2 Basic equations

In I three different types of theories have been defined which differ from each other with regard to the role of probability. The dogma underlying theories of type 1 is determinism with regard to single events; probability does not play any role. If nature behaves according to this dogma, then measurements on identically prepared individual systems yield identical results. Classical mechanics is obviously such a deterministic type 1 theory. We shall use below (as a ’template’ for the dynamics of our statistical theories) the following version of Newton’s law, where the particle momentum pk  (t) plays the role of a second dynamic variable besides the spatial coordinate xk  (t) :
 d               p   (t )      d
---x    (t)  =   --k-----,    ----p  (t )  =   F   (x,   p,  t ).
      k                             k            k
dt                 m          dt

In classical mechanics there is no restriction as regards the admissible forces. Thus, Fk is an arbitrary function of x   ,  p  ,  t
  k     k ; it is, in particular, not required that it be derivable from a potential.

Experimental data from atomic systems indicate that nature does not behave according to this single-event deterministic dogma. A simple but somewhat unfamiliar idea is, to construct a theory which is deterministic only in a statistical sense. This means that measurements on identically prepared individual systems do not yield identical results (no determinism with regard to single events) but repeated measurements on ensembles [consisting each time of a large (infinite) number of measurements on individual systems] yield identical results. In this case we have determinism with regard to ensembles (expectation values, or probabilities).

Note that such a theory is still far from chaotic even if our macroscopic anticipation of (single-event) determinism is not satisfied. Note also that there is no reason to assume that such a statistical theory for microscopic events is incompatible with macroscopic determinism. It is a frequently observed (but not completely understood) phenomenon in nature that systems with many (microscopic) degrees of freedom can be described by a much smaller number of variables. During this process of elimination of variables the details of the corresponding microscopic theory for the individual constituents are generally lost. In other words, there is no reason to assume that a fundamental statistical law for individual atoms and a deterministic law for a piece of matter consisting of, say,     23
10 atoms should not be compatible with each other. This way of characterizing a relation between two different physical theories differs from the conventional reductionistic point of view but similar positions may be found in the literature [3],[23].

As discussed in I two types (referred to as type 2 and type 3) of indeterministic theories may be identified. In type 2 theories laws for individual particles exist (roughly speaking the individuality of particles remains intact) but the initial values are unknown and are described by probabilities only. An example for such a (classical-statistical) type 2 theory is statistical thermodynamics. On the other hand, in type 3 theories the amount of uncertainty is still greater, insofar as no dynamic laws for individual particles exist any more. A possible candidate for this ’extreme’ type of indeterministic theory is quantum mechanics. The method used in I to construct statistical theories is based on three assumptions listed in the last section. The first and second of these cover type 2 as well as type 3 theories, while the third - the requirement of maximal disorder - does only hold for a single type 3 theory, namely quantum mechanics. In this sense quantum mechanics may be considered as the most reasonable theory among all statistical theories defined by the first two assumptions. There is obviously an analogy between quantum mechanics and the principle of minimal Fisher information on the one hand and classical statistical mechanics and the principle of maximal entropy on the other hand; both theories are realizations of the principle of maximal disorder.

The basic equations of I (see section 3 of I) are generalized with respect to the number of spatial dimensions and with respect to the structure of the function S  (x,  t ) . In I S  (x,  t ) was an ’ordinary’ - i.e. single-valued - function. Now, we allow for multi-valued functions S˜ (x,  t ) . This is possible because S˜ (x,  t ) itself does not appear in any physical law (see below) of the present theory. It will be shown that this ’degree of freedom’ is intimately related to the existence of gauge fields. A multi-valued S˜ (x,  t ) cannot be an observable quantity. However, all quantities derived from S˜ (x,  t ) , which occur in physical laws must be observables and must be single valued. Of particular importance are the first derivatives with respect to t and x
  k . We assume that S˜ (x,  t ) may be written as a sum of a single-valued part S  (x,  t ) and a multi-valued part N ˜ . Then, given that
                                ∂ S˜    ∂ S˜
S˜ (x,  t )   multi- valued,      -----, -------   single- valued,
                                 ∂ t   ∂ x

