Higher Spin Quantum Fields as Twisted Dirac Fields

Abstract

We pursue the idea of constructing higher spin fields as solutions to twisted Dirac operators. As general results we find that twisted prenormally hyperbolic first order operators (such as the Dirac operator) on globally hyperbolic Lorentzian spacetime manifolds have unique advanced and retarded Green's functions and their Cauchy problem is well-posed. The space of compactly supported Cauchy data can be equipped with a naturally induced Hermitian but generally indefinite product, which is shown to be independent of the underlying choice of Cauchy hypersurface. As a special result, we study one particular example of twisted Dirac operator for Fermionic fields of higher spin. It allows a canonical Hermitian product on the spinor bundle, but the resulting product on the space of compactly supported Cauchy data fails positivity. Thus, it is shown that this construction does not allow quantization by forming a C*-algebra representation of the canonical anti-commutation relations in the well known fashion generalizing the spin 1/2 construction by Dimock (1982). This document is equipped with an introduction to the formalism of 2-spinors on curved spacetimes in an invariant fashion using abstract index notation.