数理解析研究所講究録 1208 巻 2001 年 170-182

170

THE FEYNMAN PATH INTEGRAL REPRESENTATION OF
GREEN FUNCTIONS OF THE POSITION AND THE
MOMENTUM OPERATORS,$\acute{|}_{\mu}^{\check{\neq}/||^{\mathfrak{l}}|\mathrm{X}}\iota$$/\grave{\acute{\mathrm{k}}}\cdot.\ovalbox{\tt\small REJECT}\sqrt$–,$\grave{\grave{\nu}}\ovalbox{\tt\small REJECT}_{\sim}|7,\mathrm{T}\backslash$(Wataru Ichinose)

Department of Mathematical Science, Shinshu University, Matsumoto 3908621, Japan. E-mail: ichinose@math.shinshu-u.ac.jp

1. Introduction
It was an interested and important problem to give the description of quantization, ie of passing ffom classical physical systems to the corresponding
quantum ones, ffom the moment that quantum mechanics came into existence.

In the end Heisenberg and Schrod inger succeeded in giving the description
based on the notion of operators. On the other hand in 1948 Feynman pr0posed an essentially new description in [2] based on the notion of the s0-called
Feynman path integrals. His description is that the probability amplitudes canbe constructed from the classical systems in adirect way with physical meanings. In 1951 Feynman himself gave the description reformulated by means of
the path integrals in phase space in [3]. Now we know that his description isvery useful and applied to wide areas in physics (cf. [7, 20]).

Since Feynman published his paper, much work has been done by physists
and mathematicians to give the rigorous meaning to the Feynman path inteyak. Some definitions of the Feynman path integrals are proposed and proved*Research partially supported by Grant-in-Aid for Scientific N0.10640176, Ministry of
Education, Science, and Culture, Japanese Government

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to be well-defined under some assumptions. See [6, 14, 21] and their references.
Recently the author in [9-12] studied the time-slicing approximate integrals,

determined through broken line paths as oscillatory integrals, of the path integral in configuration space and also in phase space and then proved in a
general way their convergence in$L^{2}$

space. It is noted that the approximate

integrals studied in [9-12] are very familiar in physics (cf. [4, 5, 20, 21]).

Our aim in the present paper is to study the path integral representationof correlation functions of the position and also the momentum operators andthen to give arigorous meaning to their representation. It seems that there

have been no results of this problem. Our path integral representation ofcorrelation functions is defined by the limit of the time-slicing approximate integrals, determined through broken line paths as oscillatory integrals, similarly
to the path integral in [9 –12]. As is well known, correlation functions aresome of the most important quantities in quantum mechanics and quantum
field theory (cf. [16, 20]). In physics the path integral representation is wellknown of correlation functions of only the position operators, though it has
not been rigorous. We note that in the present paper amore general representation, ie of correlation functions including the momentum operators, is given

rigorously and the Feynman path integrals in phase space determined in [12]are used for obtaining our results. In addition, we note that we can give thepath integral representation of the canonical commutation relations , which

are the most fundamental in quantum mechanics.
The plan of the proof is as follows. The approximate integral of the Feyn-man path integral is determined correspondingly to each subdivision of thetime interval. We consider the family of all approximate integrals. We firstshow the uniform boundedness of the family of approximate integrals in someweighted Sobolev spaces. This result is essential in our proof. By using this re172
suit of the boundedness we show the equi-continuity w.r.t the time variable ofthe family of approximate integrals in our weighted Sobolev spaces. Then, byapplying the abstract Ascoli-Arzel\‘a theorem we can prove convergence of theapproxim. ate integrals of the Feynman path integral in our weighted Sobolev
spaces as the size of subdivisions tends to zero. We note that our method ofproving convergence is direct compared to that in [9-12], where convergencein only$L^{2}$

space was proved by using the results in [8] about solutions of thecorresponding Schrod inger equation. Convergence of the approximate integrals

of correlation functions is proved by using the result above of convergence of
the Feynman path integral in the weighted Sobolev spaces and some delicate

calculus that is special to oscillatory integrals.
In the present paper we will only state main results and some remarks,

which will be given in fi2. See [13] for their proofs.

