Würzburg Seminar on Quantum Field Theory and Gravity

In recent years, low dimensional quantum gravitational models have found fruitful application in holography due to dualities between such models and various random matrix models [1, 2]. An open problem in the generalisation of this work to higher dimensional quantum gravity is the question of how a single quantum system can produce non-factorising correlation functions like the ones found in random matrix models and expected in generic theories with quantum gravity due to the contribution from wormhole geometries.

We show that generic chaotic quantum systems, after introducing a novel dynamical average, exhibit non-factorising correlation functions that take the form of genus expansions one would expect in (2-dimensional) quantum gravity.

To this end, we introduce relevant notions from semiclassics such as Berry's [3] diagonal approximation and correlations between the actions of periodic orbits to allow the practical evaluation of periodic orbit sums [4,5], and proceed to define an average based on minimisation of a sum using only a subset of periodic orbits, from which the genus expansion is then derived.

For a specific choice of system, a particle moving geodesically on a d-dimensional hyperbolic manifold, which is described by the mathematically exact Selberg trace formula [6], we find agreement to leading order in the genus expansion of the one- and two-point correlation functions of the heat kernel with the corresponding partition functions of topological 2d gravity, the gravitational dual of the Kontsevich matrix model [7] (in the case d=3), or JT gravity (in the limit of infinite d).

[1] P. Saad, S. Shenker, D. Stanford,  arXiv:1903.11115
[2] D. Stanford, E. Witten,  Adv. Theor. Math. Phys. 24 (2020) 6
[3] M. V. Berry,  Proc. Roy. Soc. Lond. A 400 (1985)
[4] M. Sieber, K. Richter,  Phys. Scr. 128 (2001)
[5] S. Müller et al.,  Phys. Rev. E 72 (2005), 046207
[6] A. A. Bytsenko et al.,  Phys. Rept. 266 (1996)
[7] M. Kontsevich,  Commun. Math. Phys. 147 (1992)