[Paper Review] General form of the covariant field equations of arbitrary spin and the relativistic canonical quantum mechanics

В. М. Симулик|arXiv (Cornell University)|Sep 12, 2015

Algebraic and Geometric Analysis105 references3 citations

TL;DR

This paper presents a general axiomatic formulation of relativistic canonical quantum mechanics (RCQM) for arbitrary spin particles and their antiparticles, deriving manifestly covariant field equations through a generalized square-root operator formalism. It introduces a new 64-dimensional real Clifford–Dirac algebra and establishes that RCQM provides the most rigorous quantum-mechanical description, with successive transitions to canonical and covariant field theories degrading quantum-mechanical clarity.

ABSTRACT

The investigation of arXiv 1409.2766v2 [quant-ph] has been continued by the general form of the numerous equations with partial values of arbitrary spin, which were considered in above mentioned preprint. The general forms of quantum-mechanical and covariant equations for arbitrary spin together with the general description of the arbitrary spin field formalism are presented. The corresponding relativistic quantum mechanics of arbitrary spin is given as the system of axioms. Previously ignored partial example of the spin s=(0,0) particle-antiparticle doublet is considered. The partial example of spin s=(3/2,3/2) particle-antiparticle doublet is highlighted. The new 64 dimensional Clifford--Dirac algebra over the field of real numbers is suggested. The general operator, which transformed the relativistic canonical quantum mechanics of arbitrary spin into the locally covariant field theory, has been introduced. Moreover, the study of the place of the results given in arXiv 1409.2766v2 [quant-ph] among the results of other authors is started. The review of the different investigations in the area of relativistic canonical quantum mechanics is given and the brief analysis of the existing approaches to the covariant field theory of arbitrary spin is initiated. The consideration of some important details of arXiv 1409.2766v2 [quant-ph] is improved.

Motivation & Objective

• To generalize the relativistic canonical quantum mechanics (RCQM) of arbitrary spin beyond previous partial cases, including previously overlooked spin s=(0,0) and s=(3/2,3/2) doublets.
• To establish a systematic axiomatic framework for RCQM that ensures correspondence with nonrelativistic quantum mechanics and maintains clear quantum-mechanical interpretation.
• To derive manifestly covariant field equations for arbitrary spin from RCQM using a generalized square-root operator and a new 64-dimensional real Clifford–Dirac algebra.
• To clarify the hierarchy of three models—RCQM, canonical field theory, and locally covariant field theory—by showing that quantum-mechanical clarity degrades with each transition.
• To position the results within the broader context of existing approaches in relativistic quantum mechanics and field theory, reviewing key contributions and identifying gaps.

Proposed method

• Formulates RCQM as a system of axioms based on the canonical representation of the Poincaré group, using non-covariant but relativistically invariant generators.
• Applies a generalized square-root operator equation to describe the dynamics of arbitrary-spin particles, extending the Dirac-like formalism beyond spin-1/2.
• Introduces a new 64-dimensional Clifford–Dirac algebra over the real numbers to accommodate higher-spin representations and ensure closure of the algebraic structure.
• Derives manifestly covariant field equations from RCQM via a generalized Foldy–Wouthuysen-type transformation, preserving the physical interpretation of spin and statistics.
• Uses the Poincaré algebra in both non-covariant (canonical) and covariant forms to verify relativistic invariance, showing that non-covariant objects like d³x can still yield relativistic invariance.
• Compares the new 8-component equation for spin s=3/2 with the 16-component Rarita–Schwinger equation, demonstrating that the former requires no auxiliary conditions.

Experimental results

Research questions

• RQ1 How can relativistic canonical quantum mechanics be generalized to arbitrary spin, including previously neglected cases like s=(0,0) and s=(3/2,3/2)?
• RQ2 What is the role of the square-root operator in formulating a manifestly covariant field theory for arbitrary spin, and how does it unify different spin descriptions?
• RQ3 Why does the transition from RCQM to canonical field theory and then to covariant field theory degrade the quantum-mechanical clarity of the description?
• RQ4 How does the newly proposed 64-dimensional real Clifford–Dirac algebra support the description of arbitrary-spin fields and ensure algebraic consistency?
• RQ5 What is the significance of the 8-component equation for spin s=3/2 in comparison to the standard 16-component Rarita–Schwinger equation, and why is it preferable in terms of physical interpretation?

Key findings

• The relativistic canonical quantum mechanics (RCQM) of arbitrary spin is established as the most fundamental and physically interpretable framework, with the clearest quantum-mechanical meaning.
• The new 64-dimensional real Clifford–Dirac algebra provides a consistent mathematical structure for arbitrary-spin field theories, enabling the construction of manifestly covariant equations.
• The 8-component equation for spin s=3/2 is shown to be a direct analog of the Dirac equation for s=1/2, with no need for auxiliary constraints, unlike the Rarita–Schwinger equation.
• The synthesis of covariant field equations from RCQM is demonstrated to be fully analogous to the derivation of the Dirac equation from the s=1/2 RCQM, confirming the method’s consistency and generality.
• Non-covariant generators (e.g., d³x, non-covariant Poincaré generators) do not break relativistic invariance if they commute with the Dirac-like equation operator and satisfy the Poincaré algebra.
• The Fermi–Bose duality and triality properties are observed in the derived equations, suggesting deeper algebraic structures underlying arbitrary-spin field theories.