A Universal Kinematical Group for Quantum Mechanics

TL;DR

This paper introduces a universal kinematical group for quantum mechanics based on a semidirect product of scalar functions and diffeomorphisms, unifying various quantum systems and clarifying topology's role.

Contribution

It provides a foundational framework showing that this infinite-dimensional group serves as a universal kinematical group for all quantum systems with mass in any space.

Findings

Unified account of quantum kinematics across different spaces

Clarification of topology's role in quantum mechanics

Connection to classical limit without phase space quantization

Abstract

In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities; notably, anyons and nonabelian anyons in two space dimensions. We present here foundational reasons why this semidirect product group serves as a universal kinematical group for quantum mechanics. We obtain thus a unified account of all possible quantum kinematics for systems with mass in an arbitrary physical space, and clarify the role played by topology in quantum mechanics. Our development does not require quantization of classical phase space; rather, the classical limit follows from the quantum mechanics. We also consider the relationship of our development to Heisenberg quantization.