Second-order differential equations for bosons with spin $j \geq 1$ and in the bases of general 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 D (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.

Problems and Progress in Covariant High Spin Description

“…Such a formalism for spin-j in single-spin-(j, 0) ⊕ (0, j) reps has been designed in refs. [13], [14], [15], and is briefly highlighted below.…”

“…Progress in describing any spin-j in (j, 0) ⊕ (0, j) in Lorentz-tensor-, Lorentz-tensor-Dirac-spinor-, or, Weyl-Van-der-Waerden tensor-spinor bases and wave equations of second order. The spin-Lorentz group projector method The first goal of the references [13], [14], [15] has been to describe the pure spin (j, 0) ⊕ (0, j) states, be it through Lorentz-, or through Weyl-Van-der-Waerden spinor-tensors, this for the sake of constructing by simple index contractions vertexes which involve interactions of highspins with gauge fields, such as the photon, and/or spinorial targets, such as the proton, and thus avoid the introduction of the cumbersome index-matching rectangular matrices, typical for the Joos-Weinberg formalism. In order to illustrate the essentials of the method, we here bring as representative examples the two simplest cases, beginning with the description of the (1, 0) ⊕ (0, 1) field as a totally anti-symmetric Lorentz tensor of second rank, B [µ,ν] , with the brackets denoting index anti-symmetrization.…”

“…The equation (15) shows that the operator P (3/2,0)⊕(0,3/2) F has the property to nullify any irreducible representation space which is different from (3/2, 0) ⊕ (0, 3/2). Instead, for (3/2, 0) ⊕ (0, 3/2), it acts as the identity operator, meaning that P (3/2,0)⊕(0,3/2) F is a projector on this very space.…”

“…The above presentation should leave it clear that the scheme does not restrict to product spaces of the type given in (9) but extends to any product spaces containing the pure spin (j, 0) ⊕ (0, j) of interest. In particular, it applies to bases constructed as direct products of four-vectors, or, to general Weyl-Van-der-Waerden tensor-spinors, as used in the Bargmann-Wigner framework [15]. Moreover, the scheme also extends to high-spins carried by two-spin valued representation spaces of the type, (1/2, j − 1/2) ⊕ (j − 1/2, 1/2), [14], in which case the second order wave equation is not obtained from the mass shell-condition alone but from combining it by another covariant projector, P (j,m) W 2 = (−W 2 /m 2 −j(j −1)p 2 /m 2 )/(2j), which projects besides on the mass shell, also on the highest of the two spin degrees of freedom.…”

“…In parallel to (15), also the G invariant of the spin-Lorentz group algebra can be employed in the construction of projector operators within any (j 1 , j 2 ) ⊕ (j 2 , j 1 ) representations space. Such operators, here denoted by P (j 1 ,j 2 ) G and P (j 2 ,j 1 ) G , respectively, have the property to separate the reps under consideration into left (L)-and right (R)-handed degrees of freedom according to,…”