Second order differential equations for bosons with spin j > 1 and in the bases of general tensor-spinors of rank-2j

TL;DR

This paper introduces a new second order differential equation for high-spin bosons that avoids the inconsistencies of previous formalisms, ensuring causal solutions within the tensor-spinor framework.

Contribution

It proposes a novel second order differential equation for spin-j bosons that is derived from a Lagrangian and guarantees causality, combining advantages of existing formalisms.

Findings

The new equation is derivable from a Lagrangian.

Solutions do not violate causality.

The formalism is based on tensor-spinors of rank-2j.

Abstract

A boson of spin-j>1 can be described in one of the possibilities within the Bargmann-Wigner framework by means of one sole differential equation of order twice the spin, which however is known to be inconsistent as it allows for non-local, ghost and acausally propagating solutions, all problems which are difficult to tackle. The other possibility is provided by the Fierz-Pauli framework which is based on the more comfortable to deal with second order Klein-Gordon equation, but it needs to be supplemented by an auxiliary condition. Although the latter formalism avoids some of the pathologies of the high-order equations, it still remains plagued by some inconsistencies such as the acausal propagation of the wave fronts of the (classical) solutions within an electromagnetic environment. We here suggest a method alternative to the above two that combines their advantages while avoiding the related difficulties. Namely, we suggest one sole strictly (j,0)+ (0,j) representation specific second order differential equation, which is derivable from a Lagrangian and whose solutions do not violate causality. The equation under discussion presents itself as the product of the Klein-Gordon operator with a momentum independent projector on Lorentz irreducible representation spaces constructed from one of the Casimir invariants of the spin-Lorentz group. The basis used is that of general tensor-spinors of rank-2j.