4 The ’deterministic limit’ of the ’standard limit’ of quantum theory

The deterministic limit of the classical-limit PHJ of QT is of considerable interest for the present problem, despite the fact that the PHJ does no longer contain ℏ . Existence of a deterministic limit implies that ρ (x,   t) takes the form of a delta-function peaked at trajectory coordinates r   (t )
  k [which, hopefully, should then be solutions of the classical equations (2)]. Thus, adopting a standard formula, we may write ρ (x,   t) as an analytic function
                                   (                                    )
                 (      )  3-
                           2       {         ∑ 3                        }
                    -1--                  1-                          2
ρ ϵ(x,   t)  =               exp       -           [xk   -   rk  (t )]      ,
                    π ϵ            (      ϵ                             )

which represents   (3)
δ    (x   -   r (t )) under the integral sign in the limit ϵ  →    0 ,
 lim   ρ ϵ (x,  t)  =   δ    (x   -   r (t )).
ϵ →0

In order to check whether or not this deterministic representation of ρ (x,  t) is compatible with the basic equations of PHJ, we insert the Ansatz (11) into the continuity equation (8) and calculate the derivatives. After some rearrangement (8) takes the form [18]
            {                     (                            )                         }
                                                  ∂  S (x,  t )        ϵ    ∂     ∂
                                                  -------------        ----------------
ρ ϵ(x,   t)    [xk   -   rk (t ) ]   pk  (t ) -                    +                   S      =  0,
                                                      ∂ xk             2 ∂  xk  ∂  xk

where p   (t)  =   m  v˙  (t )
  k               k . At this point we recall that in the PHJ a momentum field p   (x,  t )
  k , defined by
pk (x,   t)  =                ,
                    ∂ xk

exists. The trajectory momentum pk  (t) should be clearly distinguished from this field momentum p   (x,  t )
  k .

In the limit ϵ  →    0 , ρ ϵ becomes a distribution and both sides of Eq. (13) have to be integrated over three-dimensional space in order to obtain a mathematically well-defined expression. The first term in the bracket vanishes as a consequence of the term xk   -   rk  (t) (at this point we start to disagree with Ref [18]). The second term vanishes too for ϵ  →    0 provided the second derivative of S is regular at ϵ  =   0 . But this can safely be assumed since the equation for S does not contain ρ . We conclude that the singular (deterministic) Ansatz (11) is a valid solution of PHJ for arbitrary S .

The present theory is incomplete since differential equations for the particle trajectories rk (t ) have still to be found. Generally, two conditions must be fulfilled in order to define particle coordinates in a probabilistic theory, namely (i) the limit of ’sharp’ (deterministic) probability distributions must be a valid solution, and (ii) an evolution law for the time-dependent expectation values must exist. We have just shown that the first (more critical) condition is fulfilled; Ehrenfest-like relations corresponding to the second condition exist in almost all statistical theories. For the PHJ these take exactly the same form as in QT, namely [1412]

 d             p
---------      --k-
    xk    =                                             (15)
dt             m-----------------

 d------          ∂-V--(x,---t)-
    pk    =    -                ,                       (16)
 dt                   ∂ xk
where average values such as -----
xk are defined according to the standard expression
-----               3
xk  (t ) =        d  x  ρ (x,  t)xk   .

From (15) and the continuity equation (8) we obtain the following useful relation

----                3             ∂-S--(x,--t-)
pk  (t ) =        d   x ρ (x,  t )             .
                                      ∂ xk

Since we have shown that the deterministic limit for ρ is a valid solution of PHJ, we may now use Eq. (12) and obtain in the limit ϵ  →    0 the following identification of trajectory quantities,
-----                             ----
xk  (t )  =   rk (t ),            pk  (t ) =   m   ˙rk (t ) =   pk  (t ),

from the definitions of the expectation values. The differential relations connecting these quantities, follow from Ehrenfest’s theorem and agree with the basic equations (2) of NM. A completely different type of physical law has ’emerged’ from the field theoretic relations of the PHJ theory. Thus, classical mechanics is, indeed, contained in PHJ as deterministic limit, in analogy to the second example of section 2.

Eq. (18) takes in this limit the form
             ∂--S-(x,---t) |               ∂-S-(r--(t),--t)-
pk  (t ) =                 |          =                     ,
                 ∂ xk       x=r   (t)         ∂ rk  (t )

which provides an interesting link between a particle-variable and a field-variable. We expect for consistency that this link admits a derivation of the equation for ˙p [see (2)] from the (field-theoretic) HJ-equation. This is indeed the case. We calculate the derivative of the HJ equation (10) with respect to xi , change the order of derivations with respect to x
  i and t , and project the resulting relation on the trajectory points xk   =   rk  (t ) . This leads to the equation
    |                                               2
-∂--|   ∂-S--(r-(t-),-t)-   -1--∂-S--(r-(t-),-t)--∂---S-(r-(t-),-t-)-   ∂-V--(r--(t),--t)-
    |                    +                                           +                     =   0,
∂ t  2.     ∂ ri (t )       m      ∂  rk (t )    ∂ ri (t ) ∂ rk (t )        ∂ ri (t )

where the notation indicates that the time derivative operates on the second argument of S only. Using now Eq. (20) and the definition of particle momentum we see that the first two terms of (21) agree exactly with the (total) time-derivative of pi (t) and (21) becomes the second Newton equation. This establishes the connection between the PHJ equations and trajectory differential equations mentioned in section 3 and completes our treatment of the deterministic limit of the PHJ theory. This derivation of NM seems to be new; it is based on several interesting papers [19313188] reporting important steps in the right direction.