On separable states in relativistic quantum field theory - Sanders, Ko (Journal of Physics A: Mathematical and Theoretical, 2023)
We initiate an investigation into separable, but physically reasonable, states in relativistic quantum field theory. In particular we will consider the minimum amount of energy density needed to ensure the existence of separable states between given spacelike separated regions. This is a first step towards improving our understanding of the balance between entanglement entropy and energy (density), which is of great physical interest in its own right and also in the context of black hole thermodynamics. We will focus concretely on a linear scalar quantum field in a topologically trivial, four-dimensional globally hyperbolic spacetime. For rather general spacelike separated regions A and B we prove the existence of a separable quasi-free Hadamard state. In Minkowski spacetime we provide a tighter construction for massive free scalar fields: given any R > 0 we construct a quasi-free Hadamard state which is stationary, homogeneous, spatially isotropic and separable between any two regions in an inertial time slice t = const. all of whose points have a distance >R . We also show that the normal ordered energy density of these states can be made ⩽1031m4/(mR)8e−1/4mR (in Planck units). To achieve these results we use a rather explicit construction of test-functions f of positive type for which we can get sufficient control on lower bounds on f ̂.
CRIS-Team Scopus Importer:2024-12-16
Keywords: energy density; entanglement; quantum field theory

Stress Tensor Bounds on Quantum Fields - Sanders, Ko (Communications in Mathematical Physics, 2024)
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) with F≡1 on supp(f) and any timelike smooth vector field tμ we can find constants c,C>0 such that ω(ϕ(f)∗ϕ(f))≤C(ω(Tμνren(tμtνF2))+c) for all (not necessarily quasi-free) Hadamard states 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 dimensions we also establish a bound on the pointwise quantum field, namely |ω(ϕ(x))|≤C(ω(Tμνren(tμtνF2))+c), where F≡1 near x.
CRIS-Team Scopus Importer:2024-05-24