Unimodular loop quantum gravity and the problems of time

We develop the quantization of unimodular gravity in the Plebanski and Ashtekar formulations and show that the quantum effective action defined by a formal path integral is unimodular. This means that the effective quantum geometry does not couple to terms in the expectation value of energy proportional to the metric tensor. The path integral takes the same form as is used to define spin foam models, with the additional constraint that the determinant of the four-metric is constrained to be a constant by a gauge fixing term. This extends the results of L. Smolin [Phys. Rev. D 80, 084003 (2009)] to the Hilbert space and path integral of loop quantum gravity. We review the proposal of Unruh, Wald, and Sorkin—that the Hamiltonian quantization yields quantum evolution in a physical time variable equal to elapsed four-volume—and discuss how this may be carried out in loop quantum gravity.