A Quantum Energy Inequality for a Non-commutative QFT
Harald Grosse, Albert Much
TL;DR
This work extends quantum energy inequalities to quantum field theories on non-commutative spacetimes by formulating the deformed energy density $T_{00}^{\\Theta}$ via warped convolutions and Wick ordering of NC field operators. Employing the Waldmann positivity map $S_{\\theta}$, it constructs a positive operator $W_{\\Theta}^{\\pm}$ and derives a state-independent lower bound for the smeared NC energy density, mirroring the commutative case. Crucially, the resulting NC QEI bound is independent of the non-commutative scale $\\theta$ and reduces to the standard Fewster bound as $\\theta \\\to 0$, preserving causality at macroscopic scales while retaining NC structure at short distances. The authors also outline a program to apply microlocal analysis to warped-convolution deformations to obtain further QEIs, including potential worldvolume formulations, thereby strengthening the link between quantum geometry and semiclassical locality.
Abstract
We present a quantum energy inequality (QEI) for quantum field theories formulated in non-commutative spacetimes, extending fundamental energy constraints to this generalized geometric framework. By leveraging operator-theoretic methods inspired by the positivity map of Waldmann et al. \cite{waldmannpos}, we construct linear combinations of deformed operators that generalize the commutative spacetime techniques of Fewster et al., \cite{Few98}. These non-commutative analogs enable us the derivation of a lower bound on the deformed averaged energy density, ensuring the stability of the underlying quantum field theory. Our result establishes rigorous constraints on the expectation values of the deformed (non-commutative) energy density, reinforcing the physical consistency of non-commutative models while preserving core principles of quantum field theory.