Branched Quantization

We propose a method for quantization of Lagrangians for which the Hamiltonian, as a function of momentum, is a branched function, possibly with cusps. Appropriate boundary conditions, which we identify, ensure unitary time evolution. In special cases a dual (canonical) transformation maps the problem into a problem of quantum mechanics on singular spaces, which we also develop. Several possible applications are indicated.

Figure 1

(a) A wire with kinks. (b) A wire network with two junctions. (c) Box graph for a network with four junctions and a loop.