2016/02/29 by Bianca Dittrich, Philipp A. Hoehn, Tim A. Koslowski +1 · 1 citation
Physics and Astronomy · Mathematics · #gr-qc #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1016/j.physletb.2017.02.038
published as Phys. Lett. B 769 (2017) 554-560 · 6 pages + references -- matches published version (clarifications added for why GR with cosmologically interesting matter likely fails our notion of weak-integrability)
arxiv created 2017/05/31 · arxiv updated 2017/06/01
Full general relativity is almost certainly 'chaotic'. We argue that this entails a notion of nonintegrability: a generic general relativistic model, at least when coupled to cosmologically interesting matter, likely possesses neither differentiable Dirac observables nor a reduced phase space. It follows that the standard notion of observable has to be extended to include non-differentiable or even discontinuous generalized observables. These cannot carry Poisson-algebraic structures and do not admit a standard quantization; one thus faces a quantum representation problem of gravitational observables. This has deep consequences for a quantum theory of gravity, which we investigate in a simple model for a system with Hamiltonian constraint that fails to be completely integrable. We show that basing the quantization on standard topology precludes a semiclassical limit and can even prohibit any solutions to the quantum constraints. Our proposed solution to this problem is to refine topology such that a complete set of Dirac observables becomes continuous. In the toy model, it turns out that a refinement to a polymer-type topology, as e.g. used in loop gravity, is sufficient. Basing quantization of the toy model on this finer topology, we find a complete set of quantum Dirac observables and a suitable semiclassical limit. This strategy is applicable to realistic candidate theories of quantum gravity and thereby suggests a solution to a long-standing problem which implies ramifications for the very concept of quantization. Our work reveals a qualitatively novel facet of chaos in physics and opens up a new avenue of research on chaos in gravity which hints at deep insights into the structure of quantum gravity.