2. Main Results and Remarks
We consider some charged non-relativistic particles in an electromagnetic

field. For the sake of simplicity we suppose charge and mass of every particle
to be one and $m>0$ , respectively. We consider $x\in R^{n}$ and $t\in[0,T]$ . Let$E(t, x)=(E_{1}, \cdots, E_{||})\in R^{n}$

and $(B_{jk}(t, x))_{1\leq j<k\leq n}\in R^{n(n-1)/2}$ denote electricstrength and magnetic strength tensor, respectively and $(V(t, x)$ , $A(t, x))=$$(V,A_{1}, \cdots, A_{n})\in R^{\mathrm{n}+1}$

an electromagnetic potential, ie$E=- \frac{\partial A}{\partial t}-\frac{\partial V}{\partial x}$,$d( \sum_{j=1}^{n}A_{j}dx_{j})=\sum_{1\leq j<k\leq’\iota}B_{jk}dx_{j}\wedge dx_{k}$on$R^{n}$

,

(2.1

173

where

$\partial V/\partial x=$$(\dot{x}\in R^{n})$$(\partial V/\partial x_{1}, \cdots, \partial V/\partial x_{n})$. Then the Lagrangian function.
$\mathcal{L}(t,$$x,\dot{x}$is given by$A$$\mathcal{L}(t, x,\dot{x})=\frac{m}{2}|\dot{x}|^{2}+\dot{x}\cdot$

-V.

(2.2)

The Hamiltonian function $H(t, x,p)(p\in R^{n})$ is defined through the Legendre

transformation of$\mathcal{L}$

by

.

(2.3)

$H(t,x,p)= \frac{1}{2m}|p-A|^{2}+V$

Let$T^{*}R^{n}=H_{x}^{l}\cross R_{p}^{n}$the spaces of all paths$q$$\thetaarrow(q(\theta),p(\theta))\in T^{*}R^{n}$$q\in(R^{n})^{[s,t]}$denote phase space, and:$[s, t]\ni\thetaarrow q(\theta)\in R^{n}$, respectively.

and$(R^{n})^{[s,t]}$$(T^{*}R^{n})^{[s,t]}$and $(q,p)$ :$[s, t]\ni$

The classical actions $S_{c}(t, s;q)$ forin configuration space and $S(t, s;q,p)$ for $(q,p)\in(T^{*}R^{n})^{[s,t]}$ inphase spaxie are given by$S_{c}(t, s;q)= \int_{s}^{t}\mathcal{L}(\theta, q(\theta),\dot{q}(\theta))d\theta$,$( \theta)=\frac{dq}{d\theta}(\theta)$

(2.4)

,

(2.5)

$\cdot$and$S(t, s;q,p)= \int_{s}^{t}p(\theta)\cdot\dot{q}(\theta)-H(\theta, q(\theta),p(\theta))d\theta$

respectively (cf. [1]).

Let$\Delta$

We setwhere: $0=\tau_{0}<\tau_{1}<\ldots<\tau_{\nu}=T$ be asubdivision of the interval $[0, T]$ .

$| \Delta|=\max_{1\leq j\leq\nu}(\tau_{j}-\tau_{j-1})$

compact support. Forand$f\in C_{0}^{\infty}(R^{n})$is the space of all infinitely differentiate functions in$C_{0}^{\infty}(R^{n})$$\mathrm{C}_{\Delta}(t, s)f$

. Let $0\leq s\leq t\leq T$ and$\Delta$$G_{\Delta}(t, s)f$$R^{n}$,

withabove we define the time-slicing approximate integralsof the Feynman path integrals in configuration space

and in phase space, respectively as follows.

At first we define$T$.

We take 1$t\leq\tau_{\mu}$

.

For

$y$

$\mathrm{C}_{\Delta}(t, s)f$

. We set$\leq\mu’\leq\mu\leq\nu$,$x^{(j)}$$\mathrm{C}_{\Delta}(s, s)f=f$. Let $0\leq s<t\leq$such that $\tau_{\mu’-1}\leq s<\tau_{\mu’}$ and$(j=\mu’, \mu’+1, \ldots, \mu-1)$$q_{\Delta}(\theta;y, x^{(\mu’)}, \ldots, x^{(\mu-1)}, x)\in(R^{n})^{[s,t]}$

