=1}^{n}\underline{\frac{1}{9}}2\{f(b(‘\frac{k}{\mathit{2}^{n}}t))-f(b(\frac{k-1}{2^{n}}t))\}\{$ $b_{\mu}( \frac{k}{2^{n}}t)-b_{\mu}(‘\frac{k-1}{2^{n}}t)\}$.

(2. 4)

Similarly to Lemma 2.1 ([31, p160]), it is not hard to see that the two terms on the right hand.s.of (2.3) inside (r.h.s.) of (2.4) strongly converges to the two terms on therespectively.
$\mathrm{r}.\mathrm{h}$’$L^{2}(\Omega;\mathcal{H})\square$

Renlark 2.4 One can casily.

$\mathrm{q}e(^{\supset}that$for.

$f^{\backslash }\in C_{b}^{1}(\mathbb{R}^{d};\mathcal{H})$,$s-_{narrow\infty}1 \mathrm{i}_{\mathrm{l}}\mathrm{n}\sum_{3k=}^{]}f[2’\tilde{}t(b(\frac{k^{\tau}-1}{2^{?l}}‘))\{b_{\mu}(.\frac{k}{2^{n}})-b_{\mu}(\frac{k^{\eta}-1}{2^{n}})\}=\int_{0}^{t}f(b(S))db\mu\cdot$

92

Moreover, for$f\in C_{b(\mathbb{R}^{d}}^{2};\mathcal{H}$).

$s- \lim_{narrow\infty}\sum^{]}\frac{1}{\mathit{2}}[2^{n}k=1t.\{f(b(\frac{\Lambda-1}{\underline{i)}n}))+f(b(\frac{k}{2^{n}}))\}\{$$= \int_{0}^{t}f(b(_{S}))db_{\mu}+\underline{‘.\frac{1}{)}}\int_{0}^{f}((^{-}.)l^{\mathrm{t}}f)(b(s))dS$

where$[\cdot]$

denotes the Gauss$.\backslash \cdot y’ nbol$$b_{\mu}(.

\frac{k}{\mathit{2}^{n}})-b_{l^{l}}(\frac{k-1}{\mathit{2}^{n}})\}$,.

Let be a Hilbert space andbe a family of isornetries
to , so that
where
is the identity operator in . we denoteby for simplicity. For
and the isometries , we define the -valued stochastic integralby$\mathcal{K}$$\{l_{t}\}_{t\geq}()$$\mathrm{f}\mathrm{i}\cdot \mathrm{o}\mathrm{m}\mathcal{H}$$I_{t}f$$\mathcal{H}$$I_{\mathcal{H}}$$\hat{\mathrm{J}}_{n}^{\mu}(.f, b)$$\hat{\mathrm{J}}_{n}^{\mu}(f, b\mathrm{I}=k.\sum_{=1}^{2^{\prime 1}}\int_{\frac{k-1}{2^{n}}}^{2}\urcorner\tau t)f_{-}\frac{k}{2}7\ulcorner t(b(S)db_{\mu}kt1$

Theorem 2.5 Let$f\in C_{b(\mathrm{F}^{d}}^{1},;\mathcal{H}$$||I_{t+s}^{*}$$whe\uparrow\cdot eM(f)$

)

$s$

,
)

$I_{t}^{*}I_{t}=I_{\mathcal{H}}$$f\in c_{b(}^{1}\mathrm{R}^{d};\mathcal{H}$$f_{t}$$\mathcal{K}$$I_{t}$$\mathcal{K}$ttch that for all.

$su.ffi_{C}i\epsilon ntly$

small $s\geq 0$ ,It $f(X)-f(x)||\mathcal{H}\leq sM(f)$ ,is a positive constant independent of$x\in \mathbb{R}^{d}$(2. 5)

and $t\geq 0$ . Then$s- \lim_{\infty narrow}\hat{\mathrm{J}}_{n}\mu(.f, b)\equiv\int_{0}^{t}\hat{I}_{0arrow t}f(b(S))db\mu$$\rho xistS$

in$L^{2}(\Omega;^{\kappa)}$

Proof: Fix

familyfact.

and put
is a Cauchy sequence in, we can see that$f\in C_{b}^{1}(\mathbb{P}^{d}:\mathcal{H})$$\{\hat{\mathrm{J}}_{n}^{\ell}l(f, b)\}_{n}\geq 1$$I_{t}^{*}I_{t}=I_{\mathcal{H}}$. It is sufficient to showthe
. By Lenlma 2.1.
and the.

$\mathrm{c}_{0}=\sup_{x\in 1\mathrm{R}}d||.f(X)||_{\mathcal{H}}$$L^{2}(\Omega;\mathcal{K})$$\mathrm{t}_{1}\mathrm{h}\mathrm{a}\mathrm{t}_{l}$$(‘ \mathit{2}.2)$$(\mathit{2}.5)$$||\hat{\mathrm{J}}_{n}^{\mu}(f, b)-\hat{\mathrm{J}}^{\mu}1(n+f, b)||_{L\Omega_{\backslash }}22\mathrm{t}\cdot\kappa)$$=|| \sum_{k=1}^{2^{n}}\int_{\frac{2k-1}{2n+1}}’\frac{2k}{2^{l}+1}t(f_{\frac{2k-1}{2^{n+1}}}t(b(S))-f_{\frac{2k-2}{2^{n+1}}}t(b(S)))db\mu t||_{L^{2}\Omega}^{2}(;\kappa)$$= \sum_{k=1}^{2^{n}}E(\int_{\frac{2k-1}{2^{n+1}}t}^{\frac{2k}{2^{n+1}}t}||.f_{\frac{2k-1}{\mathit{2}^{l1}+1}t}(b(s))-f\frac{2k-2}{2^{n+1}}t(b(s))||_{k}2d_{S})$$\leq 2\sum_{k=1}^{2^{n}}E(\int_{\frac{2k-1}{\underline{\mathrm{o}}n+1}t}^{\frac{2k}{2^{1}+1}t}.||(I_{\mathcal{H}}-II).f(b(s))||_{\mathcal{H}}||f(b(S))||_{\mathcal{H}}dS)\frac{*2k-1}{2^{n+1}}t\frac{2k-2}{2^{n+1}}t$$\leq‘ \mathit{2}\sum_{k=1}^{2^{n}}E(\int_{\frac{2k-1}{2^{n+1}}t}^{\frac{2k}{2^{7}+1}t}’\frac{t\Lambda I(f)}{12^{n+1}}c_{0^{d)}}S$$= \frac{\mathit{1}v\mathit{1}(f)t^{2_{C}}0}{2^{n+1}}$.

93

Thell we have,$||\hat{\mathrm{J}}_{m}^{\mu}(f.b)-\hat{\mathrm{J}}_{\mathit{1}}^{\mu}$Hence$\{\hat{\mathrm{J}}_{n}^{\mu}(f, b)\}_{\eta}\geq 1$

We call$\{I_{t}\}_{t>0’}’$$(.f. b)||_{I^{2}}, \langle\Omega;\mathcal{K})\leq t\sqrt{M(f)c_{0}}\sum_{nk=}^{m-1}(\frac{1}{\sqrt{2}})^{k}$is Cauchy in$\int_{0}^{t}\hat{I}_{0}arrow tf(b(s))db_{\mu}$.

$I^{2},(\Omega;\mathcal{K})$as required.

the “tillle-ordered$\mathcal{K}$$\square$-valued stochastic integral associated withconclucle the present, sectioll with stochastic, for
as a strong lilnit illa,s a strong limit inas follows$\}_{l}\nabla \mathrm{e}$$/_{0^{t}}.f(b(s))db_{\mu}$$\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{S}$$f\in H^{1}(\mathrm{F}^{d};\mathcal{H})$$I^{2},(\Omega;\mathcal{H})$the Wiener paths. Defining
we call define

ovel

.

$\cdot$$\int_{0}^{t}.f(\omega(s))d\omega_{\mu}$$L^{2}(\mathbb{R}^{d}\cross\Omega;\mathcal{H})$$s- \lim_{narrow\infty}\sum_{=k1}^{n}.;2(\omega(\frac{k-1}{\mathit{2}^{71}}t))\{\omega_{l^{\mathit{4}}}^{\backslash }(‘\frac{k}{2^{n}}t)-\omega_{\mu}(‘\frac{k-1}{2^{n}}t)\}\equiv\int_{0}^{t}.f(\omega(_{S}))d\omega\mu$$\ulcorner \mathrm{I}^{1}\mathrm{h}\mathrm{e}$existence of this limit can be ), we have$\Gamma \mathrm{O}1r\mathrm{e}11$$\mathrm{I}$.

(2. 6)

in the same way as in the proof of Lemma 2.1. For$f,g\in H^{1}(\mathbb{R}^{d};\mathcal{H})$$\langle\int_{0}^{t}f(\omega(s))d\omega\mu’\int_{0}^{t}g(\omega(S))d\omega\iota \text{ノ}\rangle_{L}2\mathrm{t}^{\mathrm{l}}\mathrm{R}d\cross\Omega;\mathcal{H})$denotes thewhereextend the definition of$=$$\delta_{\mu_{\mathcal{U}}}\tilde{E}(\int_{0}^{t}\langle f(\omega(s\rangle),g(\omega(s))\rangle \mathcal{H})dS$$=$$t \delta_{\mu_{\mathcal{U}}}\int_{1\mathrm{R}^{d}}\langle f(x), g(X)\rangle_{\mathcal{H}}dx$,$(‘ \mathit{2}.7)$

with respect to.
allows us toto.f such that the r.h.s. of (2.7) is finite.

$\tilde{E}$$(\mathrm{F}^{d}.’\cross\Omega, d\mu)$$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{0}11\mathrm{v}\mathrm{a}1_{11}e$$f_{0}^{t}.f(\omega(s))d\omega_{\mu}$$\mathrm{E}\mathrm{q}.(2.7)$

PROBABILISTIC DESCRIPTION OF THE TIMEZERO RADIATION FIELD WITH THE COULOMB
GAUGE

3

In this section we define a model whicll describes a quantum system of a non-relativistic
charged particle interacting with a quantized radiation field with the Coulomb gauge.
, we consider the situation where the charged particle moves
For mathematicalill
and the quantized radiatioll field is over . We define polarization vectors $e^{r}(r=$$d-1)$ as measurable functions:
such that$\mathrm{g}e\mathrm{n}\mathrm{e}\mathrm{l}\cdot \mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$$\mathbb{R}^{d}$$\mathrm{R}^{d}$$1,$$e^{r}$$e^{r}(k)\cdot e^{S}(k)=\delta_{rs}$$\ln$$1\mathrm{P}^{d}arrow \mathbb{R}^{d}$$\ldots,$,$k\cdot e^{r}(k)=0$,$a.e.k\in \mathrm{P}^{d}$.

what follows, fix the polarization vectors . We introduce two Hilbert spacesbyand
as follows. First we define two real Hilbert spaces$[\overline{\mathcal{H}}_{-2}]$$e^{r}$$\mathcal{H}_{-1}$$\mathcal{H}_{-1}\equiv\{.f\in s_{r}’(\mathbb{R}^{d})|\int_{\mathrm{R}^{d}}\frac{|\hat{f}(k)|^{2}}{|k|}dk<\infty\}$$\mathcal{H}_{-2}$,$\mathcal{H}_{-2}\equiv\{.f\in S_{r}’(\mathrm{P}d+1)|\int_{\mathrm{R}^{d+1}}\frac{|.\hat{f}(k^{n})|^{2}}{|k|^{2}},dk<\infty\}\gamma$$[\overline{\mathcal{H}}_{-1}]$

and

94

where
denotes the set of real tempered distributions on $\mathbb{P}^{n}(n=d, d+1)$
and denotes the Fourier transformation ( the inverse Fourier) fromto:$S_{r}’(\mathbb{R}^{n})$$\wedge$$\vee$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}_{0}\mathrm{r}\mathrm{m}\mathrm{a}\{_{}\mathrm{i}_{0}\mathrm{n}$$S’(\mathrm{P}^{n}\mathrm{I}$$S’(\mathbb{R}n)$$\hat{f}(k)=(27\ulcorner)^{-\frac{n}{2}}\int_{\mathrm{R}^{n}}f(x)e-ikxd_{X}$.

Put$\overline{\mathcal{H}}_{-1}$,$=$$\frac{\mathcal{H}_{-1}\oplus\ldots\oplus \mathcal{H}_{-1}}{\text{\’{e}}}$$\overline{\mathcal{H}}_{-\mathit{2}}$.

$=$$\frac{\mathcal{H}_{-2^{\oplus\ldots\oplus}}\mathcal{H}_{-2}}{d}$We introduce bilinear forms

resp.ectively, where$\mathrm{c}\mathrm{o}\mathrm{n}_{\rfloor}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{e}$and$f_{\mu}$and$(\cdot, \cdot)_{-1}$$(\cdot, \cdot)_{-2}$in$\overline{\mathcal{H}}_{-1}$and$\overline{\mathcal{H}}_{-2}$.

$(f,\subset j)-1$$=$$/’, \sum_{\nu=1}^{d}\int_{\mathrm{R}^{d}}\frac{d_{\mu_{U}}(kn)\overline{\hat{f}}_{\mu}(k\cdot)\hat{g}\nu(k)}{|k^{\tau}|}dk$$(f,g)_{-2}$$=$$‘ 2, \sum_{\nu,\nu=1}^{d}\int_{\mathrm{R}}d+1\frac{d_{\mu\nu}(k^{n}).\hat{f}_{\mu}(-k)\hat{g}_{\nu}(k)}{|k|^{2}}dk$

andare he } -th components of$\mathrm{t}$$\ell$$g_{\mu}$$d_{l^{lU}}(\Lambda\cdot)$$\equiv$$\sum_{r=1}^{d-1}\epsilon.((\ell)?ke_{\mathrm{t}\text{ノ}^{}\uparrow}.(k)$$=$$\delta_{\mu\nu}-\frac{k_{/x\iota \text{ノ}}k}{|k|^{2}}.$

We denote the associated semi-norms by$|\cdot|_{-1}$

and

alld

$f$

$|\cdot|_{-2}$$\{.f\in\overline{\mathcal{H}}_{-1}||f|_{-1}=0\}$,$N_{-\mathit{2}}$$=$$\{f\in\overline{\mathcal{H}}_{-}2||.t|-2=^{\mathrm{o}\}}$.

’$.\backslash .$and$[\overline{\mathcal{H}}_{-1}]$$=$$\overline{\mathcal{H}}_{-1}/N_{-1}$$[\overline{\mathcal{H}}_{-2}]$$=$$\overline{\mathcal{H}}_{-2}/N_{-2}$$<\cdot,$ $\cdot>_{-2}$denotes the complex

respectively and put$=$$<\cdot,$ $\cdot>_{-1}$

$-$

$g,$

.

$N_{-1}$$\rceil$,.

Then we define pre-Hilbert spaces ) the quotiellt spaces

with inner products

by

,

.

defined by$\langle_{7\mathrm{i}^{-}}-1 (.f), 7\ulcorner-1(g)\rangle_{-1}$$\equiv$$(f, \backslash c/)_{-}1$,$\langle_{7T_{-2}}(.f), \pi_{-\mathit{2}}(g)\rangle-2$$\equiv$$(.f, g)_{-2}$.

