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Parity-based formalism for high spin matter fields

Using the recent parity-based construction of a covariant basis for operators acting on the $(j,0)\oplus(0,j)$ representation of the HLG, we propose a formalism for the description of high spin matter fields, based on the projection over subspaces of well-defined parity. We identify two possibilities for the projection, on-shell and off-shell projection. For all $j$ except for $j=1/2$, we find that the projection does not completely fix the properties of the interacting theory. This freedom is related to the fact that the covariant form of parity can be written in terms of one of the symmetric traceless tensors in the covariant basis and in general allows for a free magnetic dipole term in the lagrangian. We gauge the theory and construct the charge conjugation operator. In the case of bosons, the parity invariant subspaces are also invariant under charge conjugation and time reversal and the formulation of a quantum field theory can be done using only these subspaces. As a first exhaustive example we work out the electrodynamics for $j=1$ matter bosons, rewrite the theory in terms of an antisymmetric tensor field and compare our results with existing formalisms in the literature. We find that there are three essentially different formalisms: i) formalisms equivalent to the on-shell parity projection, ii) formalisms equivalent to the off-shell parity projection and iii) the Poincaré projector formalism which describes a degenerate parity doublet. We perform a chiral decomposition of these theories and show that chiral symmetry can be realized linearly only for the theory based on the on-shell projection. Chiral symmetry forbids mass and anomalous magnetic dipole terms and in general admits six self-interaction terms. We conclude that this is the appropriate framework to attempt the incorporation of spin 1 matter bosons in chiral theories like the standard model.