_{1}

We prove that any nonrelativistic classical system must obey a statistical wave equation that is exactly the same as the Schr ödinger equation for the system, including the usual “canonical quantization” and Hamiltonian operator, provided an unknown constant is set equal to . We show why the two equations must have exactly the same sets of solutions, whereby this classical statistical theory (CST) and nonrelativistic quantum mechanics may differ only in their interpretations of the same quantitative results. We identify some of the different interpretations. We show that the results also imply nonrelativistic Lagrangian classical mechanics and the associated Newtonian laws of motion. We prove that the CST applied to a nonrelativistic rigid rotator yields spin angular momentum operators that obey the quantum commutation rules and allow both integer and half-odd-integer spin. We also note that the CST applied to systems of identical massive particles is mathematically equivalent to nonrelativistic quantum field theory for those particles.

During the latter part of the nineteenth century, Ludwig Boltzmann initiated the application of statistics to many-particle systems. He pursued this approach despite the strongly prevailing belief in a continuum structure of matter during that period. With the introduction of the ensemble concept by Willard Gibbs in 1902, the resulting statistical mechanics achieved several great successes, including the statistical definition of entropy and derivations of thermodynamics and fluid mechanics.

During the past century, the belief that quantum mechanics (QM) is the truly fundamental theory of nature has become overwhelmingly prevailing. On the basis of an enormous number of correct predictions, and apparently no incorrect ones, this belief continues to strengthen despite the puzzling fact that QM makes only statistical predictions. How can a statistical theory be fundamental? Any such theory must involve a statistical treatment of some number of underlying quantities.

Many attempts have been made to establish a classical statistical foundation for the single-particle Schrödinger or Dirac equation, e.g., Bohm’s hidden variable theory [

None of the abovementioned approaches has been shown to apply to all classical systems. The principal goals of the work reported in this paper are to develop a statistical description of the nonrelativistic classical motions of the coordinates of any system, based on the probability continuity relation in the coordinate configuration space, and to investigate how close is that description to the nonrelativistic quantum mechanics of the system.

We pursue those goals as follows: In section 2, we consider a general non- relativistic classical system involving

In this section we treat general nonrelativistic classical systems, which are invariably described in terms of

We include this subsection to establish our notation, which is the notation used by Lichnerowicz [

For such systems, the kinetic energy

where we use the extended Einstein summation convention that all repeated indices are summed over from 1 to

where

In treating classical motions statistically, one may always begin with the coor- dinate probability density. The fine-grained coordinate probability density in the

where

where

These fine-grained probability densities are almost never useful in application, because it would be virtually impossible to solve for the detailed coordinate trajectories

Note that Equation (4) must be satisfied irrespective of what stochastic process is considered, e.g., Markovian or not, and independently of what kind of stochastic dynamics is considered, e.g., the Langevin equation, the Fokker- Planck equation, etc., and independently of what kind of position-velocity or position-momentum phase space treatment may be valid. Therefore, the statistical description of a system’s classical coordinates that evolves from just this conti- nuity equation will be incomplete, but still must be obeyed.

Now we proceed by following Collins’ method [

Then

where

This relation is known as the Madelung transform [

Then, requiring that the continuity Equation (4) be satisfied yields easily

where

and

where

This

Note that we are not getting something for nothing: The quantities

Of course, one can identify

If

These identifications and the above derivation of the SWE from the probability continuity relation actually provide a derivation of classical mechanics (CM) for any nonrelativistic Lagrangian system as well, since the Hamilton-Jacobi equation implies the existence of a Hamiltonian, a Lagrangian containing the kinetic energy and metric (Equation (1)), the Euler-Lagrange equations, and thereby Newton’s laws of motion, for arbitrarily chosen potentials

Canonical quantization. We define the vector conjugate momentum operator

Then the covariant components of

and the Hamiltonian operator of Equation (13) is

Therefore, the general rule for obtaining the SWE for any nonrelativistic classical Lagrangian system is simply to write down the classical Hamiltonian and then replace the conjugate momentum

In this section, we consider three examples that should help clarify the gene- ralized-coordinate approach. One is a system of two spinless pointlike particles that may have different masses. Another is a system of arbitrarily many identical spinless pointlike particles with two-body central force instantaneous internal interactions. A third is a system of one nonrelativistic rigid rotator. In these examples, we put

