We study the oscillations of a scalar field on a noncommutative disc implementing the boundary as the limit case of an interaction with an appropriately chosen confining background. The space of quantum fluctuations of the field is finite dimensional
and displays the rotational and parity symmetry of the disc. We perform a numerical evaluation of the (finite) Casimir energy and obtain similar results as for the fuzzy sphere and torus.
The interaction of a quantum field with a background containing a Dirac delta function with support on a surface of codimension 1 represents a particular kind of matching conditions on that surface for the field. In this article we show that the worl
dline formalism can be applied to this model. We obtain the asymptotic expansion of the heat-kernel corresponding to a scalar field on $mathbb{R}^{d+1}$ in the presence of an arbitrary regular potential and subject to this kind of matching conditions on a flat surface. We also consider two such surfaces and compute their Casimir attraction due to the vacuum fluctuations of a massive scalar field weakly coupled to the corresponding Dirac deltas.
The worldline formalism has been widely used to compute physical quantities in quantum field theory. However, applications of this formalism to quantum fields in the presence of boundaries have been studied only recently. In this article we show how
to compute in the worldline approach the heat kernel expansion for a scalar field with boundary conditions of Robin type. In order to describe how this mechanism works, we compute the contributions due to the boundary conditions to the coefficients A_1, A_{3/2} and A_2 of the heat kernel expansion of a scalar field on the positive real line.
