数理解析研究所講究録 1227 巻 2001 年 140-144

140
Random Fields: Theory and Applications to Quantum Fields
TAKEYUKI HIDAMEIJO UNIVERSITY\S 1. Introduction
We shall discuss the analysis of random complex systems which have close connections
with quantum dynamics. In particular, we analyse stochastic processes $X(t)$ and random
fields $X(C)$ , in asystematic manner. Actually, our aim is to study those systems by using
white noise analysis.
The idea of the analysis is that we first provide abasic and standard system of randomvariables and to express the given system as afunction of the system provided in adavance.
Naturally follows the analysis of the function. The system of variables where we start
involves idealized elemental random variables (abbr. i.e.r.v.). To take such asystem is inline with theReductionism.
This thought seems to be similar to the atomism in physics. We may refer to the lecture. Anderson at University of Tokyo in 1999. The title of his lecture included
given byEmergence together with Reductionism.
The next step is to form afunction of the elemental elements obtained by the reduction;
namely$\mathrm{P}.\mathrm{W}$Synthesis.
The goal has to be the analysis of the function (may be called functional) to identify therandom complex system in question.
The first step of taking suitable system of i.e.r.v.’s has been influenced by the way
how to understand the notion of astochastic process. We therefore have aquick reviewof the definition of astochastic process starting from the idea of J. Bernoulli and L\’evy on
the definition of astochastic process, where we are suggested to consider the innovationof astochastic process. It is viewed as asystem of i.e.r.v. ’s, which will be specified to beawhite noise.
The analysis of white noise functional has many significant characteristics which arefitting for investigation of quantum mechnical phenomena. Thus, we shall be able to show
examples to which white noise theory is efficintly applied.
$\mathrm{s}$\S 2. Review of defining astochastic process and white noise analysis
There is atraditional, and in fact original way of defining astochastic process. Let
us refer to L\’evy’s definition of astochastic process given in his book [3] Chapt. . “unefonction al\’eatoire $X(t)$ du temps dans lequel le hasard intervient \‘a chaque instant” . The
which is independent of thehasard is expressed as an infinitesimal random variable$\zeta$$\mathrm{I}\mathrm{I}$$t$$\mathrm{Y}(t)$141observed values of $X(s)$ , $s\leq t$ , in the past. The random variable
is nothing but theinnovation of the process $X(t)$ .
Formally speaking the, which is usually an infiniotesimal random variable, contains the information that was gained by the $X(t)$ during the time interval $[t, t+dt)$ .
It would be fine if the given process is expressd as afunctional ofin the following$\mathrm{Y}(t)$$\mathrm{Y}(t)$$\mathrm{Y}(t)$manner:$X(t)=\Psi(\mathrm{Y}(s), s\leq t, t)$,where is asure (non random) function. Such atrick may be called the reductionism.
The expression is causal in the sense that the $X(t)$ is expressed as afunction of, $s\leq t$ ,and never useswith $s>t$ .
The collection
is asystem of i.e.r.v.’s so that the above expression is arealization of the synthesis. We are particularly interested in the case where the system ofi.e.r.v. ’s is taken to be awhite noise. We are now ready to discuss white noise analysis.
First we note that the white noise analysis has many advantages.
$\Psi$$\mathrm{Y}(s)$$\mathrm{Y}(s)$$\{\mathrm{Y}(s)\}$1) It is an infinite dimensional analysis. Actually, our stochastic analysis can besystematically done by taking awhite noise as asytem of i.e.r.v.’s to express the given
random complex systems. Indeed, the analysis is essentially infinite dimensional as wili
be seen in what follows.
2) Rotation group. The white noise measure supported by the spaceof generalized
functions on the parameter spaceis invariant under the rotations of. Hence a
harmonic analysis arising from the group will naturally be discussed. The group contains
significant subgroups which describes essentially infinite dimensional characters.
$E^{*}$$R^{d}$$E^{*}$

3) Random fields $X(C)$ parametrized by $C$ is discussed in the similar manner to $X(t)$so far as innovation is concerned. For concrete discussion, we assume that $C$ is aclosedsmooth convex manifold like acontour or asurface. Needless to say, $X(C)$ enjoys more
profound characteristic properties.

