[Paper Review] Formalized procedure of transition to classical limit in application to the Dirac equation

Yuri A. Rylov|ArXiv.org|Jul 26, 2005

Quantum Mechanics and Applications14 references3 citations

TL;DR

This paper introduces a dynamical disquantization method to derive a classical analog of the Dirac particle from the Dirac equation without relying on quantum axioms. By reformulating the Dirac equation using a natural parameterization that separates the quantum constant ℏ from a dynamical constant b, the authors construct a classical system with 10 degrees of freedom and internal structure, showing that the classical limit emerges when ℏ → 0 while b remains finite, resolving long-standing ambiguities in the classical-quantum transition.

ABSTRACT

Classical model S{Dcl} of the Dirac particle S_D is constructed. S_D is the dynamic system described by the Dirac equation. For investigation of S_D and construction of S_{Dcl} one uses a new dynamic method: dynamic disquantization. This relativistic purely dynamic procedure does not use principles of quantum mechanics. The obtained classical analog S_{Dcl} is described by a system of ordinary differential equations, containing the quantum constant as a parameter. Dynamic equations for S_{Dcl} are determined by the Dirac equation uniquely. The dynamic system S_{Dcl} has ten degrees of freedom and cannot be a pointlike particle, because it has an internal structure. Internal degrees of freedom appears to be described nonrelativistically. One discusses interplay between the conventional axiomatic methods and the dynamical methods of the quantum systems investigation. In particular, one discusses the reasons, why the internal degrees of freedom of the Dirac particle and their nonrelativistic character were not discovered during eighty years.

Motivation & Objective

• To resolve the ambiguity in the classical limit of quantum systems, particularly the Dirac particle, where setting ℏ = 0 in standard formulations leads to degeneracy.
• To construct a classical analog of the Dirac particle that preserves internal degrees of freedom and relativistic structure, using a purely dynamical approach.
• To demonstrate that the conventional quantum action formulation artificially mixes the quantum constant ℏ with a dynamical constant, leading to inconsistencies in the classical limit.
• To show that a natural parameterization of the wave function, introducing a separate dynamical constant b, allows a consistent classical limit when ℏ → 0.

Proposed method

• Introduces a new dynamic method called 'dynamic disquantization' that avoids axiomatic quantum mechanics and instead uses a reformulation of the Dirac equation in terms of a complex wave function Ψ_b with a separate dynamical constant b.
• Replaces the standard Schrödinger and Dirac actions by a natural action that explicitly separates the quantum constant ℏ from the dynamical constant b, ensuring the classical limit is well-defined.
• Derives the classical Dirac particle model S_Dcl as a system of ordinary differential equations with 10 degrees of freedom, where the internal structure arises from nonrelativistic internal degrees of freedom.
• Uses a transformation ψ → Ψ_b = |ψ| exp((ℏ/b) log(ψ/|ψ|)) to switch from the artificial formulation (1.1) to the natural formulation (1.7), preserving physical content while enabling a consistent classical limit.
• Applies the same method to the Dirac equation by extending the natural parameterization to four-component spinors, deriving a classical system with internal structure and nonrelativistic internal dynamics.
• Demonstrates that in the limit ℏ → 0, the natural action reduces to a classical action for a statistical ensemble of particles, while the original artificial formulation degenerates.

Experimental results

Research questions

• RQ1 Why does the standard procedure of setting ℏ = 0 in the Dirac equation fail to produce a consistent classical limit?
• RQ2 What dynamical mechanism allows the construction of a classical analog of the Dirac particle without invoking quantum axioms?
• RQ3 How can the internal degrees of freedom of the Dirac particle be revealed and described in a classical framework?
• RQ4 Why have the internal degrees of freedom of the Dirac particle remained undetected for over 80 years despite extensive study?
• RQ5 Can a consistent classical limit be derived from the Dirac equation using only relativistic dynamics and without relying on canonical quantization?

Key findings

• The classical Dirac particle S_Dcl is derived as a 10-degree-of-freedom system described by ordinary differential equations, with internal structure arising from nonrelativistic internal degrees of freedom.
• The classical limit is achieved not by setting ℏ = 0 in the original action, but by using a natural parameterization where ℏ is a separate parameter and b remains finite, avoiding degeneracy.
• The natural action (1.7) contains both b and ℏ, and when ℏ → 0, it reduces to the action for a statistical ensemble of classical particles, confirming a consistent classical limit.
• The internal structure of the Dirac particle is revealed as nonrelativistic degrees of freedom, which were previously hidden due to the artificial mixing of b and ℏ in standard formulations.
• The method explains why the internal structure of the Dirac particle was not discovered earlier: conventional formulations force b = ℏ, leading to degeneracy when ℏ → 0.
• The approach shows that the classical limit is not a limit of quantum mechanics but a dynamical process, and that the Dirac particle cannot be a pointlike particle due to its internal structure.