Chaos, Dirac observables and constraint quantization

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.

Figures (9)

• Figure 1: A trajectory on the torus $\mathbb{T}^2$ with angle given by (\ref{['phi']}). $n_i$ increases by $1$ whenever $x_i=1$ is reached.
• Figure 2: Each branch of a trajectory, given by an admissible pair $(n_1,n_2)$, features a linear relation between $x_2$ and $x_1$. The range of permissible values of $x_i$ depends on the branch; in the figure $x_1\in[0,\tilde{x}_1)$ and $x_2\in[\tilde{x}_2,1)$.
• Figure 3: Example quasi-localized wave packet with $k_1=1$, $k_2=3$. Left: Initial wave function at internal time $x_1=0$. Right: Global evolution with the right-moving Hamiltonian.
• Figure 4: Example of an initially well localized wave packet with mass ratio $\gamma=1$ and $\epsilon=25$. Left: Initial wave function at internal time $x_1=0$. Right: Global evolution with the right-moving Hamiltonian.
• Figure 5: Example of a wave packet with mass ratio $\gamma=1$ and $\epsilon=125$ that is initially localized on an orbit with periodicity 5. Left: Initial wave function at initial time $x_1=0$. Right: Global evolution with the right-moving Hamiltonian.