White Noise Approach to Feynman Path Integrals and Some of the Related Topics (Introductory Workshop on Feynman Path Integral and Microlocal Analysis)

White

Noise

Approach

to

Feynman Path Integrals and

Some

of

the Related Topics

By

Takeyuki

HIDA

*

Abstract

In thispaperthe following will bediscussed:

1. Whythe whitenoise analysis is appliedtotheFeynman path integrals. Remind the idea ofDiracand

Feynmanandseehowwe canrealizetheir ideaffom mathematics side.

2. White noise analysis, in particular the theoryofgeneralizedwhitenoise functionals, is applied. We

shallseehowtheyareintroducedsothatitis fittingfor the formulation of theFeynman integrals.

3. Actual method of applications.Examples.

4. Someof fmher developments in the present line.

\S 1.

Introduction

1.

TheIdea. The basic idea of

our

approachtothe path integrals inquantum dynamics isto

apply the white noise analysistotheconstructionof thequantummechanicalpropagators.

In fact,

an

attemptto give

a

correct

interpretation

to the Feynman integral,which had only

formal significance,

was

one

ofthe motivationofproposingthe whitenoiseanalysis. Actually,

there

were

twoproblems for

us:

one

ishowtorealize

a

“flat”

measure

on

thefunction

space,

thr function

space

itself shouldbeclarified;anotherishow tounderstandthe exponential functional of theaction.

The path integral method is,

as

is well known, viewed

as a

third method ofquantization,

which isdifferent from the formulation by W. Heisenberg and another

one

by E. Schr\"odinger.

Our methodof path integral, within the framework of white noise analysis, follows mainlythe

Feynman’s method[3]inspirit. However,

_some

otherbasic quantum mechanicalconsiderations

are

taken into account.

While

we

are

working the problem inquestion,

we

have

come

torealize that

we

should

see

the Dirac’sideas, which

are

mostly foundinhis textbook [2].

2010Mathematics SubjectClassification(s): $60H40$. Key Words: Wluite Noise Theory.

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. trajectoriesof

a

particle, and

each

trajectory

may

be viewed

as a

sum

ofthe classical

one

andfluctuation. Theclassical path,

denotedby$y$is, of

course

uniquely determinedby the Lagrangian (withboundaryvalues).

Theso-called possibletrajectories$x$inquantumdynamics

can

be expressedinthe form

$x=y+z$,

where$z$isthefluctuation(see [4]).

Our first question istodeterminethefluctuation$z$

.

We

propose

that$z$is

a

GaussianMarkov

process,

more

preciselya Brownianbridge whichis

a

linerfunctionof

a

Brownianmotion$B(t)$

.

We shall explain the

reason

why

a

Brownianbridge is fitting for

a

realizationoffluctuation in

thenextsection.

ii)_{To observe Feynman‘s expressionof propagator,}

_we

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

The integrandinvolves

a

term$B(s)^{2}$ to

express

thekinetic

energy.

The$B(s)^{2}$ is not

an

ordinary

randomfunction. To

come

tothe propagator

we

have

even

toexponentiatethe 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 ofwhitenoise.

iii) Integration

over

the function

space

$X=\{x\}$, the space of trajectories, can be defined smoothly. To thisend,

we

must specify

a measure on

$X$

:

this is

now

obvious, since

we

take

a

whitenloise,the probabilitydistribution$\mu$has automatically been introduced.

Thereis

one

problem tobe reminded. We expect the integral should be done with respectto

the uniform

measure

on

$X$

.

This problem

can

be solved alsoin \S 3.

iv)With these background

we

can

come

tothe actualcomputation of the propagators. This

can

bedone byusingthe white noise theory.

v$)$We

can

recognizethat

our

approach

can

beappliedto

a

prettylarge class of Lagrangians.

vi)Our method ofintegration

on

function

space

can

furtherbe appliedtoother problemsin

physics

as we

shall

see

inthe lasttwosections.