95, respectively.
anddenote the equivalence classes of inandHereand $<.,$We denote the norms associated with the inner products $<.,$
and, respectively. The Hilbert spaces constructed by the completions ofandandare denoted by the same symbols.
with respect tobe the Gaussian random processes
andLetsuch that the characteristic functions are given byandindexed by$\pi_{-1}(f)$$\overline{\mathcal{H}}_{-1}$$f$$\pi_{-2}(.f)$$\overline{\mathcal{H}}_{-2}$$\cdot>_{-1}$$\cdot>_{-2}\underline{\mathrm{b}\mathrm{y}}||\cdot||_{-1}$$[\mathcal{H}_{-1}]$$||\cdot||_{-2}$$[\overline{\mathcal{H}}_{-2}]$$|\underline{|\cdot}||_{-1}$$\{\phi_{-1}(T_{-1}\underline{(}f))|f\in \mathcal{H}_{-1}\}$$||\cdot||_{-\mathit{2}}$$\{\phi_{-2}(7\mathfrak{s}\cdot-2(f))|f\in\overline{\mathcal{H}}_{-2}\}$$[\overline{\mathcal{H}}_{-2}]$$[\mathcal{H}_{-1}]$$\int_{Q_{\mathrm{J}}}e^{j_{(}}p_{f(1f}\gamma J))d\}l_{j}=\rho^{-\frac{1}{4}||\mathrm{r}_{J}}’.j1f\rangle||^{2}$,$j=-1,$$-\underline{9}$,where $(Q_{-1}, d\mu-1)$ and $(Q_{-2}, d\mu-2)$ denote the underlying measure spaces of these processes,
has the orthogonal decomposition
respectively. It is well known that$L^{2}(Q_{j}, d\mu_{j})$$L^{2}(Qj, (l \mu j)=\bigoplus_{n=0}^{\infty}\mathrm{r}n([\overline{\mathcal{H}}j])$with$\Gamma_{0}([\overline{\mathcal{H}}_{j}])=\mathbb{C}$,$\Gamma_{n}([\overline{\mathcal{H}}_{j}])=L\{:d_{j(\pi}j(f1))\phi j(7\mathrm{i}^{-}j(f2)\mathrm{I}\cdots\phi j(7\tau j\langle f_{n})) :|f_{k}\in\overline{\mathcal{H}}_{j}, k=1, .., n\}^{-}$,$n\geq 1$,

closure in
where ... denotes the linear spall of the vectors in ... overby
. Suppose
and:. : the “Wick product” ([4]). We denote the complexifications of. Corresponding to each such we cantooperator fromthat is a: $L^{2}(Qi;d\mu_{i})arrow L^{2}(Q_{j\}d\mu_{j})$ by
define the contraction operator
$L\{$

$\{$

$\}$

$\}$

$L^{2}(Q_{i}, d\mu j)$$-\mathrm{t}\mathrm{h}\mathrm{e}$$\mathbb{C},$$[\overline{\mathcal{H}}_{j}]$$T$$[\overline{\mathcal{H}}_{i}]_{\mathrm{C}}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}_{1}\cdot \mathrm{a}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$[\overline{\mathcal{H}}_{j}]_{\mathrm{C}}$$T$$[\overline{\mathcal{H}}_{j}]_{\mathbb{C}}$($\Gamma(T)$$\Gamma(\tau)\Omega.i$

$=$

$0$

,

$\mathrm{I}^{\urcorner}(T):(\mathrm{i}’ i(\pi_{i}(.f1))\ldots\varphi^{!}j(7\mathrm{i}-(if\prime^{\wedge}p)):

For a nonnegative self-adjoint operator= :\phi_{j}(\tau\pi_{j(}f_{1}))\varphi_{j}(\tau_{\pi_{j(.f_{2}))}}\ldots\phi j(T\pi_{j(}fn)):$$A:[\overline{\mathcal{H}}_{i}]_{\mathrm{t}\mathrm{C}}arrow[\overline{\mathcal{H}}_{i}]_{\mathbb{C}}(i=-1, -2)$

we define$d\Gamma(A)\Omega_{\mathrm{i}}$

$=$

$0$

,

:

$=$

:

$\phi_{i}(A\pi_{i}(fi))di(7\tau i(f_{2}))\ldots(b_{i}(T_{i(}fn))$$d\Gamma(A):\phi i(\pi_{i(.f_{1}})\rangle\ldots\varphi’i(7\ulcorner i(.fn))$.

$d\Gamma(A)$:$+:\phi i(\pi i(fi))\phi i(A\pi i(f2))\ldots\phi_{i}(7\mathrm{i}^{-}i(fn))$:$+\ldots+:\phi_{\dot{\mathrm{t}}}(T_{i(f))\phi i}1(\pi_{i}(.f_{2}))\ldots\phi_{i}(A_{7}\Gamma_{i}(.f_{n})):$$\pi_{i}(.f_{k}.)\in D(\dot{A}),$ $k=1_{\backslash }\ldots.n$$L^{2}(Q_{i}, d\mu_{i})$$L^{\mathit{2}}(Qi;d\mu_{i})$$L^{2}(Q_{-1}, d\mu_{-}1)=\mathcal{F},$$L^{2}(Q_{-2\cdot l^{\mathrm{t}}-}d2)=\mathcal{E},$$\phi_{-1}(\cdot)=\phi_{F}(\cdot),$$\Omega_{-2}=\Omega_{\mathrm{c}^{c}}$$\mathcal{F}^{N}.=\bigoplus_{0\}\iota=}^{\lrcorner}\Gamma_{\mathit{7}\mathrm{t}}([\overline{\mathcal{H}}_{-1}\backslash \tau])\bigoplus_{n>N+1}\{0\}$,,. It is well known that
denotes the collstant, function 1 in
. We denote it by the same symbol
unique self-adjoint extension in
and
set
. Put
and
$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{l}\cdot \mathrm{e}\Omega_{i}$by$\phi_{-2}(\cdot)=\phi_{\mathcal{E}}(\cdot)$

has
. We

$d\Gamma(A)$$d\Gamma(A)$$\Omega_{-1}=\Omega_{F}$96

and define$\mathcal{F}^{\infty}$by.

$F^{\infty}=\cup \mathcal{F}^{N}\mathrm{N}=0\infty$

The standard Boson Fock spaceover$([28,\mathrm{X}.7])$,$\mathcal{W}=\frac{L^{2}(\mathrm{P}^{d})\oplus\ldots\oplus l^{2}(\mathrm{F}\mathrm{I}d}{d-1}$

is defined by([2,3.16]),$\mathcal{F}(\mathcal{W})=\bigoplus_{n=0}^{-}\mathcal{F}_{n}^{\cdot}(\mathcal{W})\infty$$n\geq 1$$F_{71}(\mathcal{W}’)=\otimes_{s}^{n_{\mathcal{W}}},$,$F_{0}=\mathbb{C}$

,

where
denotes the -fold symmetric tensor product. The vacuum vector
defined by$n$$\otimes_{s}^{n}$$\Omega$in$\mathcal{F}^{\cdot}(\mathcal{W})$is.

$\Omega=\{1,0,0, \ldots.\}$The Boson Fock spacedescribes a Hilbert space of state vectors for the quantizedradiation field with the Coulornb gauge. Let$F(\mathcal{W}’)$$. \mathcal{F}^{N}.(\mathcal{W})=\bigoplus_{n0}f\mathrm{v}=i\mathrm{r}_{n}(\mathcal{W})\oplus\{\mathrm{o}\}n>N+1^{\cdot}$

Then a finite particle subspace is defined by$F^{\infty}( \mathcal{W}\mathrm{I}=\bigcup_{N=}\infty 0^{\cdot}F^{N}(\mathcal{W}\mathrm{I}\cdot$

The annihilation operator $a(f)$ and .th.e
operator( $f\in$ W) ([25]) act on
the finite particle subspace and leave lnvarlant with the canonlcal commutation relations(CCR): for $f,$$0^{\mathrm{t}_{(.f)}}.$$\mathrm{c}\mathrm{r}\mathrm{e}.\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$$1\mathrm{t}$$g\in \mathcal{W}$$[Cl(.f), a\uparrow_{(}]g)--$$[a^{\mathrm{J}}(.f), a(\# g)]$where $[A, B]=AB-BA,$$\langle a(\dagger f)\Phi,$$a^{\#}$

denotes either$\Psi\rangle_{F(w)}=\langle\Phi,$$\langle\overline{f},$

$0$

$=$

$a$

or

$a^{\mathrm{t}}$$c\iota(\overline{f})\Psi\rangle_{\mathcal{F}(\mathcal{W}})$$g\rangle_{\mathcal{W}}$,,. Furthermore,,$\Phi,$$\Psi\in F^{\infty}(\mathcal{W})$.

, the ‘second quantization of ”
For any contraction operator
is a bounded operator uniquely determined by$\zeta$$A$$A:\mathcal{W}’arrow \mathcal{W}$$\Gamma_{B}(A):F(\mathcal{W})arrow \mathcal{F}(\mathcal{W})$$\Gamma_{B}(A)\Omega$$\Gamma_{B}(A)a^{\mathrm{t}_{(}}f1)a^{\mathrm{t}_{(f_{2}}\mathrm{t}_{(f_{n})\Omega}})\ldots a$

,

$=$

$0$

$=$

$a^{\mathrm{t}_{(Af_{1}\mathrm{I}\mathit{0}}\dagger\dagger}(A.f2)\ldots a(Af_{l\mathit{1}})\Omega$.

97

For a nonnegative self-adjoint operator in
defined by the infinitesimal generator of the$\sigma$$\mathcal{W}$$C_{0}$, the “second quantization of ”,-semigroup$\Gamma_{B}(c^{-t})\sigma e^{-}=td\Gamma B\mathrm{t}\sigma)$

We define the maximal multiplication operator$d\mathrm{r}_{B}(\sigma)$$\sigma$.

in$\omega_{B}$. is$\{\Gamma_{B}(e^{-t}\sigma)\}_{\geq 0;}\mathrm{f}$by$L^{2}(\mathrm{P}^{d})$$(_{\mathrm{L}\bigvee_{B}}’.f)(k\mathrm{I}=h(k)f(k\mathrm{I}\cdot$

where $h(k)=|k|$ . Put. Then$d\Gamma_{B}(\tilde{\omega}_{B})$$\tilde{\omega}_{B}=\frac{\omega_{B}\oplus\ldots\backslash I\mathrm{a}\mathrm{e}\omega_{B}}{(i-1}$

will be the free Hamiltonian of

the quantized radiation field. The second quantization of the identity operator, is called the number operator. The following inequality is well known$I_{\mathcal{W}}$on$\mathcal{W}$,$d\Gamma(I_{\mathcal{W}})$$||a^{\#}(f)\Phi||_{\mathcal{F}}1w)\leq||f||_{\mathcal{W}}\cross||(d\Gamma(Iw)+I)^{\frac{1}{2}}\Phi||_{f1w)}$

For$f\in \mathcal{H}_{-1}$,$\Phi\in \mathcal{F}^{\infty}.(\mathcal{W})$.

(3. 1)

we define the -th direction time-zero radiation field $A_{\mu}(f)(\mu=1, \ldots, d)$ by$\mu$(3. 2)$A_{\mu}(f)= \frac{1}{\sqrt{\underline{9}}}\{(\iota^{\uparrow}(L\mathrm{f}_{r=1}^{d1}’-\frac{e_{\mu}^{r}\hat{f}}{\sqrt{h}}\mathrm{I}+a(\oplus_{r1^{\frac{e_{\mu}^{r}\hat{f}\backslash }{\sqrt{h}}}}^{d-1}=1\}\tau$where $\hat{g}(k)=g(-k)$ . For$g=(g_{1}.

\ldots.g_{d})\in\overline{\mathcal{H}}_{-1}$

we put$A(.q) \equiv\sum_{\mu=1}^{d}A_{\mu}(g\mu)$by

We give connection between $F$ and$F(\mathcal{W}\rangle$.

. Here we introduce the subspace$D_{0}=L\{f^{r}=(f^{r}1, \ldots, fd|.)\in\overline{\mathcal{H}}_{-}1|f_{\mu}^{r}=(e_{\mu}^{r\sqrt{h}}\hat{f})^{\vee},\hat{f}\in c^{-}\mathrm{Y}0(\infty \mathbb{R}^{d}\backslash \{0\}),$

where
on

$\ldots,$in$\overline{\mathcal{H}}_{-1}$$d-1\}$ ,denotes the set of infinitely differentiable functions with compact support
is dense inwith respect to the-norm, which implies thatis dense in. Hence$C_{0}^{\infty}(\mathbb{R}^{d}\backslash \{\mathrm{o}\})$$\mathbb{R}^{d}\backslash \{0\}$$|\cdot|_{-1}$$r=1,$$D_{0}$. Then it can be easily seen that$\pi_{-1}(D_{0})$$L\{:\phi_{f}(\pi_{-1} (.f_{1}))\ldots\phi_{\mathcal{F}}(T_{-1}(.f_{1\iota})) :$A(p^{r})$$\mathrm{s}\mathrm{e}\mathrm{n}\dot{\mathrm{u}}$$[\overline{\mathcal{H}}_{-1}]$\Omega \mathcal{F}, \Omega_{\mathcal{F}}|f_{j}\in D_{0},j=1, \ldots, n, n\geq 1\}$is dense in F. OI1 the othel hand, choosing
out that$\cdot$$\overline{\mathcal{H}}_{-1}$$D_{0}$$p^{r}=((e_{1}^{r_{\sqrt{h}}}\hat{p})\vee,$$(e_{d}^{r}\sqrt{h}\hat{p})^{\mathrm{v}})\in D_{0}$$\ldots,$$=$$\sum_{\mu=1}^{d}A(\mu(\epsilon_{\mu}r\sqrt{h}\hat{p})\mathrm{v})$$=$$\frac{1}{\sqrt{2}}\{o^{\uparrow}(^{\frac{d-1}{0\oplus\ldots\oplus\hat{p}\ldots\oplus \mathrm{o}\vee^{\oplus}\uparrow\cdot-th}})+a(^{\frac{d-1}{0\oplus\ldots\oplus\hat{p}\oplus\ldots\oplus\vee r-\sim th0}},)\}$, it turns.

98

Then we see that
$L$

$\{:

A(f_{1})\ldots A(.fn) :\Omega, \Omega|f_{j}\in D_{0},j=1, \ldots, n.n\geq 1\}$is dense in
where:. : denotes the “WickWe define the operator inbv$\mathcal{F}(\mathcal{W}),$”$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}$$\overline{\omega.f}(k)=h(k).f\hat{\backslash }(k)$and put$\tilde{\omega}=\frac{\omega\oplus\ldots\oplus\omega}{d}$.

$\mathrm{F}\mathrm{u}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{l}\mathrm{l}01^{\cdot}\mathrm{e},$$[\tilde{\omega}]\pi-1(f)=\pi_{-}1(\tilde{\omega}.f)$

Extend:$[\tilde{\omega}]$

Then it isin$\mathrm{e}\mathrm{a}\underline{\mathrm{s}\mathrm{y}}$$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{l}\cdot \mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}$.

$I_{\mathcal{F}}$to see that Ran$([\tilde{\omega}]\pm i)=[\overline{\mathcal{H}}_{-1}](\mathrm{c}$.

is defined by$f)\in[\overline{\mathcal{H}}_{j}]|\tilde{\omega}f\in\overline{\mathcal{H}}_{-}1\}$$f_{j}\in D_{0},$$\mathcal{U}$.

, which implies that.

$[\tilde{\omega}]$

is a self-adjoint.

$j=1,$$ope\uparrow\cdot ato\Gamma \mathit{1}\mathit{4}$from$\mathcal{F}(\mathcal{W})$$\mathcal{U}\Omega=\Omega_{\mathcal{F}}$$(b)$$\mathcal{U}A(f)l\mathit{4}^{-1}=\phi_{f(}f)$$(c)$$\mathcal{U}\mathcal{F}_{n}(\mathcal{W})=\Gamma_{n}([\overline{\mathcal{H}}-1])$$(d)$$\mathcal{U}d\Gamma_{B}(\tilde{\omega}_{B})u-1=d\Gamma([\tilde{\omega}])$

,

$(e)$

$\mathcal{U}\mathrm{r}l\mathrm{r}_{B}(Iw)\mathcal{U}-1=d\Gamma(^{[_{F})}$,$\uparrow?$$\ldots,$$[\overline{\mathcal{H}}_{-1}]$

to $F$ such that,$(Cl)$

is the identity operato 7 in

Proof: For$[\tilde{\omega}]:[\overline{\mathcal{H}}_{-\iota}]arrow[\overline{\mathcal{H}}_{-1}]$$D([\tilde{\omega}])=\mathrm{f}^{\pi_{-1}}($

Theorem 3.1 Ihere exists a unitary

where

,

$.f\in\overline{\mathcal{H}}_{-1}$

,

,

.

, we define: $A($ .$f_{1})\ldots A(fn):\Omega$$\mathcal{U}\Omega$

$=$

$=$

:

$\phi_{f}(\pi_{-}1(f1))\ldots\phi_{\mathcal{F}}(\pi-1(fn)):\Omega_{F}$$\Omega_{\mathcal{F}}$,.

One can easily show can be uniquely extended to a unitary operator fromwith $(a),(b)$ and $(c)$ . We shall show $(d)$ . Let$\mathcal{U}$$X_{n}$$=$$L\{:\varphi’\mathcal{F}(\pi-1(f_{1}))\ldots\phi F(\pi_{-}1(fn)) :$Y_{n}$$=$$L$

Since, as long as$\{:

A(f_{1})\ldots A(f_{\eta}) :$\hat{\rho}\in C_{0}^{\infty}(\mathrm{P}d\backslash \{0\})$leavesX.49]). Moreover, since$\exp(-td\mathrm{r}_{B}(\tilde{\omega}_{B}))$.

,$(\pi_{-1}(.f_{1}), T-1(f2))=([\tilde{\omega}]\pi_{-1}(f1), [\tilde{\omega}]\pi-1 (.f_{2})),$ $.fi,$ $f_{2}\in\overline{\mathcal{H}}_{1}$$[\mathcal{H}_{-1}]_{\mathbb{C}}$$([\mathit{2}5,\mathrm{p}\mathit{2}26])$

as follows:$[\overline{\mathcal{H}}_{-1}]_{\mathrm{t}\mathrm{c}}arrow[\overline{\mathcal{H}}_{-1}]_{\mathrm{t}\mathrm{C}}$$[\tilde{\omega}]$in the Boson Fock space$\mathcal{H}_{-1}$$\omega$$\bigcup_{n=0}^{\infty}\}_{n}^{\overline{\prime}}$\Omega_{\mathcal{F}}|fj\in D_{()},j=1, \ldots, n\}$\Omega|f_{j}\in D_{0},j=1, \ldots, n\}$, it follows that$\bigcup_{n=0}^{\infty}Y_{n}$to$\mathcal{F}^{\cdot}$,.

