Obtaining gauge invariant actions via symplectic embedding formalism

Abstract

The concept of gauge invariance is one of the most subtle and useful concepts in modern theoretical physics. It is one of the Standard Model cornerstones. The main benefit due to the gauge invariance is that it can permit the comprehension of difficult systems in physics with an arbitrary choice of a reference frame at every instant of time. It is the objective of this work to show a path of obtaining gauge invariant theories from non-invariant ones. Both are named also as first- and second-class theories respectively, obeying Dirac's formalism. Namely, it is very important to understand why it is always desirable to have a bridge between gauge invariant and non-invariant theories. Once established, this kind of mapping between first-class (gauge invariant) and second-class systems, in Dirac's formalism can be considered as a sort of equivalence. This work describe this kind of equivalence obtaining a gauge invariant theory starting with a non-invariant one using the symplectic embedding formalism developed by some of us some years back. To illustrate the procedure it was analyzed both Abelian and non-Abelian theories. It was demonstrated that this method is more convenient than others. For example, it was shown exactly that this embedding method used here does not require any special modification to handle with non-Abelian systems.Comment: 20 pages. Two-column format. Final version to appear in Annalen der Physik. arXiv admin note: substantial text overlap with arXiv:1001.2254, arXiv:hep-th/010908

Lagrange multiplier and Wess-Zumino variable as extra dimensions in the torus universe

“…It is very important to know that quantization of gauge theories entails a particular prudence, because of the presence of gauge symmetry exist some nonphysical degrees of freedom that must be eliminated before and after the quantization is applied [25].…”

“…There are some approaches to perform such a conversion, like BFT method [30][31][32][33][34], the symplectic formalism [25,[35][36][37], and the Noether dualization technique [38][39][40]. As we mentioned before, in order to gauge a system with second-class constraints, we use the symplectic approach in order to embed a non-invariant system in an extended phasespace [41][42][43].…”

“…These null vectors are the generators of gauge symmetries, since their contraction with the gradient of the potential does not produce any constraint [25]. One can use the linear combination of these null vectors as,ñα…”

First class models from linear and nonlinear second class constraints

“…Most papers about these models are focused on the consistent canonical quantization and their quantum spectrum. This family of models were considered in several approaches including: the symplectic embedding [8,9,10,11], the BFT formalism [9,12,13,17,14,15,16], Stuckelberg field shifting [19,18] or mixed approaches based on first principles of the making gauge systems [9,18,20,21,22].…”

“…Afterward, we set the ansatz (20) in the above equation to obtain φ µ F µ (n + 1; η) = − 1 2 η a ǫ ab 1 n+1 (F µ (n; η)f bµγ φ γ −nF µ (n; η)φ µ…”