Poincare field theory for massless particles

TL;DR

This paper proposes deriving field equations for all spins, including massless particles, from Poincare group Casimir eigenvalue equations, confirming this approach for various massless fields and reproducing key results like Maxwell and Einstein equations.

Contribution

It introduces a unified method to obtain field equations for all spins from Poincare Casimir eigenvalues, extending previous work to massless fields and demonstrating consistency with known physics.

Findings

Derived Maxwell equations from Poincare eigenvalue equations.

Obtained Einstein equations in weak field approximation.

Confirmed the approach for a wide class of massless fields.

Abstract

Our main proposition is that field equations for all spins can be obtained from Casimir eigenvalue equations for Poincare group. We have already confirm that statement for massive scalar, spinor and vector fields in Ref.[1]. In the present article we are going to confirm this statement for massless vector, second and fourth rang tensor fields. In particular we will obtain Maxwell equations and Einstein equation in weak field approximation. As is well known, Wigner define a particles as irreducible representation of Poincare group [2,3]. But, as Weinberg noted in [4] irreducible representations for massless vector field with helicities $\pm 1$ do not exist. In the present article we will conform this statement for a wide class of massless fields. They are not Lorentz invariant since their Lorentz transformations have additional term in the form of gauge transformations. Such fields can appear in the theory only in the form which do not depend on corresponding gauge parameters. These forms are our equations of motion and they are by definition gauge invariant. So, the massless case is significantly different from massive one. In the end we will show that our approach can reproduce main contributions from well known articles with highest helicity.