数理解析研究所講究録
第 1797 巻 2012 年 42-50

42White Noise Approach to Feynman Path Integrals and Someof the Related Topics
By

Takeyuki HIDA *Abstract
In this paper the following will be discussed:
1. Why the white noise analysis is applied to the Feynman path integrals. Remind the idea of Dirac andFeynman and see how we can realize their idea ffom mathematics side.
2. White noise analysis, in particular the theory of generalized white noise functionals, is applied. We
shall see how they are introduced so that it is fitting for the formulation of the Feynman integrals.

3. Actual method of applications. Examples.
4. Some of fmher developments in the present line.

\S 1. Introduction
1. The Idea. The basic idea of our approach to the path integrals in quantum dynamics is to
apply the white noise analysis to the construction of the quantum mechanical propagators.
In fact, an attempt to give a correct interpretation to the Feynman integral, which had only
formal significance, was one of the motivation of proposing the white noise analysis. Actually,
there were two problems for us: one is how to realize a “flat” measure on the function space, thrfunction space itself should be clarified; another is how to understand the exponential functionalof the action.
The path integral method is, as is well known, viewed as a third method of quantization,
which is different from the formulation by W. Heisenberg and another one by E. Schr\"odinger.
Our method of path integral, within the framework of white noise analysis, follows mainly the
Feynman’s method [3] in spirit. However, some other basic quantum mechanical considerations
are taken into account.
While we are working the problem in question, we have come to realize that we should see
the Dirac’s ideas, which are mostly found in his textbook [2].
2010 Mathematics Subject Classification(s): $60H40$ .
Key Words: Wluite Noise Theory.
*Professor Emeritus of Nagoya University

43
2. We shall quickly explain our approach step by step.
i What does a path means in quantum dynamics? In quantum dynamics, following the
Lagrangian dynamics theory, there are many possible paths i.e. trajectories of a particle, andeach trajectory may be viewed as a sum of the classical one and fluctuation. The classical path,denoted by is, of course uniquely determined by the Lagrangian (with boundary values).
The so-called possible trajectories in quantum dynamics can be expressed in the form
$)$

$y$

$x$

$x=y+z$ ,where is the fluctuation (see [4]).
Our first question is to determine the fluctuation . We propose that is a Gaussian Markov
process, more precisely a Brownian bridge which is a liner function of a Brownian motion $B(t)$ .
We shall explain the reason why a Brownian bridge is fitting for a realization of fluctuation in
the next section.
ii) To observe Feynman‘s expression of propagator, we first meet the action integral
$z$

$z$

$z$

$A(t)= \int^{t}L(x,\dot{x})ds$,

The integrand involves a term
to express the kinetic energy. The
is not an ordinary
random function. To come to the propagator we have even to exponentiate the action.
Thus we shall be concemed with analysis of generalized functions, actually functionals
of $B(t)’ s$ , that is white noise. \S 4 is provided to the background for the study of generalized
functionals of white noise.
iii) Integration over the function space $X=\{x\}$ , the space of trajectories, can be definedsmoothly. To this end, we must specify a measure on : this is now obvious, since we take awhite nloise, the probability distribution has automatically been introduced.
There is one problem to be reminded. We expect the integral should be done with respect tothe uniform measure on . This problem can be solved also in \S 3.
iv) With these background we can come to the actual computation of the propagators. Thiscan be done by using the white noise theory.
v We can recognize that our approach can be applied to a pretty large class of Lagrangians.
vi) Our method of integration on function space can further be applied to other problems inphysics as we shall see in the last two sections.
$B(s)^{2}$$B(s)^{2}$

$X$

$\mu$

$X$

$)$

\S 2. Brownian Bridge and a Setup of the Propagator
First we have to explain why the Brownian bridge is involved in the class of quantum mechamical possible trajectories. In [2, \S 32], Action principle, there is a statement that $B(t,s)=$$\int^{s}L(u)du$satisfies a chain $mle$ , by which we may imagine the formula for the transition prob-

abilities of a Markov process. To fix the idea, we consider the case where the time interval is44taken to be $[0, T]$ . Now the term that expresses the quantity of fluctuation can be a Markov
process $X(t),0\leq t\leq T$ . Further assumptions on $X(t)$ can be made as follows.
$z$

is a Gaussian process, since it is a sort of noise [8].
2 As a usual requirement, the Gaussian process satisfies $E(X(t))=0$ and has the canonicalrepresentation by Brownian motion, namely

