A loop quantum gravity toy inspired by an Aharonov-Bohm ring

Question

Comparing my question to Give a description of Loop Quantum Gravity your grandmother could understand what I'm looking for here is a toy for a toddler ($\approx$ a pre-QFT graduate student).

I seek to generalize to gravity the following minimalistic model of magnetism (which is a textbook example for the phenomena of Aharonov-Bohm oscillations and persistent currents).

Take a three-state Hilbert space with a Hamiltonian matrix $h_{nm}$ with $n,m=1,2,3$. Interpreting these states as localization sites for, say, an electron in a mesoscopic ring, one may ask how to include a magnetic field piercing the ring. The answer is given by the gauge-invariant phase $\varphi =\arg (h_{12} h_{23} h_{13}^{\ast})$ which is the Aharonov-Bohm phase for a closed trajectory (1-2-3) along the ring; $\varphi$ is equal to the magnetic flux through the ring expressed in units of $\phi_0=h/e$. If $\varphi \not = 0 (\rm{mod} 2 \pi) $, then the lowest energy state carries persistent current, and the model serves as a good starting point to discuss this piece of mesoscopic physics.

I thinks it would be great to have such a toy example for gravity (classical and quantum). The can probably found in be found in a pedagogical book, so please give a reference if you know one.

More specifically, my question would be:

What is the minimal number of points to define a meaningful finite-difference analogue of Hilbert-Einstein action and how to do that? Can basic ideas of LQG quantization be demonstrated on such a finite state model?

I am looking for a simple, possibly less than $3+1$-dimensional example to demonstrate the possibility to eliminate gravity by local but not global Lorentz transformations the same way the above toy model demonstrates the relation between local and global $U(1)$ symmetry.

Answer 1

A good example for gravity along these lines is the relativistic cosmic string, or the equivalent point particle in 2+1 dimensions. The exact solution of GR in the presence of a cosmic string is a deficit angle--- meaning that spacetime is flat except for a wedge cut out of it and the two sides identified. This can be found by solving the 2+1 Einstein equations, but it is easier by just noting that in three dimensions, the Riemann tensor is determined by the Ricci tensor (it has the same number of independent components, six), so wherever the Ricci tensor is zero, spacetime is flat.

Now you can ask what is the phase for a particle to go around the cosmic string. If the particle goes two ways around the string, with momentum perpendicular to the string axis, and interferes with itself, you get an extra relative phase equal to the momentum times the impact parameter (distance to the string) times the cutout angle, which is the mass density of the string. This gives an Aharonov-Bohm like phase, equal to the 2+1 mass contained inside the interference region. If you superpose strings, you add up the deficit angle, and the deficit angle is determined by the phase independent of the location of the particle inside the loop (much like the magnetic flux).

I think this is the best analog of Aharanov Bohm in gravity. The best reference is t'Hooft's article on 2+1 gravity, which is one of these papers (I read it in a reprint collection called "Under the Spell of the Gauge Principle", and I don't remember the title, but the third article listed below is available online, and the relations inside are the ones I am referring to)

• Causality in (2+1)-dimensional gravity. Class. Quantum Grav. 9 (1992) 1335-1348
• Classical N-particle cosmology in 2+1 dimensions. Class. Quantum Grav. 10 (1993) S79-S91
• The evolution of gravitating point particles in 2+1 dimensions. Class. Quantum Grav. 10 (1993) 1023-1038. http://www.staff.science.uu.nl/~hooft101/gthpub/evolution_2plus1_dim.pdf

This is the basis of later work on the quantum version by t'Hooft, Witten and many others. It is a very active topic in matheamtical physics now, because of the intersection with topological field theory, quantum groups, and knots.