数理解析研究所講究録
第 1797 巻 2012 年 1-19

1Imaginary-Time Path Integrals for Three Magnetic
Relativistic Schrodinger Operators *By

Takashi ICHINOSE **Abstract
After brief introduction to path integral, we consider the problem with three magnetic relativistic
Schr\"odinger operators corresponding to the classical relativistic Hamiltonian symbol with magnetic vector
and electric scalar potentials. We discuss their difference in general and their coincidence in the case of
constant magnetic fields, as well as whether they are covariant under gauge transformation. Then results
are surveyed on path integral representations for their respective imaginary-time relativistic Schr\"odinger
equations, i.e. heat equations, by means of the probability path space measure coming from the L\’evy
process concemed.

Contents\S 1. Introduction\S 2.

Brief Introduction to Path Integral\S 2.1. What is path integral?
\S 2.2. How to Make It Mathematics?

\S 3. Three Magnetic Relativistic Schr\"odinger Operators\S 4. Gauge Covariance for Magnetic Relativistic Schr\"odinger Operators\S 5.

Imaginary-Time Path Integrals for Magnetic Relativistic Schr\"odinger Operators\S 6. Feynman and Dirac

References

2010 Mathematics Subject Classification(s): 81QlO;35S05;60J65;60J75;47D50;81S40;58D30.
Key Words: Feynman path integral; path integral; imaginary-time path integtral; Feynman-Kac formula; relativistic
Schrodinger operator; Feynman-Kac-It\^o formula; L\’evy process.
*based on two talks at RIMS Joint Introductory Workshop on Feynman Path Integral and Microlocal Anaysis, June
21-June 24, 2011.
**Professor emerims, Kanazawa University, Kanazawa 92kll92, Japan.

2\S 1. Introduction
In these notes, we consider the quantum Hamiltonian corresponding to the classical rela-tivistic Hamiltonian symbol$\sqrt{(\xi-A(x))^{2}+m^{2}}+V(x)$

(1.1)

,

$(\xi,x)\in R^{d}\cross R^{d}$for a spinless particle with mass , which is the sum of the kinetic energy term involving
magnetic vector potential $A(x)$ and the potential energy term of electric scalar potential $V(x)$ .
There are in the literature three kinds of quantum relativistic Hamiltonians depending on how toquantize the kinetic energy term $\sqrt{(\xi-A(x))^{2}+m^{2}}$ . We call them the relativistic Schrodinger
opemtors. We observe their difference in general, and next discuss their coincidence whenthe vector potential $A(x)$ is linear in , in particular, in the case of constant magnetic fields,
as well as handle whether they are gauge-covariant. Then, on this occasion, we would like tomake survey, which might be of some interest, on the results on path integral representations
for their respective imaginary-time unitary groups, i.e. real-time semigroups, by means of theprobability path space measure coming from the L\’evy process concemed. It will be of someinterest to collect them in one place to observe how they look like and different, though all
the three are essentially connected with the L\’evy process. Finally, an anecdote is referred to
between Feynman and Dirac conceming the subject.
We know that the authentic operator in relativistic quantum mechanics is the Dirac operator
for a spinning particle with mass , which is the first-order system of partial differential oper$m$

$x$

$m$

ators corresponding to the symbol $\sum_{j=1}^{3}\alpha_{j}(\xi_{j}-A_{j}(x))+m\alpha_{4}+V(x)$ , where$\alpha:=(\alpha_{1},\alpha_{2},\alpha 3,\alpha_{4})$-Dirac matrices. The magnetic relativistic Schr\"odinger operator without scalarpotential $(V=0)$ is considered to be the positive kinetic energy part of the Dirac operator. For
the path integral for the Dirac equation in space-dimension $d=1$ and in real time, i.e. in the
Minkowski space-time of two dimensions, we refer to [I82], [I84], [ITa84], [ITa88], [ITa87]
with its survey [I93], and [BCSS85], [CSS86], [Z88], [Z89].
The description of thses notes is of expositary character, beginning with a brief introduction
to Feynman path integral.

are the four$4\cross 4$\S 2. Brief Introduction to Path Integral\S 2.1. What is path integral?
It is a fabulous technique invented by Richard P. Feynman in his Princeton 1942 thesis (see[F05] and his 1948 paper [F48] to give altemative formulation of quantum mechanics. Its likehas never been made before or since. In fact, because of universality of its idea it has now cometo prevail over all the domains in quantum physics. He came to the idea, though, as in [F48]$)$3he himself wrote, ”suggested by some of Dirac’s remarks ([D33], [D35], [D45]) conceming therelation of classical action to quantum mechanics.” It is a special kind offunctional integml like$\int e^{(i/\hslash)S(X)}\mathcal{D}[X]$on space of paths $X:[0,t]\ni s\mapsto X(s)\in R^{d}$ with respect to a ‘measure’$\mathcal{D}[X]$on the space of

these paths, where $S(X)$ is integral of the Lagrangian $L(X),$ $S(X)= \int_{0}^{t}L(X(s))ds$, called action.

Consider the nonrelativistic Schr\"odinger equation for one particle with mass
(2.1)

$i \hslash\frac{\partial}{\partial t}\psi(t,x)=[-\frac{\hslash^{2}}{2m}\Delta+V(x)]\psi(t,x)$,$t>s$ ,

$x\in R^{d}$

$m$

:

,

where $\hslash=h/(2\pi)$ ($h>0$ :Planck’s constant). The solution is expressed as$\psi(t,x)=\int K(t,x;s,y)f(y)dy$

with integral kemel $K(t,x;s,y)$ , called fimdamental solution or propagator.
Feynman writes down this important quantity $K(t,x;s,y)$ as an ‘integral’(2.2)$K(t,x;s,y)= \int e^{iS(X)/\hslash}\mathcal{D}[X]$,

where $S(X)$ in our present case is given by(2.3)$S(X)=l^{t}[ \frac{m}{2}\dot{X}(\tau)^{2}-V(X(\tau))]d\tau$,$\dot{X}(\tau)=\frac{d}{d\tau}X(\tau)$.

stands for a uniform ’measure’, if it exists, on the space of paths
starting fromat time to arnve at position at time , formally, to be given by the product ofcontinously-many numbers of the Lebesgue measures $dX(\tau)$ on
for each individual :Hereposition$\mathcal{D}[X]$$X(\cdot)$

$s$

$y$

$x$

$t$

$R^{d}$$\tau$$\mathcal{D}[X]$$:=$‘constant”

where the “constant” should be something like$\cross\prod_{s\leq\tau\leq t}dX(\tau)$,$\prod_{s\leq\tau\leq t}\frac{m^{1/2}}{(2\pi i\hslash d\tau)^{1/2}}$, if one dares to try to writeit, wondering what it means at all. The right-hand side of (2.2) is what is called Feynman path
integml or, nowadays simply, path integml.
To explain this, Feynman put the following Two Postulates which tum out to be equivalent
to get the above expression (2.2) for $K(t,x;s,y)$ .
(i) $K(t,x;s,y)$ is the total probabilty amplitude for the event that the particle starts fromposition at time and amives at position at time . If
$y$

