Quantum inequalities in two dimensional Minkowski spacetime

TL;DR

This paper extends Ford and Roman's quantum inequality results in 2D Minkowski spacetime by calculating optimal bounds and considering arbitrary weighting functions, providing a more comprehensive understanding of energy density constraints.

Contribution

It generalizes quantum inequalities by deriving the optimal lower bounds for energy density averages with arbitrary weighting functions in two-dimensional spacetime.

Findings

Calculated the maximum possible lower bound for energy density.

Characterized the quantum state achieving the optimal bound.

Extended bounds to arbitrary smooth positive weighting functions.

Abstract

We generalize some results of Ford and Roman constraining the possible behaviors of renormalized expected stress-energy tensors of a free massless scalar field in two dimensional Minkowski spacetime. Ford and Roman showed that the energy density measured by an inertial observer, when averaged with respect to that observers proper time by integrating against some weighting function, is bounded below by a negative lower bound proportional to the reciprocal of the square of the averaging timescale. However, the proof required a particular choice for the weighting function. We extend the Ford-Roman result in two ways: (i) We calculate the optimum (maximum possible) lower bound and characterize the state which achieves this lower bound; the optimum lower bound differs by a factor of three from the bound derived by Ford and Roman for their choice of smearing function. (ii) We calculate the lower bound for arbitrary, smooth positive weighting functions. We also derive similar lower bounds on the spatial average of energy density at a fixed moment of time.