No-go theorems for pointwise-defined spinorial quantum fields

Abstract

We extend and strengthen no-go results on pointwise-defined quantum fields to cover general spinors. We show that the weak continuity of quantum fields rules out equal-time canonical conjugate (anti)commutation relations in globally hyperbolic spacetimes; for quantum fields on Minkowski spacetime, weakly continuous translation covariance enforces the needed continuity and yields the same no-go. We then prove a fermionic microcausality no-go result: a weakly continuous pointwise fermionic field satisfying local spacelike anticommutation with its adjoint field on a $C^2$ Lorentzian spacetime must vanish. We finish by generalising Wizimirski's no-go theorem to show that the existence of a Poincaré-invariant separating vacuum precludes pointwise spinorial covariance on a Minkowski background. The result applies to Weyl and Dirac multiplets and to gauge-invariant field-strength multiplets such as the electromagnetic field strength and the linearised Weyl curvature. Gauge potentials are covered only when exact tensor covariance, rather than covariance modulo gauge transformations, is imposed.

11 pages