Particle dynamics on hyperboloid and unitary representation of SO(1,N) group

TL;DR

This paper studies particle motion on hyperboloids in N+1 dimensional Minkowski space, using symmetry and quantization techniques to connect classical trajectories with unitary group representations.

Contribution

It introduces a gauge-invariant Hamiltonian reduction for particles on hyperboloids and constructs a unitary representation of SO(1,N) in quantum theory.

Findings

Classical phase space parametrizes all trajectories on the hyperboloid.

Quantum operators are ordered to yield unitary irreducible representations.

Quantization maps classical symmetries to Hilbert space representations.

Abstract

We analyze particle dynamics on $N$ dimensional one-sheet hyperboloid embedded in $N+1$ dimensional Minkowski space. The dynamical integrals constructed by $SO_\uparrow (1,N)$ symmetry of spacetime are used for the gauge-invariant Hamiltonian reduction. The physical phase-space parametrizes the set of all classical trajectories on the hyperboloid. In quantum case the operator ordering problem for the symmetry generators is solved by transformation to asymptotic variables. Canonical quantization leads to unitary irreducible representation of $SO_\uparrow (1,N)$ group on Hilbert space $L^2(S^{N-1})$.