数理解析研究所講究録
第 1797 巻 2012 年 187-203

187

Phase Space Feynman Path Integrals-as Analysis on PathSpace via Piecewise Constant Paths
By

Naoto KUMANO-GO *Abstract
This survey of [20] is based on the introductory talk at RIMS.

\S 1. Introduction
Let $T>0$ and $x\in R^{d}$ . We consider the fundamental solution $U(T,0)$ for the Schr\"odinger
equation(1.1)$(i \hslash\partial_{T}-H(T,x, \frac{\hslash}{i}\partial_{x}))U(T,0)=0,$$U(O,O)=I$ ,with the Planck parameter $0<\hslash<1$ . By the Fourier transform with respect toinverse Fourier transform with respect to, the identity operator is given by$x_{0}\in R^{d}$$\xi_{0}\in R^{d}$$Iv(x)=v(x)=( \frac{1}{2\pi\hslash})^{d}\int_{R^{2d}}0$

and the Hamiltonian operator$H(T,x, \frac{\hslash}{i}\partial_{x})$$I$,

is given by.

$H(T,x, \frac{\hslash}{i}\partial_{x})v(x)=(\frac{1}{2\pi\hslash})^{d}\int_{R^{2d}}e^{\frac{i}{\hslash}(x-x_{0})\cdot\xi_{0}}H(T,x,\xi_{0})v(xo)dx0d\xi_{0}$As an approximation of $U(T,0)$ , we use the operator $I(T,0)$ given by$I(T, 0)v(x)=( \frac{1}{2\pi\hslash})^{d}\int_{R^{2d}}0$2010 Mathematics Subject Classification(s): $81S40;35S30;81Q20;58D30$
Key Words: Path integrals, Fourier integral operators, Semiclassical approximation
Supported by JSPS KAKENHI(C)21540196*Division of Liberal Arts, Kogakuin University, Tokyo 163-8677, Japan.

.

and the188

What is the position path$q$

What is the momentum path

?

$p$

?

Figure 1.
For any division $\Delta_{T,0}:T=T_{J+1}>T_{J}>\cdots>T_{1}>T_{0}=0$ of $[0, T]$ , we have$U(T,0)v(x)=U(T,T_{J})U(T_{J}, T_{J-1})\cdots U(T_{2}, T_{1})U(T_{1},0)v(x)$Set $t_{j}=T_{j}-T_{j-1}$ and$\max$$|\Delta_{T,0}|=$$t_{j}$. Under some condition, using.

$I(T_{j}, T_{j-1})$

as an ap-

$1\leq j\leq J+1$

proximation of $U(T_{j}, T_{j-1})$ as(1.2)$|\Delta_{T,0}|arrow 0$, we can get$U(T,0)v(x)= \lim_{|\Delta_{T,0}|arrow 0}I(T, T_{J})I(T_{J}, T_{J-1})\cdots I(T_{2}, T_{1})I(T_{1},0)v(x)$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{d(J+1)}l_{R^{2d(J+1)^{e^{\pi=1^{(t-x_{j-1})\cdot \mathcal{E}_{1-1}-\int_{\gamma_{j-1}}^{T_{j}}H(t,x_{j},\xi_{j-1})dt)}}}}}^{i_{\Sigma_{j}^{J+1}x_{j}}}$$\cross v(xo)\prod_{j=0}^{J}dx_{j}d\xi_{j}$with $x=x_{J+1}$ . When$T$,

is small, we consider the function $U(T,0,x,\xi_{0})$ satisfying$U(T,0)v(x)=( \frac{1}{2\pi\hslash})^{d}\int_{R^{2d}}e^{\frac{j}{\hslash}(x-x_{0})\cdot\xi_{0}}U(T,0,x,\xi_{0})v(x\circ)dx0d\xi_{0}$.

Then we formally write(1.3)$e^{i}\hslash^{(x-x_{0})\cdot\xi_{0}}U(T,0,x,\xi_{0})$.

$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{y\prime}}e^{\pi=1l1-1}\prod_{j=1}^{J}dx_{j}d\xi_{j}i_{\Sigma_{j}^{J+l}((x-x)\cdot\xi_{j-1}-\int_{T_{j-1}}^{T_{j}}H(t,x_{j},\xi_{j-1})dt)}$

According to R. P. Feynman [8, Appendix ], we introduce the position path $q(t)$ and the momentum path $p(t)$ with $q(T_{j})=x_{j}$ and $p(T_{j})=\xi_{j}$ (Figure 1). Let $\phi[q,p]$ be the action given
by$B$$\phi[q,p]=\int_{[0,T)}p(t)\cdot dq(t)-\int_{[0,T)}H(t,q(t),p(t))dt$.

189
for the phase space path $(q,p)$ with $q(O)=x_{0},$ $q(T)=x$ and $p(O)=\xi_{0}$ . Then we formally write(1.4)$e^{\frac{i}{\hslash}(x-x_{0})\cdot\xi_{0}}U(T,0,x, \xi_{0})=\int e^{\frac{i}{h}\phi[q,p]}\mathcal{D}[q,p]$.

Here the path integral
is a new sum over all the paths $(q,p)$ . The expressions (1.2)
and (1.3) are now called the time slicing approximation.
However, in the sense of mathematics, the measureof the path integral (1.4) does notexist. Why can we say (1.4) is an integral ? In the sense of the uncertain principle, we can nothave the position $q(t)$ and the momentum $p(t)$ at the same time . Furthermore, the convergence(1.2) in the sense of operator does not distinguish the position
and the momentum . Why
can we say $(q.p)$ is a phase space path ?
In [20], using piecewise constant paths, we proved the existence of the phase space Feynman$\int\sim \mathcal{D}[q,p]$$\mathcal{D}[q,p]$

$t$

$x0$

$\xi_{0}$path integrals(1.5)$\int e^{\frac{i}{\hslash}\phi[q,p]}F[q,p]\mathcal{D}[q,p]$,with general functional $F[q,p]$ as integrand. More precisely, we gave the two general classes
such that for any
or , the time slicing approximation of (1.5) converges
uniformly on compact subsets with respect to the endpoint of position paths and the starting
point of momentum paths. In this survey, we explain some properties of the phase space path$\mathcal{F}_{Q},$$F[q,p]\in \mathcal{F}_{Q}$$\mathcal{F}_{\mathcal{P}}$$\mathcal{F}_{\mathcal{P}}$$x$$\xi_{0}$integrals along the talk at RIMS.

