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Chaos, Dirac observables and constraint quantization | | Bianca Dittrich
; Philipp A. Hoehn
; Tim A. Koslowski
; Mike I. Nelson
; | | Date: |
8 Aug 2015 | | Abstract: | There is good evidence that full general relativity is non-integrable or even
chaotic. We point out the severe repercussions: differentiable Dirac
observables and a reduced phase space do not exist in non-integrable
constrained systems and are thus unlikely to occur in a generic general
relativistic context. Instead, gauge invariant quantities generally become
discontinuous, thus not admitting Poisson-algebraic structures and posing
serious challenges to a quantization. Non-integrability also renders the
paradigm of relational dynamics cumbersome, thereby straining common
interpretations of the dynamics. We illustrate these conceptual and technical
challenges with simple toy models. In particular, we exhibit reparametrization
invariant models which fail to be integrable and, as a consequence, can either
not be quantized with standard methods or lead to sick quantum theories without
a semiclassical limit. These troubles are qualitatively distinct from
semiclassical subtleties in unconstrained quantum chaos and can be directly
traced back to the scarcity of Dirac observables. As a possible resolution, we
propose to change the method of quantization by refining the configuration
space topology until the generalized observables become continuous in the new
topology and can acquire a quantum representation. This leads to the polymer
quantization method underlying loop quantum cosmology and gravity. Remarkably,
the polymer quantum theory circumvents the problems of the quantization with
smooth topology, indicating that non-integrability and chaos, while a
challenge, may not be a fundamental obstruction for quantum gravity. | | Source: | arXiv, 1508.1947 | | Services: | Forum | Review | PDF | Favorites | |
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