and

$x$

in

$H^{\iota}$$\tau_{\mu-1}<$

let’s defineby the broken line path joining point$\mathrm{s}$

174

$y$

at ,
$s$

$x^{\mathrm{U})}$$\mathrm{C}_{\Delta}(t, s)f$

at$\tau_{j}$and$(j=\mu’, \mu’+1, \ldots, \mu-1)$

$x$

at

$t$

in order. We defineby$( \mathrm{C}_{\Delta}(t, s)f)(x)=\sqrt{\frac{m}{2\pi i\hslash(t-\tau_{\mu-1})}}^{n}\prod_{j=\mu’+1}^{\mu-1}\sqrt{\frac{m}{2\pi i\hslash(\tau_{j}-\tau_{j-1})}}^{n}\sqrt{\frac{m}{2\pi i\hslash(\tau_{\mu’}-s)}}^{n}$$\cross \mathrm{O}\mathrm{s}-\int\cdots\int(\exp i\hslash^{-1}S_{c}(t, s;q_{\Delta}))f(y)dydx^{(\mu’)}\cdots dx^{(\mu-1)}$Here$\mathrm{O}\mathrm{s}-\int\cdots\int g(y, x^{(\mu’)}, \ldots,x^{(\mu-1)})dydx^{(\mu)}’\cdots dx^{(\mu-1)}$

.

(2.6)

means the oscillatory

integral (cf. [15]).

We define$G_{\Delta}(t, s)$. For the sake of simplicity we set $s=0$ . The general casecan be defined in the same way thatwas done. We set $G_{\Delta}(0,0)f=f$ .
For $0<t\leq T$ take a $1\leq\mu\leq\nu$ such that
. For $v^{(j)}(j=$$\mathrm{C}_{\Delta}(t, s)$$\tau_{\mu-1}<t\leq\tau_{\mu}$

$0,1$

,

$\ldots$, $\mu-1$ ) in velocity space$R^{n}$

we define$v_{\Delta}(\theta;v^{(0)}, \ldots,v^{(\mu-1)})\in(R^{n})^{[0,t]}$in velocity space by the piecewise constant path taking

for

for

$(j=0,1, \ldots,\mu-2)$ and $v^{(\mu-1)}$$\tau_{j}<\theta\leq\tau_{j+1}$$v^{(0)}$

at $\theta=0$ ,$\tau_{\mu-1}<\theta\leq t$$v^{(j)}$. Let$q_{\Delta}(\theta;x^{(0)}, \ldots,x^{(\mu-1)},x)\in(R^{n})^{[0,t]}(x^{(0)}=y)$be the path in configuration space

defined above. Then we determine the path

$p_{\Delta}(\theta;x^{(0)}$$v^{(\mu-1)})\in(R^{n})^{[0,t]}$,$\ldots$,$x^{(\mu-1)}$

, ,
$x$

,

$\ldots$,

in momentum space by.

$p_{\Delta}( \theta):=\frac{\partial \mathcal{L}}{\partial i}(\theta, q_{\Delta}(\theta),v_{\Delta}(\theta))=mv_{\Delta}(\theta)+A(\theta, q_{\Delta}(\theta))$

We define$v^{(0)}$$G_{\Delta}(t, 0)f$

(2.7)

by

$-(G_{\Delta}(t,0)f)(x)=(2 \pi\hslash)^{-n\mu}\mathrm{O}\mathrm{s}-\int\cdots\int(\exp i\hslash^{-1}S(t,0;q_{\Delta},p_{\Delta}))$$\cross f(x^{(0)})dmv^{(0)}dx^{(0)}dmv^{(1)}dx^{(1)}\cdots dmv^{(\mu-1)}dx^{(\mu-1)}$.

Let $L^{2}=L^{2}(R^{n})$ be the space of all square integrable functions ininner product$| \alpha|=\sum_{j=1}^{n}\alpha_{j}$$(\cdot, \cdot)$,

and norm$||\cdot||$. For amulti-index$r_{x}=(\partial/\partial x_{1})^{\alpha_{1}}\cdots(\partial/\partial x_{n})^{\alpha_{n}}$12] we proved the following$\alpha=(\alpha_{1}, \ldots, \alpha_{n})$(2.8)$R^{n}$

withwe writeand $<x>=\sqrt{1+|x|^{2}}$ . In [9-175

Theorem A. Let $\Psi_{x}E_{j}(t, x)(j=1,2, \cdots,n),\partial_{x}^{\alpha}B_{jk}(t,x)$ and$j<k$$\leq n)$be continuous in $[0, T]\cross R^{n}$ for all$|\partial_{x}^{a}E_{j}(t, x)|\leq C_{\alpha}$

in $[0, T]$$\cross R^{n}$,$|\alpha|\geq 1$$\alpha$. We suppose, $|F_{x}B_{jk}(t,x)|\leq C_{\alpha}<x>^{-(1+\delta)}$ , $|\alpha|\geq 1(2.9)$