$\exp(-th)\hat{p}\in C_{0}^{\prime\infty}(\mathrm{F}^{d}\backslash \{\mathrm{o}\})$

invariant. Hence$\mathcal{F}^{\cdot}(\mathcal{W})$is a core of$\mathcal{U}\exp(-td\mathrm{I}^{\mathrm{t}}B(\tilde{\omega}_{B}))\mathcal{U}^{-1}=\exp(_{-t\mathcal{U}d\Gamma_{B}(}\tilde{\omega}B)u^{-1}\mathrm{I}$, one can see that$d\Gamma_{B}(\tilde{\omega}_{B})([\mathit{2}5,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}$,99

it follows that$\bigcup_{n=}^{\mathrm{o}\mathrm{c}}’ 0^{X_{n}}$is a core of$\iota \mathit{4}_{\mathrm{C}}l\Gamma_{B()\mathcal{U}^{-1}}\tilde{\omega}B$. Noting that on$\mathcal{U}d\Gamma_{B}(\tilde{\omega}B)\mathcal{U}^{-1}=d\Gamma([\tilde{\omega}])$.

Thus $(d)$ holds. The proof of $(e)$ is similar to that of $(d)$ .
. Following [We setand . For
we define the operator by
between$\mathrm{H}_{0}=d\Gamma([\tilde{\omega}]),$$\mathcal{F}^{\cdot}$$\mathrm{N}=d1^{\mathrm{t}}(I_{\mathcal{F}})$$j_{f}$$\delta_{t}$:$\mathcal{H}_{-}1-arrow \mathcal{H}_{-2}$,,$f\in \mathcal{H}_{-1}$.

is the one-dimensional delta function with mass at$( \overline{j_{t}.\mathrm{f}})(k, k_{0})=arrow(2\pi)-\frac{1}{2}\hat{f}(\vec{k})e-itk0$

whereIII], we can give connection$j_{t}$$j_{t}f=\delta_{t}\otimes f$

where$\square$$26,\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{t}\mathrm{e}\mathrm{r}$$t\in \mathrm{P}$$\mathcal{E}$$\bigcup_{n=0^{X_{n}}}^{\infty}$$(\vec{k}, k_{0})\in \mathrm{F}_{-}^{d}’\cross \mathrm{F}_{-}=\mathrm{F}^{d+1}’$. We put$[j_{t}]\sim$$Jt\sim=j_{t}\oplus\ldots\oplus j_{t}$:$\{t\}$. In momentum space,,and define,$[\overline{\mathcal{H}}_{-1}]arrow[\overline{\mathcal{H}}_{-2}]$$[jt\sim]\pi-1(f)=\pi-2(^{\sim}jtf)$.

III.2]). Hence the range
It can be easily seen thatis a linear isometry (. We denote the projection onto Ranbyofis a closed subspace of. Let$[i_{t}]\sim$$[‘ \mathit{2}6,\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$([j_{t}\sim])$$[\overline{\mathcal{H}}_{-2}]$$[\dot{\gamma}_{t}]\sim$$|_{arrow}\mathrm{r}_{[\alpha,]}b\equiv L\{\pi_{-2}(f)\in[\overline{\mathcal{H}}_{-2}]|\pi_{-2}(f)\in Ran([^{\sim}jt]),$ $a\leq$\mathrm{W}^{\tau}\mathrm{e}$denote the projection onto the closure

Proposition 3.2 ([26.
$(a)[j_{t}][j_{t}]^{*}\sim\sim=[e_{t}]$by$[e_{[a,b]}]$.

lII.3 and III. )$\mathit{4}f$.

Let $a\leq b\leq c$ . Then$[\epsilon_{a}][e_{b}][e]c=[e_{a}][e_{\mathrm{C}}]$

$(d)$

.

.

$(b)[j_{t}]^{*}[j\sim\sim s]=e-|t-s|[^{\sim}\omega]$$(c)$$P\uparrow\cdot opoSiti_{\mathit{0}}ns$$\overline{U_{1]}a,b}$t\leq b\}$$[e_{t}]$.

Let $a\leq b\leq t\leq c\leq d$ . Then$[\epsilon_{[b]}]a,[\not\in \mathrm{i}t][e[C,d]]=[e_{[a,b]}][e[c,d]]$.

Proof: (a) is straightforwardly seen. Since we have$\langle[j_{t}]^{*}[js]\pi-1(f).\pi_{-}1(g)\rangle_{-1}\sim\sim$

$=$

$=$

$=$

$\langle\pi_{-2}(j_{S}f),$$\pi-2(jtg)\rangle\sim\sim-2$$\frac{1}{\pi}.\sum_{\mu_{U=}1}^{d}\int \mathrm{R}^{d+}1\frac{\overline{\hat{f}}_{\mu}(\vec{k})\hat{g}_{\nu}(karrow)d\nu(\mu\tilde{k})ei(t-S)k\mathrm{o}}{|\vec{k}|^{2}+k_{0}^{2}},d\vec{k}dk0$,$\sum_{\mu_{U}=1}^{d}\int \mathrm{R}^{d}\frac{\overline{\hat{f}}_{\mu}(k^{\eta})\hat{g}_{\mathcal{U}}(k^{\triangleleft}arrowarrow)d_{\mu}\iota \text{ノ}(\vec{k})\epsilon^{-}|t-S||\vec{k}|}{|\vec{k}|}d\vec{k}$

,

100

(b) holds. Eq.(c) follows from (a) and (b). For any $\pi_{-2}(f)$ and $\pi_{-2}(g)$ , by the definition ofand$[e_{[a,b]}]$$[e_{[c,d]}]$, they can be presented as follows$[e_{[c.d]}]\pi-2(f)$$=$$[e_{[a.b]}]\pi_{-2}(g)$$=$$f_{n_{\alpha}}\in Ra\uparrow?([e_{t,)\alpha}]),$$\etaarrow\infty^{1}]\mathrm{i}_{11}(\sum_{\gamma=1}^{\mathrm{J}}.f\prime _{n}n\alpha$’$\lim_{marrow\infty},\sum^{1f_{m}}.f_{1?}\mathit{1}\mathit{1}\mathit{3}=1\beta$$t_{n\alpha}\in[c, d]$,$g_{n\iota_{\mathit{3}}},\in Ro|?([et,n_{\mathrm{L}};]\mathrm{I}\cdot f_{711}e\in[a, b]$’.

Hence by (c) we have$\langle[e_{1^{a,b}]}][et][e_{[}C,d]]T_{-}2(f),$ $\pi-2(g)\rangle_{-2}$$=$$\lim_{n,marrow\infty}\alpha\beta\sum_{=}^{N_{l}.’$=$$\lim_{n,marrow\infty},\sum_{\alpha,\beta 1}^{N_{n}}’ M_{1n}=\langle f_{n}\alpha’ gm_{\partial}\rangle-2$$=$$\langle[e_{[a.b]}][e_{[C},d]]\pi-2(f),$ $\pi-2(g)\rangle_{-2}$M_{m}}\mathrm{t}1\langle[\epsilon t]f_{n}\alpha’ g_{m_{\beta}}\rangle-2$

Then (d) follows.
We introduce notations;$\square$$\Gamma([e_{[,b]}]q)$

Proposition 3.3 . ($(a)J_{t}$is a linea’$(b)J_{t}J_{t}^{*}=E_{t}$$\iota som\mathrm{f}^{\supset}t’\cdot y$.

Let$\Sigma_{[a,b]}$$\equiv$$E_{[a,b]\}$$\Gamma([j_{t}])\sim$$\equiv$$J_{t}$$\Gamma([e_{t}])$$\equiv$$E_{t}$,.

(3. 3)

III.5])
from to .

$\tau h\subset ore?n$$\cdot$$(c)J_{t}^{*}J_{s}=e-|t-s|\mathrm{H}\mathrm{o}$$(d)$$[_{\sim}^{t)}\mathit{6},$.

$F$$\mathcal{E}$.

denote the a-algebra generated by$L\{\phi\epsilon(\pi-2(f))|\pi_{-2}(f)\in U_{[a,b]}\}$and the set of$\Sigma_{[a,b]}$-measurable functions in
Ran

$(e)$

(

$\mathrm{w}_{a}rkoff$$\mathcal{E}$by$\mathcal{E}_{[a,b]}$$(E_{1^{a,b}]})=\mathcal{E}_{[a,b]}$. Then.

property) Let $a\leq b\leq t\leq c\leq d$ . Then
$E_{[a.h]}\mathrm{A}_{t}^{1}E[C,d]=E_{[a,b}]E_{[C,d]}$$\mathrm{I}\mathrm{I}\mathrm{I}.8]^{f.\cdot \mathrm{s}}Proo.\mathrm{E}\mathrm{q}.(\mathrm{a}),(\mathrm{b}),(\mathrm{c})$and (e) follow$\mathrm{f}\mathrm{i}\cdot \mathrm{o}\mathrm{m}$.

Proposition 3.2. Eq.(d) follows fromAs in [26], a FKN formula follows from Proposition 3.3.

$[\mathit{2}6,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}\square$

101

Proposition 3.4 ([26, Theorem III.
bounded measurable functions on . Let$\mathit{6}f,$$FI\mathrm{i}’N$$t_{1},$$\ldots,$$s_{0}$

and$G_{0},$$G_{k}$$\ldots,$be$t_{k}\geq 0$$(_{\tau^{t}}e-t_{1}\mathrm{H}_{0}G\prime \mathcal{F}f\ldots-et_{k}\mathrm{H}0G_{k}\mathcal{F}\Omega_{F}\rangle_{\mathcal{F}}01$$=\langle\Omega\epsilon\cdot\subset_{7}^{\vee S_{0}}\ldots G^{S_{\dot{k}}}k\Omega \mathcal{E}0\rangle_{\mathcal{E}}$where$f_{n}\in\overline{\mathcal{H}}_{-1}$$f_{1},$$\ldots,$$\mathrm{F}^{d}’$$\langle\Omega_{F},$formula) Let
be given. Then,is arbitrary and$s_{j}$$G_{j}^{\mathcal{F}}$$G_{j^{j}}^{\mathrm{Y}}s$,$=$$s_{0}+ \sum_{i=1}^{j}t_{i}$$=$$G_{j}(\phi_{F}(\pi_{-1}(f_{1})), \ldots, \phi\tau(\pi-1(fn)))$$=$$G_{j}(\phi e(\pi_{-}2(j_{s_{j}}f1\sim)),$,$\phi\epsilon(\pi-2(j_{s_{J}}\sim.f_{n})))$$\ldots,$.
$\square$

The Hilbert space of state vectors in the system of a non-relativistic charged particle. The unitaryinteracting with . quantized radiation field is given by
Theorem 3.1 implements unitary equlvalence between
andoperator given$\mathcal{M}_{B}.=L^{2}(\mathbb{R}^{d})\otimes F(\mathcal{W})$$\mathrm{a}$$\mathcal{U}$$\mathcal{M}_{B}$$\ln$$\mathcal{M}=L^{2}(\mathrm{R}^{d})\otimes \mathcal{F}^{\cdot}\cong\int_{\mathrm{R}^{d}}^{\oplus}\mathcal{F}d_{X}$For an$\mathcal{H}_{-1}$-valued function on$\mathrm{p}_{\mathrm{s}’\oint)}^{d}(\cdot)$:$\mathrm{R}^{d}arrow \mathcal{H}_{-1}$, we put$\tilde{p}_{\mu}(\cdot)=\frac{d}{(0,\ldots,\rho(,\vee\mu-t)_{\backslash }\ldots,0)h}.$Then we define an operator in$\mathcal{M}$

.

.

by

$\phi_{\mathcal{F},\mu}^{\rho}=I^{\phi((}\mathrm{R}d)\oplus F\pi_{-1}p_{\mu}\sim(_{X)})d_{X}$.

$=1, \ldots, d)$ be the generalized-derivative in the -direction. Then the interaction
Let.
Hamiltonlan of the non-relativistic charg.ed particle with mass 1 and the quantized radiation”byfield isgiven as an operator$D_{\mu},$$(\mu$$‘\zeta \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$$L^{2}$$\mu$$\ln \mathcal{M}$$\mathrm{H}_{\rho}=\frac{1}{2}\sum_{\mu=1}^{d}(-iD_{\mu}\otimes I-\phi_{\mathcal{F},\mu}^{\rho})^{2}+I\otimes \mathrm{H}_{0}$

.

”
Here
means that we mention nothing about the domain of
definition will be given in the following section. We set$‘\zeta \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$$\mathrm{H}_{\rho,0}=\frac{1}{2}\sum_{\mu=1}(-iD_{\mu}\otimes I-d\phi_{\tau}\rho,)^{2}\mu$(3. 4)$\mathrm{H}_{\rho}$. The precise

102

We conclude this section with giving a typical example of the
can take$p(.\iota\cdot)=(\hat{f}(\cdot)e^{i\cdot x})^{\vee}$$\mathcal{H}_{-1}$-valued function. One(3. 5),. In thisis given by ([2.4.23])

where.f is a real-valued rapidly decreasing infinitely differentiable function oncase, the corresponding standard Boson Fock space element$\mathcal{U}^{-1}\phi f(\tilde{p}_{\mu}(x))\mathcal{U}$$p(\cdot)$$A(\tilde{\rho}_{\mu}(x))$$\mathrm{F}^{n}’$,$=$$A(\tilde{p}_{\mu}(x))$$=$$\frac{1}{\sqrt{2}}\{a^{\uparrow}(\oplus_{r1}^{d-1}=.\frac{e_{\mu}’\hat{f}e^{-}i\cdot x}{\sqrt{h}})+a(\oplus_{r1^{\frac{\epsilon_{\mu}^{r}\hat{\grave{f}}e^{i\cdot f}}{\sqrt{h}}}}^{d-1}=1\}\cdot$

Then the function serves as an ultraviolet cut-off function for photon momenta. Moreover,satisfies the Coulomb gauge condition (see (4.17)).
$f$$p_{\mu}(\sim x)$

4

FUNCTIONAL INTEGRALS

given formally by (:;.4) and derive
In this section we construct a self-adjoint extension of
for the heat semigroup associated with it. The main ideaa functional integral.
is to apply the FKN formula and the FKI formula
is a self-adjoint operator for each-valued function
For an, we can define a unitary operator
as a multiplication operator in . Then, for each
on $F$ by$\mathrm{H}_{\rho}$$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{I}1$$([31.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{r}\mathrm{n}15.3])$$\mathcal{H}_{-1}$$p,$$\mathcal{F}$$\int_{\rho}^{r}’(x, y)$Let$p_{s}(x)$$x\in \mathbb{R}^{d}$$\phi_{\mathcal{F}}(\pi_{-1}(\rho_{\mu}(\sim x)\mathrm{I})$$x,$$y\in \mathrm{P}^{d}$$\equiv$$\exp\{\underline{.\frac{1}{)}}i’(l^{)}f(_{\mu=}\sum_{1}^{d}\pi_{-1}(p\mu(L\mathrm{t}’\cdot)+\tilde{\rho}\mu(y))(_{T}\mu-y_{\mu})\sim)\}$$\equiv$$\exp(6^{\rho}(_{\mathcal{I}}.y))$.

be the heat kernel function$p_{s}(X)--(2 \pi.\mathrm{s}\mathrm{I}-\frac{d}{2}\exp(_{-\frac{1}{2_{S}}1}X|2),$$\backslash \mathrm{s}>0,$

Then we define a family of the contractive self-adjoint operators

where
see that$F(\cdot)\in \mathcal{M}$$(Q_{\rho,s}F)(x)$$=$$\int_{\mathrm{R}^{d}}p_{S}(x-y)U_{\rho}(X, y)F(y\mathrm{I}^{dy},$$(Q_{\rho,0}F)(x)$$=$$F^{\urcorner}(x)$and the integral is the$||Q_{\rho_{9}},F||_{\mathcal{M}}$$x\in \mathrm{p}_{\mathrm{c}}^{d}$.

$\{Q_{\rho,s}\}_{S\geq 0}$on$\mathcal{M}$by$s>0$ ,,$\mathcal{F}$-valued Bochner integral. Actually one can easily$\leq$$||e^{-s\mathrm{t})}- \frac{1}{2}\triangle||F(\cdot)||F||L^{2}1\mathrm{R}^{d})$$\leq$$||F||_{\mathcal{M}}$

.

