Symplectic approach to quantum constraints

TL;DR

This paper develops a symplectic formalism for constrained quantum systems, providing solutions for dynamics on algebraic submanifolds, including systems of spin-1/2 particles, with applications to energy eigenstates and disentangled states.

Contribution

It introduces a general symplectic approach for quantum constraints, offering explicit equations of motion for complex constrained quantum systems, including multi-spin systems.

Findings

Dynamics on energy eigenstate subspaces are quasi-unitary.

Explicit equations of motion are derived for disentangled state subspaces.

The formalism applies to systems of multiple spin-1/2 particles.

Abstract

A general prescription for the treatment of constrained quantum motion is outlined. We consider in particular constraints defined by algebraic submanifolds of the quantum state space. The resulting formalism is applied to obtain solutions to the constrained dynamics of systems of multiple spin-1/2 particles. When the motion is constrained to a certain product space containing all of the energy eigenstates, the dynamics thus obtained are quasi-unitary in the sense that the equations of motion take a form identical to that of unitary motion, but with different boundary conditions. When the constrained subspace is a product space of disentangled states, the associated motion is more intricate. Nevertheless, the equations of motion satisfied by the dynamical variables are obtained in closed form.