Vacuum fluctuations in theories with deformed dispersion relations

TL;DR

This paper investigates vacuum fluctuations in theories with modified dispersion relations that exhibit dimensional reduction at high energies, proposing a new quantization method and showing that such theories produce scale-invariant vacuum fluctuations in the ultraviolet regime.

Contribution

It introduces a general quantization scheme applicable to theories with deformed dispersion relations and demonstrates the universality of scale-invariant vacuum fluctuations in 2D ultraviolet regimes.

Findings

Vacuum fluctuations are scale-invariant in all theories with 2D ultraviolet behavior.

The proposed quantization method is applicable regardless of preferred frame assumptions.

Scale-invariance persists in expanding universe scenarios for these theories.

Abstract

We examine vacuum fluctuations in theories with modified dispersion relations which represent dimensional reduction at high energies. By changing units of energy and momentum we can obtain a description rendering the dispersion relations undeformed and transferring all the non-trivial effects to the integration measure in momentum space. Using this description we propose a general quantization procedure, which should be applicable whether or not the theory explicitly introduces a preferred frame. Based on this scheme we evaluate the power spectrum of quantum vacuum fluctuations. We find that in {\it all} theories which run to 2 dimensions in the ultraviolet the vacuum fluctuations, in the ultraviolet regime, are scale-invariant. This is true in flat space but also for "inside the horizon" modes in an expanding universe. We spell out the conditions upon the gravity theory for this scale-invariance to be preserved as the modes are frozen-in outside the horizon. We also digress on the meaning of dimensionality (in momentum and position space) and suggest that the spectral index could itself provide an operational definition of dimensionality.