The Quantum Vacuum of Spacetime with a Fundamental Length

A quantum theory of gravity implies a fine-grained structure of spacetime, which can be conveniently modeled as some form of pixelation at the Planck scale, with potentially observable consequences. In this work, we build upon previous results to investigate the effect of pixelation on the quantum vacuum, making use of the framework of Doubly Special Relativity (DSR). At the center of the DSR approach is an observer dependent length scale, defining the pixelation of spacetime. A key feature of quantum field theory in DSR is the dispersive nature of the vacuum state and the associated appearance of curvature in momentum space. As a result, the standard treatment of the renormalized stress-energy-momentum tensor acquires correction terms. As an illustration, we present here a calculation of the thermal vacuum and modified Casimir effect, using both modified propagators and momentum measures. We choose a consistent choice of momentum space metric that both generates the modified dispersion relations we use and preserves the Lorentz invariant character of the results obtained. Put together this constitutes a consistent calculation framework we can apply to other more complex scenarios.