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Pointwise spinor quantum fields cannot be microcausal nor Poincaré covariant

Samuel Fedida

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 Poincaré-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.

Pointwise spinor quantum fields cannot be microcausal nor Poincaré 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 Lorentzian spacetimes. We finish by generalising Wizimirski's no-go theorem to show that the existence of a Poincaré-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.

Paper Structure

This paper contains 6 sections, 8 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Canonical (Anti)Commutation Relations
  3. Microcausality
  4. Poincaré covariance
  5. No-go -- spinor covariance for nontrivial representations
  6. No-go -- spinor covariance for gravitons

Key Result

Theorem 1

Let $(\mathcal{M},g)$ be a globally hyperbolic Lorentzian spacetime and $\{\Sigma_t\} _{t \in \mathbb{R}}$ be a foliation of $\mathcal{M}$ into spacelike Cauchy hypersurfaces, and $\mathcal{D}$ be a dense subset of $\mathcal{H}$. Let $\hat{\psi} : \mathcal{M} \to \mathcal{L}(\mathcal{D},\mathcal{H}) Then for any $t \in \mathbb{R}$ and $x \in \Sigma_t$, and where these (anti)commutators are under

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem : Wightman
  • Theorem 2
  • proof
  • Corollary 2
  • proof
  • Theorem : Wizimirski
  • ...and 4 more