\S 2. BrownianBridge and

a

Setup of the Propagator

First

we

haveto explain why the Brownian bridgeis involvedinthe class ofquantum

me-chamical 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 theformula for thetransition

takento be $[0, T]$

.

Now the term $z$ that

expresses

the

quantity

of fluctuation

can

be

a

Markov

process

$X(t),0\leq t\leq T$

.

Furtherassumptions

on

$X(t)$

can

be made

as

follows.

1$)$ $X(t)$is

a

Gaussian

process,

since it is

a

sortofnoise [8].

2

$)$ As

a

usual requirement, the

Gaussian

process

satisfies $E(X(t))=0$ and has the canonical

representation byBrownianmotion,namely

$X(t)= \int_{0}^{t}F(t,u)B(u)du$

and$X(O)=X(T)=0$(bridged).

3

$)$ $X(t)$is

a

Gaussian l-ple Markov

process.

4$)$ Thenormalized

process

$Y(t)$enjoystheprojectiveinvariance under time-change.

Theorem 1. The Brownian bridge$X(t)$overthe time interval$[0, T]$ is

characterized

bythe

aboveconditions $1$)$-4)$

.

This theorem

we

have proved before and the proof is omitted here. We only note that the

canonical representationof$X(t)$isgivenby

$X(t)=(T-t) \int_{0}^{t}\frac{1}{T-u}B(u)du$,

andthe covariance$\Gamma(t,s)$is

$\Gamma(t,s)=\sqrt{\frac{s(T-t)}{t(T-s)}}$, $s\leq t$

.

Namely,

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

where$(\cdot,$

$\cdot;\cdot,$$\cdot)$ istheanharmonic 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, atthattime,

some

idea in

mind for the

use

of

a

Brownianbridge, and

we

hadpractically

many

good examplesof integrand

withvarious kinds ofpotentials, and satisfactoryresults have been obtained.

With this background

we

are

ready to

propose

how toform quantum mechanical

propaga-tors. The possiblequantummechanical 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 $z$in the earlier

Remind that the classicaltrajectory $t\in[0, T]$,is uniquely determined by the variational

principle for theaction

$A[x]= \int_{0}^{T}L(x,\dot{x})dt$,

where the Lagrangian$L(x,\dot{x})$inquestionis assumedtobeofthe 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 the}

case

where $V$ hascertain singularity,

even

time-dependent.

The actualexpressionandcomputationsof thepropagator

are

givensuccessively

as

follows:

We follow the Lagrangian dynamics. The possible trajectories

are

sample paths $y(s),s\in$

$[0,t]$,expressedin the form

(1) $y(s)=x(s)+\sqrt{\frac{\hslash}{m}}B(s)$_,

where the $B(t)$ is

an

ordinary Brownianmotion. Hence theaction $S$isexpressedin the formin

termsof quantumtrajectory$y$

:

$A= \int_{0}^{t}L(y(s),\dot{y}(s))ds$.

Note that theeffect offorming abridgeis givenbyputtingthedelta-function$\delta_{0}(y(t)-y_{2})$as a

factor oftheintegrand, where$y_{2}=x(t)$.

Nowweset

(2) $S(t_{0},t_{1})= \int_{t_{0}}^{t_{1}}L(t)dt$.

andset

$\exp[\frac{i}{\hslash}\int_{t_{0}}^{t_{1}}L(t)dt]=\exp[\frac{i}{\hslash}S(t_{0},t_{1})]=B(t_{0},t_{1})$

.

Then,

we

have(seeDirac [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 mechanicalpropagator $G(O,t;y_{1},y_{2})$ 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 $\{\}$ isunderstoodtobe the integral with respecttothe white noise

measure

$\mu$

.

\S 3. Generalized

White

Noise Functionals Revisited

It

is

well-known that there

are

two classes of generalized white

noise

functionals; $(L^{2})^{-}$ and $(S)^{*}$

.

We

use

them without discrimination except it is

necessary

to choose

one

ofthem

specifically.

