Can chaos be observed in quantum gravity?

TL;DR

The paper argues that full general relativity is likely weakly non-integrable, lacking a global set of differentiable Dirac observables and a reduced phase space, which obstructs conventional quantization. By studying a toy model of two free particles on a circle with fixed energy, the authors show that standard topology fails to produce semiclassical states and viable solutions to the quantum constraints. They then demonstrate that a refined, polymer-like topology restores a complete, continuous set of Dirac observables and yields a well-defined semiclassical limit, suggesting that the underlying topology of quantization is crucial for gravity. The results imply that quantum gravity may require topology refinement beyond standard canonical methods and could influence how we formulate observable algebras, the path integral projector, and the quantization of realistic gravitational theories.

Abstract

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.

Figures (1)

• Figure 1: A trajectory on the torus with angle given by $\gamma$. $n_i$ increases by $1$ whenever $x_i=1$ is reached.