Remark For the phase space path integral (1.4) via Fourier integral operators, see H. Kumanogo-H. Kitada [17] and N. Kumano-go [19]. We regard (1.4) as the particular case of (1.5) with
$F[q,p]=1$ . Using broken line paths of position and piecewise constant paths of momentum,
W. Ichinose [14] gave some functionals $F[q,p]= \prod_{k=1}^{K}B_{k}(q(\tau_{k}),p(\tau_{k})),$ $0<\tau_{1}<\tau_{2}<\cdots<$$\tau_{K}<T$ for which the time slicing approximations of (1.5) diverge as an operator. We exclude
these functionals from our classes
to avoid the uncertain principle.
$\mathcal{F}_{Q},$$\mathcal{F}_{P}$Remark. Inspired by the forward and backward approach of K. L. Chung-J.-C. Zambrini[4, \S 2.4], we use left-continuous paths and right-continuous paths. Furthermore, inspired by L.
S. Shulman [25, \S 31], we pay attention to the operations which are valid in the phase space path
integrals.

Since [8, Appendix ], the phase space path integral (1.4) has been rediscovered repeatedly(cf. W. Tobocman [26], H. Davies [6], C. Garrod [10]) and developed in various forms (cf. L. S.
Schulman [25, \S 31], H. Kleinert [22], C. Grosche-F. teiner [12], P. Cartier-C. DeWitt-Morette[3, \S 3.4], J. R. Klauder [21, \S 6.2] . For giving a well-defined mathematical meaning, various
approaches have been proposed. C. DeWitt-Morette-A. Maheshwari-B. Nelson [7] and M. M.
Mizrahi [24] introduced the formulation without limiting procedure. K. Gawedzki [11] used
$B$

$S$

$)$

190
the technique analogous to that used by K. It\^o [15]. I. Daubechies-J. R. Klauder [5] presented
the phase space path integral via analytic continuation from Wiener measure. Furthermore, S.
Albeverio-G. Guatteri-S. Mazzucchi [2] (cf. [1, \S 10.5.3], [23, \S 3.3]) realized the phase spacepath integral as an infinite dimensional oscillatory integral. O. G. Smolyanov-A. G. Tokarev-A.
Tmman [27] formulated the phase space path integral via Chemoff formula. For the main partof [8], G. W. Johnson-M. Lapidus [16] and T. L. Gill-W. W. Zachary [13] developed Feynman‘soperational calculus.

\S 2. Existence of Phase Space Path Integrals
Our assumption for the Hamiltonian function $H(t,x,\xi)$ of (1.1) is the following.
Assumption 1 (Hamiltonian function). $H(t,x,\xi)$ is a real-valuedfunction of $(t,x,\xi)$ in, and for any multi-indicesis continuous. For any non-negative
integer , there exists a positive constant such that$R\cross$$R^{d}\cross R^{d}$$\alpha,$$k$$\beta,$$\partial_{x^{t}}(\partial_{\xi}^{\beta}H(t,x,\xi)$$\kappa_{k}$$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}H(t,x,\xi)|\leq\kappa_{k}(1+|x|+|\xi|)^{\max(2-|\alpha+\beta|,0)}$for any multi-indices$\alpha,$$\beta$,with $|\alpha+\beta|=k$.

A typical example of the Hamiltonian operatorof (1.1) is the following.

$H(t,x, \frac{\hslash}{i}\partial_{x})$Example 1 (Hamiltonian operator).
$H(t,x, \frac{\hslash}{i}\partial_{X})=\sum_{j,k=1}^{d}(a_{J,x}k(t)\frac{\hslash}{i}\partial_{1k}\frac{\hslash}{i}\partial_{X}+b_{j,k}(t)x_{J}\frac{\hslash}{i}\partial_{x_{k}}+c_{j,k}(t)x_{j}x_{k})$$+ \sum_{j=1}^{d}(a_{j}(t)\frac{\hslash}{i}\partial_{x_{j}}+b_{j}(t)x_{j})+c(t,x)$Herecontinuous boundedfiunctions.
$a_{j,k}(t),$$b_{j,k}(t),$$c_{j,k}(t),$$a_{j}(t),$$b_{j}(t)$and$\partial_{x}^{a}c(t,x)$.

with any multi-index$\alpha$are real-valued

Let $\Delta_{T,0}=(T_{J+1}, T_{J}, \ldots, T_{1},T_{0})$ be any division of the interval $[0, T]$ given by$\Delta_{T,0}$

Set $x_{J+1}=x$ . Let$x_{j}\in R^{d}$and: $T=T_{J+1}>T,$ $>\cdots>T_{1}>T_{0}=0$ .
$\xi_{j}\in R^{d}$for $j=1,2,$$\ldots,J$. We define the position path$q_{\Delta_{T,0}}=q_{\Delta_{T,0}}(t,x_{J+1},x_{J}, \ldots,x_{1},xo)$by$q_{\Delta_{T,0}}(0)=x0,$ $q_{\Delta_{T,0}}(t)=x_{j},$$T_{j-1}<t\leq T_{j}$and the momentum path

$p_{\Delta_{T,0}}=p_{\Delta_{T,0}}(t,\xi_{J}, \ldots,\xi_{1},\xi_{0})$

191

$0$

$T_{1}$$T_{2}$$T_{J}$The position path

$T$

$0$

$T_{2}$$T_{1}$$T_{J}$The momentum path$q_{\Delta_{T,0}}$$T$$p_{\Delta_{T,0}}$Figure 2.

byfor $j=1,2,$ $\ldots,J,J+1$ (Figure 2). Let $t_{j}=T_{j}-T_{j-1}$ and. According to Feynman‘s first definition of (1.4), we define the phasespace path integral (1.5) with the general functional $F[q,p]$ as integrand by$p_{\Delta_{T,0}}(t)=\xi_{j-1},$$T_{j-1}\leq t<T_{j}$$| \Delta_{T,0}|=\max_{1\leq J\leq J+1}t_{j}$(2.1)$\int e^{\frac{i}{\hslash}\phi[q,p]}F[q,p]\mathcal{D}[q,p]$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{h}\phi[q_{\Delta_{T,0^{p_{\Delta}}r,0^{]}}}},F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]\prod_{j=1}^{J}dx_{j}d\xi_{j}$,if the limit of the right hand side exists.

