We study a simple system, a particle in an externally controlled time-independent potential , whose
motion is restricted to a single spatial dimension (coordinate
). We use the canonical formalism of classical
mechanics to describe this system. Thus, the fundamental observables of our theory are
and
and they obey the differential equations
![]() | (1) |
where . We now create statistical conditions, associated with the
type 1 theory (1), according to the method outlined in the last section. We replace the observables
and the force field
by averages
and
, and obtain
Relations (2),(3), with the definitions (4)-(6) are, to the best of my knowledge, new. They will be referred to
as “statistical conditions”. There is obviously a formal similarity of (2),(3) with Ehrenfest’s relations of quantum
mechanics, but the differential equations to be fulfilled by and
are still unknown and may well differ
from those of quantum theory. Relations (2)-(6) represent general conditions for theories which are deterministic
only with respect to statistical averages of observables and not with respect to single events. They cannot be
associated to either the classical or the quantum mechanical domain of physics. Many concrete statistical
theories (differential equations for the probability distributions) obeying these conditions may exist (see the next
section).
These conditions should be supplemented by a local conservation law of probability. Assuming that the
probability current is proportional to the gradient of a function (this is the simplest possible choice and the
one realized in Hamilton-Jacobi theory, see also section 11) this conservation law is for our one-dimensional
situation given by the continuity equation
![]() | (7) |
The derivative of defines a field with dimension of a momentum,
![]() | (8) |
Eq. (8) defines a unique number for each value of the random variable
. In the next
section we will discuss the following question: Are we allowed to identify the possible values of the
random variable
occurring in Eq. (5) with the values of the momentum field
?