Ricci Curvature of Noncommutative Three Tori, Entropy, and Second Quantization
Abstract
In noncommutative geometry, the metric information of a noncommutative space is encoded in the data of a spectral triple $(\mathcal{A}, \mathcal{H},D)$, where $D$ plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace $\textrm{Tr}(e^{-tD^2})$. In Chapter 2, we will compute the Ricci curvature of a curved noncommutative three torus. The computation is done for both conformal and a non-conformal perturbation of the flat metric. By applying Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second density of the heat trace expansion for the perturbed Laplacians on both functions and $1-$forms. On the other hand, in noncommutative geometry one also wants to get a good notion of an action functional which depends only on the spectrum of $D$, called spectral action functional. It is known that such a functional can be expressed as $\textrm{Tr(f(D))}$ for some function $f$. In chapter 3, we show that the von Neumann entropy, average energy, and negative free energy of the Gibbs state of the second quantized Dirac operator $d\Gamma D$ has a spectral action functional interpretation of the original Dirac operator $D$. To be able to carry on the computations, we have to incorporate the chemical potential $\mu$. All those spectral action coefficients can be given in terms of the modified Bessel functions.