1$)$$X(t)$$)$$X(t)= \int_{0}^{t}F(t,u)B(u)du$and $X(O)=X(T)=0$ (bridged).
3 $X(t)$ is a Gaussian l-ple Markov process.
4 The normalized process $Y(t)$ enjoys the projective invariance under time-change.
$)$

$)$

Theorem 1. The Brownian bridge $X(t)$ over the time interval $[0, T]$ is characterized by theabove conditions ) $-4)$ .
$1$This theorem we have proved before and the proof is omitted here. We only note that the
canonical representation of $X(t)$ is given by$X(t)=(T-t) \int_{0}^{t}\frac{1}{T-u}B(u)du$and the covariance$\Gamma(t,s)$,is$\Gamma(t,s)=\sqrt{\frac{s(T-t)}{t(T-s)}}$,$s\leq t$.

Namely,$\Gamma(t,s)=\sqrt{(0,T;s,t)}$where$(\cdot,$$\cdot;\cdot,$$\cdot)$,$s\leq t$,

is the anharmonic ratio.

Remark Heuristically speaking, it was 1981 when we proposed a white noise approach
to path integrals to have quantum mechanical propagators (Hida-Streit paper presented 1981
Berlin Conference on Math-Phys. Later Streit-Hida [17] . We had, at that time, some idea inmind for the use of a Brownian bridge, and we had practically many good examples of integrandwith various kinds of potentials, and satisfactory results have been obtained.
$)$

With this background we are ready to propose how to form quantum mechanical propagators. The possible quantum mechanical trajectories $x(t),t\in[0, T]$ are expressed in the form$x(t)=y(t)+\sqrt{\frac{\hslash}{m}}X(t)$,where $X(t)$ is a Brownian bridge over the time interval $[0, T]$ . The fluctuation
expression is now taken to be a Brownian bridge.

$z$in the earlier45Remind that the classical trajectory $y(t),$ $t\in[0, T]$ , is uniquely determined by the variational
principle for the action$A[x]= \int_{0}^{T}L(x,\dot{x})dt$

where the Lagrangian$L(x,\dot{x})$,in question is assumed to be of the form$L(x, \dot{x})=\frac{1}{2}m(\dot{x})^{2}-V(x)$.

The potential $V(x)$ is usually assumed to be regular, but later we can extend the theory to thecase where has certain singularity, even time-dependent.
The actual expression and computations of the propagator are given successively as follows:
We follow the Lagrangian dynamics. The possible trajectories are sample paths $y(s),s\in$$[0,t]$ , expressed in the form
$V$

(1)

,

$y(s)=x(s)+\sqrt{\frac{\hslash}{m}}B(s)$where the $B(t)$ is an ordinary Brownian motion. Hence the action
terms of quantum trajectory :$S$is expressed in the form in$y$$A= \int_{0}^{t}L(y(s),\dot{y}(s))ds$.

Note that the effect of forming a bridge is given by putting the delta-function
factor of the integrand, where $y_{2}=x(t)$ .
Now we set(2)$S(t_{0},t_{1})= \int_{t_{0}}^{t_{1}}L(t)dt$$\delta_{0}(y(t)-y_{2})$

as a

.

and set.

$\exp[\frac{i}{\hslash}\int_{t_{0}}^{t_{1}}L(t)dt]=\exp[\frac{i}{\hslash}S(t_{0},t_{1})]=B(t_{0},t_{1})$Then, we have (see Dirac [2]), for $0<t_{1}<t_{2}<\cdots<t_{n}<t$ ,$B(O,t)=B(0,t_{1})\cdot B(t_{1},t_{2})\cdots B(t_{n},t)$.

