Stress tensor bounds on quantum fields

TL;DR

This paper develops bounds on quantum fields in curved spacetime using stress tensor components instead of the Hamiltonian, enabling rigorous analysis of quantum fields in more general geometries.

Contribution

It introduces a new type of quantum energy inequality based on stress tensor components, replacing the Hamiltonian for bounds in curved spacetimes.

Findings

Established stress tensor bounds for quantum fields in globally hyperbolic spacetimes.

Derived a pointwise quantum field bound in 1+1 dimensions.

Provided a framework for stress tensor-based quantum energy inequalities.

Abstract

The singular behaviour of quantum fields in Minkowski space can often be bounded by polynomials of the Hamiltonian $H$. These so-called $H$-bounds and related techniques allow us to handle pointwise quantum fields and their operator product expansions in a mathematically rigorous way. A drawback of this approach, however, is that the Hamiltonian is a global rather than a local operator and, moreover, it is not defined in generic curved spacetimes. In order to overcome this drawback we investigate the possibility of replacing $H$ by a component of the stress tensor, essentially an energy density, to obtain analogous bounds. For definiteness we consider a massive, minimally coupled free Hermitean scalar field. Using novel results on distributions of positive type we show that in any globally hyperbolic Lorentzian manifold $M$ for any $f,F\in C_0^{\infty}(M)$ with $F\equiv 1$ on $\mathrm{supp}(f)$ and any timelike smooth vector field $t^{\mu}$ we can find constants $c,C>0$ such that $\omega(\phi(f)^*\phi(f))\le C(\omega(T^{\mathrm{ren}}_{\mu\nu}(t^{\mu}t^{\nu}F^2))+c)$ for all (not necessarily quasi-free) Hadamard states $\omega$. This is essentially a new type of quantum energy inequality that entails a stress tensor bound on the smeared quantum field. In $1+1$ dimensions we also establish a bound on the pointwise quantum field, namely $|\omega(\phi(x))|\le C(\omega(T^{\mathrm{ren}}_{\mu\nu}(t^{\mu}t^{\nu}F^2))+c)$, where $F\equiv 1$ near $x$.