the derivatives of S˜ (x,  t ) may be written in the form
∂ S˜      ∂  S                   ∂ S˜       ∂  S       e
----- =   ----- +   e Φ,        -------=    --------   --A    ,
∂  t       ∂ t                  ∂ x         ∂ x        c    k
                                    k           k

where the four functions Φ and A
   k are proportional to the derivatives of the multi-valued part N˜ with respect to t and xk respectively (Note the change in sign of Φ and Ak in comparison to [19]; this is due to the fact that the multi-valued phase is now denoted by S˜ ). The physical motivations for introducing the pre-factors e and c in Eq. (3) have been extensively discussed elsewhere [17],[19] in an electrodynamical context.

The necessary and sufficient condition for single-valuedness of a function  ˜
H   (x,  t) (in a subspace            4
G   ⊆   R ) is that all second order derivatives of H˜ (x,   t) with respect to xk and t commute with each other (in G). In this sense S˜(x,   t) is multi-valued while the four derivatives of S  (x,  t ) with respect to xk and t and the four functions Φ and Ak are single-valued. On the other hand this does not mean that the latter eight quantities must be unique. Actually it will turn out that they are not; according to the present construction only the four derivatives of  ˜
S  (x,  t ) with respect to x
  k and t are uniquely determined by the physical situation. These derivatives define four fields
                 ∂  ˜S (x,   t )                          ∂ S˜(x,   t )
˜p  (x,   t ) =   --------------,    E˜ (x,   t ) =   -   -------------, ,
  k                  ∂ x                                      ∂ t

with dimensions of momentum and energy respectively (a quantity denoted A˜ is not necessarily multi-valued; this notation is used here to indicate that it is defined with the help of a multi-valued S˜ ).

In contrast to S˜ , our second fundamental dynamic variable ρ is a physical observable (in the statistical sense) and is treated as a single-valued function. The fields S and ρ (we use the summation convention) obey the continuity equation
∂ ρ (x,  t )       ∂    ρ (x,  t ) ∂ S˜(x,   t)
-------------    -------------------------------
             +                                   =   0
    ∂ t          ∂  xk     m          ∂ xk

The statistical conditions associated with the type 1 theory (1), are obtained in the same way as in I by replacing the observables xk  (t ), pk  (t ) and the force field Fk  (x  (t ),  p(t ),  t) by averages ----- ----
xk  , pk and -----

Fk . This leads to the relations
       d-------     pk--
          xk    =
       dt           m
    pk   =  Fk   (x,   p,  t ),

where the averages are given by the following integrals over the random variables xk  ,  pk [which should be distinguished from the observables x   (t ),  p   (t )
  k          k ]:
                     -----              3
                     xk   =           d   x  ρ (x,  t ) xk
                                - ∞
                             ∫   ∞
                     p    =           d3p   w  (p,  t ) p
                       k                                  k
                       ∫        - ∞
------------------         ∞
                                  3     3
Fk   (x,   p,  t ) =            d  x  d   p W    (x,   p,  t)Fk   (x,   p,  t ).
                         -  ∞


The time-dependent probability densities W,    ρ,   w are positive semidefinite and normalized to unity, i.e. they fulfill the conditions
∫   ∞                         ∫   ∞                          ∫   ∞
           3                             3                              3     3
         d  x  ρ (x,  t ) =            d   p w  (p,  t ) =            d  x  d   p W    (x,   p,  t)  =   1

  -  ∞                          -  ∞                           -  ∞

The densities ρ and w may be derived from the fundamental probability density W by means of the relations
               ∫   ∞                                                  ∫   ∞
                          3                                                      3
ρ (x,  t ) =            d  p  W   (x,   p,  t );      w  (p,  t ) =            d   x W    (x,   p,  t).

                 -  ∞                                                   -  ∞

The present construction of the statistical conditions (6) and (7) from the type 1 theory (1) is very similar to the treatment in I. There are, however, two differences. The first is that we allow now for a p -dependent external force. This leads to a more complicated probability density W   (x,   p,   t) as compared to the two decoupled densities ρ (x,   t) and w  (p,  t) of I. The second difference, which is in fact related to the first, is the use of a multi-valued  ˜
S  (x,  t ) .