Equivalence between different classical treatments of the O(N) nonlinear sigma model and their functional Schrodinger equations

TL;DR

This paper demonstrates the equivalence of different classical Hamiltonian formulations of the O(N) nonlinear sigma model and shows that their functional Schrödinger equations are identical, supporting future quantization efforts.

Contribution

It derives and compares multiple Hamiltonian formulations of the O(N) sigma model and proves their equivalence at the quantum level through identical Schrödinger equations.

Findings

Hamiltonians for different formulations are equivalent

Functional Schrödinger equations are identical across formulations

Supports future quantization via Schrödinger representation

Abstract

In this work we derive the Hamiltonian formalism of the O(N) non-linear sigma model in its original version as a second-class constrained field theory and then as a first-class constrained field theory. We treat the model as a second-class constrained field theory by two different methods: the unconstrained and the Dirac second-class formalisms. We show that the Hamiltonians for all these versions of the model are equivalent. Then, for a particular factor-ordering choice, we write the functional Schrodinger equation for each derived Hamiltonian. We show that they are all identical which justifies our factor-ordering choice and opens the way for a future quantization of the model via the functional Schrodinger representation.