$s$

$x$

$t$

$\varphi[X]$stands for the probabilty amplitude

4
for the event that it does this motion along each individual paththeover all these paths:$X(\cdot),$$K(t,x;s,y)$ is the sum of$X(\cdot)$$\varphi[X]$(2.4)$K(t,x;s,y)= \sum\varphi[X]x:X(s)=y,X(t)=x$(ii) The contnibution$\varphi[X]$from each$X(\cdot)$.

to the total probabilty amplitude $K(t,x;s,y)$ isgiven by(2.5)$\varphi[X]=Ce^{iS(X)/\hslash}$,.
where is a constant independent of path
These two postulates can be paraphrased:
In quantum mechanics there rules Principle of Democracy that each individual pathcontributes to the total probabilty amplitude $K(t,x;s,y)$ with equal weight (absolute value inmathematics) and its personanlity is expressed by its phase (argument in mathematics).
In this respect, in classical mechanics there does not mle Principle of Democracy, because
the particle takes the particular path between two space-time points $(s,y)$ and $(t,y)$ which makes
the action $S(X)$ stationary, called classical tmjectory. It is the path determined by EulerLagrange equation or, in the present case, Newton’s equation of motion: $mX(\tau)=-\nabla V(X(\tau))$.
The most characteristic feature of these postulates lies in equation (2.5), which says that the. The phrase ”propotional to” is that which
amplitude
is propotional to the phase
Feynman determined to substitute for what Dirac had meant by the phrase ”analogous to” in[D33, D35], [D45] far before Feynman, by showing after his own analysis and deliberation thatindeed this exponentialfunction could be used in this manner directly (see Preface of [FH65]).
In \S 6 we shall come back to this subjecte again.
$C$$X(\cdot)$$X(\cdot)$$\varphi[X]$$e^{iS(X)/\hslash}$\S 2.2. How to Make It Mathematics?
Here we refer, among others, only to two methods; one is by finite-dimensional approximation, and the other by imaginary-time path integral. In fact, it is by the first method that Feynman
himself confirmed his idea of path integral. He calculated $K(t,x;s,y)$ by time-sliced approximation, making partition of the time interval $[s,t]:s=t_{0}<t_{1}<\cdots<t_{n}=t,$ $(t_{k}-t_{k-1}=t/n)$ ,$:=X(t_{k}),$ $x0=X(0)=y,$ $x_{n}=X(t)=x$ , as the limit of$x_{k}$$K_{n}(t,x;s,y):= \frac{\int_{(R)}\exp[\frac{it}{\hslash n}\sum_{k=0}^{n-1}(\frac{1}{2}(\frac{x_{k+1}-x_{k}}{t/n})^{2}-V(x_{k}))].dx_{1}\cdots dx_{n-1}}{\int(R^{d})^{n-1}\exp[\frac{it}{\hslash n}\sum_{k=0}^{n-1}\frac{1}{2}(\frac{x_{k+1}-x_{k}}{t/n})^{2}]dx_{1}\cdot\cdot dx_{n-1}}$as$narrow\infty$, to ascertain it to satisfy the Schr\"odinger equation (2.1).

Now we come to the second method, which the present note will be mainly concemed with.
First note with (2.2) that the solution of the Schr\"odinger equation (1.1) tums out to be given by5lm

Figure 1. From real time to imaginary $time-it$$t$a heuristic path integral
(2.6)

.

$\psi(t,x)=\int_{R^{d}}K(t,x;s,y)\psi(s,y)dy=\int_{\{X:X(t)=x\}}e^{iS(X)/\hslash}\psi(s,X(s))\mathcal{D}[X]$

Next, one should know thatitself does not in general exist in this situation as a countably
additive measure. Therefore we cannot go further. But if we rotate in complex t-plane by$-90^{0}:tarrow-it$ (real time
to imaginary time -it), i.e. if we go from our Minkowski spacetime to Euclidian space-time (see Figure 1), the situation changes. Before actually doing it,
for simplify put
and $s=0$ . Then our (real-time) Schr\"odinger equation (2.1) goes to theimaginary-time Schr\"odinger equation, i.e. heat equation [formally putting $u(t,x):=\psi(-it,x)$ ]$\mathcal{D}[X]$$t$$\hslash=1$(2.7)$\frac{\partial}{\partial t}u(t,x)=[\frac{1}{2m}\Delta-V(x)]u(t,x)$,$t>0$ ,

$x\in R^{d}$.

Simultaneously, the action $S(X)$ in (2.3) changes to integral of the Hamiltonian, and so $iS(X)$ totime integral of the $Hamiltonian-\int_{0}^{t}[\frac{1}{2m}\dot{X}(\tau)^{2}+V(X(\tau))]d\tau$ . Then $K(t,x;0,y)$ changes to(2.8)$K^{E}(t,x;0,y)= \int_{\{X;X(0)=y,X(t)=x\}}e^{-\int_{0}^{t}[\frac{1}{2m}X(\tau)^{2}+V(X_{0}(\tau))]d\tau}\mathcal{D}[X]$,

where the superscript
is attributed to ”Euclideian”, and $K^{E}(t,x;0,y)$ should become theheat kemel for equation (2.6). However, so as to be able to reach the so-called Feynman-Kac
$E$

”

6
fomula, we put here
then get from (2.8)(2.9)$X_{0}(\tau)$$:=X(t-\tau),$ $0\leq\tau\leq t$ , totransform paths$X(\cdot)$

to paths$K^{E}(t,x;0,y)= \int_{\{X:X_{0}(0)=x,\eta(t)=y\}}e^{-\int_{0^{[}\varpi^{\dot{\eta}(\tau)^{2}+V(\eta(\tau))]d\tau}}^{t1}}\mathcal{D}[X_{0}]$$X_{0}(\cdot)$and,so that the solution of (2.6) should be given by the following path integral(2.10)‘$u(t,x)= \int_{R^{d}}K^{E}(t,x;0,y)g(y)dy=\int_{\{X:X_{0}(0)=x\}}e$$\int_{0\pi 0\mathfrak{v}_{g(X_{0}(t))\mathcal{D}[X_{0}]}}^{t1}[\dot{X}(\tau)^{2}+V(X(\tau))]d\tau$.

