JID:PLB AID:32654 /SCO Doctopic: Theory [m5Gv1.3; v1.213; Prn:27/04/2017; 13:39] P.1 (1-7)
Physics Letters B ••• (••••) •••–•••
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Physics Letters B
www.elsevier.com/locate/physletb
Can chaos be observed in quantum gravity?
Bianca Dittrich
a,∗
, Philipp A. Höhn
b,∗
, Tim A. Koslowski
c,∗
, Mike I. Nelson
d
a
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada
b
Vienna Center for Quantum Science and Technology, and Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria
c
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, México D.F. 04510, Mexico
d
African Institute for Mathematical Sciences, P.O Box LG 197, Legon, Accra, Ghana
a r t i c l e i n f o a b s t r a c t
Article history:
Received
8 August 2016
Received
in revised form 24 January 2017
Accepted
15 February 2017
Available
online xxxx
Editor:
B. Grinstein
Full general relativity is almost certainly ‘chaotic’. We argue that this entails a notion of non-integrability:
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.
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
The canonical description of gauge theories, classical and quan-
tum
alike, with totally constrained Hamiltonian encodes the dy-
namics
of the system in ‘constants of motion’ [1,2], so-called Dirac
observables. One interprets a complete set of Dirac observables as
all that can objectively be predicted about the classical or quantum
system. Much has been written about Dirac observables for general
relativity [3–27], which involves the implementation of invariance
under spacetime diffeomorphisms.
It
has however often been overlooked that Dirac observables
may not always exist as differentiable phase space functions. This
occurs in analogy to classical chaotic systems [28–31] when the
flow generated by the Hamiltonian constraint is sufficiently com-
*
Corresponding authors.
E-mail
addresses: bdittrich@perimeterinstitute.ca (B. Dittrich),
p.hoehn@univie.ac.at (P.A. Höhn), koslowski@nucleares.unam.mx (T.A. Koslowski),
mike@aims.edu.gh (M.I. Nelson).
plicated [18–20,32–37]. Specifically, there are strong hints that full
general relativity is non-integrable or even chaotic [18,35,38–50],
and that a generic general relativistic model with cosmologically
interesting matter is likely to admit neither differentiable Dirac
observables nor a symplectic reduction. While we shall discuss ev-
idence
for the latter below, we refer the reader to [32] for a more
in-depth discussion. Tellingly, differentiable Dirac observables for
full general relativity are not known [21,27] due to the quadratic
nature of the Hamiltonian constraint [22,51] apart from bound-
ary
charges (see e.g. [51] for asymptotically flat and [52–54] for
asymptotically Anti-deSitter) or in dust filled spacetimes [55,56].
This is deeply intertwined with the absence of good (monotonic)
time or clock functions or, equivalently, good gauge fixing condi-
tions
from a generic general relativistic model [3,4,16–20,23,32].
But
if differentiable Dirac observables are absent, what is then
observable? What is the physical interpretation of such putative
observables? And what are the consequences for a quantum the-
ory?
http://dx.doi.org/10.1016/j.physletb.2017.02.038
0370-2693/
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.