4 A constraint for forces in statistical theories

Let us discuss now the nature of the macroscopic forces    (e )
F  k   (x,  p,  t) entering the expectation value on the r.h.s. of Eq. (25). In our type I parent theory, classical mechanics, there are no constraints for the possible functional form of    (e )
F      (x,  p,  t)
  k . However, this need not be true in the present statistical framework. As a matter of fact, the way the mechanical potential V   (x,  t) entered the differential equation for S (in the previous work I) indicates already that such constraints do actually exist. Let us recall that in I we tacitly restricted the class of forces to those derivable from a potential V  (x,  t ) . If we eliminate this restriction and admit arbitrary forces, with components F   (x,   t)
   k , we obtain instead of the above relation (25) the simpler relation [Eq. (24) of I, generalized to three dimensions and arbitrary forces of the form F   (x,   t)
   k ]
                          ⌊                                      ⌋
    ∫                                    (        )                     ∫
       ∞           ∂  ρ       1    ∑         ∂ S     2      ∂ S             ∞
              3   ------- ⌈ ------          -------         -----⌉
-           d   x                                       +           =            dx  ρFk    (x,  t ).
      - ∞         ∂  xk     2m              ∂ xj            ∂  t           - ∞

This is a rather complicated integro-differential equation for our variables ρ (x,  t ) and S (x,   t) . We assume now, using mathematical simplicity as a guideline, that Eq. (27) can be written in the common form of a local differential equation. This assumption is of course not evident; in principle the laws of physics could be integro-differential equations or delay differential equations or take an even more complicated mathematical form. Nevertheless, this assumption seems rather weak considering the fact that all fundamental laws of physics take this ’simple’ form. Thus, we postulate that Eq. (27) is equivalent to a differential equation
            (         )
      ∑                 2
-1----         -∂-S---         ∂-S--
                           +         +   T   =  0,
2m             ∂  xj            ∂ t

where the unknown term T describes the influence of the force F
   k but may also contain other contributions. Let us write
T   =   -  L0   +   V  ,

where L0 does not depend on Fk , while V depends on it and vanishes for Fk   →    0 . Inserting (28) and (29) in (27) yields
∫                                        ∫
       3    ∂ ρ                                3
     d  x  -------(-  L     +  V  )  =       d   x ρF     (x,  t).
           ∂ x           0                              k

For Fk   →    0 Eq. (30) leads to the relation
      3    ∂  ρ
    d   x -------L    =   0,
          ∂  x      0

which remains true for finite forces because L0 does not depend on Fk . Finally, performing a partial integration, we see that a relation
              ∂ V                 ∞
F     =   -  -------+   s   ,          d3x   ρs     =   0,
   k                      k                      k
             ∂ xk               -  ∞

exists between Fk and V , with a vanishing expectation value of the (statistically irrelevant) functions s
  k . This example shows that the restriction to gradient fields, made above and in I, is actually not necessary. We may admit force fields which are arbitrary functions of x and t ; the statistical conditions (which play now the role of a ’statistical constraint’) eliminate automatically all forces that cannot be written after statistical averaging as gradient fields.

This is very interesting and indicates the possibility that the present statistical assumptions leading to Schrödinger’s equation may also be responsible, at least partly, for the structure of the real existing (gauge) interactions of nature.

Does this statistical constraint also work in the present p -dependent case ? We assume that the force in (25) is a standard random variable with the configuration space as sample space (see the discussion in section 4 of I) and that the variable p in    (e )
F      (x,  p,  t)
   k may consequently be replaced by the field p˜(x,   t) [see (19)]. Then, the expectation value on the r.h.s. of (25) takes the form
-------------------     ∫   ∞
   (e )                                                     ∂ S˜(x,   t)
F      (x,   p, t ) =            d3x   ρ (x,  t )H    (x,   -------------, t).
   k                                                k           ∂ x
                          -  ∞

The second term on the l.h.s. of (25) has the same form. Therefore, the latter may be eliminated by writing
           ∂ S˜           e        1  ∂  ˜S                                ∂ S˜
H    (x,   -----, t)  =   --ϵ     ----------B    +   eE     +   h   (x,   -----, t),
    k                        kij               j          k       k
           ∂ x            c       m   ∂ xi                                ∂ x

with h   (x,  p,  t)
  k as our new unknown functions. They obey the simpler relations
                          ⌊                                              ⌋
    ∫                                              (        )2                   ∫
       ∞           ∂  ρ     ∂  ˜S        1   ∑          ∂ S˜                          ∞                     ∂ S˜
              3   ------- ⌈ -----     ------          -------            ⌉                 3               -----
-           d   x                 +                               +   V      =           d   x ρhk   (x,        , t).
      - ∞         ∂  xk      ∂ t      2m              ∂ xj                         - ∞                     ∂ x

On a first look this condition for the allowed forces looks similar to the p -independent case [see (27)]. But the dependence of hk on x,  t cannot be considered as ’given’ (externally controlled), as in the p -independent case, because it contains now the unknown x,  t -dependence of the derivatives of S˜ . We may nevertheless try to incorporate the r.h.s by adding a term T˜ to the bracket which depends on the derivatives of the multivalued quantity S˜ . This leads to the condition
             ˜                 ˜       ∂S˜--              ∫   ∞
          ∂ S               ∂ T  (x,   ∂x  , t)                      3
hk  (x,   -----, t)  =   -  --------------------+   sk  ,          d   x ρsk    =   0.
          ∂ x                     ∂  x
                                       k                    -  ∞

But this relation cannot be fulfilled for nontrivial h   , ˜T
  k because the derivatives of S˜ cannot be subject to further constraints beyond those given by the differential equation; on top of that the derivatives with regard to x on the r.h.s. create higher order derivatives of  ˜
S which are not present at the l.h.s. of Eq. (36). The only possibility to fulfill this relation is for constant   ˜
∂x , a special case which has in fact already be taken into account by adding the mechanical potential V . We conclude that the statistical constraint leads to hk   =   T˜  =   0 and that the statistical condition (35) takes the form
                     ⌊                                               ⌋
   ∫                                           (        )2
               ∂ ρ      ∂ S˜        1   ∑         ∂  ˜S
          3   -------⌈  -----     ------          -------            ⌉
-       d   x                 +                               +  V      =   0.
              ∂ xk       ∂ t      2m              ∂ xj

Thus, only a mechanical potential and the four electrodynamic potentials are compatible with the statistical constraint and will consequently - assuming that the present statistical approach reflects a fundamental principle of nature - be realized in nature. As is well known all existing interactions follow (sometimes in a generalized form) the gauge coupling scheme derived above. The statistical conditions imply not only Schrödinger’s equation but also the form of the (gauge) coupling to external influences and the form of the corresponding local force, the Lorentz force,
F⃗L    =   e ⃗E   +   --⃗v  ×   ⃗B,

if the particle velocity ⃗v is identified with the velocity field ⃗˜v (x,  t ) .

In the present derivation the usual order of proceeding is just inverted. In the conventional deterministic treatment the form of the local forces (Lorentz force), as taken from experiment, is used as a starting point. The potentials are introduced afterwards, in the course of a transition to a different formal framework (Lagrange formalism). In the present approach the fundamental assumptions are the statistical conditions. Then, taking into account an existing mathematical freedom (multi-valuedness of a variable) leads to the introduction of potentials. From these, the shape of the macroscopic (Lorentz) force can be derived, using the validity of the statistical conditions as a constraint.