3 Statistical conditions

We study a simple system, a particle in an externally controlled time-independent potential V  (x  ) , whose motion is restricted to a single spatial dimension (coordinate x ). We use the canonical formalism of classical mechanics to describe this system. Thus, the fundamental observables of our theory are x (t ) and p(t ) and they obey the differential equations
 d              p (t)      d
----x (t ) =    ------,    ---p (t ) =   F  (x  (t )),

dt               m         dt

where                 dV  (x )
F  (x )  =   -  ---------
                   dx . 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 x (t ),  p (t ) and the force field F  (x  (t) ) by averages ---  --
x,   p and ----

F , and obtain

d-----      -p--
   x   =                                           (2)
dt          m
    p  =    F  (x  ),                              (3)
The averages in (2),(3) are mean values of the random variables x or p ; there is no danger of confusion here, because the symbols x (t ) and p (t ) will not be used any more. In (1) only terms occur, which depend either on the coordinate or the momentum, but not on both. Thus, to form the averages we need two probability densities ρ (x,   t) and w  (p,  t) , depending on the spatial coordinate x and the momentum p separately. Then, the averages occurring in (2),(3) are given by
              ∫   ∞
      x   =            dx   ρ (x,  t)x                             (4)

              ∫  - ∞
      --          ∞

      p   =            dpw    (p,  t )p                            (5)
                 - ∞
---------         ∫   ∞
                                          dV   (x )
F  (x )   =   -            dx  ρ (x,   t) ----------.              (6)
                    -  ∞
Note that F  (x  ) has to be replaced by ---------
F  (x  ) and not by     ---
F  (x ) . The probability densities ρ and w are positive semidefinite and normalized to unity. They are time-dependent because they describe the dynamic behavior of this theory.

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 ρ and w 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 S (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
∂  ρ (x,  t)       ∂   ρ (x,  t )∂  S (x,   t)
-------------     ----------------------------
              +                                 =   0.
     ∂ t          ∂ x     m           ∂ x

The derivative of S  (x,  t) defines a field with dimension of a momentum,
                ∂ S  (x,   t)
p (x,   t)  =                 .
                     ∂ x

Eq. (8) defines a unique number p (x,   t) for each value of the random variable x . In the next section we will discuss the following question: Are we allowed to identify the possible values of the random variable p occurring in Eq. (5) with the values of the momentum field p (x,   t ) ?