Talk by Ginestra Bianconi
On May 3rd, 2023, Ginestra Bianconi (Queen Mary University of London, UK) gave a talk about "The Discrete Dirac operator and its associated field theory" as part of the online workshop on "Dirac equation between discrete and continuous: new trends and applications" organized by Ginestra Bianconi (Queen Mary University of London) and Delio Mugnolo (FernUniversität in Hagen). This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
Abstract
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We discuss briefly the properties of the discrete Dirac operator on graphs and its relevance to capture the coupled dynamics topological signals defined on nodes, links of graphs and even higher dimensional simplices of simplicial complexes.
We will show how the Dirac operator can be coupled with the algebra of gamma matrices to define a Dirac field theory in which the spinor is given a geometrical interpretation as a direct sum between 0-cochains and 1-cochains. The field theory also includes metric degree of freedom interpreted as the weights of the links for which we can define an action. The approach developed for 3+1 dimensional lattices allows to capture the matter degrees of freedom 2 of the resulting Dirac fields. We will discuss how the result obtained in the non-relativistic limit which allows us to interpret the anticommutator of the directional Dirac operator as the magnetic field while the commutator of the directional Dirac operator can be interpreted as a curvature.
