Path Integral Junctions

We propose path integral description for quantum mechanical systems on compact graphs consisting of N segments of the same length. Provided the bulk Hamiltonian is segment-independent, scale-invariant boundary conditions given by self-adjoint extension of a Hamiltonian operator turn out to be in one-to-one correspondence with N \times N matrix-valued weight factors on the path integral side. We show that these weight factors are given by N-dimensional unitary representations of the infinite dihedral group.