Fluctuating Dimension in a Discrete Model for Quantum Gravity Based on the Spectral Principle

The spectral principle of Connes and Chamseddine is used as a starting point to define a discrete model for Euclidean quantum gravity. Instead of summing over ordinary geometries, we consider the sum over generalized geometries where topology, metric, and dimension can fluctuate. The model describes the geometry of spaces with a countable number $n$ of points, and is related to the Gaussian unitary ensemble of Hermitian matrices. We show that this simple model has two phases. The expectation value $⟨n⟩$, the average number of points in the Universe, is finite in one phase and diverges in the other. We compute the critical point as well as the critical exponent of $⟨n⟩$. Moreover, the space-time dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of $⟨\delta ⟩$ is discussed and an upper bound is found, $⟨\delta ⟩<2$.