Dimensionality and boundary conditions in Hadamard regularization

In this thesis, we analyze the role of boundaries and spacetime dimensionality in the context of Hadamard regularization. We start by considering the constant density star model, whose field equation reduces to a time independent Schrödinger-like equation with a potential having a jump discontinuity. Due to the jump, which can also be expected in slightly more realistic but still simple models, there are more types of boundary conditions that one can enforce. Renormalization is what determines if a choice is physically meaningful or not, and the most general renormalization method that can be applied to a wide variety of models is the Hadamard subtraction, which relies on the universal parametrix of the Feynman propagator near coincidence limit. We attempt this renormalization procedure to flat spacetime models whose equations of motion are formally analogous to the homogeneous star case, with custom potentials. First, we apply it to the well-known case of a real massless scalar in two-dimensional flat spacetime in a Dirichlet box, and find out that it works. Then, we enhance the spacetime dimensionality to three and work with a Dirichlet spherical cavity, in which case the divergences differ from those obtained from the Hadamard parametrix. This hints to the fact that Dirichlet boundaries may not be physically meaningful in four-dimensional models. After that, we consider another variation of the first model where the Dirichlet box is substituted with a step potential with a jump discontinuity. In this case, we find that the Hadamard parametrix holds, hinting that, in two-dimensional spacetime models, the Hadamard method may work independently of the choice of boundary conditions. When possible, components of the renormalized energy-momentum tensor of the studied models are also found.