103

Let
$[C_{b}^{n}(\mathbb{R}^{d};\mathcal{H}-1)]=\{\rho(\cdot)$:$\mu=1,$$\mathrm{P}^{d}arrow \mathcal{H}_{-1}|\pi_{-1}(^{\sim}p_{\mu}(\cdot))\in C^{n}b(^{\mathrm{p}^{d}};[\overline{\mathcal{H}}_{-1}]),$

inWe define a subspace
At if and only if the following

as follows. For
hold$d^{\vee[}$$\mathcal{M}_{\rho}^{\infty}$(i)(ii) For each$F(\cdot)\in H^{2}(^{\mathrm{p}^{d}};\mathcal{F}^{\cdot})$$y\in \mathbb{R}^{d}$$\rho\in[c_{b}^{1}(\mathbb{R}^{d};\mathcal{H}_{-1})]$$d\}$$\ldots,$, we say that.

$F\in \mathrm{A}4_{\rho}^{\infty}\subset$$(\mathrm{i})-(\mathrm{i}\mathrm{i}\mathrm{i})$.

,$F^{\urcorner}(y)\in \mathcal{F}^{\infty}$,$\partial_{\mu}F^{1}(y)\in\tau^{\infty}$(iii) (Integration by parts condition) For all,$G\in \mathcal{M},$$t^{\iota=1,\ldots.d}$$x\in \mathrm{P}^{d}$(see Lemma 4.3),$\lim_{yarrow\infty}\partial_{y_{\mu}}ps(x-y)\cdot\langle F(y), U_{\rho}(X, y)G(X)\rangle F=0$,,$\lim_{yarrow\infty}p_{s}(x-y)\cdot\partial_{y\mu}\langle F(y), U_{\rho}(x, y)c\mathrm{T}(X)\rangle_{f}=^{0}$

Lemma 4.1 Let
tiable in $s>0$ ?vith$\rho\in[C_{b}^{2}(\mathbb{R}^{d};\mathcal{H}-1)]$.

$G\in \mathcal{M}$, and$F\in \mathcal{M}_{\rho}^{\infty}$.

$\mu=1,$. Then$d$$\ldots,$.

$\langle Q_{\rho,s}F, G\rangle_{\mathcal{M}}$is differen(4. 1)$\mathrm{l}\mathrm{i}\ln\underline{d}\langle Q_{\rho,s}F, G\rangle_{\mathcal{M}}=\langle-\mathrm{H}_{\rho,0}F, G^{l}\rangle \mathcal{M}^{\cdot}$$sarrow 0+ds$$\square$, analogue of (4.1) for the Schr\"odinger HamiltoFor the classical cases
constructions of path integral representations.
forimportantpotentials
isvectorwithnian
In the same way as in the classical case. however, (4.1) can not be proven directly. To verify(4.1), we prepare two fundamental lenunas (Lemmas 4.3 and 4.4) as follows. Notice that for$F,$,$([31,\mathrm{L}\mathrm{e}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{a}15.1])$$G\in \mathcal{M}$$\langle Q_{\rho.s}F, G’\rangle_{\iota 4}d=\int_{\mathrm{R}^{d}}dx/R^{d}..p_{s}(x-y)\langle U_{\rho}(x, y)F(.y).

Fubini’s lemma allows one to interchange$\int dx$c_{\mathrm{j}r}(X)\rangle fdy$.

and $fdy$ . Moreover. we have$\int_{\mathrm{R}^{d}\cross \mathrm{R}^{d}}dXdy|\frac{d}{ds}p_{S}(x-y)\langle U(\rho yX,)F(y),$$G(_{X)\rangle}F|$$\leq\int_{\mathrm{R}^{d}}(\frac{d}{\underline{9}s}+.\frac{x^{2}}{2s^{2}})p_{s}(x)dx||F||\mathcal{M}||G||_{\mathcal{M}}<\infty$,

and the integral $fdxdy$ . The following proposo that we can interchange the differential
sition is fundamental.
which is conmeasurable bounded function on
Proposition 4.2 (I) Let be atinuous at . Then$\frac{d}{ds}$$\mathrm{p}_{\mathrm{s}}^{d}$$f$$Lcb_{\mathit{6}}sg¿ e$$0$$\lim_{sarrow 0+}\int_{\mathrm{R}^{d}}p_{s}(x)f(x)dx=f(0)$(II) For any$\mathfrak{a}>0$$\lim_{sarrow 0+}\int|.\gamma\cdot|^{\alpha}pS(X)dx=0$

.

.

104

Proof: Elementary calculations.

$\square$We introduce notations and estimates. For$\phi_{\mu}^{\rho()}\dotplus^{n}(X, y)+\phi_{\mu-}\rho,(n)(x, y)$$\phi_{\mu}^{\rho,(n)}(_{X}, y)$$\equiv$$\phi_{\mu}^{\rho(n)}\dotplus(x, y)$$\equiv$$\phi_{\mu-(}^{\rho,(n)}x1y)$$\equiv$

Note that,$\frac{\dot{i}}{2}\phi_{\mathcal{F}}(-n\partial_{\mu}^{n-1}T_{-1}(\tilde{p}_{\mu}(y))-\delta_{n},1\pi_{-}1(\tilde{p}\prime^{p}(X)))$$\phi_{\mu}^{\rho,(0)}(x, y)=\phi^{\rho}(x, y)$$c_{\mu,n}(\rho)$$=$$d_{\mu,n}(\rho)$$=$$\rho_{i}\in[C_{b}^{r}\backslash (\mathrm{R}^{d}\backslash ;\mathcal{H}-1)],$$\mathrm{N}\Phi=N\Phi$, we set$\frac{i}{2}\phi_{F}(_{j=1}\sum^{d}(\partial n1T_{-}(\mu\tilde{\rho}j(y))+\delta 0\pi n,-1(p\sim j(x)))(xj-yj))$. For

In the case $n=0$ , we use notations

that for$p\in[C_{b}^{r}(\mathbb{R}^{d};\mathcal{H}_{-1})]$$p\in[C_{b}^{r}(\mathbb{R}^{d};\mathcal{H}_{-1})]$$c_{\mu,0}(\rho)=c_{0}(\rho)$$0\leq n\leq r\cdot$

.

, put

and $d_{\mu,0}(\rho)=d_{0}(p)$ . From (3.2) it follows, andsuch that$0\leq k_{i}\leq r,$ $1\leq\mu_{i}\leq d,\dot{\iota}=1,$

,

,

,

$\ldots,$$\Phi\in F^{\infty}$$n$$||\phi_{\mu_{1}}^{\rho_{1},(k_{1})}(x, y)\phi\rho\mu 2’(2(k2))x,$$y\ldots\phi\rho_{n}\mu_{n}’(kn)(x, y)\Phi||_{F}$$\leq\frac{\sqrt{2^{n}}\sqrt{N+1}\ldots\sqrt{N+n}}{2^{n}}||\Phi||_{\mathcal{F}}$$\cross\Pi_{i=}^{t\iota}\{1(1+\delta 0,ki)C_{\mu_{i}},k_{\mathfrak{i}}(\rho i)|x-y|+(k_{i}.+\delta_{k,1})id_{\mu}i,k_{i}-1(p_{i})\}$

Lemma 4.3 Let$k=1,$$r-1$ . Then$p\in[C_{b}^{r}(\mathbb{R}^{d};\mathcal{H}_{-1})]$$\ldots,$. $G\in F$ and$\langle U_{\rho}(x, y)F(y), c_{\mathrm{T}}\rangle \mathcal{F}$$\partial_{y_{\mu}}\langle U_{\rho}(x, y)F(y), G\rangle_{\mathcal{F}}$

$=$

$=$

(4. 2)

such thatis -times differentiable in . In particular,$\partial^{k}F(y)\in \mathcal{F}^{\infty}$$F\in C^{r}(\mathbb{R}^{d}\backslash \mathcal{F}^{\cdot})$

$r$

,

$y$

$c_{\tau}\rangle$yf+\langle U_{\rho}(x,x,$y)\partial_{\mu}F(y), G\rangle_{f}$ ,$\langle L^{\gamma}(\rho)\phi\rho\mu’((1)yX,)F(y),$$p\in[C_{b}^{1}(\mathbb{R}^{d};\mathcal{H}_{-}1)]$$\partial_{y_{\mu}}^{2}\langle U_{\rho}(x, y)F, G\rangle_{\mathcal{F}}$.

$\langle L_{\rho}^{\int}(X, y)\partial_{\mu}^{2}F(y),$,(4. 3)$G\rangle_{\mathcal{F}}+\mathit{2}\langle U_{\beta}(x, y)\phi\rho,(1)(x, y)\mu\partial F\mu(y),$$+\langle U_{\rho}(x, y)\{(\phi_{\mu}^{\rho,(1})(_{X}, y))2\rho+\phi_{\mu}’(2)(x, y)\}F(y),$$p\in[C_{b}^{2}\mathit{1}(\mathbb{R};\mathcal{H}_{-}d)1]$$G\rangle_{\mathcal{F}}$$G\rangle_{\mathcal{F}}$,.

(4. 4). For simplicity weput. Since, by (3.2),
is the set of the analytic vectorsof theself-adjoint operator $\phi(x, y)$ , the following equation follows

Proof: Suppose that $H\in F$ such that $\mathrm{N}H=NH$ , and$\phi^{\rho}(x, y)=\phi(x, y)$$\rho\in[C_{b}^{1}(\mathbb{R}^{d};\mathcal{H})]$$\mathcal{F}^{\infty}$$\langle U_{\rho}(x, y)H, G\rangle_{\mathcal{F}}=n=0\sum^{\infty}\frac{1}{n!}\langle\phi n(_{X}, y)H, G\rangle_{\mathcal{F}}$$([\mathit{2}5,\mathrm{X}.6])$.

(4. 5)

105

One can easily derive from (4.2) that each term on thewithrespect to$\mathrm{r}.\mathrm{h}$.s.of (4.5) is differentiable with

$y_{\mu}$$(?_{y}’\langle\mu\phi^{n}(x, y)R, (_{x}’-\rangle f=\uparrow?\langle\phi^{n-1}(_{X}, y)\phi_{\mu}^{\rho}.\mathrm{t}1)(x, y)H,$ $G\rangle_{F}$,from which and (4.2) it follows that$\sum_{n=0}^{k}.\frac{1}{\uparrow\iota!}|\langle\phi n(x, ?/)\phi\rho\mu’(\mathrm{t}1)X,$$y)H,$$G\rangle\tau|$$\leq\sum_{n=0}^{k}\frac{\sqrt{N+1}\sqrt{I\mathrm{V}+\mathit{2}}\ldots\sqrt{N+n+1}}{\sqrt{\mathit{2}^{n}}n!}$$\cross(c_{\mu,1}(\rho)|x-y|+‘ \mathit{2}d_{0}(\rho))(2c_{0}(p)|X-y|)^{n}||H||f||G||\tau$.

(4. 6)

converges uniformly in the wider sense with respect
Then the left hand side
follows.
with respect toto as $karrow\infty$ . Hence the differentiability of$(4.3)$
follows.
of and the fact
From the strong
similarly.
the remaining statements can be$(1.\mathrm{h}.\mathrm{s}.)\mathrm{o}\mathrm{f}(1.6)$$\langle U_{\rho}(x, y)H, G\rangle_{\mathcal{F}}$$y$$y_{\mu}$$\partial F(y)\in \mathcal{F}^{\infty},$$\mathit{1}^{\mathrm{i}^{\urcorner}}$$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{l}\cdot \mathrm{e}\mathrm{I}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$$\mathrm{E}\mathrm{q}.(4.4)\mathrm{a}\mathrm{n}\mathrm{d}\square$$\mathrm{s}_{1}\mathrm{h}\mathrm{o}\mathrm{w}’ \mathrm{n}$$\mathrm{L}\mathrm{e}\mathrm{n}’1\mathrm{m}\mathrm{a}4.4$

Let$\rho\in[C_{b}^{r}(\mathrm{F}^{\prime;}d\mathcal{H}_{-1})],$$0\leq k_{j}\leq r,j=1,$$\ldots,$$n$$G\in r^{[},$$F\in \mathcal{M}$. Thensuch that $F(x)\in F^{\infty}$ , and$\lim_{Xarrow 0}\int_{\mathrm{R}^{d}}\langle r\prime_{\rho}(\chi, x-x)\phi^{\rho}\mu_{\iota}’-(x.’\cdot-X)\ldots d^{\rho}’(\mu_{n-}-Xx, X)F1(k_{1})(k_{n})(x-x))(GX)\rangle_{\mathcal{F}}dx$$G(x)\rangle_{\mathcal{F}}|1-dx$$= \int_{\mathrm{R}^{d}}\langle\phi_{\mu_{1}-(x,X}^{\rho,1k}1))\ldots d_{l}\rho,(\kappa,)(\mathrm{J}^{\cdot}.X)F(X),$

Proof: One can easily see that for all$x\in \mathrm{J}\mathrm{R}^{d}$$s-\backslash ’arrow 01\mathrm{i}\mathrm{n}1L^{f_{\rho}}(x,$in$\mathcal{F}^{\cdot}$. Let$K\in.\mathcal{M}$.

$x-^{x)}=I_{\mathcal{F}}$be such that $\mathrm{N}K(x)=NK(x)$ for all$x\in \mathrm{P}^{d}$. Then$\int_{\mathrm{R}^{d}}dx|\langle(\phi_{\mu-}^{\rho,(k})(\mathcal{I}, X-X)-\phi_{\mu-(X,x}\rho,(k)))K(X),$$c(X)\rangle \mathcal{F}|$,$\leq\frac{\sqrt{\mathit{1}\mathrm{V}+1}}{\sqrt{\mathit{2}}}k\tilde{d}_{\mu},\kappa\cdot(p)|x|||K||_{\mathcal{M}}||c\tau||_{\mathcal{M}}$where. Hence by (4.2) and by$\grave{d}_{\mu,k}(\rho)=\sup_{x}||\sum^{d}j=1\partial_{j}\partial_{l^{l}}^{k1}-\pi_{-}1(\tilde{p}_{\mu}(X))||_{-1}$$\lim_{Xarrow 0}||F(\cdot-x)-F(\cdot)||_{\mathcal{M}}=0$

one can directly derive the lenrma.

Now we can prove Lemma 4.1.
Proof of Lemma 4.1.

,$\square$106

Note that$\frac{d}{c/s}p_{S}(’\cdot-y)=\frac{1}{\mathit{2}}\triangle_{y}p_{s}(x\cdot-y)$

Since$F\in \mathcal{M}_{\rho}^{\infty}$. one can use the$\frac{d}{ds}(Q_{\rho,s}F, c\tau)$$\mathrm{i}_{11\iota \mathrm{g}_{1}\cdot \mathrm{a}\mathrm{t}}\mathrm{e}\mathrm{i}\mathrm{o}\mathrm{n}$.

by parts formula. Thcll, by (4.4), we have$=$$\frac{1}{\iota 2}\sum_{\mu=1}^{d}\int \mathrm{R}^{d_{\mathrm{X}\mathrm{R}}d}.\langle\prime l_{X}d_{]}|l)(_{X}-y)\partial^{2}\lceil_{\rho}sy_{\mu}’(_{X}, y)F(y), C_{7}’(J^{\cdot})\rangle_{f}$$=$$\frac{1}{\mathit{2}}\sum_{\mu=1}^{d}\int \mathrm{R}^{d}\cross \mathrm{R}^{d})dxdyp_{S}(x-y$$G(\mu X)\rangle_{\mathcal{F}}$$\mathrm{x}\{\langle[I_{\rho}(_{X}, y)(.J2F(l^{l}y), G’(x)\rangle_{F}+\mathit{2}\langle\int I_{\rho}(x, y)(b^{\rho,1}1)(.c, y)\partial F(\mu y),$$+\langle[T_{\rho}(x.y)\{(\varphi_{\mu}^{\beta.\mathrm{t}1}()x,$ $y))2(+\phi_{\mu}^{\beta}’(2)X,$\equiv$$\frac{1}{2}\sum_{\mu=1}^{d}\int \mathrm{R}^{d_{\cross}}\mathrm{R}^{d}d_{l}.\cdot dyp_{S}(X-y)I(\mu x, y)$$=$$‘ \frac{1}{\mathit{2}}\sum_{\mu=1}^{d}J\mathrm{R}d\int p_{s}(.\mathrm{x}_{)dX}^{\Gamma}d\mathrm{R}dxI\mu(X, x-^{x})$Here, to apply Proposition 4.1, we divicle
follows;$I_{l^{l}}(X, y)$y)\}F(y\rangle, G(X)\rangle_{F}\}$.

in two components,$I_{\mu}=I_{\mu+}+I_{l^{\iota-}}$. as$.j^{1},(\mu y),$ $G(X)\rangle_{f}$$I_{\ell\iota+}(X, y)=2\langle rr_{\rho}(x, y)\phi_{\mathrm{t}+}\rho,\mathrm{t}1)(ly\rangle\partial X,$$+\langle U_{\rho}(x, y)\{(\phi_{\mu+}\rho.\mathfrak{l}1)(X.y))^{\mathit{2}}+2\phi_{\mu’}^{\rho}+(11)x,$$I_{\mu-}(x, y)=\langle U_{\rho}(x, y)\partial^{2}F(\mu y).$(-f7$(_{X}$$y$I$\phi\rho\mu.(1)-(I. y)+c^{\mathfrak{h}_{\mu}}\rho\dotplus(x, y)\}\mathfrak{l}2)f^{\urcorner}(y),$ $G(X)\rangle_{F}$)$G\prime\mathrm{t}1$$)\rangle_{F}+2\langle U_{\rho}(x,$ $y)\phi_{\mu}^{\rho}’-(x,$ $!/)\partial_{\mu}F(y),$$(x)\rangle \mathcal{F}$$+\langle U_{\rho}(x, y)\{(\phi_{\mu-(_{X},)}^{\rho,(1)}y)2x+\varphi_{\mu}-(),)\rho,12\}yF(y),$ $G’(g\cdot)\rangle_{f}$By (4.2), we have$| \int_{\mathrm{R}^{d}}dxI_{l^{A}}+(x,$$X-^{x)}|$$\leq||G||_{\mathcal{M}}\{9\frac{\sqrt{N+1}}{\sqrt{\mathit{2}}}rightarrow(c_{\mu,1}(\rho)|X|)||\partial_{\mu}F||_{\mathcal{M}}+\frac{\sqrt{N+1}\sqrt{N+\mathit{2}}}{2}$$\cross\{(C1(\mu,p)|x|)^{2}+4d_{0}(\rho)_{C_{\mu,1}}(\rho)|x|\}||F||_{\mathcal{M}}+\frac{\sqrt{N+1}}{\sqrt{2}}c_{\mu,2}(p)|x|||F||_{\mathcal{M}}\mathrm{I}$$\equiv\epsilon_{1}|X|+\epsilon 2|X|^{2}$,$| \int_{\mathrm{R}^{d}}dxI_{\mu}-(x.x-^{x)}|$

,

107

$\leq||G||_{\mathcal{M}}\{||\partial_{\mu}^{2}F||\mathcal{M}+4d_{0}(\rho)\frac{\sqrt{N+1}}{\sqrt{2}}||\partial F\mu||_{\mathcal{M}}$$+( \frac{\sqrt{N+1}\sqrt{N+2}}{2}4cl0(p)2\frac{\sqrt{N+1}}{\sqrt{2}}+2d_{\mu,1}(\rho))||F^{\iota}||_{\mathcal{M}}\}$$\equiv\epsilon_{3}$By.