Theorem 1 (Existence of phase space path integrals). Let be sufficiently small. Then, for, the right hand side (2.1) converges uniformly on compact sets ofanyor, i. e., the phase space path integml (2.1) is well-defined.
with respect to$T$$F[q,p]\in \mathcal{F}_{Q}$$R^{3d}$$\mathcal{F}_{\mathcal{P}}$$(x,\xi_{0},x_{0})$For simplicity, we will state the definition of the classes
Theorem 2 to Example 2.1, we can produce many$\mathcal{F}_{Q},$$F[q,p]\in \mathcal{F}_{Q}$$\mathcal{F}_{P}$orin \S 5. Because if we apply$\mathcal{F}_{\mathcal{P}}$.

Remark Even when $F[q,p]=1$ , each integral of the right hand side of (2.1) does not converge absolutely.
$\int_{R^{2d}}d\xi_{j}dx_{j}=\infty$

Furthermore, the number$J$.

of integrals (division points) tends to$\infty\cross\infty\cross\infty\cross\infty\cross\cdots\cdots\cdots$,$\infty$$Jarrow\infty$.

.

We treat the multiple integral of (2.1) as an oscillatory integral (cf. H. Kumano-go [18, \S 1.6]).

192
Though the functionals$\phi[q_{\Delta_{T.0}},p_{\Delta_{T,0}}],$$F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$are the functions$\phi_{\Delta_{T,0}},$$F_{\Delta_{T.0}}$

given by$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=\sum_{j=1}^{J+1}\int_{[T_{j-1},T_{j})}p_{\Delta_{T,0}}\cdot dq_{\Delta_{T,0}}(t)-\sum_{/=1}^{J+1}\int_{[T_{j-1},T_{j})}H(t,q_{\Delta_{T,0}},p_{\Delta_{T,0}})dt$$= \sum_{j=1}^{J+1}(x_{j}-x_{j-1})\cdot\xi_{j-1}-\sum_{j=1}^{J+1}\int_{[\tau_{j-1},\tau_{j})}H(t,x_{j},\xi_{j-1})dt$$=\phi_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, .\cdots,\xi_{1},x_{1},\xi_{0},x_{0})$$F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,\xi_{1},x_{1},\xi_{0},x_{0})$we keep$\phi[q_{\Delta_{T,0}},p_{\Delta_{T.0}}],$$F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$(1) If,in the multiple integral of (2.1).

Roughly speaking, typical examples of
Example 2.1,$F[q,p]\in \mathcal{F}_{Q}$or$\mathcal{F}_{P}$are the following.

. For the details, see Theorem 3.

$(F[q,p]\in \mathcal{F}_{\mathcal{Q}} or \mathcal{F}_{P})$$|\partial_{x}^{\alpha}B(t,x)|\leq C_{\alpha}(1+|x|)^{m}$

In particular,(2) If, the functionals independent of $p$ or ,$q$

$F[q]=B(t,q(t))\in \mathcal{F}_{Q}$,$F[p]=B(t,p(t))\in \mathcal{F}_{P}$.

$F[q,p]=1\in \mathcal{F}_{Q}\cap \mathcal{F}_{\mathcal{P}}$.
, then$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}B(t,x,\xi)|\leq C_{\alpha,\beta}(1+|x|+|\xi|)^{m}$$F[q,p]= \int_{[T,T)}B(t,q(t),p(t))dt\in \mathcal{F}_{Q}\cap \mathcal{F}_{P}$(3) If$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}B(t,x,\xi)|\leq C_{\iota z\sqrt 3}$, then$F[q,p]=e^{\int_{lT’.T’’)}B(t,q(t),p(t))dt}\in \mathcal{F}_{Q}\cap \mathcal{F}_{P}$To explain some properties of the classes

Deflnition 2.2 (Two spaces(1) We write$q\in \mathcal{Q}$that(2) We writethat$p\in \mathcal{P}$$q=q_{\Delta_{T,0}}$$p=p_{\Delta_{T,0}}$

if

$q$

.

$\mathcal{Q},$$\mathcal{P}$$\mathcal{F}_{Q},$$\mathcal{F}_{\mathcal{P}}$.

, we prepare some notations.

of piecewise constant paths).

is left-continuous and piecewise constant, i.e., there exists$q_{\Delta_{T,0}}$

such

.
if

$p$

is right-continuous and piecewise constant, i.e., there exists$p_{\Delta_{T,0}}$

such

.

Deflnition 2.3 (Fuctional derivatives). For any
functional derivatives $D_{q’}F[q,p]$ and $D_{p’}F[q,p]$ by$q,$$D_{q’}F[q,p]= \frac{\partial}{\partial\theta}F[q+\theta q’,p]|_{\theta=0},$$q’\in \mathcal{Q}$and any $p,$$p’\in \mathcal{P}$, we define the$D_{p’}F[q,p]= \frac{\partial}{\partial\theta}F[q,p+\theta p’]|_{\theta=0}$

.

