N. A. CHERNIKOV

A GENERALIZED PROBLEM ON THE STOCHASTIC MOTION OF A PARTICLE

(Presented by Academician V. A. Fock on 29 X 1956)

The usual formulation of the problem of the stochastic motion of a particle, in which a special role is assigned to time \(t\), is poorly adapted to the theory of relativity and, in particular, to the theory of gravitation*. The natural generalization of the usual problem proposed here removes this deficiency and gives a canonical formulation of problems analogous to the problem of the probability of collision of two particles.

In contrast to the usual case, by the state of a material point we shall mean the totality of its space-time position and velocity. We shall assume that the space \(F\) of states of a material point is a simple differentiable manifold. For example, it is such if the space of events is Galilean. The dimension of the space \(F\) is equal to 7; however, everywhere where only the fact that it is finite is important, we shall denote it by \(n\), keeping possible generalizations in mind.

The motion of a material point in a given external field of forces is specified by a system of equations:

\[ \frac{d x_i}{d \tau}=f_i(x_1,\ldots,x_n), \qquad i=1,2,\ldots,n, \tag{1} \]

where \(x_1,\ldots,x_n\) are coordinates in \(F\). The collection of functions \(\{f_1(x_1,\ldots,x_n),\ldots,f_n(x_1,\ldots,x_n)\}\) forms in \(F\) a vector field \(f(P)\), \(P\in F\). The entire manifold of possible motions of the material point for the given field \(f(P)\) is represented by the family of vector lines of this field. On each such line a distance \(\tau=\tau(P_1,P_2)\) is specified between any two of its points \(P_1\) and \(P_2\), equal to the proper time of the material point which is necessary for its state \(P_1\) to change into the state \(P_2\), with \(\tau(P_1,P_2)=-\tau(P_2,P_1)\). The problem of mechanics is posed as follows: given a field \(f(P)\) and the initial state \(P_0\) of the material point, find its state \(P=\varphi(\tau;P_0)\) after the lapse of proper time \(\tau\). This problem is equivalent to finding the solution of the system (1) with the initial conditions \(x_i|_{\tau=0}=x_i^0,\ i=1,\ldots,n\).

We shall investigate the motion of a particle of a definite kind. The particle is formed in some definite state \(P_1\in F\) and during the time of existence \(\tau>0\) “lives through” the set of states \(P=\varphi(\eta;P_1)\), \(0<\eta<\tau\), after which it decays in the state \(P_2=\varphi(\tau;P_1)\). This set of states we shall call the life curve of the particle. We shall assume that the particle, starting from an arbitrary state \(P\), decays with probability

\[ \frac{\tau}{T(p)}+o(\tau)\left(\lim_{\tau\to 0}\frac{o(\tau)}{\tau}=0\right) \]

in one of the states

\[ P'=\varphi(\eta;P), \qquad 0<\eta<\tau. \]

Let there be given a probability \(W(D)\) of formation of the particle in a state from the region \(D\subset F\), which is determined by a skew-symmetric \(n\)-linear—

* We adhere to the terminology of V. A. Fock \((^{1})\).

form \(^{(2)} Q(P;\xi_1,\ldots,\xi_n)\) according to the formula
\[ W(D)=\int_D\!\!\int\cdots\int Q(P;d_1,\ldots,d_n). \]
The substantial coefficient of this form is, generally speaking, a generalized function of the coordinates in \(F\). Let \(S\) be a piecewise-smooth hypersurface in \(F\) such that its common part with any vector line of the field \(f(P)\) is either empty, or consists of a single point, or of a single segment.

