Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism E. M. C. Abreu $^{a}$, ∗ J. Ananias Neto $^{b}$, A. C. R. Mendes$^{b}$, † C. Neves $^{c}$, ‡ and W. Oliveira b § $^{a}$Grupo de F´ısica Te´ orica e Matem´ atica F´ısica, Departamento de F´ısica, Universidade Federal Rural do Rio de Janeiro BR 465-07, 23890-971, Serop´ edica, RJ, Brazil $^{b}$Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil $^{c}$Departamento de Matem´ atica e Computa¸c˜ ao, Universidade do Estado do Rio de Janeiro Rodovia Presidente Dutra, km 298, 27537-000, Resende, RJ, Brazil November 23, 2018 The concept of gauge invariance can be considered one of the most subtle and useful concept in theoretical physics since 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 always desirable to have a bridge between gauge invariant and noninvariant 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 duality. In this paper we investigate this “duality” obtaining a gauge invariant theory starting with a noninvariant one. We analyzed both Abelian and non-Abelian theories and the procedure used is the recent dual (also called symplectic) embedding formalism. We believe that this method is the most convenient one since it is not plagued by the ambiguity problems that torments BFFT and other iterative methods. We demonstrated exactly that this “dualization” method used here does not require any special modification to handle with non-Abelian systems, which is also a new result described in this paper. To prove the gauge invariance we used just the Dirac constraint technique. It is relevant to say here that, although this work is lengthy, the majority of the results presented here are new in the literature. The results that are not new were reproduced within a new perspective. PACS numbers: 11.15.-q; 11.10.Ef; 11.30.Cp Keywords: gauge invariance, dual embedding, constrained systems This work deals with the transformation of second- class systems into gauge theories. Although there is an extensive literature about this subject, we obtain some results here that are altogether new. In a self-sustained and pedagogical way we analyze gauge invariance and dual embedding issues. A map with these conversion features can be under- stood as a kind of duality since we will show precisely that the final action have only first-class constraints. This du- ality connection characterizes both systems as represent- ing the same physical properties. We will discuss these ideas carefully later. As well known, after the procedure, having only first- class constraints. the final system is a gauge theory, Since the Hamiltonian formulation is considered by many as the more fundamental formulation of a physical theory we use the Dirac brackets in order to show exactly the gauge invariance of the final actions obtained. which can be thought as one in which the dynamical vari- ables are determined with relation to a reference frame whose choice is arbitrary at every instant of time. The relevance of a gauge theory, in few words, is that the physically important variables are those that are inde- pendent of the local reference frame [2]. Whenever a change in the arbitrary reference frame causes a trans- formation of the variables involved we have the so-called gauge transformation. This physical variables are then well known as gauge invariant variables. Such gauge the- ories and gauge transformations are the cornerstones of the construction of the standard model, to mention only one of its successful applications. We will talk with more detail about gauge theories and quantization in a mo- ment. Second-class → first-class. Gauge theories have played an important role in field theories since they are related with the fundamental physical interactions of Na- ture. In a more general sense, those theories have gauge ∗ evertonabreu@ufrrj.br † albert@fisica.ufjf.br ‡ clifford@fat.uerj.br § wilson@fisica.ufjf.br arXiv:1001.2254v2 [hep-th] 19 Jan 2010 I. INTRODUCTION