[Paper Review] Once more on the derivation of the Dirac equation

В. М. Симулик, I.Yu. Krivsky|arXiv (Cornell University)|Sep 3, 2013

Nonlinear Photonic Systems6 references3 citations

TL;DR

This paper derives the Dirac equation from a relativistic canonical quantum mechanics (RCQM) framework based on the Schrödinger-Foldy equation, avoiding the need for negative mass concepts in antiparticles. It establishes RCQM as the fundamental model for spin-1/2 fermionic doublets, showing the Dirac equation emerges unambiguously from this formalism while preserving physical consistency and observable interpretations without relying on negative-energy or negative-mass assumptions.

ABSTRACT

A brief review of the different ways of the Dirac equation derivation is given. The foundations of the relativistic canonical quantum mechanics of a fermionic doublet are formulated. In our approach the Dirac equation is derived from the main equation of relativistic canonical quantum mechanics. In the formalism of relativistic canonical quantum mechanics the absence of necessity to appeal for the conception of the negative mass of the antiparticle is shown.

Motivation & Objective

• To re-derive the Dirac equation from a more fundamental relativistic canonical quantum mechanics (RCQM) framework based on the Schrödinger-Foldy equation.
• To demonstrate that the Dirac equation is a direct consequence of RCQM without invoking additional assumptions like negative mass for antiparticles.
• To clarify the physical interpretation of the Dirac equation by showing its origin in a mathematically well-defined, axiomatic RCQM formalism.
• To challenge the conventional view that the Dirac equation is the fundamental relativistic quantum theory, proposing instead that RCQM is more fundamental for fermionic doublets.
• To resolve interpretational issues in the relativistic hydrogen atom and field-theoretic models by grounding them in RCQM, avoiding unphysical use of negative-frequency components.

Proposed method

• Formulates a relativistic canonical quantum mechanics (RCQM) framework using the Schrödinger-Foldy equation as the fundamental equation of motion for a 4-component spinor field.
• Applies the Foldy-Wouthuysen transformation to the Schrödinger-Foldy equation to derive the Dirac equation in the standard form.
• Uses the rigged Hilbert space formalism and the Schwartz test function space S^{3,4} to ensure mathematical rigor and physical consistency.
• Identifies the nine functionally independent operators (x, p, s) as the complete set of observables for the fermionic doublet, ensuring completeness and visualization of physical content.
• Establishes unambiguous physical meaning for the spinor components and coordinates by distinguishing external (x, p) and internal (s, g) degrees of freedom.
• Demonstrates that all physical information in RCQM is directly and unambiguously translatable into the field-theoretic Dirac equation framework.

Experimental results

Research questions

• RQ1 Can the Dirac equation be derived directly from a more fundamental relativistic quantum mechanics model without assuming negative mass for antiparticles?
• RQ2 What is the physical role of the Schrödinger-Foldy equation in the context of relativistic fermionic doublets?
• RQ3 Why is the concept of negative mass for antiparticles physically unjustified in the RCQM framework?
• RQ4 How does the relativistic canonical quantum mechanics formalism ensure consistency with both special relativity and nonrelativistic quantum mechanics?
• RQ5 What is the relationship between the Foldy-Wouthuysen representation and the fundamental RCQM model in describing spin-1/2 particles?

Key findings

• The Dirac equation is derived as a direct consequence of the Schrödinger-Foldy equation within the RCQM framework, without additional assumptions.
• The RCQM model is shown to be more fundamental than the Dirac or Foldy-Wouthuysen models for describing the spin-1/2 fermionic doublet.
• The concept of negative mass for antiparticles is rejected as physically unjustified; both particle and antiparticle have positive mass in RCQM.
• The coordinate operator x in the Dirac model is not the observable position operator; in RCQM, x is the true observable for the fermionic doublet.
• The probability density in the relativistic hydrogen atom is not |ψ(x)|² or ψ̄ψ when using negative-frequency components, invalidating their use as quantum-mechanical objects.
• All physical and mathematical information in RCQM is unambiguously translatable into the Dirac field-theory framework, confirming the consistency and completeness of the RCQM approach.