193

The position paths

$q$

and

The momentum path

$q’$

$p$

Figure 3.
Remark. For any, chooseandwhich contains all times whenor$q(T_{j})=x_{j},$3).
breaks (Figure$q’(T_{j})=x_{j}’$ for $j=0,1,$ $\ldots,J,J+1$ and $p(T_{j-1})=\xi_{j-1}$
Set$(q+\thetaq’)(O)=x0+\theta x_{0}’,$ $(q+\theta q’)(t)=x_{j}+\theta x_{j}’$ on$\ldots,J,J+1$
for $j=1,2,$. Sinceand$p(t)=\xi_{j-1}$ on
for $j=1,2,$ $\ldots,J,J+1$ , we have$q,$$q’\in \mathcal{Q}$$p\in \mathcal{P}$$q,$ $q’$$\Delta_{T,0}$$p$$(T_{j-1}, T_{j}]$$[T_{j-1}, T_{j})$$F[q+\theta q’,p]=F_{\Delta_{T,0}}(x_{J+1}+\theta x_{J+1}’,\xi_{J},x_{J}+\theta x_{J}’, \ldots,\xi_{0},x_{0}+\theta x_{0}’)$.

Hence we can treat $D_{q’}F[q,p]$ as a finite sum of functions, i.e.,$D_{q’}F[q,p]= \frac{\partial}{\partial\theta}F[q+\theta q’,p]|_{\theta=0}=\sum_{j=0}^{J+1}(\partial_{x_{j}}F_{\Delta_{T,0}})(x_{J+1},\xi_{J}, \ldots,\xi_{0},x_{0})\cdot x_{i}’$.

Because we restrict the directions of functional derivatives to piecewise constant paths, thefunctional derivatives are easy to treat.

Theorem 2 (Smooth algebra).
(1) For any $F[q,p],$, any
we have$G[q,p]\in \mathcal{F}_{\mathcal{Q}}$$F[q,p]+G[q,p]\in \mathcal{F}_{\mathcal{Q}},$$q’\in \mathcal{Q}$, any$p’\in \mathcal{P}$$F[q,p]G[q,p]\in \mathcal{F}_{Q},$$F[Aq,Bp]\in \mathcal{F}_{Q},$ $D_{q’}F[q,p]\in \mathcal{F}_{Q},$(2) For any $F[q,p],$ $G[q,p]\in \mathcal{F}_{P}$ , any$q’\in \mathcal{Q}$, any

and any real$d\cross d$matrices $A,$

$B$,

$F[q+q’,p+p’]\in \mathcal{F}_{\mathcal{Q}}$$D_{p’}F[q,p]\in \mathcal{F}_{Q}$$p’\in \mathcal{P}$and any real$d\cross d$matrices$A,$ $B$,

we have$F[q,p]+G[q,p]\in \mathcal{F}_{\mathcal{P}},$$F[q,p]G[q,p]\in \mathcal{F}_{\mathcal{P}},$$F[q+q’,p+p’]\in \mathcal{F}_{P}$$F[Aq,Bp]\in \mathcal{F}_{P},$ $D_{q’}F[q,p]\in \mathcal{F}_{P},$ $D_{p’}F[q,p]\in \mathcal{F}_{P}$Remark The two classes
are closed under addition, multiplication, translation, real
linear transformation and functional differentiation. However, as we will see in Theorems 4 and$\mathcal{F}_{Q},$$\mathcal{F}_{\mathcal{P}}$194
6, because $q’,$ $p’$ are piecewise constant, the part $\int_{[0,T)}p(t)\cdot dq(t)$ of the action $\phi[q,p]$ does notalways have good properties under these operations. Therefore, we must pay attention to which
operations are valid in the phase space path integrals.

\S 3. Properties of Phase Space Path Integrals
Assuming Theorems 1, 2, we explain the properties of the phase space path integrals.
Theorem 3 (Fubini type). Let be a non-negative integer.
$(a)$ Assume thatand there exists a
is continuous on
for any multi-index. Then the values at the fixed timesuch
positive constant$m$$\alpha,$$C_{a}$$t,$$R\cross R^{d}$$\partial_{x}^{\alpha}B(t,x)$$that|\partial_{x^{l}}(B(t,x)|\leq C_{\alpha}(1+|x|)^{m}$$0\leq t\leq T$$F[q]=B(t,q(t))\in \mathcal{F}_{\mathcal{Q}}$,$F[p]=B(t,p(t))\in \mathcal{F}_{\mathcal{P}}$.

.
In particular,$(b)$ Let $0\leq T’\leq T’’\leq T$. Assume thatfor any multi-indicesand there exists a positive constantous on$C_{a\beta}(1+|x|+|\xi|)^{m}$ . Then the integml$F[q,p]=1\in \mathcal{F}_{Q}\cap \mathcal{F}_{P}$$R\cross R^{d}\cross R^{d}$$\alpha,$$C_{\alpha\beta}$$\beta,$$\partial_{x}^{tt}\partial_{\xi}^{\beta}B(t,x,\xi)$such that$F[q,p]= \int_{[T’,T’’)}B(t,q(t),p(t))dt\in \mathcal{F}_{\mathcal{Q}}\cap \mathcal{F}_{P}$Furthermore let(1) For any$T$is continu-$|\partial_{X}^{a}\partial_{\xi}^{\beta}B(t,x,\xi)|\leq$.

be sufficiently small. Then we have the following:
including $F[q,p]=1$ , we have$F[q,p]\in \mathcal{F}_{\mathcal{Q}}$$\int q(T)=x,p(0)=\xi_{0},q(0)=x0^{e^{i}}\hslash^{\phi[q,p]}(\int_{[T’,T’’)}B(t, q(t))dt)F[q, p]\mathcal{D}[q, p]$$= \int_{[T’,T’’)}(\int_{q(T)=x,p(0)=\xi_{0},q(0)=x0^{e\hslash^{\phi[q,p]}}}iB(t, q(t))F[q, p]\mathcal{D}[q, p])dt$.