Theorem 2. The quantum mechanical propagator $G(O,t;y_{1},y_{2})$ is given by the following
avemge(3)$G(0,t;y_{1},y_{2})= \{N\exp(\frac{i}{\hslash}\int_{0}^{t}L(y,\dot{y})ds+\frac{1}{2}\int_{0}^{t}B(s)^{2}ds)\delta_{o}(y(t)-y_{2})\rangle$

where $N$ is the amount of multiplicative renormalization. The avemge
the integral with respect to the white noise measure .
$\mu$$\{\}$,

is understood to be

46\S 3. Generalized White Noise Functionals Revisited
It is well-known that there are two classes of generalized white noise functionals;
and. We use them without discrimination except it is necessary to choose one of themspeci fi cally.
It seems better to explain the concept of”renormalization” which makes formal but important functionals of the $B(t)$ ’s to be acceptable as generalized white noise functionals. To savetime we refer the interpretation to the literatures [11] and [16].
We should note that there are generalized white noise functionals involved in the expectationin Theorem 2. For instance, there is involved the delta function, in fact the Donsker’s delta, which is a generalized white noise functional. There is used a Gauss
function$(L^{2})^{-}$$(S)^{*}$$\delta_{o}(y(t)-y_{2})$kemel of the form $\exp[c\int_{0}^{t}B(s)^{2}ds]$ , the ideal case is$c=- \frac{1}{2}$. In general, if$c \neq\frac{1}{2}$, then it canbe a generalized functional after having the multiplicative renormalization. Now we have theexceptional case, but it can be accepted by combining with other factor of an exponential; thisis just the case. In reality, we combine it with the term that comes from the kinetic energy. The

serves as the flattening effect of the white noise measure. One may
ask why the functional does so. An intuitive answer to this question is as follows: If we writeby $dL$, the white noise measure may be
a Lebesgue measure (exists only virtually) on
factor$\exp[\frac{1}{2}\int_{0}^{t}B(s)^{2}ds]$$E^{*}$$\mu$. Hence, the the factor in question is put to makethe measure to be a flat measure $dL$ . In fact, this makes sense eventually.
Retuming to the average (3) (in Theorem 2), which is an integral with respect to the white
noise measure , it is important to note that the integrand (i.e. the inside of the angular bracket)
is integrable, in other words, it is a bilinear form of a generalized functional and a test functional.
There have to follow short notes to be reminded. They are rather cmcial. The formula (3)$f\in L^{2}(R),$C. To give a
involves a product of functionals of the form likechoice,it should be checked that it cancorrect interpretation to the expectation of (3) with thisbe regarded as a bilinear form of a pair of a test functional and a generalized functional. The
following assertion answers to this question.
expressed in the form$\exp[-\frac{1}{2}\int_{0}^{t}B(s)^{2}ds]dL$

$\mu$

$\mu$

$a\in$$\varphi(x)\cdot\delta(\langle x,f\}-a),$be a generalized white noisefunctional. Assume thatto afunctional of $f$ in $L^{2}(R)$ , in particular aisextendedthe -tmnsform, and thatis an integrable fiunction of for any fixed . If
fiunction ofisis a U-fiunctional, then the pointwise productthe transformdefined and is a generalized white noise functional.

Theorem 3 (Streit et al [19]). Let$\mathcal{T}$$(\mathcal{T}\varphi)(\xi),\xi\in E$$\xi+\lambda f$, of$\varphi(x)$$\varphi$$\lambda$$(\mathcal{T}\varphi)(\xi-\lambda f)$$\xi$$\varphi(x)\cdot\delta(\langle x,f\rangle-a)$$(\mathcal{T}\varphi)(\xi-\lambda f)$

Proof. First a formula for the -function is provided.
$\delta$$\delta_{a}(t)=\delta(t-a)=\frac{1}{2\pi}\int e^{ia\lambda}e^{-i\lambda x}d_{i}1$(in distribution sense).

