Branched Quantization

TL;DR

This paper introduces a novel quantization method for Lagrangians with branched Hamiltonians featuring cusps, ensuring unitary evolution and connecting to quantum mechanics on singular manifolds.

Contribution

It develops a new quantization approach for branched Hamiltonians and identifies boundary conditions for unitarity, also relating the problem to quantum mechanics on singular spaces.

Findings

Boundary conditions for unitarity established

Dual transformation maps to quantum mechanics on singular manifolds

Potential applications in quantum systems with branched Hamiltonians

Abstract

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