2 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism electromagnetic duality is a pedagogical and interesting example for the interested reader [14]. symmetries defined by some relations called, in Dirac’s language, first-class constraints [1]. The quantization of these theories demands a special care because the pres- ence of gauge symmetries indicate that exist some su- perfluous degrees of freedom, which must be eliminated (before or after) with the implementation of a valid quan- tization process. Recently, the so-called gauging iterative Noether dual- ization method [15] has been shown to thrive in establish- ing some dualities between models [16–18]. This method is hinges on the traditional concept of a local lifting of a global symmetry and may be realized by an iterative em- bedding of Noether counterterms. However, this method provides a strong suggestion of duality since it has been shown to give the expected result in the paradigmatic du- ality between the so-called self-dual model [19] and the Maxwell-Chern-Simons theory in three dimensions dual- ity. This correspondence was first established by Deser and Jackiw [20] and using the parent action approach [21]. The quantization of first-class systems was formulated both in Dirac’s [1] and path integral [3] point of view. Later on, the path integral analysis was extended by Batalin, Fradkin and Vilkovisky [4] in order to preserve the BRST symmetry [5]. On the other hand, the covariant quantization of second-class systems is, in general, a difficult task be- cause the Poisson brackets are replaced by Dirac brack- ets. At the quantum level, the variables become opera- tors and the Dirac brackets become commutators. Due to this, the canonical quantization process is contami- nated with serious problems such as ordering operator problems [6] and anomalies [7] in the context of non- linear constrained systems and chiral gauge theories, re- spectively. In view of this fact, it seems that it is more natural and safe to work out the quantization of second- class systems without invoking Dirac brackets. Actually, it was the strategy followed by many authors over the last decades. The noninvariant system has been embed- ded in an extended phase space in order to change the second-class nature of constraints into first-class. The paper. We have organized this paper as follows. In section 2, we review the dual embedding formalism in order to settle the notation and familiarize the reader with the fundamentals of the formalism. In section 3, we will begin to make the application of the ideas dis- cussed before in some Abelian models. We initiate with the Proca model in order to set up the general ideas dis- cussed in Section 2. We will see that a different zero- mode can bring us an equivalent action different from the literature. After that, we apply this formalism in two important models in high energy physics. The first one, in section 4, is the nonlinear sigma model (NLSM) [22], which is an important theoretical laboratory to learn the basics about asymptotically free field theories, as dynamical mass generation, confinement, and topologi- cal excitations, which is expected in the realistic world of four-dimensional non-Abelian gauge theories. The second, in section 5, is the bosonized chiral Schwinger model (CSM), which has attracted much attention over the last decade, mainly in the context of string theo- ries [27], and also due to the huge progress in under- standing the physical meaning of anomalies in quantum field theories achieved through the intensively study of this model. Through this section, we will first compute the Dirac brackets among the phase space fields and af- ter that, the gauge invariant version of the model will be obtained. Later on, the gauge symmetry will be in- vestigated from the Dirac point of view. Section 6 is devoted to an application of the formalism to the non- Abelian Proca model. In this section, we will show that this gauge-invariant formalism does not require modifi- cations to deal with non-Abelian models as demanded by the BFFT method. In section 7, we analyze the symme- tries of the rotational fluid model with a new extra term, which introduce a dissipative force into the system. The objective is to promote an approximation to reality where we always have dissipation. Our concluding observations and final comments are given in Section 8. In spite of the great success achieved by these meth- ods, they have an ambiguity problem [13]. This problem naturally arise when the second-class constraints are con- verted into first class ones with the introduction of WZ variables. Due to this, the process of constraint conver- sion process may become a hard task, as shown in [13]. Duality. Duality is a very useful concept in field theory and statistical mechanics since there are very few analytic tools available for studying non-perturbative properties of systems with many degrees of freedom. Always stud- ied as one of the first applications of duality basics, the In this way, the entire machinery [5, 8] for quantiz- ing first class systems can be used. To implement this concept, Faddeev [9] suggests to enlarge the phase space with the introduction of new variables to linearize the system, which were named after, as the Wess-Zumino (WZ) variables. This idea has been embraced by many authors and some methods were proposed and some constraint conversion formalisms, based on the Dirac method [1], were constructed. Among them, the Batalin- Fradkin-Fradikina-Tyutin (BFFT) [10] and the iterative [11] methods were strong enough to be successfully ap- plied to a great number of important physical systems. Although these techniques share the same conceptual ba- sis [9] and follow the Dirac framework [1], these constraint conversion methods were implemented following differ- ent directions. Historically, both BFFT and the iterative methods were introduced to deal with linear systems such as chiral gauge theories [11, 12] in order to eliminate the gauge anomaly that hampers the quantization process.