47
Hence, for$\varphi\in(S)^{*}$

and $f\in L^{2}(R)$ we have$\mathcal{T}(\varphi(x)\delta(\langle x,f\}-a))\xi)=\frac{1}{2\pi}\int e^{ia\lambda}e^{-i\lambda\langle x,f\rangle}e^{i\langle x,\xi\rangle}\varphi(x)d\mu(x)d\Lambda$(4)$= \frac{1}{2\pi}\int e^{ia\lambda}(\mathcal{T}\varphi)(\xi^{\lambda}f)d\lambda$.

By assumption this determines a U-functional, which means the productmakes sense and it is a generalized white noise functional.

$\varphi(x)\cdot\delta(\{x,f\rangle-a)$$\square$Example 1. The above theorem can be applied to a Gauss kemel $\varphi_{c}(x)=N\exp[c\int x(t)^{2}dt]$,

with.
i The case where is real and $c<0$ . We have$c \neq\frac{1}{2}$

$)$

$c$

$( \mathcal{T}\varphi)(\xi-jtf)=\exp[\frac{c}{1-2c}\int(\xi(t)-\Lambda f(t))^{2}dt]$$= \exp[\frac{c}{1-2c}(\Vert\xi\Vert 2-2$ ノ $t(\xi,f)+\Lambda^{2}\Vert f\Vert^{2}])$.

This is an integrable function of real . Hence, by the above Theorem 3, we have a generalized
white noise functional.
ii) The case where $c= \frac{1}{2}+ia,$ $a\in R-\{0\}$ .
The same expression as in i) is given.
$\lambda$

Example 2. In the following case, exact values of the propagators can be obtained and, ofcourse, they agree with the known results.
i Free particleii) Harmonic oscillator.
iii) Potentials which are Fourier transforms of measures (the the Albeverio-Hohkron poten$)$tial).

\S 4. Some of Further Developments and Related Topics
In addition to Example 2, we have some more interesting potentials,including those aremuch singular and time depending. There are satisfactory results in the recent developments.
[I]

Example 3. Streit et al have obtained explicit formulae in the following cases:
1 a time depending Lagrangian of the form$)$$L(x(t), \dot{x}(t),t)=\frac{1}{2}m(t)\dot{x}(t)^{2}-k(t)^{2}x(t)^{2}-\dot{f}(t)x(t)$where $m(t),k(t)$ and $f(t)$ are smooth funtions.

,48
2 A singular potential $V(x)$ of the form$)$$V(x)= \sum_{n}c^{-n^{2}}\delta_{n}(x)$,$c>0$ ,and others.

The Hopf equation.
There are many approaches to the Navier-Stokes equation.

$[n]$$u_{(\chi,f}+u_{\beta^{u}(t,\rho=-p\cdot\alpha+\mu u_{\iota r\beta\beta}}$,where $\alpha,\beta=1,2,3$ , and where the following notations are used:$f_{\alpha,t}= \frac{\partial f_{\alpha}}{\partial t}$$f_{\alpha\beta}= \frac{\partial f_{\alpha}}{\partial x_{\beta}}$

,
,

and
$f_{\alpha\sqrt 3\gamma}= \frac{\partial^{2}f_{\alpha}}{\partial x_{\beta}\partial x_{\gamma}}$.

There is an approach to this equation by using the characteristic functional$P(du)$ defined on the phase space $\{u=(u,u,u3)\}$ :$\Phi(\xi,t)=\int e^{i<\xi,u>}P(du)$E. Hopf shows that the characteristic functional
ferential equation, called Hopf equation:$\Phi(\xi, ,t)$$\Phi$of the measure.

satisfies the following functional dif-$\frac{\partial\Phi}{\partial t}=\int_{R}\xi_{\alpha}(x)[i\frac{\partial}{\partial x_{\beta}}\frac{\partial^{2}\Phi}{\partial\xi_{\beta}(x)dx\partial\xi_{\alpha}(x)dx}+\mu\Delta_{x}\frac{\partial\Phi}{\partial\xi_{\alpha}(x)dx}-\frac{\partial\Pi}{\partial_{X_{\alpha}}}]dx$

Studying this approach, we may think of the two matters. One is a similarity to the Feynman
integral in the sense that both cases deal with functional obtained in the form
$E(\exp[f(u)])$where $f(u)$ is a function of a path (trajectory) . The expectation is taken with respect to theprobability measure introduced on the path space.
, that come from the Hopf
As the second point, one may think of equations forequation and the Fock space expansion of generalized white noise functionals. In this case we
expect that the calculus can be done in a similar manner to the white noise calculus. We may
remind an interesting approach to the Navier-Stokes equation by A. Inoue (see [13]).
$u$$\Phi_{n},n\geq 0$

49\S 5. Concluding Remarks(1) There appears a particular quadratic form in the white noise analysis, i.e.
$\int:B(t)^{2}:dt$.

