Noncommutativity and Duality through the Symplectic Embedding Formalism

Everton M. C. Abreu ; Albert C. R. Mendes ; Wilson Oliveira

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

This work is devoted to review the gauge embedding of either commutative and noncommutative (NC) theories using the symplectic formalism framework. To sum up the main features of the method, during the process of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily and directly chosen. Among other advantages, this enables a greater control over the final Lagrangian and brings some light on the so-called “arbitrariness problem”. This alternative embedding formalism also presents a way to obtain a set of dynamically dual equivalent embedded Lagrangian densities which is obtained after a finite number of steps in the iterative symplectic process, oppositely to the result proposed using the BFFT formalism. On the other hand, we will see precisely that the symplectic embedding formalism can be seen as an alternative and an efficient procedure to the standard introduction of the Moyal product in order to produce in a natural way a NC theory. In order to construct a pedagogical explanation of the method to the nonspecialist we exemplify the formalism showing that the massive NC $U(1)$ theory is embedded in a gauge theory using this alternative systematic path based on the symplectic framework. Further, as other applications of the method, we describe exactly how to obtain a Lagrangian description for the NC version of some systems reproducing well known theories. Naming some of them, we use the procedure in the Proca model, the irrotational fluid model and the noncommutative self-dual model in order to obtain dual equivalent actions for these theories. To illustrate the process of noncommutativity introduction we use the chiral oscillator and the nondegenerate mechanics.

Détail

Keywords: noncommutativity; symplectic embedding mechanism; gauge theories.

Bibliographie

[1] Seiberg N., Witten E., “String theory and noncommutative geometry”, J. High Energy Phys., 1999:9 (1999), 032, 93 pp., arXiv: hep-th/9908142 | DOI | MR

[2] Deriglazov A. A., Neves C., Oliveira W., Abreu E. M. C., Wotzasek C., Filgueiras C., “Open string with a background $B$ field as the first order mechanics, noncommutativity, and soldering formalism”, Phys. Rev. D, 76 (2007), 064007, 8 pp., arXiv: 0707.1799 | DOI | MR

[3] Connes A., Douglas M. R., Schwarz A., “Noncommutative geometry and matrix theory: compactification on tori”, J. High Energy Phys., 1998:2 (1998), 003, 35 pp., arXiv: ; Konechny A., Schwarz A., “Introduction to M(atrix) theory and noncommutative geometry”, Phys. Rep., 360 (2002), 353–465, arXiv: ; Harvey J. A., Komaba lectures on noncommutative solitons and D-branes, arXiv: ; Nekrasov N. A., Trieste lectures on solitons in noncommutative gauge theories, arXiv: hep-th/9711162hep-th/0012145hep-th/0102076hep-th/0011095 | DOI | MR | DOI | MR | Zbl | MR

[4] Amorim R., Barcelos-Neto J., “Hamiltonian embedding of the massive noncommutative $\mathrm{U}(1)$ theory”, Phys. Rev. D, 64 (2001), 065013, 8 pp., arXiv: (and references therein) hep-th/0101196 | DOI | MR

[5] Arefeva I. Y., Belov D. M., Giryavets A. A., Koshelev A. S., Medvedev P. B., NC field theories and (super) string field theories, arXiv: hep-th/0111208

[6] Szabo R. J., “Quantum field theory on NC spaces”, Phys. Rep., 378 (2003), 207–299, arXiv: ; Douglas M. R., Nekrasov N. A., “Noncommutative field theory”, Rev. Modern Phys., 73 (2001), 977–1029, arXiv: ; Szabo R. J., “Quantum gravity, field theory and signatures of NC spacetime”, Gen. Relativity Gravitation, 42 (2010), 1–29, arXiv: ; Szabo R. J., “Symmetry, gravity and noncommutativity”, Classical Quantum Gravity, 23 (2006), R199–R242, arXiv: hep-th/0109162hep-th/01060480906.2913hep-th/0606233 | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | Zbl

[7] Dayi O. F., Kelleyane L. T., “Wigner functions for the Landau problem in noncommutative spaces”, Modern Phys. Lett. A, 17 (2002), 1937–1944, arXiv: hep-th/0202062 | DOI | MR | Zbl

[8] Horváthy P. A., Plyushchay M. S., “Anyon wave equations and the noncommutative plane”, Phys. Lett. B, 595 (2004), 547–555, arXiv: ; Horváthy P. A., Plyushchay M. S., “Nonrelativistic anyons and exotic Galilean symmetry”, J. High Energy Phys., 2002:6 (2002), 033, 11 pp., arXiv: hep-th/0404137hep-th/0201228 | DOI | MR | DOI | MR

[9] Deriglazov A. A., “Quantum mechanics on noncommutative plane and sphere from constrained systems”, Phys. Lett. B, 530 (2002), 235–243, arXiv: ; Deriglazov A. A., “Noncommutative relativistic particle on the electromagnetic background”, Phys. Lett. B, 555 (2003), 83–88, arXiv: hep-th/0201034hep-th/0208201 | DOI | MR | Zbl | DOI | MR | Zbl

[10] Deriglazov A. A., NC version of an arbitrary nondegenerate mechanics, arXiv: hep-th/0208072