Remarkable is that N. Wiener, already around 1923, had constructed a countably additive, for each $x\in R^{d}$ , on the space $C_{x};=C_{x}([0,\infty)arrow R^{d})$ of the continuous paths
measureis called(Bmwnian motions) $B:[0,\infty)arrow R^{d}$ starting ffom $B(O)=x$ at time $t=0$ . This
Wiener measure, which is a probability measure on with characteristic function$\mu_{x}(X_{0})$$\mu_{x}(\cdot)$$C_{0}$$\exp[-t\frac{\xi^{2}}{2m}]=\int_{C_{X}([0,\infty)arrow R^{d})}e^{iB(t)\xi}d\mu_{x}(B)$.

Around 1947, Mark Kac, who had been at Comell University as Feynman and seemed tohave heard his lecture, used the Wiener measure to represent the solution $u(t,x)$ of the Cauchy
problem of the heat equation (2.6) with initial data $u(O,x)=f(x)$ as a genuine functional integral(2.11).

$u(t,x)= \int K^{E}(t,x;0,y)f(y)dy=\int_{C_{x}([0,\infty)arrow R^{d})}e^{-\int_{0}^{t}V(B(s))ds}f(B(t))d\mu_{x}(B)$This is the Feynman-Kac formula [K66,80] already mentioned above. Thus, identify the pathin , thenappearing on the right-hand side of $(2.9)/(2.10)$ with the continuous pathonthe Wiener measuretums out to be constructed from the factor
the right-hand side of $(2.9)/(2.10)$ .
$B(\cdot)$$X_{0}(\cdot)$$C_{x}$“ $e^{-\int_{0R^{B(\tau)^{2}d\tau}}^{t1}}\mathcal{D}[B]$ ”$\mu_{x}(\cdot)$\S 3. Three Magnetic Relativistic Schr\"odinger Operators
We consider the quantized operator $H:=H_{A}+V$ corresponding to the classical Hamiltonian(3.1)$\sqrt{(\xi-A(x))^{2}+m^{2}}+V(x)$,$(\xi,x)\in R^{d}\cross R^{d}$,for a relativistic particle of mass under magnetic vector potential $A(x)$ and electric scalarpotential $V(x)$ . This $H$ is used for a spinless particle in electromagnetic fields in the situation
where we may ignore quantum-field theoretic effect like particles creation and annihilation but
should take relativistic effect into consideration.
andIn this note, we pay attention to the following three quantized operators
corresponding to the classical relativistic Hamiltonian symbol (3.1). Their difference is in how, corresponding to the symbol $\sqrt{(\xi-A(x))^{2}+m^{2}}$ .
to define the first term on the right,
For simplicity, it is assumed here and throughout this note that $A(x)$ is smooth and $V(x)$ isbounded below.
$m$$H^{(1)},$ $H^{(2)}$$H_{A}$$H^{(3)}$

7$:=H_{A}^{(1)}+V$ is defined with the first term on the
Definition 3.1. The firstrightbeing the Weyl pseudo-differential operator through mid-point prescription (e.g. [ITa86, I89,
$H^{(1)}$$H_{A}^{(1)}$I95] :$)$$(H_{A}^{(1)}f)(x):= \frac{1}{(2\pi)^{d}}\int\int_{R^{d}\cross R^{d}}e^{i(x-y)\cdot\xi\sqrt{(\xi-A(\frac{x+y}{2}))^{2}+m^{2}}f(\mathcal{Y})d_{\mathcal{Y}}d\xi}$(3.2)$= \frac{1}{(2\pi)^{d}}\int\int_{R^{d}\cross R^{d}}e^{i(x-y)\cdot(\xi+A(\frac{x+y}{2}))}\sqrt{\xi^{2}+m^{2}}f(y)dyd\xi$$:=H_{A}^{(2)}+V$ is defined with term
Definition 3.2. The second
being the pseudodifferential operator modified by Iftimie-Purice $[IfMP07],$ $[IfMP08],$ $[IfMP10]$ :$H^{(2)}$$H_{A}^{(2)}$$-M\dot{a}ntoiu$(3.3)$(H_{A}^{(2)}f)(x):= \frac{1}{(2\pi)^{d}}\int\int_{R^{d}\cross R^{d}}e^{i(x-y)\cdot(\xi+\int_{0}^{1}A((1-\theta)x+\theta y)d\theta)}\sqrt{\xi^{2}+m^{2}}f(y)dyd\xi$.

Here the integrals in (3.2), (3.3) on the right-hand side are oscillatory integrals with $f$ beinga function in
or in $S(R^{d})$ .
$C_{0}^{\infty}(R^{d})$$:=H_{A}^{(3)}+V$ is defined with term
Definition 3.3. The thirdof the nonnegative selfadjoint operator $(-i\nabla-A(x))^{2}+m^{2}$ :$H^{(3)}$(3.4)$H_{A}^{(3)}:=\sqrt{(-i\nabla-A(x))^{2}+m^{2}}+V(x)$$H_{A}^{(3)}$being the square root.

This
does not seem to be defined as a pseudo-differential operator corresponding to acertain tractable symbol. So long as it is defined through Fourier and inverse-Fourier tansforms,
the candiadte of its symbol will not be $\sqrt{(\xi-A(x))^{2}+m^{2}}$.
The last
is used, for instance, to study “stability of matter” in relativistic quantum
mechanics in E. Lieb and R. $Se\ddot{m}nger$ [LSei10].
Needles to say, we can show these three relativistic Schr\"odinger operators
anddefine selfadjoint operators in, which are bounded from below and, in general,
different from one another. In fact further, the three magnetic relativistic Schr\"odinger operators
andare bounded from below by the same lower bound . This was shown forin [I89] with the aid of its expression (5.5) in \S 5 instead of (3.2) and similarly can be forwith the aid of (5.15) instead of (3.3), while it is trivial for.
$H_{A}^{(3)}$$H^{(3)}$$H^{(1)},$$H^{(3)}$$H_{A}^{(1)},$$H^{(2)}$$L^{2}(R^{d})$$H_{A}^{(2)}$$H_{A}^{(3)}$$m$$H_{A}^{(1)}$$H_{A}^{(2)}$$H_{A}^{(3)}$\S 4. Gauge Covariance for Magnetic Relativistic Schrodinger Operators
Among these three magnetic relativistic Schr\"odinger operators, the Weyland(in general, the Weyl pseudo-differential operator) is compatible wellquantized one likewith path integral. But it is pity that, for general vector potential $A(x)$ it is in general notcovariant under gauge transformation, namely, there exists a real-valued function
for whichit fails to hold that.
$H_{A}^{(1)},$$H_{A}^{(2)}$$H_{A}^{(3)}$$H_{A}^{(1)}$$\varphi(x)$$H_{A+\nabla\varphi}^{(1)}=e^{i\varphi}H_{A}^{(1)}e^{-i\varphi}$