(4. 7)$\mathrm{p}\mathrm{r}\mathrm{o}_{11}$)$\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}4.(\mathrm{I}\mathrm{I})$, we have$\lim_{sarrow 0+}|\int dXp_{9}(X)\int dxI\mu+(X, x-X)|\leq\lim_{sarrow 0+}\int dXp_{s}(\lrcorner\kappa)(\epsilon_{1}|X|+\epsilon_{2}|X|^{2})=0$

Thus it is enough to analyze the$I_{\mu}(\cdot, \cdot-X)$.

component. By Lenlrna 4.4, it follows that$\lim_{Xarrow 0}\int_{\mathrm{R}^{d}}I_{\mu-}(_{X,.r}’-x)dx=\int_{\mathrm{R}^{d}}I_{\mu-}(x, x)dX$.

(4. 8)By (4.7) and (4.8), Proposition 4.1 (I) yields$\lim_{sarrow 0+}\frac{d}{ds}\langle Q_{\rho.s\backslash }FG\rangle_{\mathcal{M}}$$=$$\lim_{sarrow 0+}\mu 1\sum_{=}\frac{1}{2}\int_{\mathrm{R}}dddXp_{s}(x)\int \mathrm{R}^{d}I-(\mu Xx,-x)d_{X}$$=$$\underline{‘\frac{1}{)}}\sum_{1\mu=}^{d}\int \mathrm{R}^{d})dxI_{\mu}-(x,x$$=$$-\langle \mathrm{H}_{\rho,0}F, G\rangle \mathcal{M}^{\cdot}$$\square$Lemma 4.5 Let

differentiable$p\in[C_{b}^{2}(\mathrm{F}^{d}’;\mathcal{H}-1)]$$c\iota t\llcorner \mathrm{s}=0$

with

.

$\frac{d}{ds}\langle Q_{\rho},SF^{1}.

$F\in \mathcal{M}_{\rho}^{\infty}$, and$G\in \mathcal{M}$. ThenG\rangle \mathcal{M}|_{s=}0+=-\langle \mathrm{H}_{p},0F, G\rangle_{\mathcal{M}}$$\langle Q_{\rho,s}F, G\rangle_{\mathcal{M}}$is right side.

(4. 9)

Proof: We have that$\langle Q_{\rho,s}F, G\rangle_{\mathrm{t}4}p$$=$$J_{\mathrm{R}^{d}}p_{S}(x_{)d}X \int_{\mathrm{R}^{d}}\langle U_{\rho}(x, x-x)F(x-X), G(x)\rangle_{F}dx$$\langle Q_{\rho}.0F, G\rangle/\vee\{$$=$$\int_{\mathrm{R}^{d}}\langle F(X), G(X)\rangle_{F}dx$.

Hence, similarly to the proof of Lenuma 4.1, it follows thatin at $s=0$ . Thus by the
expansion we can see that$\langle Q_{\rho,s}F_{\backslash }G\rangle_{\lambda\rho}$$s$

is right continuous$\mathrm{T}\mathrm{a}_{u}\backslash r101^{\cdot}$$\lim_{\epsilonarrow 0+}\underline{\langle Q_{\rho}.\epsilon F\backslash G\mathrm{v}\rangle_{\mathcal{M}}-\langle F,c\tau}(\rangle_{\mathcal{M}}$$=$$\lim_{srightarrow 0+}\frac{d}{ds}\langle Q\rho,sF, G\rangle_{\mathcal{M}}$$=$$-\langle \mathrm{H}_{\rho,0}F, G\rangle_{\mathcal{M}}$.

Hence (4.9) follows.

Following $[8,31]$ , we shall construct a contraction

simplicity we put $2^{n}=7l*$ .

$\square$$C_{0}$-semigroup from$\{Q_{\rho,n}\}_{n\geq 1}$

. For

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Lemma 4.6 Let$\rho\in[c_{b(\mathrm{F}^{d};}^{2}\backslash ’ \mathcal{H}_{-1})]$. Then, for all $t\geq 0$ , the stro

$7\iota g$limit$s- \lim_{\uparrow 1arrow \mathrm{x}}Q_{p.\frac{t}{11t}}^{n}*\equiv G_{\rho}(t)$exists. Moreover,$G_{\rho}(t)$has the following functional integral$\uparrow\cdot ep?\cdot escntation$

with$\Gamma^{\mathrm{t}},$$H\in \mathcal{M}$$\langle F, G_{p}(t)H\rangle \mathcal{M}$$= \int_{\mathrm{R}^{d}\cross\Omega}d\mu\int_{Q_{-}}1\mu d-1c\dot{\prime}C^{)}F(\Sigma_{\mu=1}^{d}(\int_{0}^{\iota}\pi_{-1}(p\sim\cdot)\mu 1b(9)+x)db+\frac{1}{2}\int\mu\partial\pi \mathrm{o}t\mu-1(\rho\sim\mu\{|)(s)+x))d_{S}))$$\cross\overline{F}(b(t)+x)H(x)$.
(4. 10)

Proof: To prove the existence, we show that$\{Q_{\rho_{\backslash }\frac{t}{n*}}^{n*}\}_{n\geq 1}$

is a Cauchy sequence in M. We see

that$||Q_{\rho,\frac{t}{1\tau*}}^{n*}F-Q_{\rho}7)l, \frac{*t}{m}.F||_{\mathcal{M}}^{2}$$=\langle F.Q_{\rho.\frac{*t}{n*}}^{2n}F\rangle,\mathrm{t}+\langle F,$ $Q_{\rho,\frac{t*}{m}}^{2m}.F\rangle \mathcal{M}^{-}‘ \mathit{2}\Re\langle F,$ $Q_{\rho,\frac{t}{\mathfrak{n}*}}^{n*}Q_{\rho,\frac{*t}{m}}^{m}.F\rangle_{\mathcal{M}}$The last term on the$\langle F,$$\mathrm{r}.\mathrm{h}$.

(4. 11).s.of (4.11) is$Q^{n*}.Q^{m}.F \rangle\rho,\frac{t}{n}\rho.\frac{*t}{m}\mathrm{A}4$$= \int_{\mathrm{R}^{d}}\langle F(x),$$(Q_{\frac{n*t}{n*}}Q_{\frac{m*t}{m*}}F)(x)\rangle_{\mathcal{F}}dx$$= \int_{\mathrm{R}^{d}\cross \mathrm{R}^{n}\cross}*d\mathrm{R}m\mathrm{s}d\cdot-1^{-}x,.X-y_{1})..\mathcal{P}\frac{t}{m*}(ym*-1-y_{m*})p_{\frac{t}{n}}(x-x1)..p_{\frac{t}{i1*}}(x)n*n*p_{\frac{t}{n}}(n*$$\mathrm{x}\langle F(x), U(p1)x, x..U_{\rho}(x_{n*-}1\cdot X)n*U\rho(x_{n*}, y1)..[f_{\rho}(ym*-1, ym*)F(y_{m*})\rangle_{\tau}dxd^{\prec_{d^{arrow}}}X.y$$= \int_{\mathrm{R}^{d}}dx\langle F(b(2t)+x),$$e^{ib_{\mathcal{F}}}( \sum d\mu=1\pi-1(\square _{\mu,m,n}(x))\mathrm{I}\Gamma\rangle(x)L^{2}1\Omega:\mathcal{F})$

,

where
.

$\square _{\mu,m,n}(x)$$=$$\sum_{k=1}^{*}n\iota\{\tilde{p}_{\mu}(b(\frac{t}{?n}k^{\tau})*+x)+p_{\mu}\sim(b(\frac{t}{\gamma\eta}(k-\lrcorner*)\mathrm{I}+x)\}$$\cross\{b_{\mu}(\frac{t}{1n}k)*-b_{\mu}(_{7*}\frac{t}{n}(k-1))\}$$+$$\sum_{k=1}^{n*}\{\tilde{\rho}_{\mu}(b(\frac{t}{n}k*+t)+x)+p_{\mu}\sim(b(_{*}\frac{t}{n}(k-1)+t)+x)\}$$\cross\{b_{\mu}(\frac{t}{n}k*+t)-b_{\mu}(\frac{t}{n}(k-1*)+t)\}$From Lemma 2.3 it follows that for each.

$x\in \mathbb{R}^{d}$$s- \lim_{\infty marrow n}1\mathrm{i}\mathrm{n}1,-1(\square arrow\infty^{\pi}(\mu,m,nX))$$= \int_{0}^{2t}\pi_{-1}(p_{\mu}(\sim x+b(\backslash .\sigma.)))db_{\mu}+\frac{1}{2}\int_{0}^{2t}\partial_{\mu}T-1(\rho_{\mu}(\sim b(s)+x))ds$

109

in

$L^{2}(\Omega, [\overline{\mathcal{H}}_{-1}-])$$L^{2}(\Omega;[\mathcal{H}_{-1}])$. One can easily see that the strong convergence of

implies that for each’$x\in \mathrm{F}_{\backslash }^{d}$and$\Phi\in \mathcal{F}$in$\pi_{-1}(\square _{\mu,\iota,n}(t\gamma x))$,$s- \lim_{marrow\infty},1n\underline{\mathrm{i}\mathrm{n}}1\exp\infty(id_{\mathcal{F}}(_{\mu=}\sum_{1}^{d}\pi-1(\square _{\mu,m,n}(X))))\Phi$$= \exp(i\phi_{F}(\sum_{\mu=1}^{d}\int \mathrm{o}l\pi_{-1}(^{\sim}p\mu(.\cdot+b(_{S)))\frac{1}{2}}db_{\mu}+\sum_{\mu=1}^{d}\mathit{1}^{2}2t\cdot t)0\partial\mu\pi-1(\tilde{\rho}\mu(x+b(S))dS))\Phi$.

(4. 12)in$L^{2}(\Omega;\mathcal{F})$. On the other hand we have$|\langle F(b(2t)+\cdot),$ $e^{i}F(\cdot)\rangle\phi_{f}(\Sigma_{\mu=1}d)7\ulcorner-1(\square _{\mu,m},n(\cdot))L^{2}(\Omega,\mathcal{F})|$$\leq||F(b(2t)+\cdot)||_{L^{2}()}\Omega;\mathcal{F}||F(\cdot)||_{L^{2}}(\Omega;\mathcal{F})\in L^{1}(\mathrm{P}^{d})$.

Hence, by the Lebesgue dominated convergence theorem, we have$\lim_{narrow\infty}\varliminf_{m\infty}\langle F,$.

$Q^{n*}p, \frac{t}{nn}Q^{7?l}\rho,\frac{*t}{m*}F^{\urcorner}\rangle_{\mathcal{M}}$$= \int_{\mathrm{R}^{d}}dx\langle F(b(2t)+x)$$\cdot$,$e^{i\phi_{f}}( \Sigma_{\mu}^{d}=1(\int_{0}^{2}t(^{\sim}\rho_{\mathit{1}^{4}}\mathrm{t}b(s)+x))db_{\mu}+\frac{1}{2}\int_{0}^{2t}\partial\mu 1(\pi_{-}\rho_{\mu}"\langle b(s)+x))ds)\pi_{-1})F(x)\rangle_{L\Omega}2(;\mathcal{F})$

Similarly it can be easily seen that$\langle F,$$Q_{\rho}^{2n}.

\frac{*t}{n}.\rangle_{\lambda 4}F$

and$\langle F,$of (4.13) as $n,$ $marrow\infty$ , respectively. Then it follows thateasily follows from (4.13).
Lemma 4.7 Let
on.

$p\in[C_{b}^{2}(\mathbb{R};\mathcal{H}_{-1}d)]$. Then the family$Q_{\rho}^{2m}.

. $(4.13)$

converge to the r.h.s.
.
a Cauchy.

\frac{t*}{m*}F\rangle_{\lambda}r$$\mathrm{E}\mathrm{q}.(4.10)$$\{Q_{p,\frac{t}{n}}^{n*}.\}_{n\geq 01\mathrm{s}}$$\{G_{\rho}(t)\}_{t}\geq 0$$\square$is a contraction$C_{0}$-semigroup$\mathcal{M}$

Proof. By the definition of$\cdot$$G_{\rho}(t)$and the proof of Lemma.4.6, it holds that
(4. 14)$G_{\rho}(t\mathrm{I}^{C_{\tau^{\gamma}}}\beta(S)=G_{\rho}(t+s).$

We show the strong right continuity at $t=0$ . Because of (4.14), the weak continuity implies
the strong continuity. Hence it is enough to show that$\lim_{tarrow 0+}\langle G_{\rho}(t)F, H\rangle_{\mathcal{M}}=\langle\epsilon_{\tau_{\rho}}^{t}(0)F, H\rangle_{\mathcal{M}}$,$F\in \mathcal{M},$ $H\in \mathcal{M}$

.

110

From (4.10), we have$\langle F,$$G_{p}(t)H)_{\mathcal{M}}-\langle F, (^{\wedge}\tau^{\mathrm{t}}\rho(0)H\rangle_{\mathcal{M}}$$= \int_{\mathrm{R}^{d}}dx\langle F(b(t)+x)-F(.1^{\cdot})$,$e^{i\emptyset r(\Sigma_{\mu}}d=1( \int_{0^{\pi_{-\mathrm{J}}}}^{\mathrm{f}}(\overline{\rho}\mu(b(.9)+J’))db+\mu\frac{1}{2}\int^{t}0\partial_{\mu}\pi_{-1}(^{\sim}\rho\mu(b(s)+x))d\mathrm{s}))H(x)\rangle Ij2(\Omega;f)$$+ \int_{\mathrm{R}^{d}}dx\langle F^{\urcorner}(_{X)_{\}}$$(e^{i\phi \mathcal{F}(\Sigma_{\mu}^{d}}=1( \int_{0}^{\iota\sim}\pi-1(\rho_{\mu}(b(s)+x))db_{\mu}+\frac{1}{2}\int_{0}^{\mathrm{t}}\partial_{\mu}\pi-1(\rho_{\mu}\sim)(b(s)+x)ds)\mathrm{I}-I\mathrm{I}^{H}(x)\rangle_{L^{2}(}\Omega;f)$(4. 15)

The first summand on the r.h.s.of (4.15) can be evaluated as follows$| \int_{\mathrm{R}^{d}}dx\langle F(b(t)+x)-F(.x\cdot)$,$e^{i\emptyset r(\Sigma_{\mu 1}^{d}(}) \mathrm{I}H=\int_{0^{\pi_{-1}(^{\sim}\mathfrak{s}+}}^{t}\rho\mu(J(S)x))db\mu+_{\mathit{4}0^{\partial(}}t\mu\pi_{-}1\rho\sim\mu(b(s)+x\mathrm{I})d_{S}(X)\rangle L2(\Omega;f)|$$\leq||H||_{\mathcal{M}}(\int_{\Omega}||F(b(t\rangle+\cdot)-F(\cdot)||2\mathcal{M})^{\frac{1}{2}}d\mu$Since$\lim_{tarrow 0}||F(b(f)+\cdot)-F(\cdot)||_{\mathcal{M}}=0$,$a.s.b\in\Omega$,

the first summand on the.s.of (4.15) converges to . On the second summand on the. of (4.15), one can see that by Remark 2.2 (3)$0$$\mathrm{r}.\mathrm{h}$$\mathrm{r}.\mathrm{h}.\mathrm{s}$$s- \lim_{tarrow 0}(\int_{0}^{t}\pi_{-1}(\tilde{p}_{\mu}(b(S)+x))db_{\mu}+\frac{1}{2}\int_{0}^{t}\partial_{\mu-1}\pi(\tilde{\rho}_{\mu}(b(s)+.\gamma\cdot)\mathrm{I}ds)=0$in$L^{2}(\Omega;[\overline{\mathcal{H}}_{-1}])$. As in the case of$(/\mathrm{t}.1\mathit{2})$, we see that for $\Phi\in F$$s- \lim_{tarrow 0}\exp(i\phi_{F}(_{\mu}\sum_{=1}^{d}(\int_{0}^{t}\pi_{-}1(\tilde{p}_{\mu}(b(S)+x))db_{\mu}+\frac{1}{2}\int_{0}^{t}\partial\pi\mu-1(\tilde{\rho}(b(s)+x))dS)))\Phi=\Phi$in$L^{2}(\Omega;\mathcal{F}^{\cdot})$.