It

seems

bettertoexplain the conceptof”renormalization” whichmakes formal but

impor-tant functionals of the $B(t)$’s _tobe acceptable

as

generalized white noise functionals. To

save

time

we

refer theinterpretationtotheliteratures [11] and [16].

Weshouldnotethat there

are

generalizedwhitenoise functionals involved intheexpectation

in Theorem 2. For instance, there is involved the delta function, in fact the Donsker’s delta

function $\delta_{o}(y(t)-y_{2})$, which is

a

generalized white noise functional. There is used

a

Gauss

kemel of theform$\exp[c\int_{0}^{t}B(s)^{2}ds]$,theideal

case

is$c=- \frac{1}{2}$

.

Ingeneral, if$c \neq\frac{1}{2}$,then it

can

be

a

generalized functional after having the multiplicative renormalization. Now

we

have the exceptional case,but it

can

beaccepted bycombining with other factorof

an

exponential; this

is justthe

case.

In reality,

we

combineitwith thetermthat

comes

fromthekinetic

energy.

The

factor$\exp[\frac{1}{2}\int_{0}^{t}B(s)^{2}ds]$

serves

as

the flattening effect of the white noise

measure.

One

may

askwhythe functionaldoes

so.

An

intuitive

answer

tothis questionis

as

follows: If

we

write

a

Lebesgue

measure

(exists only virtually)

on

$E^{*}$ by $dL$_, the white noise

measure

$\mu$

may

be

expressed in the form$\exp[-\frac{1}{2}\int_{0}^{t}B(s)^{2}ds]dL$

.

Hence, the the factor inquestionisput tomake

the

measure

$\mu$tobe

a

flat

measure

$dL$

.

Infact,this makes

sense

eventually.

Retuming to the

average

(3) (in Theorem2), which is

an

integral with respect to the white

noise

measure

$\mu$,itis importantto notethattheintegrand (i.e.the inside of the angularbracket) is integrable, inotherwords,it is

a

bilinear form of

a

generalizedfunctional and

a

testfunctional.

There have tofollow shortnotes to bereminded. They

are

rather cmcial. The formula(3) involves

a

productoffunctionals of theformlike$\varphi(x)\cdot\delta(\langle x,f\}-a),$$f\in L^{2}(R),$$a\in$ C. Togive

a

correct interpretationtotheexpectationof(3)with this choice,it should be checked that it

can

be regarded

as a

bilinear form of

a

pair of

a

test functional and

a

generalized functional. The

followingassertion

answers

tothis

question.

Theorem3(Streitetal [19]). Let$\varphi(x)$beageneralizedwhite

noisefunctional.

Assumethat the$\mathcal{T}$

-tmnsform

$(\mathcal{T}\varphi)(\xi),\xi\in E$,

of

$\varphi$ isextendedto

afunctional of

$f$in $L^{2}(R)$, inparticulara

fiunction of

$\xi+\lambda f$, and that $(\mathcal{T}\varphi)(\xi-\lambda f)$ is an integrable

fiunction of

$\lambda$

for

any

fixed

$\xi$

. If

the

_transform

$(\mathcal{T}\varphi)(\xi-\lambda f)$ is

a

U-fiunctional, then thepointwiseproduct $\varphi(x)\cdot\delta(\langle x,f\rangle-a)$ is

defined

and isageneralizedwhite noise

_functional.

Proof.

First

a

formula for the$\delta$-functionis provided.

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 product $\varphi(x)\cdot\delta(\{x,f\rangle-a)$

makes

sense

andit is

a

generalized whitenoisefunctional. $\square$

Example 1. Theabove theorem

can

be appliedto

a

Gauss kemel$\varphi_{c}(x)=N\exp[c\int x(t)^{2}dt]$,

with$c \neq\frac{1}{2}$

.

i$)$The

case

where

$c$is real and$c<0$

.