8However,$H_{A}^{(2)}$(and so$H^{(2)}$) and$H_{A}^{(3)}$(and so$H^{(3)}$) are gauge-covariant, though these three

are not equal in general. Let us observe these facts in the following.
First, why $H_{A}^{(3)}=\sqrt{(-i\nabla-A(x))^{2}+m^{2}}$ is gauge-covariant is because the selfadjoint operis gauge-covariant. Next, as in the following proposition,
ator $(-i\nabla-A(x))^{2}+m^{2}$ inside
is gauge-covariant. This property was emphasized in
it is easy to show that the modified$[IMP07],$ $[IfMP08],$ $[IfMP10]$ in contrast to.
$\sqrt{}$$H_{A}^{(2)}$$H_{A}^{(1)}$

Proposition 4.1.
$S(R^{d})$that$\mathscr{A}_{A}^{2)}$is covariant under gauge transformation, $i.e$. it followss for any.
. Therefore, so is$\varphi\in$$H^{(2)}$$H_{A+\nabla\varphi}^{(2)}=e^{i\varphi}H_{A}^{(2)}e^{-i\varphi}$

The proof is due to the mean value theorem.

Theorem 4.2. $IfA(x)$ is linear inandconstant matrix, then
$H_{A}^{(1)},$$x,$$H_{A}^{(3)}$$H_{A}^{(2)}$fields for $d=3$ .

.

real symmetricwith being any
coincide. In particular, this holds for uniform magnetic$i.e$$\iota fA(x)=A\cdot x$$A$$d\cross d$

Proof is omitted.

\S 5. Imaginary-Time Path Integrals for Magnetic Relativistic Schrodinger Operatorsin
Now, let $H$ be one of the magnetic relativistic Schr\"odinger operators
Definitions 3.1, 3.2, 3.3. In the same way as in the nonrelativistic case, start ffom (real-time)$H^{(1)},$ $H^{(2)},$$H^{(3)}$. Rotate it by-90’ from real time torelativistic Schr\"odinger equation
imaginary $time-it$ in complex t-plane, we amive at the imaginary-time relativistic Schr\"odinger
equation, “heat equation” for $H-m$ [formally putting $u(t,x):=\psi(-it,x)$ ]:$i \frac{\partial}{\partial t}\psi(t,x)=H\psi(t,x)$

(5.1)

$t$

$\{\begin{array}{ll}\frac{\partial}{\partial t}u(t,x)=-[H-m]u(t,x), t>0,u(O,x)=g(x), x\in R^{d}.\end{array}$

The semigroup $u(t,x)=(e^{-t[H-m]}g)(x)$ gives the solution of this Cauchy problem. We want
to deal with path integral representation for each $e^{-[H^{(j)}-m]}g(j=1,2,3)$ . The relevant path
integral is connected with the Le pmcess $[IkW81,89; Ap09]$ on the space $D_{x}:=D_{x}([0, \infty)arrow$of the “c\‘adlag paths”, i.e. right-continuous paths $X:[0,\infty)arrow R^{d}$ having left-hand limits,
and with $X(O)=x$. The associated path space measure is a probability measure , for each$x\in R^{d}$, on $D_{x}([0,\infty)arrow R^{d})$ whose characteristic function is given by$vy$$R^{d})$$\lambda_{x}$(5.2)$e^{-t[\sqrt{\xi^{2}+m^{2}}-m]}= \int_{D_{X}([0,\infty)arrow R^{d})}e^{i(X(t)-x)\cdot\xi}d\lambda_{x}(X)$,$t\geq 0$,$\xi\in R^{d}$.

We are going to start on task of representing the semigroup $e^{-t[H-m]}g$ by path integral.
But before that, we want briefly touch on what a kind of path integral expression emerges

9heuristically in the present relativistic case (cf.[I93, pp. 26-29, \S 5]), to compare it with thenonrelativistic case $(2.9)/(2.10)$ , which together with $(2.2)/(2.6)$ is called configumtion spacepath integml. Though taking the same procedure as before to find it, we tum out to leam it to
be given by phase space path integml.
However, to see it, as the general case for $H$ is dependent on which of the three relativis$:=\sqrt{-\Delta+m^{2}}+V(x)$tic Schr\"odinger operators is dealt with, so we do only with the casewithout vector potential $A(x)$ . Then we have for the solution of (5.1) within place of $H$$H_{0}$$H_{0}$$u(t,x)=(e^{-t[H_{0}-m]}g)(x)$

(5.3)

Here

.

$= \int\int 0_{X(0)=x\}}^{t}$$D[P]\mathcal{D}[X]$$:= \prod_{0\leq\tau\leq t}\frac{dP(\tau)dX(\tau)}{(2\pi)^{d}}$is a ‘measure’ on the space of the phase space paths (i.e.

momentum and position paths) $(P,X);[0,t]\ni s\mapsto(P(s),X(s))\in R^{d}\cross R^{d}$ with $X(O)=x$ and,for each fixed
the measure$\tau,$

is the Lebesgue measure on $R^{2d}=R^{d}\cross R^{d}$ . It will tum out thatis to be constmcted from the factor$dP(\tau)dX(\tau)$$\lambda_{x}(\cdot)$$( \int_{\{P:arbitary\}}e^{-\int_{0}^{t}\{iP(s)dX(s)+[\sqrt{P(s)^{2}+m^{2}}-m]ds\}}\mathcal{D}[P])\mathcal{D}[X]$”on the right-hand side of (5.3), so that we have a correct functional integral representaion quite
similar to the nonrelativistic case (2.11):
(5.4)

.

$u(t,x)=(e^{-t[H_{0}-m]}g)(x)= \int_{D_{x}([0,\infty)arrow R^{d})}e^{-\int_{0}^{t}V(X(s))ds}g(X(t))d\lambda(X)$Now we tum to come to the situation involving also the vector potential $A(x)$ .
(1) First consider the case for the Weyl pseudo-differential operator $H^{(1)}=H_{A}^{(1)}+V$ in Definition 3.1. The partcan be rewritten as the integral operator:$H_{A}^{(1)}$.

$([H_{A}^{(1)}-m]f)(X)=- \int_{|y|>0}\tau$$=- \lim_{r\downarrow 0}\int_{|y|\geq r}[e^{-iy\cdot A(x+\not\in)}f(x+y)-f(x)]n(dy)$(5.5)$=-$ p.v. $\int|y|>0^{[e^{-iy\cdot A(x+^{y})}}zf(x+y)-f(x)]n(dy)$.