The probability \(w(S)\) that the life curve of the particle intersects the hypersurface \(S\) is expressed by the integral
\[ w(S)=\int_S\cdots\int \Omega_0(P;d_1,\ldots,d_{n-1}) \]
over \(S\) of the \((n-1)\)-linear skew form
\[ \Omega_0(P;\xi_1,\ldots,\xi_{n-1})=A_0(P;f(P),\xi_1,\ldots,\xi_{n-1}). \]
The substantial coefficient of the \(n\)-linear skew form \(A_0(P;\xi_1,\ldots,\xi_n)\) is equal to the curvilinear integral
\[ A^0_{12\ldots n}(x_1,\ldots,x_n) = \int_0^\infty Q_{12\ldots n}(x'_1,\ldots,x'_n) \frac{\partial(x'_1,\ldots,x'_n)}{\partial(x_1,\ldots,x_n)} \exp\left[-\int_0^\tau \frac{d\eta}{T(\varphi(-\eta;P))}\right]d\tau, \tag{2} \]
where \(x_1,\ldots,x_n\) and \(x'_1,\ldots,x'_n\) are the coordinates of the points \(P\) and \(\varphi(-\tau;P)\).

Having decayed, the particle may be formed again. Let the conditional probability of restoring the particle to a state from the region \(D\subset F\), provided that it has decayed in the state \(P^*\), be equal to the integral
\[ \int_D\!\!\int\cdots\int K(P;d_1,\ldots,d_n;P^*). \]
The skew form \(K(P;\xi_1,\ldots,\xi_n;P^*)\) is a scalar function of the point-parameter \(P^*\). The particle may decay and be restored an unlimited number of times. If the process of decay and restoration does not change the quantities \(f(P)\), \(T(P)\), and \(K(P;\xi_1,\ldots,\xi_n;P^*)\), then the probability of \(k\)-fold restoration is determined by the skew form
\[ Q_k(P;\xi_1,\ldots,\xi_n) = \int_F\!\!\int\cdots\int \frac{K(P;\xi_1,\ldots,\xi_n;P^*)}{T(P^*)} A_{k-1}(P^*;d^*_1,\ldots,d^*_n), \tag{3} \]
where the skew form \(A_{k-1}(P^*;d^*_1,\ldots,d^*_n)\) determines the probability that the life curve of a particle restored \(k-1\) times intersects the hypersurface \(S\). The form \(A_{k-1}(P;\xi_1,\ldots,\xi_n)\) depends on \(Q_{k-1}(P';\xi'_1,\ldots,\xi'_n)\) in the same way as \(A_0(P;\xi_1,\ldots,\xi_n)\) depends on \(Q(P';\xi'_1,\ldots,\xi'_n)\). Hence, from (2) it follows that the skew form
\[ A(P;\xi_1,\ldots,\xi_n;\lambda)=\sum_{k=0}^{\infty}\lambda^k A_k(P;\xi_1,\ldots,\xi_n) \]
satisfies the kinetic equation:
\[ \Omega'(P;\xi_1,\ldots,\xi_n) + \frac{A(P;\xi_1,\ldots,\xi_n)}{T(P)} = \lambda \int_F\!\!\int\cdots\int \frac{K(P;\xi_1,\ldots,\xi_n;P^*)A(P^*;d^*_1,\ldots,d^*_n)}{T(P^*)} + Q(P;\xi_1,\ldots,\xi_n) \tag{4} \]
with the condition
\[ \lim_{\tau\to\infty}\Omega(\varphi(-\tau;P);\xi_1,\ldots,\xi_{n-1};\lambda)=0, \]
\(\Omega'(P;\xi,\xi_1,\ldots,\xi_{n-1};\lambda)\) being the exterior derivative of the \((n-1)\)-linear skew form
\[ \Omega(P;\xi_1,\ldots,\xi_{n-1};\lambda) = A(P;f(P),\xi_1,\ldots,\xi_{n-1};\lambda). \]
The substantial coefficient of the derivative form is
\[ \Omega'_{12\ldots n}(x_1,\ldots,x_n) = \sum_{i=1}^n \frac{\partial}{\partial x_i} \bigl[A_{12\ldots n}(x_1,\ldots,x_n)f_i(x_1,\ldots,x_n)\bigr]. \]