4) The s0-called -transform applied to white noise functionals provides abridgeconnecting white noise functionals and classical functionals of ordinary functions. We
can therefore appeal to the classical theory of functionals established in the first halfof the twentieth century. Differential and integral calculus of white noise functionals,
often generalized functionals, harmonic analysis including Fourie analysis, Laplacians,complexification and other theories are refered to the monograph [13] and others.
$S$\S 3. Relations to Quantum Dynamics
We now explain briefiy some topics in quantum dynamics to which white noise theory
can be applied.
1) Representation of the canonical commutation relations for Boson field. This topic
denote the-derivative. Then it
be awhite noise and let
is well known. Let$\dot{B}(t)$$\partial_{t}$$\dot{B}(t)$142
is an annihilation operator and its dual operator
the commutation relations$\partial_{t}^{*}$

stands for the creation. They satisfy$[\partial_{t}, \partial_{s}]=[\partial_{t}^{*}, \partial_{s}^{*}]=0$$[\partial_{t}, \partial_{s}^{*}]=\delta(t-s)$,.

From these, arepresentation of the canonical commutation relations hold for Bosonicparticle.
2) Reflection positivity ( -positivity). Astationary multiple Markov (say, TV-pleMarkov) Gaussian process has aspetral density functionof particular type. Namely,$\mathrm{T}$$f(\lambda)$$f( \lambda)=\sum_{1}^{N}\frac{c_{k}}{\lambda^{2}+a_{k}^{2}}$.

On the other hand, it is proved thatProposition. The covariance function
is expressed in the form$\gamma(h)$of astationary -positive Gaussian process$\mathrm{T}$$\gamma(h)=\int_{0}^{\infty}\exp[-|h|x]dv(x)$where$v$,

is apositive finite measure.

By applying this assertion to the -ple Markov Gaussian process we claim that Tpositivity requires $c_{k}>0$ for every . Note that in the strictly -ple Markov case thiscondition is not satisfied. It is our hope that this result would be generalized to the casesof general stochastic processes of multiple Markov properties.
$\mathrm{N}$$k$$\mathrm{N}$3) Apath integral formulation. One of the realization of Dirac-Feynman ’s idea ofthe path integral may be given by the following method using generalized white noisefunctionals. First we establish aclass of possible trajectories when aLagrangian $L(x,\dot{X})$is given. Let be the classical trajectory determined by the Lagrangian. As soon as wecome to quantum dynamics we have to consider fluctuating paths . We propose thatthey are given by
$x$

$y$

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

The average over the paths is replaced with the expectation with respect to the probability
ismeasure for which Brownian motion $B(t)$ is defined. Thus, the propagator
given by.
$G(y_{1}, y_{2}, t)$$E \{N\exp[\frac{i}{\hslash}\int_{0}^{t}L(y,\dot{y})ds+\frac{1}{2}\int_{0}^{t}\dot{B}(s)^{2}ds]\delta(y(t)-y_{2})\}$

With this setup, actual computations have been done to get exact formulae of the propagators. (L. Streit et al.)4) Dirichlet forms in infinite dimensions. With the help of positive grneralized white
noise functionals we prove criteria for closability of energy forms

1435) Random fields $X(C)$ . We assume that $X(C)$ has acausal representation in termsof white noise.
5.1) Markov property and multiple Markov properties. We are suggested by Dirac’s
paper [1] to define Markov property. For Gaussian case we are given reasonable definition(see [14]) by using the canonical representation in terms of white noise. Some attempthave been made for some non Gaussian fields. It is an interesting question to fineconditions related to multiple Markov properties.
$\mathrm{s}$5.2) Stochastic variational equations of Langevin type. Let $C$ runs through aclassof concentric circles. The equation is$\delta X(C)=-\lambda X(C)\int_{C}\delta n(s)ds+X_{o}\int_{C}v(s)\partial_{s}^{*}\delta n(s)ds$

The explicit solution is given by using thetionals.

$S$$\mathrm{C}$.

-transform and the classical theory of func-5.3) We have made an attempt to define arandom field $X(C)$ ,conformal invariance. Reversibility can also be discussed.

$C\in \mathrm{C}$which satisfies

Q4. Concluding remarks

Some of future directions are proposed.
1. One is concerned with good applications of the L\’evy Laplacian. Its significance is that
it is an operator that is essentially infinite dimensional.

2. Atwo dimensional Brownian path is considered to have some optimality in occupying
the territory. This property should reflect to the construction of amodel of physicalphenomena.

3. Systematic approach to invariance of random fields under transformation group will
be discussed. The reversibility of arandom field discussed in this line would suggest ageneralization of the path integral method discussed in 3) of \S 3.
Acknowledgement. The author is grateful to the organizer Professor N. Obata who hasgiven the author an opportunity to deliver alecture on this meeting at RIMS.

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[16] L. Accardi and Si Si, Innovation approach to multiple Markov properties of some nonGaussian fields. to appear