[Paper Review] A Note on the Majorana-Oppenheimer Quantum Electrodynamics

В. В. Варламов|arXiv (Cornell University)|Jun 6, 2002

Algebraic and Geometric Analysis26 references3 citations

TL;DR

This paper presents a group-theoretical formulation of Majorana-Oppenheimer quantum electrodynamics, where the electromagnetic field is described via a complex spinor wave function ψ = E − iB, leading to Dirac-like equations for photons. The key contribution is a rigorous Lagrangian formulation using the Gel’fand-Yaglom formalism, which naturally incorporates observable fields and avoids gauge redundancies, offering a unified, group-theoretically consistent framework for all physical fields without distinguishing between 'gauge' and 'matter' fields.

ABSTRACT

A group theoretical description of the Majorana-Oppenheimer quantum electrodynamics is considered. Different spinor realizations of the Maxwell and Dirac fields are discussed. A representation of the Majorana-Oppenheimer wave equations in terms of the Gel'fand-Yaglom formalism is given.

Motivation & Objective

• To address the foundational difficulties in Gupta-Bleuler quantization of the electromagnetic field, such as unphysical degrees of freedom and indefinite metric.
• To reformulate quantum electrodynamics using the Majorana-Oppenheimer approach, which treats the photon as a massless particle described by a spinor wave function ψ = E − iB.
• To provide a group-theoretically consistent framework based on the Lorentz and Poincaré groups, avoiding the gauge-field dichotomy of the Standard Model.
• To establish a Lagrangian formulation of the Majorana-Oppenheimer equations via the Gel’fand-Yaglom formalism, enabling a natural field-theoretic description.

Proposed method

• Represent the electromagnetic field via a complex vector spinor ψ = E − iB, transforming the Maxwell equations into a Dirac-like form (W − α·p)ψ = 0 with transversality condition p·ψ = 0.
• Use the matrices α¹, α², α³ defined by the angular-momentum commutation rules to realize the Dirac-like structure for the photon.
• Construct the Lagrangian L_M = −½(ψ*α_μ ∂ψ/∂x_μ − ∂ψ*/∂x_μ α_μ ψ) for the massless case, with α₀ = I.
• Derive the Euler-Lagrange equations from the Lagrangian, yielding α_μ ∂ψ/∂x_μ = 0 and αᵀ_μ ∂ψ*/∂x_μ = 0, equivalent to the original Majorana-Oppenheimer equations.
• Utilize the tensor product of biquaternion algebras C₂ and C̄₂ to construct higher-spin spaces S_{2^{k+r}}, enabling a unified representation of field states.
• Apply the Gel’fand-Yaglom formalism to ensure a consistent Lagrangian formulation and to connect the theory to representation theory of the Lorentz group.

Experimental results

Research questions

• RQ1 Can the Majorana-Oppenheimer formulation of electrodynamics be consistently embedded within a group-theoretical framework based on the Poincaré group and spinor representations?
• RQ2 How can the Gel’fand-Yaglom formalism be adapted to describe the photon as a massless particle via a complex spinor wave function ψ = E − iB?
• RQ3 What is the role of the Lagrangian formulation in ensuring a consistent and observable field theory free from gauge redundancies?
• RQ4 How does the Majorana-Oppenheimer approach eliminate the distinction between 'gauge' and 'matter' fields present in the Standard Model?
• RQ5 Can the formalism naturally accommodate the photon’s self-conjugacy (ψ = ψ*), implying it is its own antiparticle, through group-theoretic structures?

Key findings

• The Majorana-Oppenheimer formulation leads to Dirac-like equations (W − α·p)ψ = 0 and (W + α·p)ψ* = 0, which describe the photon as a truly neutral, massless particle with helicity ±1.
• The Lagrangian L_M = −½(ψ*α_μ ∂ψ/∂x_μ − ∂ψ*/∂x_μ α_μ ψ) yields the correct equations of motion, confirming the consistency of the field-theoretic formulation.
• The theory naturally incorporates the photon’s self-conjugacy, as ψ and ψ* describe the same particle, implying photons are their own antiparticles.
• The current j₀ = ψ*α₀ψ is proportional to the electromagnetic field energy E² + B², indicating a physical, observable charge density.
• The Gel’fand-Yaglom formalism provides a systematic, group-theoretically grounded Lagrangian formulation that avoids the need for gauge fixing and unphysical degrees of freedom.
• The construction of higher-spin spaces via tensor products of C₂ and C̄₂ algebras enables a unified description of field states within the spinor representation framework of the Lorentz group.