Quantum $L^p$ inequalities in thermal states
Henning Bostelmann, Daniela Cadamuro, Leonardo Sangaletti
TL;DR
The paper develops a modular-theoretic framework for local Liouvillian densities in thermal quantum field theory and proves that, in typical linear models, these densities satisfy nontrivial quantum $L^4$ inequalities (lower bounds in the noncommutative $L^4$ norm) while not admitting corresponding upper bounds. It formalizes a split between system and bath densities via $h$ and $ ilde{h}$, and defines strong and weak notions of density that connect local densities to the global Liouvillian generator $L$. The authors verify the theory with explicit calculations for a real scalar field on Minkowski space in a thermal state, including the Rindler wedge where the Unruh temperature arises, and they show both positive $L^4$ bounds and nontrivial violations of more stringent bounds in certain regimes. The results illuminate how passivity and modular structure constrain local energy-like densities in thermal and wedge contexts, with potential implications for relativistic quantum information and extensions to interacting theories.
Abstract
In thermal quantum field theory, the global Liouvillian (the generator of time translations) is passive. How is this reflected in the properties of its local density, a quantum field? We propose that the locally averaged density is bounded below, but not above, with respect to the noncommutative $L^4$ norm. This is analogous to the known quantum energy inequalities in the vacuum situation. Our examples include thermal equilibrium on Minkowski space, and the Unruh effect on the Rindler wedge, both for the real scalar free field.