Hence, by the Lebesgue dominated convergence theorem, we can derive that the second. The strong continuity in the case $t>0$ is proven similarly.
converges to as$0$$tarrow \mathrm{O}$$\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{m}\square \mathrm{a}\mathrm{n}\mathrm{d}$By Lemma 4.7 andunique positive self-adjoint operator$\mathrm{H}\mathrm{i}\rfloor 1\mathrm{e}- \mathrm{Y}_{\mathrm{o}\mathrm{s}}\mathrm{h}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{s}$theorem, for eachinsuch that$\overline{\mathrm{H}}_{\rho,0}$$\mathcal{M}$$\subset_{\tau_{\rho}}’(t)=e^{-t\tilde{\mathrm{H}}_{\rho,0}}$.

$\rho\in[(_{b}^{2}’(\mathbb{R}^{d};\mathcal{H}_{-}1)]$, there exists a

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Lenlma 4.8 Let
extension of.

$\mathrm{H}_{\rho,0}|_{\mathcal{M}_{\rho}^{\infty}}$

Proof: Let. Then the self-adjoint operator$\rho\in[C_{b}^{2}\text{ノ}(\mathbb{R}^{d};\mathcal{H}-1)]$$F\in D(\overline{\mathrm{H}}_{\rho,0})$and$\langle\frac{1}{t}(e^{-t\tilde{\mathrm{H}}_{\rho,0}}-I)G,$$G\in \mathcal{M}_{\rho}^{\infty}$$p’\rangle \text{ノ}\iota$$\overline{\mathrm{H}}_{\rho,0}$

is a self-adjoint. Then we have$=$$\varliminf_{n\infty}\langle\frac{1}{t}(Q^{n*}p,\frac{t}{1)*}-\tau)G,$$=$$\lim_{narrow\infty}\sum_{j=0}^{*}\frac{1}{?l}n-1*\langle\frac{n*}{t}(Q\rho,\frac{\mathrm{t}}{n*}-I\mathrm{I}^{G,Q^{n*}}\rho.\frac{\perp nt}{n}.\cdot F\rangle_{\mathcal{M}}$$=$$\lim_{narrow\infty}\int_{0}^{1}\langle\frac{n*}{t}(Q_{\rho,\frac{t}{n*}}-I)G^{1},$ $Q_{\rho}[n, \frac{*s]\mathrm{t}}{n*}F\rangle \mathcal{M}d_{S}$$f\mathrm{i}’\rangle J\mathrm{t}4$.

Since, by Lemma 4.5,$u)- \varliminf_{n0}\frac{1?*}{t}(Q_{\frac{t}{n*}}-I)G=-\mathrm{H}_{\rho,0}G$

the norm$|| \frac{n*}{t}(Q_{\frac{t}{n}}.

is unifornlly bounded in-I)G||_{\mathcal{M}}$$n$$s- \lim_{narrow\infty}Q_{\rho,\frac{*S\mathrm{t}*}{n}}^{[n}]=G_{\rho}(t_{S})$,. By Remark 2.4, we can see that.

Then we have$\langle\frac{1}{t}(e^{-t\tilde{\mathrm{H}}_{\rho,0}}-I)G,$A,s$tarrow \mathrm{O}$$F \rangle Ju=\int_{0}^{1}ds\langle-\mathrm{H}_{\rho,0}c\tau_{\}e-tS\tilde{\mathrm{H}}\rho,0F\rangle \mathcal{M}$on the both sides, we get$\langle G,\overline{\mathrm{H}}_{\rho.0}F\rangle_{\mathcal{M}}=\langle \mathrm{H}_{\rho,0}G, F\rangle\lambda 4$

which implies that

$G\in D(\overline{\mathrm{H}}_{\rho.0})$,

and$\overline{\mathrm{H}}_{p.0}G=\mathrm{H}_{\rho},0G$.

Thus the proof is complete.
We denote thein terms of the form$\square$$\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{i}_{0}\mathrm{n}\overline{\mathrm{H}_{\rho,0}.}$$\mathrm{H}_{\rho}$by the same symbol;
and$\mathrm{s}\mathrm{u}\mathrm{m}+\mathrm{o}\mathrm{f}\mathrm{H}_{\rho.0}$$\mathrm{H}_{\rho,0}$. We give a rigorous definition of$I\otimes \mathrm{H}_{0}$$\mathrm{H}_{\rho}=\mathrm{H}_{\rho},0\dotplus I\otimes \mathrm{H}0$.

Next we study functional integral representations concerning
plication operator inby$L^{2}(\mathrm{p}^{dc_{)}};\llcorner$,$\phi_{\mathcal{E}’.\mu}^{\rho}s\equiv\int_{\mathrm{R}^{d}}^{\oplus}\phi e(\pi_{-2}(j_{S}\tilde{\rho}_{\mu}(x)\mathrm{I})\sim d_{X}$.

$e^{-t\mathrm{H}_{\rho}}$. We introduce a multi-

112

We define an operator acting inby$L^{\mathit{2}}(\mathrm{F}^{d}’;\mathcal{E})$$\mathrm{H}_{\rho,0,s}=\frac{1}{2}\sum_{\mu=1}^{d}(-iD_{\mu}\otimes I-\phi^{\rho,s}\mathcal{E},\mu)2$[$c_{b(\begin{array}{l}\mathrm{P}^{d}.,\mathcal{H}_{-1}fi\end{array})\mathrm{e}\mathrm{n}\mathrm{e}}^{2}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\rho\in$a]$s\geq 0$.

.

$\pi-(\tilde{j}\tilde{\rho}\mu \mathrm{i}^{s}\mathrm{n}\mathrm{f},\mathrm{o}\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{H}(\cdot))\in c_{b}2(\mathbb{R},\rho d.,,0^{-\sim}s\mathrm{a}\mathrm{I}\mathrm{n}\mathrm{e}\mathrm{W}\mathrm{a}S\mathrm{n}),\mathrm{w}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}j\tilde{\rho}_{l}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{S}\mathrm{y}\mathrm{a}\mathrm{S}\mathrm{H}_{\rho,0}.\mathrm{T}\mathrm{h}’ \mathrm{e}\mathrm{n},\mathrm{t}r(\cdot)\in[C_{b(\mathrm{R}^{d}}2.\mathcal{H}-24^{]}\mathrm{e}$$\mathrm{s}\mathrm{e}\mathrm{i}\mathrm{f}_{-}\mathrm{a}\mathrm{d}^{2}\mathrm{j}\mathrm{o}$following equation holds for$\langle F,$,$F^{1}.H\in L^{2}(\mathrm{F}^{d}’.$;$e^{-t\mathrm{H}}\rho,0,kR\rangle_{\epsilon}$$= \int_{\mathrm{R}^{d}\cross\Omega}d\mu\int_{Q_{-}}2d\mu_{-2}ei\emptyset\epsilon(\Sigma_{\mu=1}^{d}(\int_{0}^{t}\pi-2(^{\sim\sim}j_{k\rho}\mu(b(s)+x))db+\frac{1}{2}\mu f_{0^{\partial\pi}}^{t}\mu-2(j_{k}\sim\sim\rho_{\mu}(b(s)+x)d_{S)))}$$\cross\overline{F}(b(t)+x)H(x)$Lemma 4.9 Let.
. Then the following equation holds on$\rho\in[C_{b}^{2}(\mathrm{p}^{d};\mathcal{H}_{-}1)]$$S^{*}s$$\rho,$$J_{S}e-t\mathrm{H}_{\rho},0,J=Ee^{-}t\mathrm{H}\mathrm{o},\underline{\epsilon}$

where$J_{s}$

and$E_{s}$are defined in

Proof: Note that for any$(^{\circ},\mathit{3}^{\mathit{0}}.\mathit{3})$.

$J_{s}e^{i1A)}J^{*}\phi_{\mathcal{F}}s=F_{\lrcorner}e^{i\phi_{\mathcal{E}}1}$$s$

From (4.10) it follows that for $F,$$\langle F,$ES ,$j_{\grave{b}}^{\sim_{A)}}$$A\in\overline{\mathcal{H}}_{-1},$$L^{2}(\mathrm{R}^{d};\mathcal{E})$$H\in L^{2}(\mathbb{R}^{d};\mathcal{E})$,E s , and$(J_{S}^{*}F)(x)=J_{s}^{*}(F^{\mathrm{t}}(x))$.

$J_{s}e-t\mathrm{H}_{\rho,JH}0*s\rangle_{L(\epsilon)}2\mathrm{R}d$;$= \int_{\mathrm{R}^{d}\cross\Omega}d\mu\langle(J_{S^{*}}F)(b(t)+X)$,$e^{i\tau} \varphi_{\mathrm{j}}^{\prime(}\Sigma d\mu=1(\int_{0}^{t}\pi_{-}1(^{\sim}\rho_{\mu}\langle b1S^{t})+x))db_{\mu}+\frac{1}{2}\int^{t}0\mu-1\pi(\partial\rho\mu(b\{S)’+x)ds’\sim))\mathrm{I}_{(}J*H)(x)\rangle_{\mathcal{F}}s$$= \int_{\mathrm{R}^{d}\cross\Omega}d\mu\langle F(b(t)+X)$,$E_{s}e^{i\phi_{\mathcal{E}}(\sum jS()} \mu=1d\sim\int_{0}^{t}\pi-1(^{\sim}\rho\mu(b\langle s’)+x))db+\frac{1}{2}\int\mu 0\mu t\partial\pi_{-1}(^{\sim}\rho_{\mu}\mathrm{t}b(_{S’)+)s’}xd))E_{s}H(x)\rangle_{\mathcal{E}}$$=\langle F,$

Since$E_{sS}e^{-t\mathrm{H}}\rho,0,sEH\rangle_{L^{2}}(\mathrm{R}d\epsilon;)$.

are arbitrary, the proof is complete.
Now we are ready to state the nlain theorem in this section.
$F,$ $H\in L^{2}(\mathbb{R}^{d};\mathcal{E})$

Theorem 4.10 Let$F,$$G\in \mathcal{M},$$V\in c_{b}(\mathbb{R}^{d})$andsuch th at$p\in[C_{b}^{2}(\mathrm{P}\mathrm{c};\mathcal{H}d)-1]$$\sup_{x\in \mathrm{R}}||d[\tilde{\omega}]\pi_{-}1(\tilde{p}_{\mu}(X))||-1=\sup_{x\in \mathrm{R}^{d}}|\tilde{\omega}\rho\mu(\sim X)|_{-1}<\infty$$||_{\mu=} \sum_{1}^{d}\partial_{\mu}\pi_{-1}(\tilde{p}\mu(x))||_{-1}=0$$\square$,$\mu=1,$, (the Coulomb gauge).

$d$$\ldots,$,(4. 16)(4. 17)

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Then$\langle F,$$e^{-t()}G \mathrm{H}_{\rho}+^{\mathrm{v}}r\mathrm{t}\rangle_{\mathcal{M}}=\int_{\mathrm{R}^{d}\cross\Omega}d\mu\int_{Q_{-2}}d\mu-2\exp(-\int_{0}^{t}V(b(s)+x)dS)$$\cross\exp(id\epsilon(\sum_{\mu=1}^{d}\int^{t}\mathrm{o}\mathrm{o}[jarrow t]\sim\pi_{-}1(\tilde{p}_{\mu}(b(s)+x))db_{\mu}\mathrm{I})J_{t}\overline{F}(b(t)+x)J_{0}G(x),$$(4.18)$where$I_{0}^{t}[\tilde{j}0arrow \mathrm{p}]\pi-1(\tilde{p}\mu(b(S)+x))db_{\mu}$is the$time- O\Gamma dered\underline{[}\overline{\mathcal{H}}_{-}2$]-valued stochastic integral associated with the family of isometries

.
to
from
Proof: By the strong Trotter product formula [18] and Proposition 4.3 (c), we see that$[\overline{\mathcal{H}}_{-1}]$$[j_{t}]\sim$$[\mathcal{H}_{-2}]$$\langle F_{\backslash }e^{-t(}\mathrm{H}+\rho)VG\rangle_{\mathcal{M}}$$= \lim_{narrow\infty},$$\langle F,$$= \lim_{narrow_{\mathrm{c}}\mathrm{t}}\langle F,$$(e- \frac{\mathrm{t}}{n*}\mathrm{H}_{\rho},0\epsilon-\frac{t}{n*}\mathrm{H}0\frac{t}{n}e^{-}\cdot V)n*G\rangle_{\mathcal{M}}$$J_{t}^{*}(J_{t}e^{-\frac{t}{n*}}J_{t}^{*}0) \mathrm{H}_{\rho},e^{-\frac{t}{n}}.V(J_{t-\frac{t}{n*}}e^{-\frac{t}{n}\mathrm{H}}.\rho,0J_{t-}*\frac{t}{n}.)e^{-}\frac{t}{n\mathrm{s}}V\ldots\ldots$........

$(J_{\frac{t}{n*}} \epsilon^{-\frac{t}{n*}\mathrm{H}_{\rho,J}}0\frac{*t}{n*})e^{-\frac{t}{1l*}}J_{0}G\rangle_{\mathcal{M}}V$,

fronl which and Lemma 4.9 it follows that$=$$n\infty 1\underline{\mathrm{i}\mathrm{n}}\mathrm{l}\langle F,$...
$\equiv$

Let$\frac{t}{n*}=s$$J_{t}^{*}(E_{t}e^{-\frac{t}{n*}\mathrm{H}} \rho,0,{}^{t}E_{t)e}-\frac{\mathrm{t}}{n}.v(E_{t.t-\frac{t}{n*})}-\frac{t}{n}e^{-\frac{t}{n}\mathrm{H}}\rho,0,t-\frac{t}{n*}Ee^{-\frac{t}{n}}.v$$e^{-\frac{t}{1?*}\mathrm{L}’}(E_{\frac{t}{n*}}C^{-\frac{t}{l)*}\mathrm{H}_{\rho}}’ n \cdot E0,\perp.)\frac{t}{n}e^{-\frac{t}{n*}\mathrm{v}_{J_{0}c}}\rangle_{\mathcal{M}}\gamma$$\lim_{narrow\infty}1\sigma_{n*}’$.

. From the definition of$S_{n*}$$=$$\lim_{karrow\infty}\langle F.J_{t}^{*}(E_{t}Q_{\rho,\frac{s}{\lambda*}}^{k*},tEt\mathrm{I}e^{-}SV(E_{t-S}Q_{\rho}^{k*},\frac{\grave{\delta}}{k}.\tau t-sEt-s)e-sV\ldots$...
$\equiv$

where$Q_{\rho,t,t’}$and Lemma 4.6 it follows that$\mathrm{H}_{\rho,0,t’}$$e^{-sV}(E_{s}Q_{\rho,\frac{s}{k*},S}^{k*}ES\mathrm{I}e-sVJ_{0\rangle}G’ \mathcal{M}$$\lim_{karrow\infty}S_{n*,k}*$ ’is defined by operators on$L^{2}(\mathbb{R}^{d};\mathcal{E})$such that$(Q_{\rho,t,t’}F)(x)$$=$$/\mathrm{R}^{d}.p_{t}(x-y)U_{p},t’(x, y)F(y)dy$$(Q_{\rho,0.t^{\prime F)(}}x)$$=$$\Gamma^{t}(x)$,$t>0$ ,,

and
$U_{\rho,t’}(_{X}, y)= \exp\{.\frac{1}{2}i\phi_{\mathcal{E}}(_{\mu=1}\sum^{d}[^{\sim}j_{t}’]\pi-1(^{\sim}p\mu(_{X})+p\mu\sim(y))(_{X_{\mu}}-y_{\mu})\mathrm{I}\}\cdot$

114

One can see that$s_{n*,k*\int_{\mathrm{R}^{d}\mathrm{p}\cross}.\vec{x}}=\cross\underline{dk*\chi\ldots \mathbb{R}}{}_{dk*}P9(X_{1})arrow\ldots P_{s}(n*)\epsilon’-s\Sigma_{j}n=1v*(x_{j}^{k}.)$$\cross\langle F^{\urcorner}(x),$...