for some constants $\delta>0$ and$C_{\alpha}$, where 6is independentof . Let $(V, A)$ be an arbit rary potential such that ,$V$$\alpha$$(j, k =1,2, \cdots, n)$$\partial A_{j}/\partial x_{k}$Letare continuous inbe small Then both of$|\Delta|$$\mathrm{C}_{\Delta}(t, s)$and$\partial V/\partial x_{j}$,$\partial A_{j}/\partial t$and. Then we have: (1)are well-definedon$[0, T]\cross R^{n}$$G_{\Delta}(t, s)$$C_{0}^{\infty}$. They are equal to one other.
be small Then there exists a constant $K\geq 0$ independent ofand can be extended to bounded operators on(2) Let$\partial_{t}B_{jk}(t, x)(1($$L^{2}$$\Delta$$|\Delta|$

such that$||\mathrm{C}_{\Delta}(t, s)f||\leq e^{K(t-s)}||f||$

for all$f\in L^{2}$. (3)$As|\Delta|arrow \mathrm{O}$, C&(t, )
$s$

$f$

,

(2.10)$0\leq s\leq t\leq T$

for$f\in L^{2}$

converges in$L^{2}$

unifomly

in $0\leq s\leq t\leq T$ and this limit satisfies the Schrodinger equation$i \hslash\frac{\partial}{\partial t}u(t)=H(t)u(t)$(2.11)$u(s)=f$ ,

,

where

.

(2.12)$H(t)= \frac{1}{2m}\sum_{j=1}^{n}(\frac{\hslash}{i}\frac{\partial}{\partial x_{j}}-A_{j})^{2}+V$

We write $\int(\exp i\hslash^{-1}S_{c}(t, s;q))f(q(s))Dq$ and $\int\int(\exp i\hslash^{-1}S(t, s;q,p))$, respectively.
as$\cross f(q(s))DpDq$ for the limit ofand$\mathrm{C}\mathrm{A}(\mathrm{t}, s)f$

$G_{\Delta}(t, s)f$$|\Delta|arrow 0$

Remark 2.1. In (2.8) we make the change of variables:$v^{(\mu-1)})arrow(p^{(0)}, \ldots,p^{(\mu-1)})\in R^{n\mu}$$mv^{(j)}+A(\tau_{j}, x^{(j)})$. Then, setting$G_{\Delta}(t, 0)f$$R^{n\mu}\ni(v^{(0)},$ $\ldots$,$p^{(j)}=\partial \mathcal{L}(\tau j, q\Delta(\tau j),v_{\Delta}(\tau j))/\partial\dot{x}=$is written$(G_{\Delta}(t, 0)f)(x)=(2 \pi\hslash)^{-n\mu}\mathrm{O}\mathrm{s}-\int\cdots\int(\exp i\hslash^{-1}S(t, 0;q_{\Delta},p_{\Delta}))$$\cross f(x^{(0)})dp^{(0)}dx^{(0)}dp^{(1)}dx^{(1)}\cdots$ $dp^{(\mu-1)}dx^{(\mu-1)}$

176

in the form of an integral on the product space of phase space.

Remark 2.2. In Theorem Aonly smooth electromagnetic fields are considered. We can apply Theorem Aas follows to the case that electromagnetic

fields have singularities. For example consider atomic Hamiltonians
H
where
$1,2$

,

$=- \frac{\hslash^{2}}{2m}\sum_{j=1}^{n}\Delta_{j}-\sum_{j=1}^{n}\frac{n}{|x^{(j)}|}+\sum_{1\leq j<k\leq n}\frac{1}{|x^{(j)}-x^{(k)}|}$$x^{(j)}\in R^{3}$$\ldots$

and$\Delta_{j}$denotes the Laplacian operator in) be real valued infinitely differentiable functions in$|P_{x}\chi\iota(x)|<\infty$for$R^{3}$$x^{(j)}$

,

. Let

such that$\chi_{l}(l=$$\sup_{x\in R^{3}}$and$|\alpha|\geq 2$$\lim_{larrow\infty}\chi\iota(x)=-\frac{1}{|x|}$$.\mathrm{n}L^{2}(R^{3})+L^{\infty}(R^{3})$.

We set$H_{l}=- \frac{\hslash^{2}}{2m}\sum_{j=1}^{n}\Delta_{j}+\sum_{j=1}^{n}n\chi\iota(x^{(j)})-.\sum_{\lrcorner 1<<k\leq n}\chi_{l}(x^{(j)}-x^{(k)})$We know that$e^{-:\hslash^{-1}(t-\epsilon)H_{l}}$converges to$e^{-\dot{|}\hslash^{-1}(t-\epsilon)H}$strongly in$L^{2}$.
as$\mathit{1}arrow\infty$.