Here $n(dy)=n(y)dy$ is an m-dependent measure on(5.6)$R^{d}\backslash \{0\}$, called L\’evy measure with density$n(y)=\{\begin{array}{ll}2 (\frac{m}{2\pi})^{(d+1)/2}\frac{K_{(d+1)/2}(m|y|)}{|y|^{(d+1)/2}}, m>0,\frac{\Gamma((d+1)/2)}{\pi^{(d+1)/2}}\frac{1}{|y|^{d+1}}, m=0.\end{array}$10$n(dy)$(5.7)

appears in the L\’evy-Khinchin formula:$\sqrt{\xi^{2}+m^{2}}-m=-\int_{|y|>0}[e^{iy\cdot\xi}-1-i\xi\cdot yI_{\{|y|<1\}}]n(dy)=-\lim_{rarrow 0+}\int_{|z|\geq r}[e^{iz\cdot\xi}-1]n(dz)$.

Pmofof (5.5). By the L\’evy-Khinchin formula (5.7),$(H_{A}^{(1)}f)(x)=(2 \pi)^{-d}\int\int e^{i(x-y)\cdot(\xi+At^{x}\not\simeq^{+}))(m-\lim_{rarrow 0+}\int_{|z|\geq r}[e^{iz\cdot\xi}-1]n(dz))f(y)dyd\xi}$$=(2 \pi)^{-d}[m\int\int e^{i(x-y)\cdot\xi}e^{i(x-y)\cdot A(^{X}\not\simeq^{+})}dyd\xi$$- \lim_{rarrow 0+}\int\int\int_{|z|\geq r}(e^{i(x-y+z)\cdot\xi}-e^{i(x-y)\cdot\xi})n(dz)e^{i(x-y)\cdot At^{X}\not\simeq^{+})}f(y)dyd\xi]$$=m \int\delta(x-y)e^{i(x-y)\cdot At^{x}\not\simeq^{+})}f(y)dy$$- \lim_{rarrow 0+}\int\int_{|z|\geq r}(\delta(x-y+z)-\delta(x-y))n(dz)e^{i(x-y)\cdot A(^{X}\not\simeq^{+})}f(y)dy$.

$=mf(x)- \lim_{rarrow 0+}\int\int_{|z|\geq r}(e^{-iz\cdot A(x+\xi)}f(x+z)-f(x))n(dz)$$\square$

To represent $e^{-t[H^{(1)}-m]}g$ by path integral, we need some further notations from L\’evy pro-cess.
(dsdy) denotes the counting measure on
For each path
, i.e.
the number of discontinuiies of$X(\cdot),$ $N_{X}$$[0,\infty)\cross(R^{d}\backslash \{0\})$to count$X(\cdot)$(5.8)$N_{X}((t,t’]\cross U):=\#\{s\in(t,t’];0\neq X(s)-X(s-)\in U\}$ ,where $0<t<t’$ and$U\subset R^{d}\backslash \{0\}$$\overline{N}_{X}(dsdy)$$:=N_{X}(dsdy)-dsn(dy)$ , which may be thought of as a renormalization of $N_{X}(dsdy)$

Then any path
(5.9)

is a Borel set. It satisfies $\int_{D_{X}}N_{X}(dsdy)d\lambda_{x}(X)=dsn(dy)$ . Put$X\in D_{x}([0,\infty)arrow R^{d})$can be expressed with

$N_{x}(\cdot)$

and$\overline{N}_{X}(\cdot)$as$X(t)=x+ \int_{0}^{t+}\int_{|y|\geq 1}yN_{X}(dsdy)+\int_{0}^{t+}\int_{0<|y|<1}y\overline{N}_{X}(dsdy)$

Now we have the following path integral representation for $e^{-t[H^{(1)}-m]}g$ .

Theorem 5.1 ([ITa86], [I95]).
,$(e^{-t[H^{(1)}-m]}g)(x)= \int_{D_{x}}e^{-S^{(1)}(t,X)}g(X(t))d\lambda_{x}(X)$

.

.

11
$S^{(1)}(t,X)=i \int^{t+}\int A(X(s-)+\frac{y}{2})\cdot yN_{X}(dsdy)+i\int^{t+}\int A(X(s-)+\frac{y}{2})\cdot y\overline{N}_{X}$(5.10)$+i \int_{0}^{t}ds$(dsdy)

p.v. $\int_{0<|y|<1}A(X(s)+\frac{y}{2})\cdot yn(dy)+\int_{0}^{t}V(X(s))ds$ .

Pmof. We only give a sketch. Put(5.11)$(T(t)g)(x):= \int_{R^{d}}e^{-iA(^{x}\not\simeq^{+})\cdot(y-x)-V(X}\not\simeq^{+})t$

where $k_{0}(t,x-y)$ is the integral kemel of$e^{-t(\sqrt{-\Delta+m^{2}}-m)}$,. Then we can rewrite it as$(T(t)g)(x)= \int_{D_{x}}e^{-iA(\frac{x+X(t)}{2})\cdot(X(t)-x)-V(\frac{x+X(t)}{2})t}g(X(t))d\lambda_{x}(X)$with partition of $[0,t]:0=t_{0}<t_{1}<\cdots<t_{n}=t,$ $t_{j}-t_{j-1}=t/n$ ,(5.12)$S_{n}(x0, \cdots,x_{n}):=i\sum_{j=1}^{n}A(\frac{x_{j-1}+x_{j}}{2})\cdot(x_{j}-x_{j-1})+\sum_{j=1}^{n}V(\frac{x_{j-1}+x_{j}}{2})\frac{t}{n}$,where $x_{j}=X(t_{j})(j=0,1,2, \ldots,n);x=x_{0}=X(t_{0}),$ $y=x_{n}=X(t_{n})\equiv X(t)$ .
Substitute these $n+1$ points of path $x_{j}=X(t_{j})$ into
to get$S_{n}(x_{0}, \cdots,x_{n})$$S_{n}(X):=S_{n}(X(t_{0}), \cdots,X(t_{n}))$$=i \sum_{j=1}^{n}A(\frac{X(t_{j-1})+X(t_{j})}{2})\cdot(X(t_{j})-X(t_{j-1}))+\sum_{j=1}^{n}V(\frac{X(t_{j-1})+X(t_{j})}{2})\frac{t}{n}$

Then
$n$

times

$= \int_{D_{x}}e^{-S_{n}(X)}g(X(t))d\lambda_{x}(X)$.

We can show

Proposition 5.2.

$T(t/n)^{n}garrow e^{-t[H^{(1)}-m]}g$in$L^{2}(R^{d})$,$narrow\infty$.

