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 \(f,F\in C_0^{\infty }(M)\) $f,F\in C_0^{\infty }(M)$ with \(F\equiv 1\) $F\equiv 1$ on \(\textrm{supp}(f)\) $\text{supp}(f)$ and any timelike smooth vector field \(t^{\mu }\) $t^{\mu }$ we can find constants \(c,C>0\) $c,C>0$ such that \(\omega (\phi (f)^*\phi (f))\le C(\omega (T^{\textrm{ren}}_{\mu \nu }(t^{\mu }t^{\nu }F^2))+c)\) $\omega (\varphi {(f)}^{\ast }\varphi (f))\le C(\omega (T_{\mu \nu }^{\text{ren}}(t^{\mu }{t}^{\nu }{F}^2))+c)$ for all (not necessarily quasi-free) Hadamard states \(\omega \) $\omega$ . This is essentially a new type of quantum energy inequality that entails a stress tensor bound on the smeared quantum field. In \(1+1\) $1+1$ dimensions we also establish a bound on the pointwise quantum field, namely \(|\omega (\phi (x))|\le C(\omega (T^{\textrm{ren}}_{\mu \nu }(t^{\mu }t^{\nu }F^2))+c)\) $|\omega (\varphi (x))|\le C(\omega (T_{\mu \nu }^{\text{ren}}(t^{\mu }{t}^{\nu }{F}^2))+c)$ , where \(F\equiv 1\) $F\equiv 1$ near x.