See Example 2of \S X.2 in [18] and Theorems VIII.21, VIII.25 in [17] and alsosee [22]. It follows from Theorem Ain the present paper that$e^{-:\hslash^{-1}(t-s)H}{}^{\mathrm{t}}f$

for $f\in L^{2}$ can be written in the form of our path integrals. So we see thatcan be written in the form of the limit of our path integrals. The
same argument can be applied to the general case of electromagnetic fields$e^{-:\hslash^{-1}(t-\epsilon)H}f$

having singularities.

Let$B^{a}$$(a=1,2, \ldots)$ be the weighted Sobolev space $\{f\in L^{2};||f||_{B^{a}}:=$$||f||+ \sum_{|\alpha|=a}(||x^{\alpha}f||+||\Psi_{x}f||)<\infty\}$and$B^{-a}$its dual space. We write $B^{0}=L^{2}$ .

As the first result in the present paper we have

Theorem 1. Besides the assumption of Theorem A we suppose$|P_{x}A_{j}|\leq C_{\alpha}$,$|\alpha|\geq 1$,$|F_{x}V|\leq C_{\alpha}<x>$,$|\alpha|\geq 1$

(2.13

177

in $[0, T]$$\cross R^{n}$. Let $a=0,1$ ,$\ldots$. Then we have: (1) Let$|\Delta|$

be small Then

there eists a constant $K_{a}\geq 0$ such that$||\mathrm{C}_{\Delta}(t, s)f||_{B^{a}}\leq e^{K_{a}(t-s)}||f||_{B^{a}}$. In addition,
for allfunction in $0\leq s\leq t\leq T$ . (2) As
uniforrmly in $0\leq s\leq t\leq T$ .
$f\in B^{a}$$\mathrm{C}_{\Delta}(t, s)f$for,

$f\in B^{a}$(2.14)$0\leq s\leq t\leq T$

is continuous as a $B^{a}$ -valued$|\Delta|arrow 0_{f}\mathrm{C}_{\Delta}(t, s)f$for

converges in$f\in B^{a}$$B^{a}$

Remark 2.3. Suppose that

$E$

and

$B_{jk}$satisfy the assumption of Theorem

A. We remark that then, we can find apotential $(V, A)$ satisfying (2.13), which

was proved in Lemma 6.1 of [10]. In addition, we can easily prove Theorem
Afrom Theorem 1where $a=0$ by using the gauge transformation as in the
proof of Theorem of [10].

Remark 2.4. Let$B^{a+2}$$\mathcal{E}_{t,s}^{0}([0, T];B^{a+2})\cap \mathcal{E}_{t,s}^{1}([0, T];B^{a})$-valued continuous and$B^{a}$

denote the space of all-valued continuously differentiable functions in$0\leq s\leq t\leq T$. Suppose (2.13) and consider the Schrodinger equation (2.11)

for. Then uniqueness of the solutions in$f \in\bigcup_{a=0}^{\infty}B^{a}$$\mathcal{E}_{t,s}^{1}([0, T];B^{a})$$\bigcup_{a=-\infty}^{\infty}\mathcal{E}_{t,s}^{0}([0, T];B^{a+2})\cap$has been proved in [8]. So we write the solution of (2.11) as$U(t, s)f$ hereafter. As was noted in introduction, Theorem 1is proved directly

without the use of the results in [8]. We also note that we can prove uniquenessstated above of the solutions of (2.11) from Theorem 1as in the proof of
Theorem in [8].

Let$v^{(0)}$,$\ldots$$\Delta$,

be subdivision and$v^{(\nu-1)}$$(q_{\Delta}(\theta;x^{(0)}, \ldots, x^{(\nu-1)}, x),p\Delta(\theta;x^{(0)}$) $)\in(T^{*}R^{n})^{[0,T]}$ the path determined before for$\Delta$,$\ldots$. Let,$x^{(\nu-1)}$

