23 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism We also have proposed a gauge-invariant version for the rotational fluid model. As a consequence, we have demonstrated that the hidden symmetry found is unique. Although we have studied the rotational fluid model with an extra term, which introduces dissipation into the model, the results are also valid without viscosity ( k = 0). We have noted that, although having dissipa- tion, this fluid model presents hidden symmetry, which does not belong to the other symmetries group obtained for the irrotational fluid model [41]. Due to this, the lo- cal version of the extra global symmetries [42] does not exist in the rotational fluid model, with dissipation or not. Furthermore, the physical meaning of the hidden symmetry can be interpreted. Consider a flow of fluid with viscosity in a tube. It is well known that there are many layers [46], each layer flowing with a specific velocity: the velocity of the layer at the center of the tube is maximum and, distant from the center, the ve- locity decreases in accordance with the increasing of the distance from the center. It also happens when vortices are present. The are many layers in a vortex and the one that is near to the center of the vortex has increased its velocity. Due to this, the velocity of each layer can be transformed into the velocity of the other layer by using the equation s in equation (7.25). Therefore, the dynamics of the fluid is preserved and governed by both Lagrangian and Hamiltonian which present the hidden symmetry. the parent one through the elimination of the WZ terms [1, 2]. This means a new interpretation of the method. However, as recently demonstrated by some of us [47] the choice of the zero-mode must obey some “boundary conditions”. In other words, we can say that the phys- ical coherence must guide us to choose the correct (or convenient) zero-mode. But it is not an impediment to obtain, as mentioned just above, a whole family of dual equivalent actions. Speaking in another way, we can say that this particu- lar mapping between the parent action and the respective final gauge invariant theory can be interpreted as a kind of duality. The gauge invariance of the final actions ob- tained here was demonstrated via the Dirac analysis. The characterization of the final action as a first-class system corroborates the success of this process of “dualization”. Firstly in this paper we used a kind of toy model, the Proca model, to illustrate the procedure. Although be- ing a toy model, the resulting “dual” action is new. Af- ter that, we apply the formalism to the non-linear sigma model and to the chiral Schwinger model. In the NLSM, a hidden symmetry lying on the original phase space was disclosed, oppositely to other approaches [13, 29, 30], where the symmetry resides on the extended WZ phase space. In the CSM, the chiral anomaly was eliminated and the gauge symmetry was recovered. It is important to notice that this result was achieved introducing one WZ field while other schemes thrive with the introduction of two or more WZ fields, which is the origin of the ambiguity problem. Besides, we showed in the context of a non-Abelian model (the non-Abelian Proca model) that the dual em- bedding formalism can be used without any restrictions with the algebra obeyed by the noninvariant model, while other constraint conversion techniques work since the al- gebra was previously and necessarily taken into account. The authors would like to thank CNPq, FAPEMIG, FAPERJ and FAPESP, Brazilian Research Agencies, for financial support. EMCA would like to thank the kind- ness and hospitality of Departamento de F´ısica of Univer- sidade Federal de Juiz de Fora, where part of this work has been accomplished. IX. ACKNOWLEDGMENTS [6] N. Banerjee, B. Banerjee and S. Ghosh, Ann. Phys. 241 (1995) 237, and references therein. [7] R. Rajaraman, Phys. Lett. B 154 (1985) 305. [8] I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B 102 (1981) 27; Phys. Lett. B 120 (1983) 166; Phys. Rev. D 28 (1983) 2587; Phys. Rev. D 30 (1984) 508; Nucl. Phys. B 234, 106 (1984); J. Math. Phys. 26 (1985) 172. [9] L. Faddeev and S. L. Shatashivilli, Phys. Lett. B 167 (1986) 225. [10] I. A. Batalin and E. S. Fradkin, Nucl. Phys. B 279 (1987) 514; Int. J. Mod. Phys. A (1991) 3255. [11] C. Wotzasek, Int. J. Mod. Phys. A 5 (1990) 1123. [12] M. Moshe and Y. Oz, Phys. Lett. B 224 (1989) 145; T. Fujiwara, Y. Igarashi and J. Kubo, Nucl. Phys. B 341 (1990) 695. [13] J. Barcelos-Neto, Phys. Rev. D 55, (1997) 2265. [14] C. Montonen, D. I. Olive, Phys. Lett. B 72, 117 (1977); L. Alvarez-Gaume and F. Zamora, Duality in quantum [5] C. Becchi, A. Rouet and R. Stora, Ann. Phys. [N.Y.] 98 (1976) 287; Phys. Lett. B 52 (1974) 344; I.V. Tyutin, Lebedev-preprint 39/1975. [4] E. S. Fradkin, G. A. Vilkovisky, Phys. Lett. B 55 (1975) 224; I. A. Batalin, G. A. Vilkovisky, Phys. Lett. B 69 (1977) 309. [3] L. D. Faddeev, Theor. Math. Phys. 1 (1970) 1. [2] M. Henneaux and C. Teiltelboim, Quantization of Gauge Systems , Princeton University Press, 1992. [1] P. A. M. Dirac, Proc. Roy. Soc. A 257 (1960) 32; P. A. M. Dirac, Lectures on Quantum Mechanics , Yeshiva University Press, New York, 1964; A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems , Academia Nazionale dei Lincei, Roma 1976; K. Sun- dermeyer, Constrained Dynamics, Lectures Notes in Physics , Springer, New York, 1982, vol 169; P. G. Bergmann, Phys. Rev. 75, (1949) 680; P. G. Bergmann, Phys. Rev. 89 (1953) 4.