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

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.

From physical symmetries to emergent gauge symmetries

“…Indeed, it is possible to construct a quite illuminating counterexample to this premature identification, using the simple setting of a complex Klein-Gordon field φ. We take this example from the work of Aldaya and collaborators [12]. Let us consider the Lagrangian density…”

Jet-Gauge Groups as Basic Symmetries of Non-Linear Sigma and Yang–mills Models and Quantization

“…It must be clearly stated from now on that x ∈ Σ plays the role of an index, so that the space derivative ∂ i on φ moves this index. The time component of φ µ , however, does not refer to a (time) derivative of a given φ and, rather, corresponds to a different family of group parameters, that is, the momenta (see in this respect [2]). Thus, an element g of the Klein-Gordon-field group, G K−G , is parametrized by (φ( x), φ µ ( x), ζ), as well as ordinary parameters on the Poincaré group, (a ν , Λ µσ ), in the case that the space-time symmetry were explicitly considered.…”

“…In this paper we briefly report on this situation and analyze the specific case of the group quantization of non-Abelian massless and massive gauge theories. Firstly, we start with the simplest example of the centrally extended group describing linear fields (see [2] and references therein). Then we face the obstruction that appears when attempting to generalize such a symmetry to the case of fields living on a semisimple group G like SU (n), that is to say, Non-Linear Sigma fields, or fields parametrizing the corresponding gauge group G(M ) on the Minkowski space-time [3].…”

Symmetry group of massive Yang–Mills theories without Higgs and their quantization

“…The time component of φ µ , however, does not refer to a (time) derivative of a given φ and, rather, corresponds to a different family of group parameters, that is, the momenta (see in this respect Ref. [23].…”

“…where all fields are assumed to be defined on the Cauchy surface Σ, so that, the time translation can not be directly implemented, in contrast with the case of free fields [23]. However, as in the Abelian case, we shall construct an explicit Hamiltonian operator to account for the time evolution on the quantum states (see below) 3 .…”