, ,
$x$

$0\leq t_{1}\leq$

178. For $z=q$ or we write$t_{2}\leq\ldots\leq t_{k}\leq T$$p$$\int\int(\exp i\hslash^{-1}S(T, 0;q_{\Delta},p_{\Delta}))(z_{\Delta})_{j_{k}}(t_{k})\cdots(z_{\Delta})_{j_{1}}(t_{1})f(q_{\Delta}(0))Dp_{\Delta}Dq_{\Delta}$$:= \mathrm{O}\mathrm{s}-\int\cdots\int(\exp i\hslash^{-1}S(T, 0;q_{\Delta},p_{\Delta}))(z_{\Delta})_{j_{k}}(t_{k})\cdots(z_{\Delta})_{j_{1}}(t_{1})$$f(x^{(0)})(2\pi\hslash)^{-n\nu}dmv^{(0)}dx^{(0)}dmv^{(1)}dx^{(1)}\cdots dmv^{(\nu-1)}dx^{(\nu-1)}$$\cross$(2.15)

and$\int(\exp i\hslash^{-1}S_{c}(T, 0;q_{\Delta}))(q_{\Delta})_{j_{k}}(t_{k})\cdots(q_{\Delta})_{j_{1}}(t_{1})f(q_{\Delta}(0))Dq_{\Delta}$$:= \mathrm{O}\mathrm{s}-\int\cdots\int(\exp i\hslash^{-1}S_{c}(T, 0;q_{\Delta}))(q_{\Delta})_{j_{k}}(t_{k})\cdots(q_{\Delta})_{j_{1}}(t_{1})f(x^{(0)})$,(2.16)$\cross\prod_{j=1}^{\nu}\sqrt{\frac{m}{2\pi i\hslash(t_{j}-t_{j-1})}}ndx^{(0)}dx^{(1)}\cdots dx^{(\nu-1)}$

where$(z_{\Delta})_{j}$is the -th component of$\mathrm{j}$Theorem 2. Let$z_{\Delta}\in(R^{n})^{[0,\eta}$$0\leq t_{1}\leq t_{2}\leq\ldots\leq t_{k}\leq T$

assumption of Theorem 1we have: (1) Let(2.15) on$B^{a+k}$

into$C_{0}^{\infty}$$B^{a}$.

and $a=0,1$ ,$\ldots$. Under theis small. Then the operator$|\Delta|$

is well-defined and can be extended to a bounded operator from. In more detail, we have$|| \iint(\exp i\hslash^{-1}S(T,0;q_{\Delta},p_{\Delta}))(z_{\Delta})_{j_{k}}(t_{k})\cdots$$\cross$where$C_{a}$$(z_{\Delta})_{j_{1}}(t_{1})f(q_{\Delta}(0))Dp_{\Delta}Dq_{\Delta}||_{B^{l}}\leq C_{l}||f||_{B^{a+k}}$

is a constant independent of$t_{:}\neq t_{j}(i\neq j)$. Then as$|\Delta|arrow 0$$\Delta,t_{1}$,$\ldots$,$t_{k-1}$

and$t_{k}$,(2.17). (2) We assume, (2.15) for $f\in B^{a+k}$ converges in$B^{a}$, whichwe write $\iint(\exp i\hslash^{-1}S(T, 0;q,p))z_{j_{k}}(t_{k})\cdots z_{j_{1}}(t_{1})f(q(0))DpDq$. This limit isequal to $U(T,tk)zjhU\{tk,$operator$x_{j}$$\mathrm{t}\mathrm{k}-\mathrm{i}$)$\cdots\hat{z}_{j_{1}}U(t_{1},0)f$when $z=q$ and denotes, where$i^{-1}\hslash\partial_{x_{j}}$$\hat{z}_{j}$denotes a multiplication

when $z=p$ . (3) Let $t\in[0, T]$

179
and $f\in B^{a+2}$ . We take afor each Aso that$\mu$$\tau_{\mu-1}<t\leq\tau_{\mu}$. Then we have$\lim_{|\Delta|arrow 0}\int\int(\exp i\hslash^{-1}S(T, 0;q_{\Delta},p_{\Delta}))(p_{\Delta})_{k}(t)(q_{\Delta})_{j}(t)f(q_{\Delta}(0))Dp_{\Delta}Dq_{\Delta}$(2.18)$=U(T,t) \hat{q}_{j}\hat{p}_{k}U(t,0)f+\frac{\hslash}{i}\delta_{jk}\lim_{|\Delta|arrow 0}(\frac{\tau_{\mu}-t}{\tau_{\mu}-\tau_{\mu-1}})U(T,0)f$in$B^{l}$, where $6jk$ is the Kronecker delta. It is noted that the right-hand side

above is divergent if $j=k$ . (4) Here we don’t assume $t_{:}\neq t_{j}(i\neq j)$ . Let
be small. Then the operator (2.16) on
is well-defined and is equal to(2.15) where $z=q$ . In addition, in this case,allfor $f\in B^{a+k}$$|\Delta|$$C_{0}^{\infty}$$ie$converges in$B^{a}$, as$|\Delta|arrow 0$$z=q(\mathit{2}.\mathit{1}\mathit{5})$.

