Stress Tensor Bounds on Quantum Fields

Abstract

AbstractThe 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 ∈ C 0 ∞ ( M ) with $$F\equiv 1$$ F ≡ 1 on $$\textrm{supp}(f)$$ supp ( f ) and any timelike smooth vector field $$t^{\mu }$$ t μ we can find constants $$c,C>0$$ c , C > 0 such that $$\omega (\phi (f)^*\phi (f))\le C(\omega (T^{\textrm{ren}}_{\mu \nu }(t^{\mu }t^{\nu }F^2))+c)$$ ω ( ϕ ( f ) ∗ ϕ ( f ) ) ≤ C ( ω ( T μ ν ren ( t μ t ν F 2 ) ) + c ) for all (not necessarily quasi-free) Hadamard states $$\omega $$ ω . 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 $$|\omega (\phi (x))|\le C(\omega (T^{\textrm{ren}}_{\mu \nu }(t^{\mu }t^{\nu }F^2))+c)$$ | ω ( ϕ ( x ) ) | ≤ C ( ω ( T μ ν ren ( t μ t ν F 2 ) ) + c ) , where $$F\equiv 1$$ F ≡ 1 near x.