Hidden symmetries in the two-dimensional isotropic antiferromagnet

TL;DR

This paper explores hidden symmetries in the two-dimensional isotropic antiferromagnet by applying gauge invariance and symplectic embedding, revealing how second class constraints generate gauge symmetries and deriving equivalent Schrödinger equations.

Contribution

It introduces a systematic method to convert second class constraints into gauge-invariant form using symplectic embedding, uncovering hidden symmetries in the model.

Findings

Second class constraints generate hidden gauge symmetries.

The gauge-invariant model yields the same Schrödinger equations as the original.

The approach justifies factor ordering choices in quantization.

Abstract

We discuss the two-dimensional isotropic antiferromagnet in the framework of gauge invariance. Gauge invariance is one of the most subtle useful concepts in theoretical physics, since it allows one to describe the time evolution of complex physical systesm in arbitrary sequences of reference frames. All theories of the fundamental interactions rely on gauge invariance. In Dirac's approach, the two-dimensional isotropic antiferromagnet is subject to second class constraints, which are independent of the Hamiltonian symmetries and can be used to eliminate certain canonical variables from the theory. We have used the symplectic embedding formalism developed by a few of us to make the system under study gauge-invariant. After carrying out the embedding and Dirac analysis, we systematically show how second class constraints can generate hidden symmetries. We obtain the invariant second-order Lagrangian and the gauge-invariant model Hamiltonian. Finally, for a particular choice of factor ordering, we derive the functional Schr\"odinger equations for the original Hamiltonian and for the first class Hamiltonian and show them to be identical, which justifies our choice of factor ordering.