We write $\int(\exp i\hslash^{-1}S_{c}(T, 0;q))qjk(tk)\cdots q_{j_{1}}(t_{1})f(q(0))Dq$ for the limit of(2.16) as$|\Delta|arrow 0$

tum mechanics,$|f$, $t>*$ , where. Let’s use the notations of the Heisenberg picture of quan-$\hat{z}_{j}$$g^{*}$(t)=%(t)$0)^{-1}\hat{z}_{j}U(t, 0)$

,

$|f$

, $t>=\%(\mathrm{t})0)^{-1}f$ and $<f,t|=$is the complex conjugate of .
$g$

Corollary. Under the assumption of Theorem 1we have: (1) Let $0\leq t_{1}<$$t_{2}<\ldots<t_{k}\leq T,g\in L^{2}$

and $f\in B^{k}$ . Then we obtain the path integral

representation of correlation functions$<g,T|\hat{z}_{j_{k}}(t_{k})\cdots\hat{z}_{j_{1}}(t_{1})|f$, $0>(:=(|g, T>,\hat{z}_{j_{k}}(t_{k})\cdots\hat{z}_{j_{1}}(t_{1})|f, 0>))$$=(g, \int\int(\exp i\hslash^{-1}S(T, 0;q,p))z_{j_{k}}(t_{k})\cdots z_{j_{1}}(t_{1})f(q(0))DpDq)$.
(2.19)We also have$<g,T|\hat{q}_{j_{k}}(t_{k})\cdots\hat{q}_{j_{1}}(t_{1})|f$, $0>$$=(g, \int(\exp i\hslash^{-1}S_{c}(T,0;q))q_{j_{k}}(t_{k})\cdots q_{j_{1}}(t_{1})f(q(0))Dq)$

.

180

(2) Let $0\leq t$ $<t\leq T$ and $f\in B^{2}$ . Then we have for , $k=1,2$ ,$j$$\ldots$,$n$$, \lim_{tarrow t}\iint(\exp i\hslash^{-1}S(T, 0;q,p))(p_{j}(t)q_{k}(t’)-q_{k}(t)p_{j}(t’))f(q(0))DpDq$(2.21)$= \frac{\hslash}{i}\delta_{jk}\iint(\exp i\hslash^{-1}S(T, 0;q,p))f(q(0))DpDq$

in$L^{2}$.

Proof. Since$U(T,tk)zhU(tk, t_{k-1})\cdots\hat{z}_{j_{1}}U(t_{1},0)f=U(T,0)\hat{z}_{j_{k}}(tk)\cdots$ $\hat{z}_{j_{1}}(t_{1})f$, (2.22)

we can easily prove (2.19) and (2.20) from the assertions (2) and (4) of Theorem
2. It follows ffom the assertion (2) of Theorem 2that the left-hand side of(2.21) is equal to.

$\lim_{tarrow t}(U(T,t)\hat{p}_{j}U(t,\#)\hat{q}_{k}U(t’,0)f-U(T,t)\hat{q}_{k}U(t, t’)\hat{p}_{j}U(t’, 0)f)$

Here let’s use the fact that$B^{a}$

is continuous as a

$B^{a}$$||U(t, s)g||_{B^{t}}\leq \mathrm{e}^{K_{a}(t-\iota)}||g||_{B^{a}}$and $U(t, s)g$ for $g\in$-valued function in $0\leq s\leq t\leq T$ , which follows from

Theorem 1. Then$||U(t, t)\hat{q}_{k}U(t’, 0)f-\hat{q}_{k}U(t’, 0)f||_{B^{1}}$$\leq \mathrm{e}^{K_{1}(t-t’)}||\hat{q}_{k}(U(t, \mathrm{O})-U(t, 0))f||_{B^{1}}+||U(t, t’)\hat{q}_{k}U(t, 0)f-\hat{q}_{k}U(t, 0)f||_{B^{1}}$and so$\lim_{arrow t},\mathrm{U}(\mathrm{t},t)\hat{q}_{k}U(t, 0)f=\hat{q}_{k}U(t, 0)f$in$B^{1}$. Consequently we have$\lim_{\nuarrow t}U(T,t)\hat{p}_{j}U(t, t’)\hat{q}_{k}U(t, 0)f=U(T,t)\hat{p}_{j}\hat{q}_{k}U(t, 0)f$in$L^{2}$. Hence we can prove (2.21).

