On the Application of Bandlimitation and Sampling Theory to Quantum Field Theory

Abstract

It is widely believed that combining the uncertainty principle with gravity will lead to an effective minimum length scale. A particular challenge is to specify this scale in a coordinate-independent manner so that covariance is not broken. Here we will consider bandlimitation and sampling theory as a means to model Planck-scale modifications to spacetime within quantum field theory. Two different cases will be considered.

The first is the case of Euclidean-bandlimitation, which imposes a notion of minimal length while preserving Euclidean symmetries. This leads to a sampling theory where one can represent fields as equivalently living on either continuous space or on lattices. We will discuss how this leads to a regulation of the information density in quantum fields. We then proceed to quantify notions of localization and density of degrees of freedom within these fields.

We then turn to the case of Lorentzian-bandlimitation. Quantum fields bandlimited in this way have reconstruction properties which are qualitatively different than the Euclidean case. Nevertheless, here we will examine what impacts this has on the structure of quantum field theory with such a bandlimit imposed. In particular, we will investigate which quantities are and are not regulated by the Lorentzian bandlimit in both free and interacting field theories.