SL(2,R) model with two Hamiltonian constraints

TL;DR

This paper introduces a simple dynamical model with two noncommuting Hamiltonian constraints, mimicking general relativity's structure, and analyzes its classical and quantum dynamics governed by SL(2,R) symmetry.

Contribution

It presents a solvable model with two Hamiltonian constraints, exploring its classical and quantum properties, including gauge invariance and evolving constants, to illustrate gauge-invariant evolution.

Findings

Classical solutions with SO(2,2) algebra of observables

Quantum states and inner product determined by reality conditions

Construction of classical and quantum evolving constants

Abstract

We describe a simple dynamical model characterized by the presence of two noncommuting Hamiltonian constraints. This feature mimics the constraint structure of general relativity, where there is one Hamiltonian constraint associated with each space point. We solve the classical and quantum dynamics of the model, which turns out to be governed by an SL(2,R) gauge symmetry, local in time. In classical theory, we solve the equations of motion, find a SO(2,2) algebra of Dirac observables, find the gauge transformations for the Lagrangian and canonical variables and for the Lagrange multipliers. In quantum theory, we find the physical states, the quantum observables, and the physical inner product, which is determined by the reality conditions. In addition, we construct the classical and quantum evolving constants of the system. The model illustrates how to describe physical gauge-invariant relative evolution when coordinate time evolution is a gauge.