Chaos in Constrained System

Abstract 

Chaos poses technical challenges to constrained Hamiltonian systems. This is an important topic for discussion, because general relativity in its Hamiltonian formulation is a constrained system, and there is strong evidence that it exhibits chaotic features. We review concepts in gauge systems and their association with Hamiltonian constraints, relational Dirac observables as gauge-invariant encodings of physical information, and chaos in unconstrained Hamiltonian systems. We then construct a non-integrable, ergodic toy model, and with it explicitly illustrate the non-existence of a maximal set of Dirac observables and a solution space that fails to be a manifold. The potential consequences of these qualitative features of a chaotic constrained Hamiltonian system for general relativity and the quest for its quantum theory are deliberated.