[Paper Review] An Extended Poincare Algebra for Linear Spinor Field Equations

James Lindesay|ArXiv.org|Aug 14, 2003

Quantum and Classical Electrodynamics3 references3 citations

TL;DR

This paper extends the Poincaré algebra to include additional generators that allow for linear spinor field equations with non-trivial commutation relations between energy-momentum and gamma matrices. It introduces a new operator, 𝒢, interpreted as a transverse mass operator, enabling finite-dimensional unitary representations and a linear wave equation 𝛾𝜇(−i∂𝜇)ψ = λψ, which supports cluster-decomposable scattering formalisms for arbitrary spin systems.

ABSTRACT

When utilizing a cluster decomposible relativistic scattering formalism, it is most convenient that the covariant field equations take on a linear form with respect to the energy and momentum dispersion on the fields in the manner given by the Dirac form for spin ${1 \over 2}$ systems. The general spinor formulation for arbitrary spins given in a previous paper is extended to include momentum operators. Unitary quantum mechanical representations are developed for these operators, and physical interpretations are suggested.

Motivation & Objective

• To develop a manifestly cluster-decomposable relativistic scattering formalism by extending the Poincaré algebra to include momentum operators and non-vanishing commutation relations between energy-momentum and Γμ operators.
• To construct finite-dimensional unitary representations of the extended Poincaré group using the little group and complementary transformations on standard state vectors.
• To introduce a new operator 𝒢 required for algebraic closure, which is interpreted as a transverse mass operator mixing with the physical mass under finite group transformations.
• To derive a linear wave equation 𝛾𝜇(−i∂𝜇)ψ = λψ that supports conserved currents and allows for off-shell, off-diagonal dynamics in non-interacting clusters.
• To explore physical interpretations of the extended algebra, particularly the mixing of mass parameters and the role of 𝒢 in finite transformations of energy-momentum eigenvalues.

Proposed method

• Extends the Lorentz group by including generators for space-time translations, resulting in an extended Poincaré algebra with non-vanishing commutators between Γμ and Pμ, and between Γμ and Kν.
• Introduces a new operator 𝒢 to close the algebra, satisfying commutation relations [𝒢, Pβ] = −iηβνPνPν and [𝒢, Γμ] = −iPμ.
• Constructs finite-dimensional unitary representations using spinor formalism, with eigenstates defined via raising/lowering operators ΔJ± and spinor basis states χ±.
• Derives a linear differential wave equation in configuration space: Γμ(−i∂μ)ψ = λψ, with λ = γμ, where μ is a mass-like parameter.
• Analyzes finite group transformations using exponential maps, showing that time-like transformations mix P₀ (mass) and 𝒢 eigenvalues, leading to oscillatory mass behavior.
• Demonstrates that the operator ΓμPμ is a scalar under the extended group, ensuring invariance of the conserved current ∂μ(ψ̄Γμψ) = 0.

Experimental results

Research questions

• RQ1 How can the Poincaré algebra be extended to include momentum operators while preserving unitary representations for arbitrary spin systems?
• RQ2 What algebraic closure conditions are required when introducing non-vanishing commutators between energy-momentum and Γμ operators?
• RQ3 What physical interpretation can be assigned to the new operator 𝒢 introduced to close the algebra?
• RQ4 How do finite transformations in the extended group parameters affect the eigenvalues of mass and transverse mass operators?
• RQ5 Can a linear wave equation be constructed that supports cluster-decomposable scattering formalisms and conserves a current without requiring the standard probability interpretation?

Key findings

• The extended algebra requires the addition of a new operator 𝒢 to close the commutation relations, with [𝒢, Pβ] = −iηβνPνPν and [𝒢, Γμ] = −iPμ.
• The Casimir operator CΛ = J·J − K·K + Γ⁰Γ⁰ − Γ·Γ is invariant under the extended group, ensuring consistency of the representation theory.
• Finite-dimensional unitary representations are constructed using spinor basis states, with Γ⁰ eigenvalues labeled by γ and Jz eigenvalues by M, satisfying Ĥψ = γμψ.
• The wave equation Γμ(−i∂μ)ψ = λψ is derived, with λ = γμ, and implies a conserved current ∂μ(ψ̄Γμψ) = 0, independent of the standard probability current.
• Finite transformations via e^{iωμΓμ} mix P₀ (mass) and 𝒢 eigenvalues, leading to oscillatory behavior of mass parameters under continuous group parameters.
• The operator 𝒢 is interpreted as a transverse mass operator, and its non-vanishing expectation value prevents the existence of finite-dimensional unitary representations unless group parameters are discrete.