Can chaos be observed in quantum gravity?

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.

6 pages + references -- matches published version (clarifications added for why GR with cosmologically interesting matter likely fails our notion of weak-integrability)

References in corpus (6)

• Polymer Quantum Mechanics and its Continuum Limit
• An effective approach to the problem of time
• Effective approach to the problem of time: general features and examples
• Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?
• Gauge invariant perturbations around symmetry reduced sectors of general relativity: applications to cosmology
• Chaos, Dirac observables and constraint quantization

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• Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
• Quantum gravity in timeless configuration space
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• Perspective-neutral approach to quantum frame covariance for general symmetry groups
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• Freeze-free cosmological evolution with a non-monotonic internal clock
• Algebraic properties of quantum reference frames: Does time fluctuate?
• Incompleteness Theorems for Observables in General Relativity
• Benchmarking quantum chaos from geometric complexity
• Relational Dynamics with Periodic Clocks
• Gravitational null rays: Covariant Quantization and the Dressing Time
• Quantum cosmology in teleparallel gravity with a boundary term
• The Limitations of the Notion of `Observable' in Diffeomorphism-Invariant Models
• Does quantum gravity relate the constants of nature?
• Large effects from quantum reference frames
• Diffeomorphism-Invariant Quantities in Phase Space: More than Correlations