Q.E.D.

Remark 2.5. (i) The path integral representation (2.20) of correlation functions of the position operators is well known in physics, though it has not been
rigorous ([16, 20]). It is noted that our result (2.19) gives amore general rep-resentation of correlation functions including the momentum operators, (ii) It

181
follows from Theorem 2and (2.22) that the equation (2.21) is equivalent to(2.22),$\lim_{\nu\nearrow t}(\hat{p}_{j}(t)\hat{q}_{k}(t’)f-\hat{q}_{k}(t)\hat{p}_{j}(t’)f)=\frac{\hslash}{i}\delta_{jk}f$

ie the canonical commutation relations.

Example 2.1. Let $(V, A)$ be an electromagnetic potential such that$|\partial_{x}^{\alpha}V|+<x>^{1+\delta}|\partial_{x}^{a}A|\leq C_{\alpha}$

in $[0, T]$and$\cross R^{n}$,$|\alpha|\geq 2$,$|\partial_{x}^{\alpha}\partial_{t}A|\leq C_{\alpha}$,$|\alpha|\geq 1$

for some constant $\delta>0$ . Then since $E_{j}=-\partial A_{j}/$ リー$B_{jk}=\partial A_{k}/\partial x_{j}-\partial A_{j}/\partial x_{k}$$V/\partial x_{j}$from (2.1), we can see that the assumption ofTheorems 1and 2is satisfied.

References[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978.
[2] R. P. Feynman, “Space time approach to non-relativistic quantum mechanics”, Rev.

Mod. Phys. 20 (1948) 367-387.

$\mathrm{t}$[3] R. P. Feynman, ”An operator calculus having applications in quantum electrodynamics”, Phys. Rev. 84 (1951) 108128.

[4] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill,

New York, 1965.
[5] C. Garrod, ”Hamiltonian path integral methods”, Rev. Mod. Phys. 38 (1966) 483494.
[6] D. Fujiwara and T. Tsuchida, ”The time slicing approximation of the fundamental

solution for the Schrodinger equation with electromagnetic fields

J. Math. Soc. Japan49 (1997) 299-327.
[7] K. Huang, Quantum Field Theory: From Operators to Path Integrals, John Wiley andSons, New York, 1998.
[8] W. Ichinose, “Anote on the existence and -dependency of the solution of equations in$\hslash$quantum mechanics”, Osaka J. Math. 32 (1995) 327-345.
[9] W. Ichinose, “On the formulation of the Feynman path integral through broken line

paths”, Cornrnun. Math. Phys. 189 (1997) 17-33.
[10] W. Ichinose, “On convergence of the Feynman path integral formulated through broken

line paths”, Rev. Math. Phys. 11 (1999) 1001-1025

182[11] W. Ichinose, ”On convergence of the Feynman path integral in phase spac\"e, preprint.
[12] W. Ichinose, “The phase space Feynman path integral with gauge invariance and its

convergenc\"e, Rev. Math. Phys. 12 (2000) 1451-1463.
[13] W. Ichinose, “A rigorous proof of the path integral representation of correlation functions ”, preprint.

[14] G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman’s Operational
Calculus, Oxford Univ. Press, Oxford, 2000.

[15] H. KumanO-go, PseudO-Differential Operators, MIT Press, Cambridge, 1981.
[16] J. M. Rabin, “Introduction to quantum field theory for mathematicians”, pp. 183269,

in Geometry and Quantum Field Theory, e&.

$\mathrm{D}.\mathrm{S}$. Freed and$\mathrm{K}.\mathrm{K}$. Uhlenbeck, Amer.

Math. Soc., 1995.
[17] M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional Anal$\cdot$

ysis, Rev. and $enl$. $ed$, Academic Press, New York, 1980.
[18] M. Reed and B. Simon, Methods of Modern Mathematical Physics$II.\cdot$

Fourier Analysis,

Self-Adjointness, Academic Press, New York, 1975.
[19] M. Reed and B. Simon, Methods of Modern Mathematical Physics$IV.\cdot$

Analysis ofOperators, Academic Press, New York, 1978.
[20] L. H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, 1985.
[21] L. S. Schulman, Techniques and Applications of Path Intelration, John Wiley and Sons,

New York, 1981.
[22] K. Yajima, “Existence of solutions for Schrodinger evolution equations”, Commun.

Math. Phys. 110 (1987) 415-426