Stress tensor bounds on quantum fields

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 $x$.