$J_{t}*(B_{\mathrm{t}}^{1}l^{l}tx^{arrow}/J,t(1)E_{t})(E_{t_{-s^{\zeta_{-}}\rho},t-s}T(.x_{2})arrow F_{\lrcorner}t-s)\ldots$$(F_{\lrcorner}’ Is\rho_{S}(;.x_{n*})arrow Es)\prime J0G(.\gamma_{n*}’ k)\rangle_{\mathcal{F}}$(1. 19)

,

where
$=$

$p_{s}(X_{j1^{-}}^{k}-x_{j}*1)p_{s}(Xj1-x_{j}^{2})\ldots p_{S}(x^{k}j*-\mathrm{l}-x_{j}^{k*})$$U_{p,\alpha}(X_{j})arrow$$=$$U_{p,\alpha}(x_{jj}k*1-1’ x)l^{\gamma}(p,\alpha x_{j}, x12j)\ldots U(p,\alpha)X_{j}, X_{j}k*-1k*$$=$$\exp\{.\frac{1}{2}i\phi_{\mathrm{c}^{C}}(\sum_{i=1}^{k*}.[j\alpha]\sum_{\mu=1}^{f}\pi_{-1}(p_{\mu}(\sim x_{j}^{i1}-)+\tilde{\rho}_{\mu}(x_{j}^{i})\mathrm{I}^{(,x}xj-\sim(-i1i\mu j,\mu)\mathrm{I}\}$$x_{j}^{0}=x_{j1}^{k*}-,$$x_{0}^{\Lambda\cdot*}\equiv x,$

By Proposition 3.3 one can neglect$‘ g_{n*,k*}$,$P_{s}(x_{j}^{arrow})$$\int_{\mathrm{R}^{d}}d_{T}\langle F(b(t)+x)$$=$$E_{J}^{1}$$j=1,$$\ldots,$

$n*$

,
,

.

in (4.19), so that,,

$J_{t}^{*} \exp(id_{\mathcal{E}}(\sum_{\mu=1j}^{d}\sum_{=0}^{n}[i_{j}S]\pi_{-}1*-1\sim(\square _{\mu,j,k}(x)))-s.\sum^{n*}V7=1(b(jS)+x))J_{0}C_{\mathrm{J}}(x)\rangle$$L^{2}\langle\Omega;\mathcal{F})$where$\square _{\mu,j,k}(x)$$=$$\sum_{71\mathrm{t}=}^{*}k1\{\tilde{\rho}_{\mu}(b(\frac{\prime?l}{L,*}S+js)+x)+\tilde{p}_{\mu}(b(\frac{\prime n-1}{k*}S+js)+x)\}$$\mathrm{x}\{b_{\mu}(\frac{m}{k*}.9+js)-b_{\mu}(\frac{?1?-1}{k*}s+j_{6})\},$$j=0,$$\ldots,$$n*-1$ .

As in the case of (4.12), by the Coulomb gauge condition (4.17), we see that for each,and$x\in \mathbb{R}^{d}$$\Phi\in \mathcal{E}$$s- \lim_{karrow\infty}\exp(i\phi_{\mathcal{E}}(\sum_{\mu=1}^{d}\sum_{j=0}^{n*}[j_{js}]\pi-1-1\sim(\coprod\mu,j,k(X))))\Phi$$= \exp(i\phi_{\mathcal{E}}(\sum_{\mu=1}^{d}\sum^{n*-1}\int jsb_{\mu}[^{\sim}j_{j}s]\pi_{-1}(p\mu(x+b(s)\prime \mathrm{I})d\mathrm{I}j=0(j+1)s\sim)\Phi$in$L^{2}(\Omega;\mathcal{E})$. On the other hand, we have$|\langle J_{t}F(b(t)+\cdot),$ $\exp(i\phi\epsilon(\sum_{\mu=1}^{d}\sum_{j=0}^{?\iota*}[jj_{s}]\pi-1(\square \mu-1\sim,j,k(\cdot)))-s\sum_{j=1}^{n*}V(b(j_{S})+\cdot))J_{0}G(\cdot)\rangle|L^{2}\mathfrak{l}\Omega;\epsilon_{)}$$\leq\exp(-s\inf_{x}V(x))||F(b(t)+\cdot)||_{L^{2}\mathrm{t}\Omega;\tau)}||G(\cdot)||L2(\Omega;\mathcal{F})\in L^{1}(1\mathrm{R}^{d})$

.

115

Hence, by the Lebesgue dominated convergence theorem, we have$\int_{\mathrm{R}^{d}}dx\langle J_{t}F(b(t)+x)$$=$$\lim_{karrow\infty}S_{n*,k}*$,,$\exp(i\phi_{\mathcal{E}}(\sum_{\mu=1}^{d}\sum^{n*-1}\square (X))j=0\mu,j-s\sum_{j=1}^{n*}V(b(js)+x))J_{0}G(x)\rangle$$L^{2}(\Omega$

;

where

.

$\square _{\mu,j}(x)=\int_{js}^{\mathrm{t}j+}1)s_{\sim,[j})js]\pi_{-1}(\tilde{\rho}\mu(b(S’+x))db\mu$Note that for sufficiently small$\epsilon>0$$||[j_{t}+\epsilon]^{*}[jt]\pi-1(\tilde{p}\mu(x))-T-1(\rho\mu(\sim\sim\sim X))||e^{-\epsilon}\pi-1([^{\sim}\omega](\tilde{\rho}_{\mu}X))-\pi_{-1}(\tilde{\rho}\mu(X))||-1$$\leq$$\epsilon\sup_{x\in \mathrm{R}^{d}}|\tilde{\omega}\tilde{\rho}\mu(_{X)|}-1^{\cdot}$

Hence by Theorem 2.5 and (4.16), we see that for$s- \lim_{\infty narrow}j=\sum^{n*}-10\square \mu,j(x)$in$L^{2}(\Omega;[\overline{\mathcal{H}}_{-2}])$$x\in \mathrm{p}_{\mathrm{c}}^{d}$$=$$s- \lim_{\infty narrow}\sum_{j=0}^{n*-1}\int^{\frac{j+1}{n}}\lrcorner_{\frac{t}{1}}.n^{-]}.-1(p_{\mu}(b(s)+X))d[j\perp^{\mathrm{p}}\pi bnt_{\sim}\sim\mu$$=$$\int_{0}^{t}[\overline{j}0arrow t]\pi_{-}1(\tilde{\rho}\mu(X+b(s)))db_{\mu}$. Then again as in the case of (4.12), for each,$x\in \mathbb{R}^{d}$and$\Phi\in \mathcal{E}$$s- \lim_{\mathrm{R}narrow}\exp(id\epsilon(\sum_{\ell_{l=}1}^{d}\sum^{*}\int^{\mathit{1}4}\lrcorner_{\frac{t}{}}.\cdot.b[j_{\frac{jt}{n}}]T_{-1}(\tilde{\rho}\mu((s)+x))db_{\mu}\mathrm{I}nj=0-1nn^{\frac{1}{.}t_{\sim}})\Phi$$= \exp(i\phi_{\mathcal{E}}(\sum_{\mu=1}^{d}\int^{t_{\sim}}0[j0-t]\pi_{-}1(\tilde{\rho}_{\mu}(x+b(s)))db_{\mu}\mathrm{I})\Phi$(4. 20). Passing to the subsequences for (hereafter denoted again by n), (4.20) holds
inin the strong topology of . Since $V(b(s)+x)$ is continuousandfor each$a.s.b\in\Omega$
we haveandin for each$L^{2}(\Omega;\mathcal{E})$$n$$x\in \mathrm{P}^{d}$$s$$x\in \mathrm{P}^{d}$$\mathit{0}.s.b\in\Omega$$\mathcal{E}$.

$\lim_{narrow\infty}\frac{t}{n}\sum_{=j0}^{1}V*n*-(b(\frac{jt}{n})*+x)=\int_{0}^{t}V(b(S)+x)ds$,$x\in \mathrm{F}_{-}^{d}$’a. $s.b\in\Omega$ .

Furthermore,$| \exp(-s\sum_{j=1}^{*}Vn(b(jS)+\cdot))\langle J_{t}F(b(t)+\cdot),\exp(i\phi_{\mathcal{E}}(_{\mu=}\sum_{1}^{d}\sum_{j=0}^{\iota}\square _{\mu,j}(\cdot))n*-)J_{0}G(\cdot)\rangle_{\mathcal{E}}|$$\leq\exp(-s\inf_{x}V(X))||F^{1}(b(t)+\cdot)||_{\mathcal{F}}||F(\cdot)||_{F}\in L^{1}(\mathbb{R}^{d}\cross\Omega;d\mu)$

.

116

Hence, again by the Lebesgue dominated convergence theorem we get (4.18).
, we have an interest in extending
Similarly to the classical caseand satisfying(4.18) to more general potentials. From (4.18) it follows that forthe conditions stated in Theorem 4.10$\square$$[31,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}15.5]$$V\in C_{\text{ノ}}b(\mathrm{p}^{d})$$|\langle F,$$p$(4. 21)$e^{-}t(\mathrm{H}_{\rho}+V)G|\rangle_{J}\vee\{|\leq\langle||F||\tau,$ $e-t(- \frac{1}{2}\Delta+V)||C_{7}||_{f}\rangle_{L^{2}\mathrm{R}^{d})}$(We define for$G\in \mathcal{M}$$\frac{G(x)}{||G(x)||_{\mathcal{F}}}$$(signG)(_{X})\equiv\{$
$0$

,

,

$||G(X)||_{\mathcal{F}}\neq 0$,$||G(X)||\mathcal{F}=0$.

Lemma 4.11 $Let|V|$ be a multiplication operator-form bounded with relative
bound . Then for satisfying the condition in Theorem 4.10, $|V|$ is-form bounded with.
relative bound$u$) $hichis- \frac{1}{2}\triangle$$\mathrm{H}_{\rho}$$p$$\epsilon$$\leq\epsilon$

Proof: Substituting $V=0$ and $F=sign$$(e^{-t\mathrm{H}_{\rho}}G)\cdot\psi$, where$\psi\in C_{0}^{\infty}(\mathrm{P}_{\searrow)}^{d}$and$\psi\geq 0$(4.21), we have$\langle\psi,$$||e^{-}\rho G|t\mathrm{H}|F\rangle L2(\mathrm{R}^{d})\leq\langle\psi,$ $e^{-t(-\frac{1}{2}}c\tau\Delta \mathrm{I}_{||||}\rangle_{L^{2}(\mathrm{R}^{d})}\mathcal{F}$Hence it follows that$||(e^{-t\mathrm{H}_{\rho}}G)(x)||_{F} \leq(e^{-t(-\Delta)}\frac{1}{2}||G(\cdot)||\tau)(x)$,$a.e.x\in$ IR

$d$

.

Since
$(( \mathrm{H}_{\rho}+E)^{-\frac{1}{2}}G)(x)=\Gamma(\frac{1}{2})/()\infty e^{-Et}t^{-\frac{1}{2}}(e^{-t\mathrm{H}_{\rho}}G)(x)dt$,$E>0$ ,

$a.e.x\in \mathbb{R}^{d},$one can see that$||(( \mathrm{H}_{p}+E)^{-\frac{1}{2}}C\mathrm{T})(x)||_{\mathcal{F}}\leq((-.\frac{1}{2}\triangle+E)^{-\frac{1}{2}}||G(\cdot)||_{f}\mathrm{I}(x),$$a.e.x\in \mathbb{R}^{d}$Then we have$\int_{\mathrm{R}^{d}}\{|V(X)|^{\frac{1}{2}}||((\mathrm{H}_{p}+E)^{-\frac{1}{2}}G)(x)||_{\mathcal{F}}\}^{2}dx$$\leq\int_{\mathrm{R}^{d}}\{|V(x)|^{\frac{1}{2}}((-\frac{1}{2}\triangle+E)^{-\frac{1}{2}}||G(\cdot)||_{\mathcal{F}}\mathrm{I}(x)\}^{2}dX$.

Thus$\frac{|||V|^{\frac{1}{2}}(\mathrm{H}+\rho)^{-}\frac{1}{2}G|E|\mathcal{M}}{||G||l},\leq\frac{|||V|^{\frac{1}{2}(\mathrm{I}^{-}|G(}-\frac{1}{2}\triangle+F\lrcorner|\cdot)\frac{1}{2}||f||_{L^{2}}(\mathrm{R}^{d})}{||||G(\cdot)||\mathcal{F}||_{L(\mathrm{R})}2d}$

,

.

, in

117

which implies that the following operator norm estimate holds(4. 22)$|||V|^{\frac{1}{2}}( \mathrm{H}+pE)^{-\frac{\mathrm{J}}{2}}||,1\leq|||V|^{\frac{1}{2}}(-\frac{1}{2}\triangle+E)^{-\frac{1}{2}}||_{L^{2}(\mathrm{R}^{d})}$

Since
,

$\lim_{Earrow\infty}|||V|^{\frac{1}{2}}(-\frac{1}{2}\triangle+E)^{-\frac{1}{2}}||_{L^{2}()}\mathrm{R}^{d}=\epsilon$

the lemma follows.
For a multiplication operator V. we set $V_{+}= \max\{0, V\}$ and $V_{-}= \max\{0, -V\}$ . Let us
introduce a class of potentials.
$\square$$P$

is in the set
Definition 4.12 A potential-form bounded with relative bound $<1$ .
$V$

if and only if$P$$V_{+}\in L_{l_{o\mathrm{C}}}^{1}(\mathbb{R})d$and$V_{-}$

is$- \frac{1}{2}\triangle$$\square$For $V\in P$ , we define a quadratic form$t(\Gamma^{;}, F)=\langle \mathrm{H}^{\frac{1}{\rho 2}}F,$

$t$

by

$\mathrm{H}^{\frac{1}{\rho 2}}F\rangle_{\mathrm{A}\mathrm{t}}+\langle V_{+}^{\frac{1}{2}}F,$$Q(t)=Q(\mathrm{H}_{p})\cap Q(V_{+})$$V+F \frac{1}{2}\rangle \mathcal{M}^{-}\langle V_{-}^{\frac{1}{2}}F,$$V_{-^{F\rangle}}^{\frac{1}{2}}\mathcal{M}$’,of a positive self-adjoint operator , i.e. $Q(A)=$where $Q(A)$ denotes the. By Lenrma 4.11, is positive closed form on $Q(t)$ . We denote the positive selfadjoint operator associated with$A$$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}\urcorner \mathrm{d}\mathrm{o}3\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{l}$$D(A^{\frac{1}{2}})$$t$$t\}_{)}\mathrm{y}$$\mathrm{H}_{\rho}\dotplus V_{+^{-V_{-}}}^{\cdot}$so that$Q(\mathrm{H}_{\rho}\dotplus \mathrm{T}/’-+V-)=Q(t)$.

.

Theorem 4.13 Let $V\in P$ andreplaced byholds with
Then$p\in[C_{b(\mathrm{F}_{-}^{d};}^{\mathrm{Y}2}’ \mathcal{H}_{-1,\wedge})]$$\mathrm{H}_{\rho}+V$$(\mathit{4}\cdot \mathit{1}\mathit{8})$Proof: As in

holds forsatisfy the conditions in Theorem 4.10.

$\mathrm{H}_{\rho}+V_{+^{-}}V_{-}$.

, one can easily see that by an approximation argument, (4.18)
set of bounded functions). Fix $V\in P$ . We set$[31,\mathrm{T}\mathrm{h}\mathrm{e}d\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{l}6.2]$$V\in L^{\infty}(\mathbb{R}^{d})$($\mathrm{t}\mathrm{h}\mathrm{e}$$(x)$$V_{+n}(x)=\{$$nV_{+},$,$V_{-}(x)$$V_{+(_{X})\geq?\mathrm{z}}V_{+}(x)<n,$$7V_{-m}(x)=\{$
$m$

,

,

$V_{-}(_{X})<m$,$V_{-}(x)\geq m$

Then we have$\langle F,$$e^{-t(-V_{-}}n \mathrm{H}_{\rho}+V+’?’)G\rangle \mathcal{M}=/\mathrm{R}^{d}\cross\Omega d\mu\exp(-\int_{0}^{t}(V_{+n}-V_{-})m(b(s)+x)dS)$$\cross\langle J_{t}F(b(t)+x\rangle, \exp(i\phi_{\mathrm{c}^{C}}(\sum_{\mu=1}^{d}\int_{0}t_{\sim,[j\mathrm{o}}]arrow tT_{-1}(p_{\mu}(\sim b(s)+x))db_{\mu}\mathrm{I})J_{0}G(x)\rangle_{\mathcal{E}}$(4. 23)

118

Define closed quadratic forms by$t_{n,n\iota}(\Gamma^{J}, F)$$=$$\langle \mathrm{H}^{\frac{1}{\rho 2}}$,$\mathrm{H}^{\frac{1}{\rho 2}}\rangle_{\mathcal{M}}+\langle V_{+n}^{\frac{1}{2}}F,$$Q(t_{1},,)\uparrow|\iota=Q(\mathrm{H})\rho$$t_{n,\infty}(F^{\urcorner}, F)$$=$$\langle \mathrm{H}^{\frac{1}{\frac{?}{p}}}$,$=$$\langle \mathrm{H}^{\frac{1}{p2}},$$V_{-^{F}\rangle} \frac{1}{2}\mathcal{M}$,’$\mathrm{H}^{\frac{1}{p2}\rangle_{J}}\backslash \Lambda+\langle V_{+}^{\frac{1}{\underline{9}}}$F.

