Green's Function Approach to Entanglement Entropy on Fuzzy Spaces

In this dissertation, we introduce a novel Euclidean Green function approach to compute Rňyi entropy on lattices and fuzzy spaces. Rňyi entropy for an arbitrary subset of coupled harmonic oscillators is written as a zero temperature partition function generated by an Euclidean action with n-fold step potential. The associated Green's function is explicitly constructed and an alternative new formula for Rňyi entropy is obtained. The developed approach allows one to go beyond the Gaussian case and systematically investigate interacting theories, which represents a real advance and paves the way for the investigation of entanglement entropy on lattices and fuzzy spaces for interacting theories. This approach is further applied to several systems with a focus on 1+1 scalar field theory. The case of half space entanglement was obstructed by the necessity to invert a special class of Toeplitz matrices. An asymptotic inverse for this class of matrices is conjectured and several analytical and numerical tests are in favor of its truthfulness. We finally outline how this approach can be used to investigate entanglement entropy for free and interacting scalar field theory on fuzzy spaces.