Wehave

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

$= \exp[\frac{c}{1-2c}(\Vert\xi\Vert 2-2$ノ$t(\xi,f)+\Lambda^{2}\Vert f\Vert^{2}])$

.

Thisis

an

integrable functionof real$\lambda$. Hence, bythe above Theorem3,

we

have

a

generalized whitenoise functional.

ii)The

case

where$c= \frac{1}{2}+ia,$_{$a\in R-\{0\}$}

.

The

same

expression

as

ini)is given.

Example

2.

In the following case, exactvalues of thepropagators

can

beobtained and, of course, they

agree

with the known results.

i$)$Free particle

ii)Harmonicoscillator.

iii) _{Potentials which}

_are

_{Fourier transforms of}

_measures

(thethe Albeverio-Hohkron

poten-tial).

\S 4. Someof Further Developments and Related Topics

[I] In addition to Example 2,

we

have

some

more

interesting potentials,including those

are

muchsingular andtimedepending. There

are

satisfactory results inthe recentdevelopments. Example

3.

Streitetal have obtained explicit formulae inthe following

cases:

1$)$

a

timedepending Lagrangian of the form

$L(x(t), \dot{x}(t),t)=\frac{1}{2}m(t)\dot{x}(t)^{2}-k(t)^{2}x(t)^{2}-\dot{f}(t)x(t)$,

2

$)$A singular potential $V(x)$of the form

$V(x)= \sum_{n}c^{-n^{2}}\delta_{n}(x)$, $c>0$,

andothers.

$[n]$ The Hopf equation.

There

are

manyapproaches tothe Navier-Stokesequation.

$u_{(\chi,f}+u_{\beta^{u}(t,\rho=-p\cdot\alpha+\mu u_{\iota r\beta\beta}}$,

where $\alpha,\beta=1,2,3$, and wherethefollowing 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 $\Phi$of the

measure

$P(du)$_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 $\Phi(\xi, ,t)$ satisfies the following functional

dif-ferential equation,called Hopfequation:

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

Studying this approach,

we may

think of the two matters. One is

a

similarity tothe 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) $u$

.

The expectation is taken with respect to the probability

measure

introduced

on

the path

space.

As the secondpoint,

one may

think ofequations for $\Phi_{n},n\geq 0$, that

come

from the Hopf

equation and the Fock

space

expansion ofgeneralized whitenoise functionals. Inthis

case

we

expect that the calculus

can

be done in

a

similar

manner

to the white noise calculus. We

may

\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 quadraticform

$\int f(t):B(t)^{2}:dt+\int\int F(u,v);B(u)B(v);dudv$

whichis _{called normalfunctional, the firsttermiscalled the singularpart and the second term} isthe regularpart. The two terms

can

becharacterized from

our

viewpointand play significant

roles,_{respectively. Remind the role of singularpartin the path integral.}

(2)_{Our method of path integrals enables}

_us

_{todealwith the}

_case

_of

_very

_{irregular potentials}

tohave thepropagator,byL. Streitandothers.

(3) Some other approaches: Itis significant to

see

the results by C. C. Bemido and M. V.

Carpio Bemido [1]. _They

are

_using

_our

_{method of pathintegral toinvestigatethe entanglement}

probabilities oftwo chain like macro-molecules where

one

polymer lies

on a

plane and the other perpendicularto it. The entanglement probabilities

are

calculated and the result shows

a

characteristic ofthepolymer.

We also should liketo note that Masujima [14] has published a beautiful monograph

col-lecting various approachestopathintegrals,

Acknowlegements The authoris grateful to _{Professor N. Kumano-go, the workshop}

orga-nizer, who hasgiven theauthor theopportunity todeliver

a

talk

on

pathintehrals.

References

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_of

QuantumMechanics,4th ed. OxfordUniv. Press. 1958.

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20(1948)_367-387.

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a

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Pro-cessesandApplications16$(1983),55-69$.

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[19] Streit,L. etal,Feynmanintegra]sfor Aclassofexponentiallygrowing potentials, J. Math. phys.39