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∈ C₀∞(M) f , F ∈ C 0 ∞ ( M ) with F≡ 1 F ≡ 1 on supp(f) supp ( f ) and any timelike smooth vector field tμ t μ we can find constants $$c,C>0$$ c , C > 0 such that ω (φ (f)^*φ (f))≤ C(ω (Tʳᵉⁿμ ν(tμtνF²))+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 ω 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 |ω (φ (x))|≤ C(ω (Tʳᵉⁿμ ν(tμtνF²))+c) | ω ( ϕ ( x ) ) | ≤ C ( ω ( T μ ν ren ( t μ t ν F 2 ) ) + c ) , where F≡ 1 F ≡ 1 near x .