Proof is omitted.
Now we are in a position to complete the proof of Theorem 5.1. By Proposition 5.2, we see
the left-hand side of (5.13) converges to $e^{-t[H^{(1)}-m]}g$ as. On the other hand, we see by’s formula [ $see*)$ below] that the right-hand side converges to $\int_{D_{x}}e^{-S(X)}g(X(t))d\lambda_{x}(X)$ by
Lebesgue convergence theorem.
$narrow\infty$$It\hat{o}$$\square$

12
$*)$

For instance, in $t_{j-1}\leq s<t_{j}$ , we have by It\^o’s formula,$A( \frac{X(t_{j-1})+X(t_{j})}{2})\cdot(X(t_{j})-X(t_{j-1}))$$= \int_{t_{j}-1}^{t_{j}+}\int_{|y|>0}[A(\frac{X(s-)+X(t_{j-1})+yl_{|y|\geq 1}(y)}{2})\cdot(X(s-)-X(t_{j-1})+yI_{|y|\geq 1}(y))$$-A( \frac{X(s-)+X(t_{j-1})}{2})\cdot(X(s-)-X(t_{j-1}))]N_{X}(dsdy)$$+ \int_{t_{j}-1}^{t_{j}+}\int_{|y|>0}[A(\frac{X(s-)+X(t_{j-1})+yI_{|y|<1}(y)}{2})\cdot(X(s-)-X(t_{j-1})+yI_{|y|<1}(y))$$-A( \frac{X(s-)+X(t_{j-1})}{2})\cdot(X(s-)-X(t_{j-1}))]\overline{N}(dsdy)$$+ \int_{t_{j}-1}^{t_{j}}\int_{|y|>0}[A(\frac{X(s)+X(t_{j-1})+yI_{|y|<1}(y)}{2})\cdot(X(s)-X(t_{j-1})+yI_{|y|<1}(y))$$-A( \frac{X(s)+X(t_{j-1})}{2})\cdot(X(s)-X(t_{j-1}))$$-I_{|y|<1}(y) \{(\frac{1}{2}(y\cdot\nabla)A)(\frac{X(s)+X(t_{j-1}}{2})\cdot(X(s)-X(t_{j-1}))$$+y \cdot A(\frac{X(s)+X(t_{j-1})}{2})\}]dsn(dy)$.

(2) Next we come to the case for the pseudo-differential operator modified by Iftimie-

Mantoiu-Purice: $H^{(2)}:=H_{A}^{(2)}+V$ in Definition 3.2. By exactly the same argument as used toshow (5.5), we can show that$([H_{A}^{(2)}-m]f)(x)=- \int_{|y|>0}[e^{-iy\cdot\int_{0}^{1}A(x+\theta y)d\theta}f(x+y)-f(x)$$-I_{\{|y|<1\}}y\cdot(\nabla-iA(x))f(x)]n(dy)$$=- \lim_{r\downarrow 0}\int_{|y|\geq r}[e^{-iy\cdot\int_{0}^{1}A(x+\theta y)d\theta}f(x+y)-f(x)]n(dy)$(5.13)$=-$ p.v. $\int_{|y|>0}[e^{-iy\cdot\int_{0^{1}}A(x+\theta y)d\theta}f(x+y)-f(x)]n(dy)$ .

Theorem 5.3. $[ImP07, IfMP08, IfMP10]$$(e^{-t[H^{(2)}-m]}g)(x)= \int_{D_{X}([0,\infty)arrow R^{d})}e^{-S^{(2)}(t,X)}g(X(t))d\lambda_{x}(X)$,$S^{(2)}(t,X)=i \int_{0}^{t+}\int_{|y|\geq 1}(\int_{0}^{1}A(X(s-)+\theta y)\cdot yd\theta)N_{X}(dsdy)$$+i \int_{0}^{t+}\int_{0<|y|<1}(\int_{0}^{1}A(X(s-)+\theta y)\cdot yd\theta)\overline{N}_{X}(dsdy)$(5.14)$+i \int_{0}^{t}ds$p.v. $\int_{0<|y|<1}(\int_{0}^{1}A(X(s)+\theta y)\cdot yd\theta)n(dy)+\int_{0}^{t}V(X(s))ds$ .

The proof of Theorem 5.3 will be done in exactly the same way as that of Theorem 5.1.
.
Indeed, we have only to replace $A(X(s-)+\not\in)\cdot y$ by$\int_{0}^{1}A(X(s-)+\theta y)\cdot yd\theta$

13(3) Finally, we consider the case for the operator defined, in Definition 3.3, with the square$:=H_{A}^{(3)}+V$ .
root of a nonnegative selfadjoint operator,
On the one hand, we can determine by functional analysis, namely, by theory of fractional$H^{(3)}$

powers (e.g. [Y68, Chap.IX,11, pp.259-261])

from the nonnegative selfadjoint operator $S:=(-i\nabla-A(x))^{2}+m^{2}=:2mH_{A}^{NR}+m^{2}$ where
stands for the magnetic nonrelativistic Schr\"odinger operator
without scalar potential. Indeed, we have$e^{-t[H_{A}^{(3)}-m]}$$H_{A}^{NR}$$\frac{1}{2m}(-i\nabla-A(x))^{2}$$e^{-t[H_{A}^{(3)}-m]}g=\{\begin{array}{ll}e^{mt}\int_{0}^{\infty}f_{t}(\lambda)e^{-\lambda S}gd\lambda, t>0,0, t=0\end{array}$(5.15)$f_{t}(\lambda)=\{\begin{array}{ll}(2\pi i)^{-1}\int_{\sigma-i\infty}^{\sigma+i\infty}e^{z\lambda-tz^{1/2}}dz, \lambda\geq 0,0, \lambda<0 (\sigma>0).\end{array}$Here we quickly insert the $Feynman-Kac$-It\^o formula (e.g. [S05]) for the magnetic non$:=H_{A}^{NR}+V$ $:= \frac{1}{2m}(-i\nabla-A(x))^{2}+V(x)(m>0)$, a more
relativistic Schr\"odinger operator
general formula than the Feynman-Kac formula (2.11):$H^{NR}$(5.16)$(e^{-tH^{NR}}g)(x)$$= \int_{C_{x}([0,\infty)arrow R^{d})}e0z^{i}g(B(t))d\mu_{x}(B)$.

$\equiv\int_{C_{x}(l0,\infty)arrow R^{d})^{e^{-[i\int_{0}^{t}A(B(s))odB(s)+\int_{0}^{t}V(B(s))ds]}}}g(B(t))d\mu_{x}(B)$This can provide a kind of path integral representation forwith the Wiener measure,$t=2m\lambda$
by substituting the $Feynman-Kac$-It\^o formula (5.17) for $V=0$ with
into $e^{-t(S-m^{2})}j=$in the integrand of equation (5.16) for. Then, to represent $e^{-t[H^{(3)}-m]}g$
for $V\neq 0$ , we might apply the Trotter-Kato product formula$e^{-t[H_{A}^{(3)}-m]}g$$e^{-2m\lambda H_{A}^{NR}}$(5.17)$e^{-t[H_{A}^{(3)}-m]}g$$e^{-t[H^{(3)}-m]}= s-\lim_{narrow\infty}(e^{-(t/n)[H_{A}^{(3)}-m]}e^{-(t/n)V})^{n}$,to the sum $H^{(3)}-m=(H_{A}^{(3)}-m)+V$ to express the semigroup $e^{-t[H^{(3)}-m]}$ as a “limit”, whereconvergence of the right-hand side usually takes place in strong sense as indicated, but now
even, in operator norm, by the recent results on operator norm convergence [ITOI], [ITTZOI](cf. [IT04], [IT06]). However it is not clear whether this procedure could further yield a path