There are somewhat general quadratic form$\int f(t):B(t)^{2}:dt+\int\int F(u,v);B(u)B(v);dudv$

which is called normalfunctional, the first term is called the singular part and the second termis the regular part. The two terms can be characterized from our viewpoint and play significant
roles, respectively. Remind the role of singular part in the path integral.
(2) Our method of path integrals enables us to deal with the case of very irregular potentials
to have the propagator, by L. Streit and others.
(3) Some other approaches: It is significant to see the results by C. C. Bemido and M. V.
Carpio Bemido [1]. They are using our method of path integral to investigate the entanglement
probabilities of two chain like macro-molecules where one polymer lies on a plane and theother perpendicular to it. The entanglement probabilities are calculated and the result shows acharacteristic of the polymer.
We also should like to note that Masujima [14] has published a beautiful monograph collecting various approaches to path integrals,
Acknowlegements The author is grateful to Professor N. Kumano-go, the workshop organizer, who has given the author the opportunity to deliver a talk on path intehrals.

References[1] Bemido, C. C. and Carpio-Bemido, M. V, Whited noise functional approach to polymer entangle-

ments. Proc. Stochastic Analysis: Classical and Quantum World Sci. Pub. Co. 2005.

[2] Dirac, P. A. M., The Principles of Quantum Mechanics, 4th ed. Oxford Univ. Press. 1958.
[3] Feynman, R. P., Space-time approach to non-relativistic quantum mechanics. Rev. of Modem Phys.

20 (1948) 367-387.
[4] Feynman, R.P. and Hibbs, A. R., Quanm Mechanics and Path integmls, McGraw-Hill Inc. 1965.
[5] Hida, T., Analysis of Brownian functionals, Carleton Math. Notes no. 13. 1975.
[6] –, Brownian Motion, Iwanami Pub.Co. 1975; English Transl., Springer-Verlag, 1980.
[7] –, White noise approach to Feynman integrals, J. Korean Math. Soc. 38 (2001), 275-281.

[8] –, The birth of noises, Preprint 2011.
[9] –, The dawning of white noise theory, Lec. Notes, Tokyo Univ, of Sci. June, 2011.
[10] Hida, T. and Si Si, Innovation approach to random fields, An Application
White Noise Theory,

Wold Sci. Pub. Co. 2004.

of

50[11] –, Lectures on White Noise Functionals, World Sci. Pub. Co. 2008.
[12] Hopf, E., Statistical hydromechanics and functional calculus. J. Rational Mechanics and Analysis,
1, (1952), 87-128.
[13] Inoue, A., Strong and classical solutions of the Hopf equation-An example of functional derivativeequation of second order, Tohoku Math J39 (1987), 115-144[14]

Masujima, M., Path Integrals and Stochastic Processes in Theoretical Physics, Feshbach Pub. LLC.,

2007.

[15] Si Si, A new noise depending on a space parameter and its application. QBIC2011 Conf.2011.
[16] –, Introduction to Hida distributions. World Sci. Pub. Co. 2011.
[17] Streit, L. and Hida, T., Generalized Brownian functionals and the Feynman integral, Stochastic Pro-cesses and Applications 16 $(1983),55-69$ .

[18] Grothaus, M., Khandekar, D. C., da Wilva, J. L. and Streit, L., The Feynman integral for timedependent anharmonic oscillators. J. Math. Phys. 38 (1997), 3278-3299.

[19] Streit, L. et al, Feynman integra]s for A class of exponentially growing potentials, J. Math. phys. 39(1998), 4476A49l.