TY - JOUR
AB - 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.
The canonical description of gauge theories, classical and quantum alike, with totally constrained Hamiltonian encodes the dynamics 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 complicated [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 evidence 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 boundary 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 conditions 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 theory?
Given that chaotic gravitational models turn out to be analytically too intricate, we address these crucial questions in the probably simplest non-trivial toy model which, however, qualitatively mimics dynamical properties of a chaotic cosmological model. The employed model is the reparametrization-invariant description of two free particles on a circle with fixed energy, in which the angular momentum-like Dirac observable is discontinuous in the standard topology. We show that a quantization using standard techniques precludes a semiclassical limit or even any solutions to the quantum constraints. This confirms heuristic worries in the older quantum gravity literature [33–37] and mimics the breakdown of semiclassical wavepackets for the Wheeler–DeWitt equation of the chaotic Friedmann–Robertson–Walker model with a massive scalar field reported in [18,40,57,58] (see also references therein). However, we also identify the root of the problem: the topology underlying the quantization. If chosen unsuitably, it may not admit sufficiently many solutions to the quantum constraints in order to allow for semiclassical states. This aspect has been neglected in older heuristic discussions [33–37] and leads to our proposal for a resolution: to adapt the method of quantization to the (dis-)continuity of the observables which one wishes to represent in the quantum theory. We show that, basing the quantization on a polymer-like topology (see e.g. [66–68]), similar to the one used in loop quantum cosmology [69–71] and a novel representation of loop quantum gravity [72,73], resolves the continuity problem and the topology refined quantization admits the expected semiclassical limit. This leads to the main conclusion of the paper: The quantization of systems with Hamiltonian constraints needs to be based on a topology that is fine enough to allow for a complete set of continuous gauge invariant observables.
Our results suggest profound repercussions for non-integrable constrained systems and, given the evidence of non-integrability in general relativity, thereby also for the quantization of gravity.
AU - Dittrich, B.
AU - Höhn, P. A.
AU - Koslowski, T. A.
AU - Nelson, M. I.
DA - 2017/06/10/
DO - 10.1016/j.physletb.2017.02.038
ET - 2017/10/12/
JF - Physics Letters B
PY - 2017
SE - 2017/02/15/
SP - 554-560
TI - Can chaos be observed in quantum gravity?
UR - http://www.sciencedirect.com/science/article/pii/S037026931730151X
VL - 769
ER -