integral representation for $e^{-t[H^{(3)}-m]}g$ .
On the other hand, it does not seem possible to represent $e^{-t[H^{(3)I}-m]}g$ by path integral
through directly applying L\’evy process, as we saw in the cases for $e^{-t(H^{(1)}-m)}g$ and $e^{-t(H^{(2)}-m)}g$ ,becausedoes not seem to be explicitly expressed by a pseudo-differential operator of a certain tractable symbol. It was in this situation that the problem of path integral representation for
was studied first by DeAngelis-Serva and Rinaldi $[AnSe90, AnRSe91]$ with useof $subordination/time$-change of Brownnian motion, and recently more extensively in [HIL09]$H_{A}^{(3)}$$e^{-t[H^{(3)\mathfrak{l}}-m]}g$

14
not only for the magnetic relativistic Schr\"odinger operator
but also for Bemstein functionsof the magnetic nonrelativistic Schr\"odinger operator even with spin. To proceed, let us explain
about subordination.
Letbe the one-dimensional standard Brownian motion being a function in $C_{0}([0,\infty)arrow$$H_{A}^{(3)}$$B^{1}(t)$R with $B^{1}(0)=0$ , so that
$C_{0}([0,\infty)arrow R)$ . Put$)$(5.18)

with$e^{-t_{T}^{\mathcal{E}^{2}}}= \int_{C_{0}([0,\infty)arrow R)}e^{i\xi B^{1}(t)}d\mu_{0}^{1}(B^{1})$$T(t)$ $:= \inf\{s>0 ; B^{1}(s)+\sqrt{m}s=\sqrt{m}t\}$,$\mu_{0}^{1}$

the Wiener measure on$t\geq 0$.

Then $T(t)$ is a monotone, non-decreasing function on $[0,\infty)$ with $T(O)=0$ , belonging to$D_{0}([0, oo)arrow R)$ and so becoming a one-dimensional L\’evy process, called subordinator. Let
be the associated probability measure on $D_{0}([0,\infty)arrow R)$ .
$v0$

Proposition 5.4. (e.g. [Ap09, p.54, Example 1.3.21])(5.19)$e^{-t[\sqrt{2m\sigma+m^{2}}-m]}= \int_{D_{0}([0,\infty)arrow R)}e^{-T(t)\sigma}dv_{0}(T)$,$\sigma\geq 0$This proposition implies that the chacteristic function of the measure$e^{-t\phi(p)}= \int_{D_{0}([0,\infty)arrow R)}e^{iT(t)\rho}dv_{0}(T)$,$\rho\in R$$v_{0}$.

is given by,$\phi(p)=(\frac{m}{2})^{1/2}\frac{\sqrt{m^{2}+\rho^{2}}-m}{(\sqrt{m^{2}+\rho^{2}}+m)^{1/2}+\sqrt{2}m^{1/2}}-\frac{(2m)^{1/2}\rho}{(\sqrt{m^{2}+\rho^{2}}+m)^{1/2}}i$

To see this, first analytically extend.

to the right-half complex plane $z:=\sigma+i\rho,$$\phi(p)=\lim_{crarrow+0}\sqrt{2m(\sigma+i\rho)+m^{2}}-m$$0,\rho\in R$ , and then we have
, of which the right-handside is calculated as above.
We are in a position to give a path integral representation for $e^{-t[H^{(3)}-m]}g$ .
$\sqrt{2m\sigma+m^{2}}$Theorem 5.5. $[AnSe90,$ $AnRSe91$ ; HIL09$\sigma>$$]$$(e^{-t[H^{(3)l}-m]}g)(x)= \int\int_{C_{X}([0,\infty)arrow R^{d})}e^{-S^{(3)}(t,B,T)}g(B(T(t)))d\mu_{X}(B)dv_{0}(T)\cross D_{0}([0,\infty)arrow R)$ ’$S^{(3)}(t,B, T)=i \int_{0}^{T(t)}A(B(s))dB(s)+\frac{i}{2}\int_{0}^{T(t)}divA(B(s))ds+\int_{0}^{t}V(B(T(s)))ds$,(5.20)

where,$\equiv i\int_{0}^{T(t)}A(B(s))\circ dB(s)+\int_{0}^{t}V(B(T(s)))ds$$\mu_{x}$is the Wiener measure on $C_{x}([0,\infty)arrow R^{d})$.

Proofof Theorem 5.5. (Sketch) We use Proposition 5.4 and the $Feynman-Kac$-It\^o formula(5.17). Note that $H_{A}^{(3)}=\sqrt{2mH_{A}^{NR}+m^{2}}$. By Spectral Theorem for the nonnegative selfadjoint

15
operator$H_{A}^{NR}$, we have$H_{A}^{NR}= \int_{Spec(H_{A}^{NR})}\sigma dE(\sigma)$. Then for$f,$ $g\in L^{2}(R^{d})$$\{f,e^{-t[H_{A}^{(3)}-m]}g\}=\int_{Spec(H_{A}^{NR})}e^{-t[\sqrt{2m\sigma+m^{2}}-m]}\{f,dE(\sigma)g\}$where
stands for the inner product of the Hilbert spaceagain by Spectral Theorem,$\{\cdot,$$\cdot\}$$L^{2}(R^{2})$,. By Propositopn 5.4 and$\{f,e^{-t[H_{A}^{(3)}-m]}g\}=\int_{Spec(H_{A}^{NR})}\int_{D_{0}([0,\infty)arrow R)}e^{-T(t)\sigma}dv_{0}(T)\langle f,dE(\sigma)g\}$$= \int_{D_{0}([0,\infty)arrow R)}\langle f,e^{-T(t)H_{A}^{NR}}g\rangle dvo(T)$Applying the $Feynman-Kac$-It\^o formula (5.17) (with $V=0$) to.

