Spin Geometry and Quantum Diffusion

Author: Gakkhar, Sitanshu

Year: 2025

Degree: Dissertation (Ph.D.)

Advisor: Marcolli, Matilde

Committee Members: Makarov, Nikolai G.; Gukov, Sergei; Ćaćić, Branimir Josip; Marcolli, Matilde

Option: Mathematics

DOI: 10.7907/ded0-hn95

Abstract

This thesis studies diffusion processes on spinor endomorphism algebras. The spinor and connection laplacian generated heat semigroups are shown to quantum dynamical semigroups, and after spectral truncation the existence of Evans-Hudson flows is established. The vacuum state expectation of the process is related to the spectral action principle in noncommutative geometry. Examples where the flow is proven to exist for untruncated laplacians are given. Convergence of finite dimensional approximations, through discretization and truncation, to spectral triples encoding Riemannian geometry and their statespaces as quantum metric spaces is also considered.

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