Relativistic Quantum Geometries for Spin-0 Massive Fields

Abstract

The formulation of the canonical quantization procedure for spin-1/2 quantum fields proceeds in curved spacetime in very much the same manner as for the spin-0 case described in Sec. 7.1, as can be best seen from the comparative summary of the two respective procedures provided by Gibbons (1979). It is based on the curved spacetime counterpart of the Dirac equation, obtained by replacing in (6.1.13) the partial derivatives ∂μ by the covariant derivatives ∇μ, and then adopting as an inner product in the single fermion-antifermion space H the counterpart of (6.1.16) along the spacelike reference hypersurfaces σ t in a foliation (5.4.7) of a globally hyperbolic spacetime (M, g). As opposed to the situation in the spin-0 case, that inner product is positive definite in the spin-1/2 case. However, that does not help with the difficulties of the canonical approach described in Secs. 7.1-7.3, since an unambiguous decomposition H = H (+) ⊕ H (−) into a subspace H (+) of positive-frequency solutions representing fermion state vectors, and a subspace H (−) of negative-energy solutions representing antifermion state vectors, does not exist in the spin-1/2 case either, due to the absence of global Poincare invariance