  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 worldline 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.