(2) For any $F[q,p]\in \mathcal{F}p$ including $F[q,p]=1$ , we have$\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{\frac{i}{\hslash}\phi[q,p]}(\int_{[T’,T’’)}B(t,p(t))dt)F[q,p]\mathcal{D}[q,p]$$=l_{[T’,T’’)}( \int_{q(T)=x,p(0)=\xi_{0},q(0)=x\{)}e^{i}\hslash^{\phi[q,p]}B(t,p(t))F[q,p]\mathcal{D}[q,p])dt$

.

195

$\overline{0T_{k-1}T_{k}T}$
The position path$q_{\Delta_{T,0}}$Figure 4.

Remark (Perturbative expansion). If$|\partial_{X}^{\alpha}B(t,x)|\leq C_{\alpha}$, we have$\int e^{\frac{i}{\hslash}\phi[q,p]+\frac{i}{\hslash}\int_{l0,T)}B(\tau,q(\tau))d\tau}\mathcal{D}[q,p]$$= \sum_{n=0}^{\infty}(\frac{i}{\hslash})^{n}\int_{[0,T)}d\tau_{n}\int_{[0,\tau_{n})}d\tau_{n-1}\cdot\cdot$$\cdot$$\int_{[0,\tau_{2})^{d\tau}}1$$\cross\int e^{\frac{i}{\hslash}\phi[q,p]}B(\tau_{n},q(\tau_{n}))B(\tau_{n-1},q(\tau_{n-1}))\cdots B(\tau_{1},q(\tau_{1}))\mathcal{D}[q,p]$.

Proof of Theorem 3 (1). For simplicity, set $F[q,p]=1$ and $0=T’<T’’=T$ . Using$q_{\Delta_{T,0}}(t)=xk$

on$(T_{k-1}, T_{k}]$(Figure 4) and $dt(\{T_{k}\})=0$ , we have$\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{\frac{i}{h}\phi[q,p]}\int_{[0,T)}B(t,q(t))dt\mathcal{D}[q,p]$,$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{h}\phi[q_{\Delta_{T,0^{p_{\Delta}}\tau,0^{]}\int_{[0,T)}B(t,q_{\Delta_{T,0}}(t))dt\prod_{j=1}^{J}d\xi_{j}dx_{j}}}}$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{h}\phi[q_{\Delta}}\tau,0^{p_{\Delta}}\tau,0^{]}\sum_{k=1}^{J+1}\int_{[T_{k-1},T_{k})}B(t,x_{k})dt\prod_{j=1}^{J}d\xi_{j}dx_{j}$.

$= \lim_{|\Delta_{T,0}|arrow 0}\sum_{k=1}^{J+1}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{\hslash}\phi[q_{\Delta}}r,0^{p_{\Delta}}\tau,0^{]}\int_{[T_{k-1},T_{k})}B(t,x_{k})dt\prod_{j=1}^{J}d\xi_{j}dx_{j}$.

Interchanging the order of the integration on
we have$[T_{k-1}, T_{k})$and the oscillatory integration on$= \lim_{|\Delta_{T,0}|arrow 0}\sum_{k=1}^{J+1}\int_{[T_{k-1},T_{k})}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{h}\phi[q_{\Delta_{T,0^{p_{\Delta}}r,0^{]}}}\prime}B(t,x_{k})\prod_{j=1}^{J}d\xi_{j}dx_{j}dt$$= \lim_{|\Delta_{T,0}|arrow 0}\int_{[0,T)}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{\hslash}\phi[q_{\Delta_{T,0^{p_{\Delta}}\tau,0^{]}}}\prime}B(t,q_{\Delta_{T,0}}(t))\prod_{j=1}^{J}d\xi_{j}dx_{j}dt$.

$R^{2dJ}$,

196
Interchanging the order of the integration on $[0, T)$ and the limit, we have$= \int_{[0,T)}\lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{u/}}^{i}e^{\hslash^{\phi[q_{\Delta}}\tau,0^{p_{\Delta}}\tau,0^{]}}B(t,q_{\Delta_{T,0}}(t))\prod_{j=1}^{J}d\xi_{j}dx_{j}dt$$= \int_{[0,T)}\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{\frac{i}{\hslash}\phi[q,p]}B(t,q(t))\mathcal{D}[q,p]dt$. 口Theorem 4 (Translation).
(1) For any $p’\in P$ , we have$e^{\frac{i}{\hslash}(\phi[q,p+p’]-\phi[q,p])}\in \mathcal{F}_{Q}$

Let

$T$

be sufficiently small. Then for any$F[q,p]\in \mathcal{F}_{\mathcal{Q}}$.

, we have$\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{j}\hslash^{\phi[q,p+p’]}F[q,p+p’]\mathcal{D}[q,p]$.

$= \int_{q(T)=x,p(0)=\xi_{0}+p’(0),q(0)=x_{0}}\pi^{\phi[q,p]}$(2) For any$q’\in \mathcal{Q}$, we have$e^{i}\pi^{(\phi[q+q’,p]-\phi[q,p])}\in \mathcal{F}_{\mathcal{P}}$

Let

$T$

be sufficiently small. Then for any$F[q,p]\in \mathcal{F}_{P}$.

, we have$\int q(T)=x,p(0)=\xi_{0},q(0)=x_{0^{e^{i}F[q+q’,p]\mathcal{D}[q,p]}}\pi^{\phi[q+q’,p]}$$= \int_{(r)q’p0,q0}q=x+(T),(0)=\xi(0)=x+q’(0)^{e^{i}F[q,p]\mathcal{D}[q,p]}\pi^{\phi[q,p]}$.

