[Paper Review] Sound relativistic quantum mechanics for a strictly solitary nonzero-mass particle, and its quantum-field reverberations

Steven Kenneth Kauffmann|ArXiv.org|Sep 22, 2009

Quantum Electrodynamics and Casimir Effect1 references3 citations

TL;DR

This paper proposes a sound relativistic quantum mechanics framework for a solitary nonzero-mass particle by extending the natural square-root Hamiltonian to include electromagnetic interactions via Lorentz-covariant four-momentum techniques and rest-frame nonrelativistic dynamics. It avoids the problematic negative-energy solutions of Klein-Gordon and Dirac theories by independently quantizing particle and antiparticle wavefunctions, yielding consistent relativistic dynamics without field quantization or reinterpretation postulates.

ABSTRACT

It is generally acknowledged that neither the Klein-Gordon equation nor the Dirac Hamiltonian can produce sound solitary-particle relativistic quantum mechanics due to the ill effects of their negative-energy solutions; instead their field-quantized wavefunctions are reinterpreted as dealing with particle and antiparticle simultaneously--despite the clear physical distinguishability of antiparticle from particle and the empirically known slight breaking of the underlying CP invariance. The natural square-root Hamiltonian of the free relativistic solitary particle is iterated to obtain the Klein-Gordon equation and linearized to obtain the Dirac Hamiltonian, steps that have calculational but not physical motivation, and which generate the above-mentioned problematic negative-energy solutions as extraneous artifacts. Since the natural square root Hamiltonian for the free relativistic solitary particle contrariwise produces physically unexceptionable quantum mechanics, this article focuses on extending that Hamiltonian to describe a solitary particle (of either spin 0 or spin one-half) in relativistic interaction with an external electromagnetic field. That is achieved by use of Lorentz-covariant solitary-particle four momentum techniques together with the assumption that well-known nonrelativistic dynamics applies in the particle's rest frame. Lorentz-invariant solitary particle actions, whose formal Hamiltonization is an equivalent alternative approach, are as well explicitly displayed. It is proposed that two separate solitary-particle wavefunctions, one for a particle and the other for its antiparticle, be independently quantized in lieu of "reinterpreting" negative energy solutions--which indeed don't even afflict proper solitary particles.

Motivation & Objective

• To resolve the foundational inconsistencies in standard relativistic quantum mechanics, particularly the ill-posed negative-energy solutions of the Klein-Gordon and Dirac equations.
• To develop a physically consistent relativistic quantum theory for a solitary particle (spin 0 or 1/2) interacting with an external electromagnetic field.
• To eliminate the need for field quantization and reinterpretation of negative-energy states by treating particle and antiparticle as distinct entities from the outset.
• To ensure Lorentz covariance and consistency with nonrelativistic dynamics in the particle's rest frame.
• To provide a Hamiltonian formulation that correctly reproduces the relativistic Lorentz force law and avoids violations of special relativity and Newton's first law.

Proposed method

• Uses the natural square-root Hamiltonian $ H = \sqrt{m^2c^4 + |c\mathbf{p}|^2} $ as the starting point for a solitary relativistic particle.
• Extends this Hamiltonian to electromagnetic interactions by expressing the canonical momentum $ \mathbf{p} $ as a function of the total four-momentum $ \mathbf{P} $, incorporating vector and scalar potentials.
• Applies the assumption that nonrelativistic electromagnetic interactions are exact in the particle's rest frame to derive the relativistic form.
• Imposes Lorentz covariance through four-momentum techniques, ensuring manifest invariance under Lorentz transformations.
• Derives the relativistic Hamiltonian $ H^{\mathrm{(REL)}}_{\mathrm{EM;}{\scriptscriptstyle\frac{1}{2}}} $ up to first order in spin coupling, including spin-electric field coupling via $ s^{\mu\nu} \partial_\mu A_\nu $.
• Constructs a candidate Lorentz-invariant action for consistency checks, though the four-momentum method is preferred for deriving the Hamiltonian.

Experimental results

Research questions

• RQ1 Can a relativistic quantum mechanics for a solitary particle be formulated without introducing unphysical negative-energy solutions?
• RQ2 How can the natural square-root Hamiltonian be consistently extended to include electromagnetic interactions while preserving Lorentz covariance?
• RQ3 Why do the standard Klein-Gordon and Dirac formulations fail to describe solitary particles despite their success in quantum field theory?
• RQ4 What is the correct relativistic Hamiltonian for a spin-1/2 particle in an external electromagnetic field that avoids the issues of the Dirac equation?
• RQ5 How does the spin of a moving particle couple to an external electric field, and what is its effect on the energy spectrum?

Key findings

• The proposed Hamiltonian $ H^{\mathrm{(REL)}}_{\mathrm{EM;}{\scriptscriptstyle\frac{1}{2}}} $ correctly reduces to the nonrelativistic Pauli Hamiltonian in the $ c \to \infty $ limit.
• The Hamiltonian yields a spin-electric field coupling term proportional to $ (g e / (m^2 c^3)) s^{\mu\nu} (\mathbf{P} - e\mathbf{A}/c) \partial_\mu A_\nu $, arising from the particle's motion in an electric field.
• The model avoids negative-energy solutions entirely by treating particle and antiparticle as independent degrees of freedom from the start.
• The classical limit of the Hamiltonian reproduces the relativistic Lorentz force law exactly, ensuring consistency with special relativity.
• The model predicts that a moving spin-1/2 particle couples to an external electric field via its magnetic moment in its rest frame, due to the field transformation under Lorentz boosts.
• The framework provides a consistent derivation of relativistic corrections to the hydrogen atom spectrum without relying on field quantization or reinterpretation of negative-energy states.