1 Introduction

Recently several authors have proposed searches for quantum gravity effects in the propagation of light; for instance, according to Ellis, Mavromatos and Nanopoulos [1], the vacuum of quantum gravity should be dispersive and this should show up in the arrival times of energetic photons from distant sources like gamma-ray bursters, active galactic nuclei, and gamma-ray pulsars[2]. Others [3,4] have proposed searches in large earth-based interferometers, like those used for gravitational wave detection, starting from the considerations suggested by the work of Wigner [5] and Wigner and Salecker [6], who - more than forty years ago - tried to clarify the interplay between general relativity and quantum physics using the uncertainty principle. Wigner and Salecker showed that quantum phenomena might set severe limits on the concept of spacetime on short length scales and gave a lucid analysis of our understanding of these concepts.

Here we propose yet another heuristic treatment of the fluctuations of the gravitational energy density and gravitational field, based on the direct use of the uncertainty principle for energy and time, with additional assumptions on the dynamic and statistical behaviour of the resulting quantum fluctuations. Afterwards we use this model to evaluate the spectral properties of the energy density fluctuations and to derive the behaviour of light in a generic two-arm interferometer.

Accordingly, this paper is organized as follows: in section 2 we introduce the heuristic calculation of quantum fluctuations of the energy density from a simple statistical point of view, assuming that the uncertainty principle and some crude form of relativistic causality are satisfied; we also assume specific dynamics for the fluctuations. In section 3 we derive the fluctuations in the gravitational potential induced by the energy fluctuations. Since our description is statistical in nature, the propagation of light in this model must be described by the space-time correlation functions of the gravitational potential and we devote section 4 to the estimation of the second order correlations. In section 5 we show how to use these correlation functions to study the light fluctuations in a two-arm interferometer. We make some additional remarks and conjectures in section 6 and finally we summarize our conclusions in section 7.

We also wish to remark that - contrary to common usage - we do explicitly retain all factors $\hbar$ and $c$ in the subsequent formulas, since we want to stress the role of the fundamental constants.