Chaos in Constrained Systems

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 which 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.