# Derivation of Hamilton-like equations on a non-Cauchy hypersurface and their expected connection to quantum gravity theories

## TL;DR

This paper proposes a Hamilton-like formalism for classical fields on non-Cauchy hypersurfaces, addressing causality issues in quantum gravity contexts and suggesting potential implications for quantum gravity theories.

## Contribution

It introduces a method to define causal classical brackets on non-Cauchy hypersurfaces and demonstrates Hamilton-like evolution for fields in 3rd spatial direction.

## Key findings

- Classical brackets can be made causal on non-Cauchy hypersurfaces.
- Hamilton-like equations can describe field evolution in non-Cauchy settings.
- Relevance to quantum gravity theories is discussed.

## Abstract

Recently it was found that quantum gravity theories may involve constructing a quantum theory on non-Cauchy hypersurfaces. However this is problematic since the ordinary Poisson brackets are not causal in this case. We suggest a method to identify classical brackets that are causal on 2+1 non-Cauchy hypersurfaces and use it in order to show that the evolution of scalars and vectors fields in the 3rd spatial direction can be constructed by using a Hamilton-like procedure. Finally, we discuss the relevance of this result to quantum gravity.