$e^{-T(t)H_{A}^{NR}}g$

on the right, we have$\langle f,e^{-t[H_{A}^{(3)}-m]}g\rangle$$= \int D_{0([0,\infty)arrow R)^{v0}}d(T)\int_{R^{d}}dx\overline{f(B(0))}\int_{C_{X}([0,\infty)arrow R^{d})}e^{-i\int_{0^{T(t)}}A(\beta(s))\circ dB(s)}g(B(T(t)))d\mu_{x}(B)$$= \int_{R^{d}}d_{X}\overline{f(x)}\int\int_{\cross D_{0}([0,\infty)arrow R)}C_{x}(l0,\infty)arrow R^{d})^{e^{-i\int_{0^{T(t)}}A(B(s))\circ dB(s)}}g(B(T(t)))d\mu_{x}(B)d|nu_{0}(T)$,where note $B(O)=x$ . This proves the assertion when $V=0$ .
When $V\neq 0$ , with partition of $[0,t]:0=t_{0}<t_{1}<\cdots<t_{n}=t,$ $t_{j}-t_{j-1}=t/n$ , we can

express $e^{-t[H^{(3)}-m]}g=e^{-t[(H_{A}^{(3)}-m)+V]}$ by the Trotter-Kato formula (5.18). Rewrite the productof these operators by path integral with respect to the product of two probability measuresand note that $T(O)=T(t_{0})=0,$ $B(O)=B(T(t_{0}))=x$ , then we have$n$$vo(T)\cdot\mu_{x}(B)$$\{f, (e^{-(t/n)[H_{A}^{(3)}-m]}e^{-(t/n)V})^{n}g\}$$= \int_{R^{d}}dx\int_{D_{0}([0,\infty)arrow R)}dv_{0}(T)\int_{C_{x}([0,\infty)arrow R^{d})}\overline{f(B(0))}$$\cross e^{-i\Sigma_{j=1}^{n}\int_{\tau t_{j-1})}^{T(t_{j})}A(B(s))\circ dB(s)_{e^{-\Sigma_{j=1}^{n}V(B(T(t_{j}))\frac{t}{n}}}}tg(B(t_{n}))d\mu_{x}(B)$.

, that the left-hand side converges to
We see, as, and the right-hand side
also converges to the goal formula by the Lebesgue theorem, as integral by.
Hence or similarly we can also get (5.21).
Finally, as summary, we will collect the three path integral representation formulas in Theorems 5.1, 5.3, 5.5, below, so as to be able to easily see x-dependence. To do so, make changeof space, probablity measure and paths by translation:$\langle f,e^{-t[H_{A}^{(3)}-m]}g\rangle$$narrow\infty$$dx\cdot v_{0}(T)\cdot\mu_{x}(B)$$\square$16$D_{x}arrow D_{0},$(5.10):$\lambda_{x}arrow\Lambda_{0},$$X(s)arrow X(s)+x,$ $B(s)arrow B(s)+x,$ $B(T(s))arrow B(T(s))+x$ , then$(e^{-\prime[H^{(1)}-m]}g)(x)= \int_{D_{0}([0,\infty)arrow R^{d})}e^{-S^{(1)}(t,X)}g(X(t)+x)dA_{0}(X)$$S^{(1)}(t,X)=i \int_{0}^{t+}\int_{|y|\geq 1}A(X(s-)+x+\frac{y}{2})\cdot yN_{X}$,(dsdy)$+i \int_{0}^{t+}\int_{0<|y|<1}A(X(s-)+x+\frac{y}{2})\cdot y\overline{N}_{X}(dsdy)$$+i \int_{0}^{t}dsp.v.\int_{0<|y|<1}A(X(s)+x+\frac{y}{2})\cdot yn(dy)$$+ \int_{0}^{t}V(X(s)+x)ds$(5.15):;$(e^{-t[H^{(2)}-m]}g)(x)= \int_{D_{0}([0,\infty)arrow R^{d})}e^{-S^{(2)}(t,X)}g(X(t)+x)dA_{0}(X)$,$S^{(2)}(t,X)=i \int_{0}^{t+}\int_{|y|\geq 1}(\int_{0}^{1}A(X(s-)+x+\theta y)\cdot yd\theta)N_{X}(dsdy)$$+i \int_{0}^{t+}\int_{0<|y|<1}(\int_{0}^{1}A(X(s-)+x+\theta y)\cdot yd\theta)\tilde{N}_{X}(dsdy)$$+i \int_{0}^{t}dsp.v.\int_{0<|y|<1}(\int_{0}^{1}A(X(s)+x+\theta y)\cdot yd\theta)n(dy)$$+ \int_{0}^{t}V(X(s)+x)ds$(5.21):;,$(e^{-t[H^{(3)\mathfrak{l}}-m]}g)(x)= \int\int_{\cross D_{0}([0,\infty)arrow R)}C_{0}([0,\infty)arrow R^{d})e^{-S^{(3)}(t,B,T)}g(B(T(t))+x)d\mu_{0}(B)dv_{0}(T)$$S^{(3)}(t,B, T)=i \int_{0^{A(B(s)+x)\cdot dB(s)}}^{T(t)}+\frac{i}{2}\int_{0^{d}}^{T(t)}ivA(B(s)+x)ds$$+ \int_{0}^{t}V(B(T(s))+x)ds$,$\equiv i\int_{0}^{T(t)}A(B(s)+x)\circ dB(s)+\int_{0}^{t}V(B(T(s))+x)ds$\S 6. Feynman and Dirac
Finally, I would like to close thes notes to write something about Feynman and Dirac.
In \S 1, we observed Feynman’s Two Postulates equivalent to “path integral”. In them, equation(2.5) saying thatis the pivotal point. As he himself wrote
is “proportional to”
in his celebrated paper [F48], “this formulation was suggested by some of Dirac’s remarks$\varphi[X]$$e^{S(X)/\hslash}$

17([D33, D35], [D45])”, though. For what Dirac had criptically remarked as ”analogous to”

there, Feynman believed to be able to substitute ”proportional to” (see Preface of [FH65]).

There has recently been published a book entitled The Strangest Man: The Hidden Lifeof Paul Dimc, Quantum Genius, by Graham Farmelo [Faber and Faber Ltd, London, 2009;
paperback ed. 2010]. This volume describes in detail the life of Dirac from his birth to deathwith much favor and affection. From it I have leamed something novel which lets me thinkagain about how it was when Feynman had met Dirac, and how Feynman had been thinking
afterwards.
*Time: September 1946
*Place: Conference on ’The Future of Nuclear Science’, Princeton‘s Graduate College.
Feynman was Chairman to introduce Dirac to the audience. The following 12lines are citedfrom this book by Graham Farmelo, Chap. 24, p. 333.
Feynman described in his pmblem to Dimc and came to crunch:
FEYNMAN. Did you know that they were pmpotional ?
DIRAC: Are they ?
FEYNMAN: Yes they are.
DIRAC: That’s interesting.
Dirac then got up and walked away. Feynman subsequently became famous for his versionof quantum mechanics but thought the credit was undeserved. The more closely he looked atthe ’little paper ‘, the more he realized that he had done nothing new. He later said, repeatedly,‘I don’t know what all thefu ss is about–Dirac did it all before me.’[Intervie with Freeman Dyson, 27 June 2005. Dyson noted that Feynman madethe point repeatedly.]$\cdot$$w$

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