S21 DYNAMICS: DISCRETE QUANTUM GRAVITY ON A FINITE VACUUM MANIFOLD

The unification of general relativity and quantum mechanics remains the central open problem in fundamental physics. General relativity describes gravity as the curvature of a dynamical spacetime geometry, while quantum field theory operates on a fixed background spacetime with quantized matter fields. When one attempts to quantize gravity perturbatively—treating the metric as a quantum field on flat spacetime—the theory is non-renormalizable: ultraviolet divergences proliferate at each loop order, requiring infinitely many counterterms with no predictive power. This perturbative failure has motivated several non-perturbative approaches to quantum gravity. Loop quantum gravity (LQG) reformulates general relativity in terms of connection variables and quantizes the resulting phase space, leading to discrete spectra for geometric operators such as area and volume. Causal dynamical triangulations (CDT) regularize the gravitational path integral by restricting to piecewise-flat manifolds built from equilateral simplices. Causal set theory takes discreteness as fundamental, positing that spacetime is a locally finite partial order whose elements are the fundamental 'atoms' of spacetime. Each of these approaches confronts a common challenge: the state space is infinite or unbounded, and exact analytical results are difficult or impossible to obtain. What is missing from the landscape of quantum gravity approaches is a finite, exactly solvable model—a 'theoretical laboratory' in which the concepts of discrete quantum gravity can be tested with complete analytical control.