Symmetries of Non-Linear Systems: Group Approach to their Quantization

TL;DR

This paper introduces a symmetry-based approach to quantize non-linear systems, offering a non-perturbative alternative to traditional methods, with applications to particle interactions and non-linear sigma models.

Contribution

It develops a group-theoretic quantization scheme that improves geometric quantization and applies it to complex non-linear quantum field theories.

Findings

Quantization of non-linear SU(2) sigma model demonstrated

Algebraic version of the equivalence principle derived

Method applicable to electromagnetic and gravitational interactions

Abstract

We report briefly on an approach to quantum theory entirely based on symmetry grounds which improves Geometric Quantization in some respects and provides an alternative to the canonical framework. The present scheme, being typically non-perturbative, is primarily intended for non-linear systems, although needless to say that finding the basic symmetry associated with a given (quantum) physical problem is in general a difficult task, which many times nearly emulates the complexity of finding the actual (classical) solutions. Apart from some interesting examples related to the electromagnetic and gravitational particle interactions, where an algebraic version of the equivalence principle naturally arises, we attempt to the quantum description of non-linear sigma models. In particular, we present the actual quantization of the partial-trace non-linear SU(2) sigma model as a representative case of non-linear quantum field theory.