[Paper Review] Galilei invariant theories. II. Wave equations for massive fields

J. Niederle, A. G. Nikitin|ArXiv.org|Jul 23, 2007

Geophysics and Sensor Technology10 references3 citations

TL;DR

This paper derives and classifies Galilei-invariant wave equations for massive fields of spins 0, 1/2, 1, and 3/2 using the Bhabha approach and representations of the homogeneous Galilei group. It shows that a broad class of non-equivalent, physically consistent wave equations exists, resolving limitations of prior relativistic-inspired models in non-relativistic settings.

ABSTRACT

Galilei invariant equations for massive fields with various spins are found and classified. They have been obtained directly, i.e., by using requirement of Galilei invariance and the facts on representations of the Galilei group deduced in our previous paper de Montigny M, Niederle J and Nikitin A G, J. Phys. A {\bf 39}, 1-21, 2006 . It is shown that the collection of non-equivalent Galilei-invariant wave equations for vector and scalar fields is very broad and describes many physically consistent systems.

Motivation & Objective

• To systematically derive Galilei-invariant wave equations for massive fields with various spins using the Galilei group's representations.
• To overcome shortcomings of prior relativistic-inspired models that violate causality or predict incorrect gyromagnetic ratios in non-relativistic settings.
• To classify all non-equivalent Galilei-invariant wave equations for scalar and vector fields, extending beyond standard formulations.
• To establish a consistent framework for constructing quantum mechanical and field-theoretical models of interacting particles with spin 0, 1/2, 1, and 3/2 under Galilean invariance.
• To provide a complete description of the matrix structure of the wave equation coefficients (βμ, β4) under Galilei symmetry constraints.

Proposed method

• Applies the Bhabha approach by extending the relativistic method of first-order linear partial differential equations to the Galilean setting.
• Uses the Lie algebra of the homogeneous Galilei group HG(1,3) with generators P₀, Pₐ, Jₐ, Gₐ, and imposes invariance under Galilean transformations.
• Imposes transformation rules for the spinor field Ψ under Galilean boosts and rotations, with phase factors involving mass m and velocity v.
• Derives the matrix structure of β₄ and βμ by requiring invariance of the equation (βμpμ + β₄m)Ψ = 0 under the Galilei group action.
• Classifies solutions via submatrices R and E derived from β₄, based on the spin content (m, n, λ) of the representation.
• Relies on previously established classification of finite-dimensional indecomposable representations of HG(1,3) that decompose into spin 0, 1/2, and 1 representations upon restriction to the rotation subgroup.

Experimental results

Research questions

• RQ1 What is the complete set of non-equivalent Galilei-invariant wave equations for massive scalar and vector fields?
• RQ2 How can the Bhabha approach be adapted to construct Galilei-invariant equations for particles with spin 0, 1/2, 1, and 3/2?
• RQ3 Why do standard relativistic wave equations (e.g., Proca, Rarita-Schwinger) fail in non-relativistic settings, and how can they be corrected via Galilei invariance?
• RQ4 What is the role of projective and finite-dimensional indecomposable representations of the homogeneous Galilei group in constructing consistent non-relativistic field theories?
• RQ5 Can a unified framework be established for Galilei-invariant equations that avoids causality violations and predicts correct gyromagnetic ratios?

Key findings

• A broad and complete classification of non-equivalent Galilei-invariant wave equations for scalar and vector fields is achieved, showing that such equations are not unique but form a rich family.
• The matrix β₄ in the wave equation (βμpμ + β₄m)Ψ = 0 is fully determined by the spin content (m, n, λ) and decomposes into submatrices R and E that classify the system's structure.
• For spin 1/2, the equations reproduce the most general Pauli interaction with an external electromagnetic field, consistent with Galilean invariance.
• The method successfully constructs equations for spin 3/2 fields that avoid the causality and g-factor problems of earlier tensor-spinorial formulations.
• The classification reveals that for certain spin configurations (e.g., m=n=1, λ=0), the matrix E or R may not exist, indicating constraints on possible realizations.
• The framework provides a consistent foundation for non-relativistic quantum field theories of interacting particles with spins 0, 1/2, 1, and 3/2 under Galilean symmetry.