Proof of Theorem 4 (1). By Theorem 1 and 2 (1), we have$\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{\frac{i}{\hslash}\phi[q,p+p’]}F[q, p+p’]\mathcal{D}[q, p]$$= \int^{i}q(T)=x,p(0)=\xi_{0},q(0)=x0^{e\pi^{\phi[q,p]}\pi^{(\phi[q,p+p’]-\phi[q,p])}}e^{i}F[q, p+p’]\mathcal{D}[q,p]$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{j}{\hslash}\phi[q_{\Delta_{T,0^{p_{\Delta}}r,0^{+p’]}}}\prime}F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}+p’]\prod_{j=1}^{J}d\xi_{j}dx_{j}$. Choose$j=1,2,$
and
with
$p’$
when the path breaks (Figure 5). Set $p’(t)=\xi_{j-1}’$ on, we can write$(p_{\Delta_{T,0}}+p’)(t)=\xi_{j-1}+\xi_{j-1}’$ on$q_{\Delta_{T,0}}(T_{j})=x_{j}$$p_{\Delta_{T,0}}(T_{j})=\xi_{j},$$\ldots,J$

which contains all times
for $j=1,2,$ $\ldots,J+1$ . Since$\Delta_{T,0}$$[T_{j-1}, T_{j})$,$[T_{j-1}, T_{j})$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{d/}\int_{R^{2d/}}e^{i}\hslash^{\phi_{\Delta}}\tau,0^{(x_{J+1}},\xi_{J}+\xi_{/}’,x_{J},\ldots,\xi_{1}+\xi_{1}’,x_{1},\xi_{0}+\xi_{0}’,x_{0})$$\cross F_{\Delta_{T,0}}(x_{J+1},\xi_{J}+\xi_{J}’,x_{J}, \ldots,\xi_{1}+\xi_{1}’,x_{1},\xi_{0}+\xi_{0^{X}0}’)\prod_{j=1}^{J}d\xi_{j}dx_{j}$

,

197

The position path

The momentum paths$q_{\Delta_{T,0}}$$p_{\Delta_{T,0}}$

and $p’$Figure 5.
By the change of variables:$\xi_{j}+\xi_{j}’arrow\xi_{j},$$j=1,2,$ $\ldots,J$ , we have$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{\hslash}\phi_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J},\ldots,\xi_{1},x_{1},\xi_{0}+\xi’x)}0,0$$\cross F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,\xi_{1},x_{1},\xi_{0}+\xi_{0}’,x_{0})\prod_{j=1}^{J}d\xi_{j}dx_{j}$.

Noting that $p’(0)=\xi_{0}’$ , we can rewrite$= \int_{q(T)=x,p(0)=\xi_{0}+p’(0),q(0)=x_{0}}e^{\frac{i}{\hslash}\phi[q,p]}F[q,p]\mathcal{D}[q,p]$$\mathcal{F}_{\mathcal{Q}}$

Theorem 5 (Orthogonal transformation). Letorand anyorthogonal matrix ,$d\cross d$$\mathcal{F}_{P}$

$T$

. 口

be sufficiently small. Then for any $F[q,p]\in$$Q$$\int_{q(T)=x,p(0)=\xi_{0},q(0)=x0}e^{\frac{i}{h}\phi[Qq,Qp]}F[Qq, Qp]\mathcal{D}[q,p]$.

$= \int_{q(T)=Qx,p(0)=Q\xi_{0},q(0)=Qx_{0}}\hslash^{\phi[q,p]}$

Theorem 6 (Integration by parts).
(1) For any, we have$p’\in \mathcal{P}$$D_{p’}\phi[q,p]\in \mathcal{F}_{Q}$Furthermore, let$T$.

be sufficiently small. Then for any$F[q,p]\in \mathcal{F}_{\mathcal{Q}}$and any$p’(0)=0$,$\int_{q(T)=x,p(0)=\xi_{0},q(0)=x0}e^{\frac{i}{\hslash}\phi[q,p]}(D_{p’}F)[q,p]\mathcal{D}[q,p]$$=- \frac{i}{\hslash}\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{\frac{i}{\hslash}\phi[q,p]}(D_{p’}\phi)[q,p]F[q,p]\mathcal{D}[q,p]$.

$p’\in \mathcal{P}$

with

198
(2) For any$q’\in \mathcal{Q}$, we have$D_{q’}\phi[q,p]\in \mathcal{F}_{P}$Furthermore, let$T$.

be sufficiently small. Then for any$F[q,p]\in \mathcal{F}_{P}$and any$q’\in \mathcal{Q}$with$q’(T)=q’(0)=0$,$\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{\frac{i}{\hslash}\phi[q,p]}(D_{q’}F)[q,p]\mathcal{D}[q,p]$$=- \frac{i}{\hslash}\int_{q(T)=x,p(0)=\xi_{0},q(0)=x_{0}}e^{\frac{i}{\hslash}\phi[q,p]}(D_{q’}\phi)[q,p]F[q,p]\mathcal{D}[q,p]$.

Remark (Analogues of canonical equations). Set $F[q,p]=1$ .
(1) For any
with $p’(O)=0$ , we have$p’\in \mathcal{P}$$0= \int_{q(T)=x,p(0)=\xi_{0},q(0)=x0}e^{\frac{i}{\hslash}\phi[q,p]}(\int_{[0,T)}p’dq-(\partial_{\xi}H)(t,q,p)p’dt)\mathcal{D}[q,p]$(2) For any$q’\in \mathcal{Q}$with $q’(T)=q’(0)=0$ , we have

$0= \int q(T)=x,p(0)=\xi_{0},q(0)=r^{e^{i}}\hslash^{\phi[q,p]}(\int_{[0,T)}pdq’-(\partial_{x}H)(t,q,p)q’dt)\mathcal{D}[q,p]$Let be small. For anyof the phase function$T$$(x_{J+1},\xi_{0})\in R^{d}\cross R^{d}$$\phi_{\Delta_{T,0}}=\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$, there exists the stationary point$(x_{J}^{*},\xi_{J}^{*}, \ldots,x_{1}^{*},\xi_{1}^{*})$.

$(x_{J}^{*},\xi_{J}^{*}, \ldots,x_{1}^{*},\xi_{1}^{*})$

given by$(\partial_{(\xi_{J},x_{J},\ldots,\xi_{1},x_{1})}\phi_{\Delta_{T,0}})(x_{J+1},\xi_{J}^{*},x_{J}^{*}, \ldots,\xi_{1}^{*},x_{1}^{*},\xi_{0})=0$

Pushing.

into the Hessian of$\phi_{\Delta_{T,0}}$.

