  The worldline formalism is a useful scheme in Quantum Field Theory which has
also become a powerful tool for numerical computations. It is based on the
first quantisation of a point-particle whose transition amplitudes correspond
to the heat-kernel of the operator of quantum fluctuations of the field theory.
However, to study a quantum field theory in a bounded manifold one needs to
restrict the path integration domain of the point-particle to a specific subset
of worldlines enclosed by those boundaries. In the present article it is shown
how to implement this restriction for the case of a spinor field in a
two-dimensional curved half-plane under MIT bag boundary conditions, and
compute the first few heat-kernel coefficients as a verification of the
proposed construction. This construction admits several generalisations.
