In these continuation papers (VI and VII) we are interested in approach the problem of spin from a classical point of view. In this first paper we will show that the spin is neither basically relativistic nor quantum but reflects just a simmetry property related to the Lie algebra to which it is associated. The classical approach will be paraleled with the usual quantum one to stress their formal similarities and epistemological differences. The important problem of Einstein-Bose condensation for fermions will also be addressed.
We continue in this paper our program of rederiving all quantum mechanical formalism from the classical one. We now turn our attention to the derivation of the second quantized equations, both for integral and half-integral spins. We then show that all the quantum results may be derived using our approach and also show the interpretation suggested by this derivation. This paper may be considered as a first approach to the study of the quantum field theory beginning by the same classical ideas we are supporting since the first paper of this series.
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schreodinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schroedinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.
As part of a probabilistic reconstruction of quantum theory (QT), we show that spin is not a purely quantum mechanical phenomenon, as has long been assumed. Rather, this phenomenon occurs before the transition to QT takes place, namely in the area of the quasi-classical (here better quasi-quantum) theory. This borderland between classical physics and QT can be reached within the framework of our reconstruction by the replacement
p→M(q,t), where
p is the momentum variable of the particle and
M(q,t) is the momentum field in configuration space. The occurrence of spin, and its special value
1/2 , is a consequence of the fact that
M(q,t) must have exactly three independent components
Mk(q,t) for a single particle because of the three-dimensionality of space. In the Schrödinger equation for a "particle with spin zero", the momentum field is usually represented as a gradient of a single function
S. This implies dependencies between the components
Mk(q,t) for which no explanation exists. In reality,
M(q,t) needs to be represented by three functions, two of which are rotational degrees of freedom. The latter are responsible for the existence of spin. All massive structureless particles in nature must therefore be spin-one-half particles, simply because they have to be described by
4 real fields, one of which has the physical meaning of a probability density, while the other three are required to represent the momentum field in three-dimensional space. We derive the Pauli-Schrödinger equation, the correct value
g=2 of the gyromagnetic ratio, the classical limit of the Pauli-Schrödinger equation, and clarify some other open questions in the borderland between classical physics and QT.
The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result. The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory. The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.