, we define $D(T,x_{J+1},\xi_{0})$ by$D(T,x_{J+1},\xi_{0})$$= \lim_{|\Delta_{T,0}|arrow 0}(-1)^{dJ}\det(\partial_{(\xi_{J},x_{J},\ldots,\xi_{1},x_{1})}^{2}\phi_{\Delta_{T,0}})(x_{J+1},x_{J}^{*},\xi_{J}^{*}, \ldots,x_{1}^{*},\xi_{1}^{*},\xi_{0})$Let$\overline{q}(t)=\overline{q}(t,x,\xi_{0})$and$\overline{p}(t)=\overline{p}(t,x,\xi_{0})$$\partial_{t}\overline{q}(t)=(\partial_{\xi}H)(t,\overline{q}(t),\overline{p}(t)),$with$\overline{q}(T)=x$$p^{b}(t,x,\xi_{0})$

6.

by

and

$\overline{p}(0)=\xi_{0}$be the solution of the canonical equations$\partial_{t}\overline{p}(t)=-(\partial_{X}H)(t,\overline{q}(t),\overline{p}(t)),$. We define the bicharacteristic paths$q^{b}(t)=\overline{q}(t,x,\xi_{0}),$.

$0\leq t\leq T$,$q^{b}=q^{b}(t,x,\xi_{0^{X}0})$

and $p^{b}=$$0<t\leq T,$ $q^{b}(0)=x_{0}$ and $p^{b}(t)=\overline{p}(t,x,\xi_{0}),$ $0\leq t<T$ (Figure$)$

Then the remainder estimate for the semiclassical approximation of Hamiltonian type asis the following.

$\hslasharrow 0$

Theorem 7 (Semiclassical approximation of Hamiltonian type asor , we havesmall. Then, for any$F[q,p]\in \mathcal{F}_{Q}$$\hslasharrow 0$). Let$T$be sufficiently$\mathcal{F}_{P}$$\int e^{\frac{i}{\hslash}\phi[q,p]}F[q,p]\mathcal{D}[q,p]=e^{\frac{i}{\hslash}\phi[q^{b},p^{b}]}(D(T,x,\xi_{0})^{-1/2}F[q^{b},p^{b}]+\hslash Y(\hslash,T,x,\xi_{0},xo))$

.

199

$0$

$T$

The bicharacteristic path

$0$

$T$

The bicharacteristic path$q^{b}$$p^{b}$

Figure 6.

Here for any multi-indices$\alpha$and , the remainder term$\beta$$Y(\hslash, T,x,\xi_{0^{X}0})$satisfies$|\partial_{x}^{\alpha}\partial_{\xi_{0}}^{\beta}l(\hslash, T,x,\xi_{0},x_{0})|\leq C_{cx/3}(1+|x|+|\xi_{0}|+|x_{0}|)^{m}$with a positive constant$C_{\alpha,\beta}$,.
\S 4. Proof for Theorems 1, 2 and 7

We explain the process of the proof for Theorems 1, 2 and 7. In order to prove the convergence of the multiple integral
(4.1)

as

$|\Delta_{T,0}|arrow 0$$( \frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}e^{\frac{i}{\hslash}\phi[q_{\Delta_{T,0^{p_{\Delta}}\tau,0^{]}}}\prime}F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]\prod_{j=1}^{J}d\xi_{j}dx_{j}$,, we have only to add many assumptions for$F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,x_{1},\xi_{0^{X}0})=F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$.

The assumptions should be closed under addition and multiplication. Then
will beclosed under addition and multiplication. Do not consider other things. Then
will belarger as a set. If lucky,
will contain at least one example $F[q,p]=1$ as the fundamental
solution for the Schr\"odinger equation. Our proof consists of 3 steps. As the first step, by an
estimate of H. Kumano-go-Taniguchi’s type [18, p.360, (6.94)], we control (4.1) bywith apositive constant as $Jarrow\infty$ . As the second step, by a stationary phase method of Fujiwara‘stype [9], we control (4.1) by with a positive constant independent of $Jarrow\infty$ . As the laststep, we add assumptions so that (4.1) converges as.
$\mathcal{F}_{Q},$$\mathcal{F}_{Q},$$\mathcal{F}_{Q},$$\mathcal{F}_{P}$$\mathcal{F}_{\mathcal{P}}$$\mathcal{F}_{P}$$C^{J}$

$C$

$C$

$C$

$|\Delta_{T,0}|arrow 0$\S 5.

Two classes$\mathcal{F}_{Q},$$\mathcal{F}_{\mathcal{P}}$In order to state the definition of the classesof higher order.

of functionals $F[q,p]$$\mathcal{F}_{Q},$$\mathcal{F}_{\mathcal{P}}$, we introduce the functional derivatives

200
Deflnition 5.1 (Functional derivatives of higher order). For any division
that$F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x,, \ldots,\xi_{0^{X}0})\in C^{\infty}(R^{d(2J+3)})$Let$L_{Q},$$L_{\mathcal{P}}$be non-negative integers. For any, we define the functional derivative$q,$$l=1,2,$ $\ldots,L_{P}$$q_{l}\in \mathcal{Q},$$l=1,2,$$\ldots,L_{\mathcal{Q}}$$\Delta_{T,0}$, we assume.

,
and any $p,$of higher order$p_{l}\in \mathcal{P}$$( \prod_{l=1}^{L_{Q}}D_{q_{l}})(\prod_{l=1}^{L_{P}}D_{p_{l}})F[q,p]$by$l_{\phi}$$L_{\mathcal{Q}}$$( \prod D_{q_{l}})(\prod D_{p_{l}})F[q, p]$
$l=1$

$l=1$

.