$V_{+}^{\frac{1}{2}}t^{1\rangle_{\mathcal{M}}-}\langle\dagger\nearrow_{-}^{\frac{1}{2}}F^{1},$$Q(t_{\infty,\infty})=Q(\mathrm{H}_{\rho})\cap Q(V_{+})$We denote the self-adjoint operator associated with$t_{n,\infty}$$t_{n,m}\geq t_{\mathit{7}_{\dot{l}}},m+1\geq t_{n,m+2}\geq....$$l_{-}^{\prime\Gamma\{\rangle_{\mathcal{M}}} \frac{1}{2}$.
by. We have$\mathrm{H}_{\rho}+V_{+n}-V_{-}$$\geq t_{n.\infty}$

andin the sense of quadratic form on. Since, by the monotoneontheorem for forms ([17,VIII.
assoclated positive self-adjoint operators satisfy$\bigcup_{m}Q(t)7\iota,\eta \mathrm{t}=Q(\mathrm{H}_{p})$$t_{n,m}arrow t_{n,\infty}$$Q(.\mathrm{H}_{\rho})$,’$\mathrm{H}^{\frac{1}{\rho 2}}\rangle_{J\vee \mathrm{t}}+\langle V_{+n+7}^{\frac{1}{2}}F,$$1,I \frac{1}{2}F\rangle_{\mathcal{M}^{-}}?\langle V_{-}^{\frac{1}{2}}F^{\urcorner},$$Q(t_{n.\infty})=Q(\mathrm{H})\rho$$t_{\infty,\infty}(ft^{1}, F)$$V \frac{1}{2}F\rangle_{\mathrm{A}\mathrm{t}}+n-\langle\}_{-\uparrow?\iota-ln}^{r^{\frac{1}{2}}}F,$$\}’\frac{1}{2}Fr\rangle_{\mathcal{M}}$$t_{n,\infty}$

is closed), the$\ulcorner \mathrm{I}^{\urcorner}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}3.11]$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{V}\mathrm{e}\mathrm{l}\cdot \mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$$\mathrm{H}_{\rho}+V_{+’ \mathit{1}^{-}}V-marrow \mathrm{H}_{\rho}+V_{+n}-V_{-}$

in the strong resolvent sense, which implies that for all $t\geq 0$ ,$\exp(-t(\mathrm{H}_{\rho}+V+n-V_{-m}))arrow\exp(-t(\mathrm{H}_{\rho}+V_{+n}-V_{-)})$

strongly. Similarly, we have$t_{n,\infty}\leq t_{n+1.\infty}\leq t_{n+2,\infty}\leq\ldots\leq t_{\infty,\infty}$and$t_{n,\infty}arrow t_{\infty,\infty}$in the sense of$\mathrm{q}_{11}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}}$form on$\{\Gamma^{r}\in\bigcap_{n}Q(\dagger_{n,\infty \mathrm{I}}|\sup_{?l}t\infty n,(F, F)<\infty\}=Q(\mathrm{H}_{p})\cap Q(\dagger \text{ノ_{}+}’\mathrm{I}\cdot$

Hence the monotone convergence theorem for forms ( $[17,111$ . Theorem $3.13],[30]$ ), we get$e\mathrm{x}\mathrm{p}(-t(\mathrm{H}_{\rho}\dotplus V_{+\iota-}\uparrow-V\mathrm{I})arrow\exp(-t(\mathrm{H}_{p}\dotplus V_{+^{-}}V-))$,$t\geq 0$,

strongly. On the other hand, the 1.h.s.of (4.23) converges to$\int_{\mathrm{R}^{d}\cross\Omega}d\mu\exp(-\int_{0}^{t}(V_{+n}-\iota\nearrow_{-)}(b(s)+x)dS)$$\cross\langle J_{t}F(b(t)+x),\exp(i\varphi_{\mathcal{E}}^{\partial}(\sum_{\mu=1}^{(}\int_{0}t][j_{0t}arrow\phi_{-}1l\sim(\tilde{p}_{\mu}(b(s)+x))db_{\mu}\mathrm{I})J_{0}G(X)\rangle\epsilon$(4. 24)as $marrow\infty$ by the monotone convergence theorem for integrals. Also (4.24) converges to ther.h.s of (4.18) as $?arrow\infty$ by the Lebesgue dominated convergence theorem. Thus theis complete.

$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{o}_{\square }\mathrm{f}$119

INEQUALITIES5

In this section we shall derive some inequalities similar to classical models from the functionalbe the spectrum of . For
integral representation constructed in Section IV. Let
simplicity, we put for $V\in P$$A$$\sigma(A)$$\mathrm{H}_{\rho}\dotplus\dagger\nearrow-+\cdot V-$$\equiv$$-. \frac{1}{2}\triangle\dotplus V_{+}-\cdot V_{-}$$\equiv$,$\mathrm{H}_{\rho}+V$$- \frac{1}{2}\triangle+\mathrm{I}^{r}$.

Theorem 5.1 (Diamagnetic inequality)
satisfy the conditions in Theorem 4.10. Then
Let $V\in P$ and$\rho\in[C_{b}^{2}(\mathbb{R}^{d};\mathcal{H}_{-}1)]$

inf$a(- \underline{\frac{1}{9}}\triangle+V)\leq\inf\sigma(\mathrm{H}_{\rho}+V)$

Proof: Similarly to (4.21), we can also$‘ \mathrm{s}\mathrm{e},\mathrm{e}$.

(5. 1)

that for $V\in P$(5. 2)$|\langle F_{6^{-t\langle V}},p+)G\mathrm{H}\rangle_{1},|\leq\langle||F||_{\mathcal{F}},$ $e^{-t(-} \frac{1}{2}\Delta+V)||G||_{f}\rangle L2(\mathrm{R}^{d})$Fixsuch that
denotes the spectral projection of$G\in \mathcal{M}$$F_{\lrcorner \mathrm{H}_{\rho}+V}$$E_{\mathrm{H}_{\rho}+1^{r}},([E_{0}, E_{0}+\epsilon))G\neq 0$$\inf\sigma(\mathrm{H}_{p}+V)$. for all$\mathrm{H}_{\rho}+V$

and$0<\epsilon<\epsilon_{0}$$=$$\varliminf_{t\infty}-\frac{1}{t}\log\langle G.e^{-t1^{\mathrm{H}_{\rho}+}V}G)\rangle_{\lambda 4}$$\geq$$t arrow\infty 1\mathrm{i}\mathrm{n}\urcorner-\frac{1}{t}\log\langle||G||\mathcal{F},$$\geq$

linl$\dagger-\infty$$\geq$

with some. Then$\epsilon_{0}>0$$e^{-}(- \frac{1}{2}\Delta+V)|t|G||_{\mathcal{F}}\rangle_{L^{2}(\mathrm{R}^{d})}$$\underline{1}\log(||G||_{\mathcal{F}}^{2}e-t\inf\sigma(-\frac{1}{2}\Delta+V)\mathrm{I}$$t$$\inf a(-\frac{1}{2}\triangle+V)$.

Thus (5.1) follows.
Theorem 5.2 (Abstract Kato’s
Let $V\in P$ andSuppose that

, where$E_{0}= \inf a(\mathrm{H}_{\rho}+V)$$\square$$\uparrow nequality$$\rho\in[C_{b}^{2}(\mathbb{R}^{d};\mathcal{H}-1)].;atisfy$)

the conditions in Theorem 4.10.
and $G\in D((\mathrm{H}_{\rho}+V))$ .

$\psi\in D((-\frac{1}{2}\triangle+V)^{\frac{1}{2}})_{\mathrm{t}}?\ell)\geq 0$

Then$||G( \cdot)||_{F}\in D((-\frac{1}{2}\triangle+V)^{\underline{\frac{1}{9}}})$and the following inequality holds$\Re((signG)\psi$ ” $( \mathrm{H}_{\rho}+V)G\rangle_{\mathcal{M}}\geq\langle(-\frac{1}{\mathit{2}}\Delta+V)^{\frac{1}{2}}\psi,$$(- \frac{1}{2}\Delta+V)^{\frac{1}{2}}||G||\mathcal{F}\rangle_{L^{2}}(\mathrm{R}^{d})$(5. 3)

120

Proof: The proof is a slight modification of that of [15]. By (4.21), we$\langle||F||_{\mathcal{F}},$$\mathrm{I}_{1}\mathrm{a}\mathrm{v}\mathrm{e}$(5. 4)$\frac{I-\epsilon^{-t(\Delta V)}\frac{1}{2}+}{t}||F||\mathcal{F}\rangle_{L()}2\mathrm{R}^{d}\leq\langle F,$ $\frac{I-e^{-t\{+\mathrm{I}}\mathrm{H}_{\rho}\mathrm{t}\prime}{t}F1\rangle_{\mathcal{M}}$Putting$s_{\mathrm{t}}(\phi, \phi)\equiv\langle\phi,$

one can see that$\{s_{t}\}_{t\geq 0}$$\frac{I-\epsilon^{-t()}\frac{1}{2}\Delta+V}{t}\phi\rangle_{L(\mathrm{R})}2d$$\phi\in L^{2}(\mathrm{F}^{d}’)$,is a family of positive closed quadratic forms such that$\backslash \backslash _{t}\leq s_{k}$,$t\leq k$.

Thus the monotone convergence theorem for forms, we see that $s_{\infty}(G’, G) \equiv\sup_{t}s_{t}(G, G)$ is
a closed quadratic form on$R_{\infty},$$= \{F\in \mathcal{M}|\sup(tF, F)<\infty\}$.

moreover the corresponding positive self-adjoint operators to converge tothe strong resolvent sense. Since, by (5.4), $F\in D(\mathrm{H}_{\rho}+V)$ implies that
have$||F||_{f} \in D((-\frac{1}{2}\triangle+V)^{\frac{1}{2}})$$\Re\langle F,$. By$(\overline{:_{\mathrm{J}}}.2)$$||F||_{\mathcal{F}}\in R_{\langle\infty}$, we have$\frac{1}{t}(e^{-t\mathrm{t}V}-\mathrm{H}_{\rho}+)I)C\tau\rangle \mathcal{M}\leq\langle||F||f,$ $\frac{1}{t}e^{-t(\frac{1}{2}}-\Delta+V)||G||_{\mathcal{F}}\rangle L2\mathrm{t}\mathrm{R}^{d})-\frac{1}{f_{J}}\Re\langle F, c\rangle_{\lambda 4}$

Substituting. $(5.5)$, into (5.5). we have

$F=(signG^{\mathrm{t}}\mathrm{I}\uparrow f$$\Re\langle(signG’)\psi,$

Then, taking

in
, we

$(- \frac{1}{2}\triangle+V)^{\frac{1}{2}}$$s_{t}$$\frac{1}{t}(e^{-t1^{\mathrm{H}_{\rho}+})}-\mathrm{L}\gamma I)G\rangle_{\mathcal{M}}\leq\langle\psi,$$tarrow 0+\mathrm{w}\mathrm{e}$$\lim_{tarrow 0+}\Re\langle(signG)\psi,$$\frac{1}{t}(e^{-t(-\frac{1}{2}\Delta})-I+^{\iota^{\mathit{7}}})||G||_{\mathcal{F}}\rangle_{L}2(\mathrm{R}d)$see that$\frac{1}{t}(e^{-t()}-\mathrm{H}_{\rho}+\backslash ^{\mathit{7}}I)G\rangle \mathcal{M}=-\Re\langle(signG)\psi, (\mathrm{H}_{\rho}+V)G\rangle_{\mathcal{M}}$,an d$\lim_{tarrow 0+}\langle\uparrow l$”$\frac{1}{t}(e^{-t(-\frac{1}{2}}$)$-\Delta+VI)||C_{I}||f\rangle_{L^{2}(}\mathrm{R}^{d}$)$=-\langle(-\underline{\frac{1}{9}}\triangle+V)^{\frac{1}{2}}\psi,$

Thus (5.3) follows.

$(- \frac{1}{2}\Delta+V)\frac{1}{2}||G’||\rangle L2(\mathrm{R}^{d})$$\square$

121

REMARKS6

Any concrete core of the Hamiltonian
defined in Section IV is not known. Ill [23],the case when the coupling cconstant is sufficiently small, the authors proved essentiallyself-adjointness of.
(2) In the FKI formula, the Wiener path measure
is more useful than the Brownian pathmeasure $Db$ . For the Schr\"odinger Hamiltonian$.(1)$$\mathrm{H}_{\rho}$$\ln$$\mathrm{H}_{\rho}$$d\mu$$\mathrm{H}_{cl}=.-\text{可}(-iD_{\mu}-1A\mu)^{2}+V$

the FKI formula is, first, established for a magnetic vector potential$\mu=$, and after that it is extended toby a limiting argument $([31])$ .
In the model which we consider,corresponds to the classical magnetic vector potential. But we have no strategy of limiting argument used in the classical model. Then
it is necessary to deal with the Hilbert space-valued stochastic integral forsuch that
directly (not using limiting arguments as in the classical case).
Then it can not be assumed thati.e., one can not define (see(2.6)
$A_{\mu}(\cdot)\in C_{0}^{\infty}(\mathbb{R})d,$

$1,$

$d$

$A_{\mu}(\cdot)\in L_{l_{oC}}^{2}(\mathbb{R}d)$$\ldots,$$\phi_{\mathcal{F},\mu}^{\rho}$$A_{\mu}(\cdot)$$\phi_{F,\mu}^{\rho}$$\pi_{-1}(\tilde{\rho}_{\mu}(\cdot))\in C_{b}^{2}(\mathrm{p}^{d};[\overline{\mathcal{H}}_{-1}])$$\pi_{-1}(\tilde{\rho_{\mu}.}(\cdot.).)\in H^{2}(\mathbb{R}^{d};[\overline{\mathcal{H}}_{-1}])$$)$$\int_{0}^{t}\phi \mathcal{F}(\pi_{-}1(\tilde{\rho}_{\mu}(\omega(s\mathrm{I})))d\omega_{\mu}$.

Therefore we consider the Hilbert) space-valued stochastic integral not on the Wiener pathbut on the Brownian path.
(3) The FKI formula holds without the Coulomb gauge condition
However, in ourmodel, if one does not assume the Coulomb gauge condition (4.17), then the integrand on
the r.h.s.of (4.18), the factor$([31]).\cdot$$\ln$(6. 1)$\exp(-\frac{1}{\mathit{2}}\sum_{k=1}^{2^{n}}\int_{\frac{\lambda-1}{2^{n}}}\frac{k}{2^{l}},t_{\sim,[j\frac{k}{2}-_{\tau t}]}\pi 1\sum^{d}\partial_{\mu-1}(\tilde{p}\mu(b(_{S))})ds\mathrm{I}t\urcorner\mu=1$

appears. It is not clear how to show the convergence of (6.1) as $narrow\infty$ in the strong.
topology in(4) In scalar field theory ([21,22,26]), the range of the projection(notations follow [26])can be characterized by some support properties(
III.4]), i.e.
$L^{2}(\Omega;[\overline{\mathcal{H}}_{-2}])$$e_{[\alpha,b]}$$[\mathit{2}6,\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$Ran $(e_{[a,b]})=\{f\in N|supp.f\subset(a, b)\cross \mathbb{R}^{d}\}^{-}$ .
In particularRan.

$(e_{t})=\{f\in N|suppf\subset\{t\}\cross \mathrm{F}_{\mathrm{A}}^{d}’\}^{-}$

However, the corresponding projection

be characterized in such a way. For$[e_{[a,b]}]$, which we introduce in Proposition 3.2, can not$\mathrm{e}\mathrm{x}\mathrm{a}\ln_{\mathrm{P}^{1}}\mathrm{e}$$Ran.([e_{t}])\neq\{\pi_{2}(f)\in[\overline{\mathcal{H}}_{-2}]|suppf\subset\oplus_{\mu=1}^{d}\{t\}\cross \mathbb{R}^{d}\}^{-}$

.

122

7

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