| Summary: | This thesis presents aspects of noncommutative spectral geometry as an approach to formulate a model of gravity and particle physics, while addressing open issues associated with this approach. We propose a novel de nition of the bosonic spectral action using zeta function regularisation, in order to address the issues of renormalisability, ultraviolet completeness and spectral dimensions. We compare the zeta spectral action with the usual (cuto based) spectral action and discuss its purely spectral origin, predictive power, stressing the importance of the issue of the three dimensionful fundamental constants, namely the cosmological constant, the Higgs vacuum expectation value, and the gravitational constant. We emphasise the fundamental role of the neutrino Majorana mass term for the structure of the bosonic action. We subsequently show that the regularised zeta spectral action gives a stable linearised gravitational theory despite being a 4th-order derivative theory. Afterwards, we explore the notion of Lorentzian noncommutative geometry, where the bosonic action is not well-de ned. However, in such a case, the dynamics of fermions is still well-de ned. We have shown that one could give a geometrical meaning to the energy-momentum dispersion relation of fermions.
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