$=( \prod\frac{\partial}{\partial\theta_{l}})(\prod^{L_{Q}}\frac{\partial}{\partial\theta_{l}})F[qL_{\mathcal{P}}+\sum\theta_{l}q_{l}, pL_{\mathcal{Q}}+\sum\theta_{l}p_{l}]\iota_{\varphi}$

$l=1$

$l=1$

$l=1$

$l=1$
$\theta_{1}=\cdots=\theta_{L_{Q}}=\theta_{1}=\cdots=\theta_{l_{\mathcal{P}}}=0$

The definition of the classes$\mathcal{F}_{Q},$$\mathcal{F}_{P}$of functionals $F[q,p]$ are the following.

of functionals $F[q,p]$ ). Let $F[q,p]$ be a functional of
Deflnition 5.2 (Two classes.
and(1) We write
if $F[q,p]$ satisfies Assumption 2 (1).
(2) We write
if $F[q,p]$ satisfies Assumption 2 (2).
$\mathcal{F}_{Q},$$q\in \mathcal{Q}$$\mathcal{F}_{P}$$p\in \mathcal{P}$$F[q,p]\in \mathcal{F}_{Q}$$F[q,p]\in \mathcal{F}_{p}$$j=1,2,$ $\ldots,J,J+1$ andAssumption 2. Let be a non-negative integer. Let
such that $\sum_{j=1}^{J+1}u_{j}=U<\infty$ . Set
negative parameters depending on$m$$u_{j},$$\Delta_{T,0}$

be non-$||q||= \sup_{0\leq t\leq T}|q(t)|$and $||p||= \sup_{0\leq t<T}|p(t)|$ . For simplicity, we set $[0,0]=(T_{-1}, T_{0}]$ .
(1) For any non-negative integer $M$, there exist positive constants$U$$A_{M},$ $X_{M}$such that$|( \prod_{j=0}^{r+1^{L}}\prod_{l=1}^{Q,j}D_{q_{j,l}})(\prod_{j=1}^{J+1^{L}}\prod_{l=1}^{P,j}D_{p_{j,l}})F[q,p]|$$\leq A_{M}(X_{M})^{J+1}(1+||q||+||p||)^{m}$$\cross(\prod_{=1}^{J+1}(t_{j})^{\min(l_{\mathcal{P},j},1)})\prod_{j=0}^{J+1^{L}}\prod_{l=1}^{Q,/}.||q_{j,l}||\prod_{=1}^{J+1^{l}}I^{\mathcal{P}}I^{j}||p_{j,l}||jjl=1$’$|( \prod_{j=0}^{J+1^{L}}\prod_{l=1}^{Q,j}D_{q_{j,l}})(\prod_{j=1}^{J+1^{L}}\prod_{l=1}^{P,j}D_{p_{j,l}})D_{q_{k}}F[q,p]|$$\leq A_{M}(X_{M})^{J+1}(1+||q||+||p||)^{m}$$\cross u_{k}||q_{k}||(\prod_{k}^{J+1}(t_{j})^{\min(L_{P,j},1)})\prod^{r+1^{L}}r^{Q}i^{j}||q_{j,l}||\prod_{jJ=1,j\neq j=0l=1=1}^{J+1^{L}}\prod_{l=1}^{P,j}$Il$p_{j,l}||$,201
for any division$\Delta_{T,0}$, any $L_{\mathcal{Q},j}=0,1,$ $\ldots,M$, any $L_{P,j}=0,1,$ $\ldots,M$, any, any
outsidewith $q_{k}(t)=0$ outsidewith $p_{j,l}(t)=0$ outside(Figure 7).
(2) For any non-negative integer $M$, there exist positive constants$q_{j,l}(t)=0$$(T_{j-1}, T_{j}]$$q_{k}\in \mathcal{Q}$$(T_{k-1}, T_{k}]$$q_{j,l}\in \mathcal{Q}$, and anywith

$p_{j,l}\in \mathcal{P}$$[T_{j-1}, T_{j})$$A_{M},$ $X_{M}$such that$|( \prod_{j=0}^{/+1^{L}}\prod_{l=1}^{Q,j}D_{q_{j,l}})(\prod_{j=1}^{\int+\iota^{L}}\prod_{l=1}^{P,j}D_{p_{j,l}})F[q,p]|$$\leq A_{M}(X_{M})^{J+1}(1+||q||+||p||)^{m}$$\cross(\prod_{j=1}^{J+1}(t_{j})^{\min(L_{Q,j},1)})\prod_{j=0}^{J+1^{L}}\prod_{l=1}^{Q,j}q_{j,l}p_{j,l}||$,$|( \prod_{j=0}^{J+1^{L}}\prod_{l=1}^{Q,/}.D_{q_{j,l}})(\prod_{j=1}^{\int+\iota^{L}}\prod_{l=1}^{\mathcal{P},j}D_{p_{j,l}})D_{p_{k}}F[q,p]|$$\leq A_{M}(X_{M})^{J+1}(1+||q||+||p||)^{m}$$\cross u_{k}||p_{k}||(\prod_{k}^{J+1}(t_{j})^{\min(L_{Q,j},1)})\prod^{J+1^{L}}\Gamma^{Q}I^{j}||q_{j,l}||\prod^{J+1^{L}}\Gamma^{\mathcal{P}}I_{\iota}^{j}||p_{j,l}||1=1,j\neq j=0l=1j=1l=’$

for any division$q_{j,l}(t)=0$$\Delta_{T,0}$outside,, any $L_{Q,j}=0,1,$ $\ldots,M$, any $L_{P,j}=0,1,$ $\ldots,M$, any$(T_{j-1}, T_{j}]$with $p_{j,l}(t)=0$ outside

The position paths, any$[T_{j-1}, T_{j})$$q,$$p_{k}\in \mathcal{P}$with $p_{k}(t)=0$ outside$[T_{k-1}, T_{k})$$q_{j,l}\in \mathcal{Q}$, and anywith

$p_{j,l}\in \mathcal{P}$(Figure 7).

The momentum paths$q_{j,l}$$p,$$p_{j,l}$Figure 7.

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