Pointwise spinor quantum fields cannot be microcausal nor Poincar\'e covariant

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 also extend Wightman's no-go theorem to show that the weak continuity of quantum fields rules out fermionic microcausality in $C^2$ Lorentzian spacetimes. We finish by generalising Wizimirski's no-go theorem to show that the existence of a Poincar\'e-invariant vacuum precludes pointwise spinorial covariance on a Minkowski background -- ruling out, in particular, pointwise covariance for Weyl and Dirac fermions, for photons and gravitons -- which further highlights the difficulty of quantising gravity pointwisely.