10 The classical limit of quantum theory is a statistical theory

The classical limit of Schrödinger’s equation plays an important role for two topics discussed in the next section, namely the interpretation of QT and the particular significance of potentials in QT; to study these questions it is sufficient to consider a single-particle ensemble described by a single state function. This ’classical limit theory’ is given by the two differential equations

                                         (                    )

                    ∂--ρ-     --∂----ρ--    -∂-S---    e-
                          +                        -     Ak        =    0,  (118)
                     ∂ t      ∂ x   m       ∂ x         c
                               (  k             k    )
                         ∑                              2
∂--S-              --1---          ∂--S---    e-
      +   e ϕ  +                          -     Ak        +   V    =    0,  (119)
 ∂ t               2m              ∂ xk       c
which are obtained from Eqs. (50) and (51) by performing the limit ℏ   →    0 . The quantum mechanical theory (50) and (51) and the classical theory (118) and (119) show fundamentally the same mathematical structure; both are initial value problems for the variables S and ρ obeying two partial differential equations. The difference is the absence of the last term on the l.h.s. of (51) in the corresponding classical equation (119). This leads to a decoupling of S and ρ in (119); the identity of the classical object described by S is no longer affected by statistical aspects described by ρ .

The field theory (118), (119) for the two ’not decoupled’ fields S and ρ is obviously very different from classical mechanics which is formulated in terms of trajectories. The fact that one of these equations, namely (119), agrees with the Hamilton-Jacobi equation, does not change the situation since the presence of the continuity equation (118) cannot be neglected. On top of that, even if it could be neglected, Eq. (119) would still be totally different from classical mechanics: In order to construct particle trajectories from the partial differential equation (119) for the field S  (x,  t ) , a number of clearly defined mathematical manipulations, which are part of the classical theory of canonical transformations, see [20], must be performed. The crucial point is that the latter theory is not part of QT and cannot be added ’by hand’ in the limit ℏ   →    0 . Thus, (118), (119) is, like QT, an indeterministic theory predicting not values of single event observables but only probabilities, which must be verified by ensemble measurements.

Given that we found a solution S (x,   t) , ρ (x,  t) of (118), (119) for given initial values, we may ask which experimental predictions can be made with the help of these quantities. Using the fields ⃗p˜(x,   t) , E˜ (x,  t ) defined by Eqs. (19), (18), the Hamilton-Jacobi equation (119) takes the form
⃗˜p2  (x,  t)
------------                      ˜
             +  V   (x,  t)  =   E  (x,  t ),

The l.h.s. of (120) depends on the field ⃗˜p in the same way as a classical particle Hamiltonian on the (gauge-invariant) kinetic momentum ⃗p . We conclude that the field ⃗p˜(x,  t ) describes a mapping from space-time points to particle momenta: If a particle (in an external electromagnetic field) is found at time t at the point x , then its kinetic momentum is given by ⃗
˜p (x,  t) . This is not a deterministic prediction since we can not predict if a single particle will be or will not be at time t at point x ; the present theory gives only a probability ρ (x,   t) for such an event. Combining our findings about ⃗˜p (x,  t) and x we conclude that the experimental prediction which can be made with the help of S  (x,  t ) , ρ (x,  t) is given by the following phase space probability density:
                               (3 )         ∂-S-(x,---t)-
w (x,   p, t ) =   ρ (x,   t)δ     (p  -                 ).
                                                ∂ x

Eq. (121) confirms our claim that the classical limit theory is a statistical theory. The one-dimensional version of (121) has been obtained before by means of a slightly different method in I. The deterministic element [realized by the delta-function shaped probability in (121)] contained in the classical statistical theory (118), (119) is absent in QT, see I.

Eqs. (118), (119) constitute the mathematically well-defined limit ℏ  →     0 of Schrödinger’s equation. Insofar as there is general agreement with regard to two points, namely that (i) ’non-classicality’ (whatever this may mean precisely) is expressed by a nonzero ℏ , and that (ii) Schrödinger’s equation is the most important relation of quantum theory, one would also expect general agreement with regard to a further point, namely that Eqs. (118), (119) present essentially (for a three-dimensional configuration space) the classical limit of quantum mechanics. But this is, strangely enough, not the case. With a few exceptions, see [68][59][7][62][32], most works (too many to be quoted) take it for granted that the classical limit of quantum theory is classical mechanics. The objective of papers like [57][69][37][3] devoted to “..the classical limit of quantum mechanics..“ is very often not the problem: ”what is the classical limit of quantum mechanics ?” but rather: “how to bridge the gap between quantum mechanics and classical mechanics ?”. Thus, the fact that classical mechanics is the classical limit of quantum mechanics is considered as evident and any facts not compatible with it - like Eqs. (118), (119) - are denied.

What, then, is the reason for this widespread denial of reality ? One of the main reasons is the principle of reductionism which still rules the thinking of most physicists today. The reductionistic ideal is a hierarchy of physical theories; better theories have an enlarged domain of validity and contain ’inferior’ theories as special cases. This principle which has been extremely successful in the past dictates that classical mechanics is a special case of quantum theory. Successful as this idea might have been during a long period of time it is not necessarily universally true; quantum mechanics and classical mechanics describe different domains of reality, both may be true in their own domains of validity. Many phenomena in nature indicate that the principle of reductionism (alone) is insufficient to describe reality, see [38]. Releasing ourselves from the metaphysical principle of reductionism, we accept that the classical limit of quantum mechanics for a three-dimensional configuration space is the statistical theory defined by Eqs. (118), (119). It is clear that this theory is not realized in nature (with the same physical meaning of the variables) because ℏ is different from zero. But this is a different question and does not affect the conclusion.