The designation “pointlike” does not mean that the particles are actual points; instead, it means that the model particles considered are allowed no coordinates other than their CM coordinates. In this example, let the masses be

where the index

with other components zero. Since

where

For this example, we consider the unperturbed central force case, by choosing

is the distance between the particle CM’s. The classical Hamiltonian is

At this point, one may go to the conventional notation

One reason for choosing this particular example is that it is probably the simplest two-particle example of the general method derived in section 2. Another reason is to emphasize that what you get in the Hamiltonian operator in the derived SWE is exactly what you have included in the classical Hamiltonian. For example, it is clearly physically incorrect to choose

Consider the extension of the two-particle system above to

where terms with

where

For the identical particles in this example, the total Hamiltonian is invariant under all pair interchanges of particle indices. This invariance leads immediately to the result that the total wavefunction solution of the general many-particle SWE must either change sign under each pair interchange, yielding Fermions, or not change sign, yielding Bosons. As discussed in detail by Schweber [

Many authors have considered classical spinning top models and their possible connections to quantum spin and magnetic moment, e.g. [

We include this subsection to establish our notation and method. For a rigid rotator with a fixed CM, the coordinates are a set of Euler angles,

where the orthogonal matrices

Thus, the complete rotation is specified by the orthogonal matrix

The angular velocity 3-vector can be found from the relations defining rigidly rotating Cartesian coordinates,

where × is the cross-product, and

where we have specified the time-dependent trajectories of the Euler angles by

In this introductory work, we treat a very simple model rotator, a non-translating but freely rotating rigid extended symmetric object having only the attributes of mass

The metric of the Euler angle 3-space is easily identified as

We may define the conjugate momentum 3-vector by

where

The (angular) momenta conjugate to the angles are

Contraction with

where the

The relevant SWE for any nonrelativistic system having three coordinates is the three-dimensional version of the general SWE (12). In this case, the fields

where

just as postulated by some of the authors mentioned above. Writing Equation (31) in summation notation and making these substitutions yields

where, from Equations (33) and (35),

It is not difficult to show that the operators

where

Equations (28) and (39) then yield easily

Note the minus sign, compared to Equation (38). These “left-handed” commu- tation rules must be obeyed by the rotating system Cartesian components of

Note that

In a paper to follow, we will derive several relevant detailed results for a charged nonrelativistically spinning top of arbitrary shape and structure immersed in a magnetic field, including the following: 1) The commutation rules of Equations (38) and (40) are unchanged, whereby the commuting operators

In this work, using a derivation from first principles with no approximations, we proved that any nonrelativistic classical physical system must obey a statistical wave equation (SWE) that has the same form and the same quantitative solu- tions as the Schrödinger equation (SEQ) for the system. In the non-statistical (“classical”) limit, the SWE yields a system Hamiltonian and Lagrangian and thus the Euler-Lagrange equations and Newton’s laws of motion for the system coordinates. The SWE also yields quantum spin and many-body quantum field theory for nonrelativistic systems. On the basis of these results, should we not conclude that the classical statistical theory (CST) developed in this work actually provides a derivation of nonrelativistic classical and quantum dynamics for all Lagrangian systems composed of massive particles?

Our answer to that question is “not yet, and maybe never”. In earlier work [

Detailed analysis of such profound interpretational differences is well beyond the scope of this introductory paper. Our results would require an ensemble interpretation of QM, as well as other re-interpretations; see e.g. work by Ballentine [

The CST derived herein makes sense only if there are reasons why even a one-particle classical system might require a statistical treatment. One possible reason is the known fact that any classical system is continually bombarded by thermal and other highly fluctuating background radiation fields. One model background field used in Stochastic Electrodynamics [

In conclusion: Before we are willing to conclude that the CST actually provides a derivation of a good part of modern physics, we feel that at least two things must be accomplished. One is a statistical treatment of relativistic classical systems, with a closer correspondence to our nonrelativistic treatment herein than our work reported in 2010 [

The author would like to thank Stephen Pate, Michael Engelhardt, and Stefan Zollner for helpful discussions and assistance in preparing the manuscript.

Goedecke, G.H. (2017) Statistical Description of Nonrelativistic Classical Systems. Journal of Modern Physics, 8, 786-802. https://doi.org/10.4236/jmp.2017.85050

As mentioned above, one objection to a nonrelativistic extended spinning electron model having semidefinite charge density is that it seems to require tangential linear speeds

Evidently we should have used relativistic expressions for the momenta. For this model rotator, any infinitesimal ring segment of restmass

It is interesting to to evaluate the magnitude

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