The probability of intersection with the hypersurface \(S\) by the life curve of the particle, without taking into account how many times the particle decayed and was restored, is determined by the formula
\[ \Omega(P;\xi_1,\ldots,\xi_{n-1}) = \sum_{k=0}^{\infty}\Omega_k(P;\xi_1,\ldots,\xi_{n-1}) = \Omega(P;\xi_1,\ldots,\xi_{n-1};1). \]
The usual formulation of the problem is obtained from the one set forth if \(S\) is part of the hypersurface \(t=\mathrm{const}\). In the formulation of the problem set forth, gravitation is easily taken into account; however, its advantages over the usual one are obvious

and in the theory of relativity, since the hypersurface \(t=\mathrm{const}\) is not invariant with respect to Lorentz transformations. In the latter case, as coordinates in \(F\) one may choose the Galilean coordinates \(x,y,z,t\) of the particle and the coordinates \(u_1,u_2,u_3\) of its velocity, equal to the ratios of the spatial components of its momentum to the rest mass. As the volume element in \(F\) it is convenient to take the skew form \(\varepsilon(P;d_1,\ldots,d_7)\), which on the vectors of elementary displacements along the coordinate lines takes the value

\[ \varepsilon(P;d_1,\ldots,d_7)=dx\,dy\,dz\,dt\,\frac{du_1du_2du_3}{u_4}, \qquad u_4=\sqrt{1+\frac{u_1^2+u_2^2+u_3^2}{c^2}}, \]

and as the area element on the hypersurface \(S\), the 6-linear skew form \(\varepsilon(P;f(P),d_1,\ldots,d_6)\).

Since
\[ f(P)=\{u_1,u_2,u_3,u_4,\omega_1(P),\omega_2(P),\omega_3(P)\}, \]
the area element on the hypersurface \(t=\mathrm{const}\) is equal to
\[ \varepsilon(P;f(P),d_1,\ldots,d_6)=dx\,dy\,dz\,du_1\,du_2\,du_3. \]
Any 7-linear skew form \(B(P;\xi_1,\ldots,\xi_7)\) differs from \(\varepsilon(P;\xi_1,\ldots,\xi_7)\) only by a scalar factor:
\[ B(P;\xi_1,\ldots,\xi_7)=B(P)\varepsilon(P;\xi_1,\ldots,\xi_7). \]
The scalar function \(Q(P)\) obtained from this formula is the probability density that, in the state \(P\), the particle under consideration is formed. The scalar function \(A_k(P)\) on any hypersurface \(S\) is the probability density for the intersection of the world line of the particle, reconstructed \(k\) times, with this hypersurface (at the point \(P\)). Passing from skew forms to scalar functions, we obtain the kinetic equation in the theory of relativity in the form:

\[ u_4\frac{\partial A(P)}{\partial t} +u_1\frac{\partial A(P)}{\partial x} +u_2\frac{\partial A(P)}{\partial y} +u_3\frac{\partial A(P)}{\partial z} + \]

\[ +u_4\sum_{l=1}^{3}\frac{\partial}{\partial u_l} \left[\frac{\omega_l(P)A(P)}{u_4}\right] +\frac{A(P)}{T(P)} = \]

\[ =\lambda\int_{-\infty}^{+\infty}\!\!\int\cdots\int \frac{A(P^*)K(P;P^*)}{T(P^*)}\, dx^*\,dy^*\,dz^*\,dt^*\, \frac{du_1^*du_2^*du_3^*}{u_4^*} +Q(P) \tag{5} \]

with the condition
\[ \lim_{\tau\to+\infty} A(\varphi(-\tau;P))=0. \]

Usually
\[ T^{-1}(P^*)K(P;P^*)= L(P;a^*)\delta(x-x^*)\delta(y-y^*)\delta(z-z^*)\delta(t-t^*), \]
where \(a^*\) is the velocity of the particle in the state \(P^*\). The invariance with respect to Lorentz transformations under this approach to the problem is obvious.

The author is deeply grateful to Acad. V. A. Fock for valuable comments.

Laboratory of Theoretical Physics
Joint Institute for Nuclear Research

Received
13 X 1956

REFERENCES

1. V. A. Fock, The Theory of Space, Time and Gravitation, 1955.
2. P. K. Rashevskii, Geometric Theory of Partial Differential Equations, 1947.