Consistency Conditions for Fundamentally Discrete Theories

TL;DR

This paper investigates the conditions that difference equations must satisfy to be consistent with their continuum differential equation counterparts, especially in the context of quantum gravity and loop quantum cosmology.

Contribution

It derives specific consistency conditions for fundamental difference equations, providing a criterion to select acceptable quantizations in theories like loop quantum cosmology.

Findings

Conditions are restrictive and serve as a selection criterion.

Difference equations cannot be chosen freely but must derive from fundamental theories.

Implications for acceptable quantizations in quantum gravity.

Abstract

The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfill certain conditions to ensure that the numerical solutions can reliably be used as approximations to solutions of the differential equation. There are, however, also systems where a difference equation is deemed to be fundamental, mainly in the context of quantum gravity. Since difference equations in general are harder to solve analytically than differential equations, it can be helpful to introduce an approximating differential equation as a continuum approximation. In this paper implications of this change in view point are analyzed to derive the conditions that the difference equation should satisfy. The difference equation in such a situation cannot be chosen freely but must be derived from a fundamental theory. Thus, the conditions for a discrete formulation can be translated into conditions for acceptable quantizations. In the main example, loop quantum cosmology, we show that the conditions are restrictive and serve as a selection criterion among possible quantization choices.