3 The ’standard limit’ of quantum theory

Let us now compare QT and NM [Eqs. (1) and (2)] in the light of the above examples. Both theories differ obviously with respect to their mathematical structure; this indicates the possibility to obtain NM from QT by means of a ’deterministic’ limiting process. However, in addition, a new fundamental constant (the number ℏ ) appears in QT; this indicates the possibility to obtain NM from QT by means of a ’standard’ limiting process. This shows that the limiting process which leads from QT to NM is either nonexistent or more complex than any one of the above examples.

Let us try, as a first step, to perform the ’standard limit’ of QT - as defined by the first example in section 2. Performing the limit ℏ  →    0 in Eq. (1) produces a nonsensical result. This indicates that the real and imaginary parts of ψ are not appropriate variables with regard to this limiting process; probably because they become singular in the limit ℏ  →    0 . Thus, a different set of dynamical variables should be chosen, which behaves regular in this limit. There is convincing evidence, from various physical contexts, that appropriate variables, denoted by ρ and S , are defined by the transformation
        √  -- i S-
ψ   =      ρe   ℏ .

This transformation has been introduced by Madelung [17]. Note that using these variables in a meaningful limiting process requires that the modulus of ψ remains regular while its phase diverges like ℏ -  1 for small ℏ . This singular behavior, which has been noted very early [?], is the behavior of the majority of ’well-behaved’ quantum states. Other, more singular, states may, however, behave in a different manner and will then require a different factorization in terms of ℏ . This may also lead to different equations in the limit ℏ   →    0 ; an example will be given in section 6.

In terms of the new variables Schrödinger’s equation takes the form of two coupled nonlinear differential equations. The first is a continuity equation which does not contain ℏ ,
∂  ρ        ∂    ρ   ∂ S
----- +   ----------------- =   0.
 ∂ t      ∂ x   m   ∂  x
              k          k

The second equation contains ℏ as a proportionality factor in front of a single term,
                      (         )                         √  --
                ∑                  2                2
∂-S--    --1---          -∂-S---                 -ℏ----△-----ρ--
      +                              +   V   -          √   --   =   0.
∂ t      2m              ∂ xk                    2m         ρ

Eq. (9) is referred to as quantum Hamilton-Jacobi equation (QHJ). The ℏ -dependent ’quantum term’ in (9) describes the influence of ρ on S (It is frequently denoted as ”quantum potential”, which is an extremely misleading nomenclature because a potential is, as a rule, an externally controlled quantity). Its physical meaning, as interpreted by the present author, has been discussed in more detail elsewhere [11].

In the limit ℏ   →    0 the quantum term disappears. Thus the ’standard limit’ of QT is given by two partial differential equations, the continuity equation (8), which depends on ρ and S , and the Hamilton-Jacobi (HJ) equation,
                       (        )
∂  S        1   ∑          ∂ S     2
-----     ------          -------
      +                               +   V   =   0,
 ∂ t      2m              ∂ xk

which depends only on S . The two equations (8) and (10) which will be referred to as probabilistic Hamilton-Jacobi theory (PHJ) constitute the classical limit of Schrödinger’s equation or ’single-particle’ QT, respectively. Clearly, this limit does not agree with the trajectory equations (2) of NM.

Much confusion has been created by the fact, that the Hamilton-Jacobi formulation of classical mechanics allows the determination of particle trajectories with the help of the HJ equation. From the fact that this equation can be obtained from QT in the limit ℏ  →    0 it is often concluded, neglecting the continuity equation, that classical mechanics is the ℏ  →    0 limit of QT. However, the limit ℏ  →     0 of QT does not provide us with the theory of canonical transformations, which is required to actually calculate particle trajectories. Note also that for exactly those quantum-mechanical states which are most similar to classical states (e.g. coherent states, see section 6) the classical limit of QHJ differs from HJ. There is in fact a connection between the PHJ and NM but this requires a second limiting process, as will be explained in section 4.

Both the PHJ and its (standard) covering theory QT are probabilistic theories, which provide statistical predictions (probabilities and expectation values) if initial values for S and ρ are specified. Although we have now partial differential equations, the relation between QT and PHJ resembles in essential aspects the relation between relativistic mechanics and NM.