[11] Chakraborty B., Gangopadhyay S., Saha A.,, “Seiberg–Witten map and Galilean symmetry violation in a non-commutative planar system”, Phys. Rev. D, 70 (2004), 107707, 4 pp., arXiv: ; Scholtz F. G., Chakraborty B., Gangopadhyay S., Hazra A. G., “Dual families of non-commutative quantum systems”, Phys. Rev. D, 71 (2005), 085005, 11 pp., arXiv: ; Gangopadhyay S., Scholtz F. G., “Path-integral action of a particle in the noncommutative plane”, Phys. Rev. Lett., 102 (2009), 241602, 4 pp., arXiv: hep-th/0312292hep-th/05021430904.0379 | DOI | MR | DOI | MR | DOI | MR

[12] Sheikh-Jabbari M. M., “$C$, $P$, and $T$ invariance of noncommutative gauge theories”, Phys. Rev. Lett., 84 (2000), 5265–5268, arXiv: hep-th/0001167 | DOI | MR

[13] Garcia-Bondía J. M., Martín C. P., “Chiral gauge anomalies on noncommutative $\mathbb R^4$”, Phys. Lett. B, 479 (2000), 321–328, arXiv: ; Ardalan F., Sadooghi N., “Axial anomaly in noncommutative QED on $\mathbb R^4$”, Internat. J. Modern Phys. A, 16 (2001), 3151–3177, arXiv: hep-th/0002171hep-th/0002143 | DOI | MR | DOI | MR | Zbl

[14] Dayi O. F., “BRST-BFV analysis of equivalence between noncommutative and ordinary gauge theories”, Phys. Lett. B, 481 (2000), 408–412, arXiv: hep-th/0001218 | DOI | MR | Zbl

[15] Girotti H. O., Gomes M., Rivelles V. O., da Silva A. J., “A consistent noncommutative field theory: the Wess–Zumino model”, Nuclear Phys. B, 587 (2000), 299–310, arXiv: ; Grisaru M. T., Penati S., “Noncommutative supersymmetric gauge anomaly”, Phys. Lett. B, 504 (2001), 89–100, arXiv: hep-th/0005272hep-th/0010177 | DOI | MR | Zbl | DOI | MR | Zbl

[16] Minwalla S., Van Raamsdonk M., Seiberg N., “Noncommutative perturbative dynamics”, J. High Energy Phys., 2000:2 (2000), 020, 31 pp., arXiv: hep-th/9912072 | DOI | MR

[17] Hayakawa M., Perturbative ultraviolet and infrared dynamics of noncommutative quantum field theory, arXiv: ; Aref'eva I. Y., Belov D. M., Koshelev A. S., “Two-loop diagrams in noncommutative $\phi^4_4$ theory”, Phys. Lett. B, 476 (2000), 431–436, arXiv: ; Martín C. P., Sánchez-Ruiz D., “The one loop UV divergent structure of $\mathrm U(1)$ Yang–Mills theory on noncommutative $\mathbb R^4$”, Phys. Rev. Lett., 83 (1999), 476–479, arXiv: ; Bonora L., Salizzoni M., “Renormalization of noncommutative $\mathrm U(N)$ gauge theories”, Phys. Lett. B, 504 (2001), 80–88, arXiv: hep-th/0009098hep-th/9912075hep-th/9903077hep-th/0011088 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[18] Gomis J., Mehen T., “Space-time noncommutative field theories and unitarity”, Nuclear Phys. B, 591 (2000), 265–276, arXiv: hep-th/0005129 | DOI | MR | Zbl

[19] Arfaei H., Yavartanoo M. H., Phenomenological consequences of non-commutative QED, arXiv: hep-th/0010244

[20] Álvarez-Gaumé L., Meyer F., Vázquez-Mozo M. A., “Comments on noncommutative gravity”, Nuclear Phys. B, 753 (2006), 92–117, arXiv: ; Calmet X., Kobakhidze A., “Noncommutative general relativity”, Phys. Rev. D, 72 (2005), 045010, 5 pp., arXiv: ; Harikumar E., Rivelles V. O., “Noncommutative gravity”, Classical Quantum Gravity, 23 (2006), 7551–7560, arXiv: ; Müller-Hoissen F., “Noncommutative geometries and gravity”, Recent Developments in Gravitation and Cosmology, AIP Conf. Proc., 977, Amer. Inst. Phys., Melville, NY, 2008, 12–29, arXiv: hep-th/0605113hep-th/0506157hep-th/06071150710.4418 | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl | DOI | MR

[21] Banerjee R., “A novel approach to noncommutativity in planar quantum mechanics”, Modern Phys. Lett. A, 17 (2002), 631–645, arXiv: hep-th/0106280 | DOI | MR | Zbl

[22] 't Hooft G., “Quantum gravity as a dissipative deterministic system”, Classical Quantum Gravity, 16 (1999), 3263–3279, arXiv: gr-qc/9903084 | DOI | MR | Zbl

[23] Jackiw R., Nair V. P., “Relativistic wave equations for anyons”, Phys. Rev. D, 43 (1991), 1933–1942 ; Plyuschay M. S., “Relativistic particle with torsion, Majorana equation and fractional spin”, Phys. Lett. B, 262 (1991), 71–78 ; Chow C., Nair V. P., Polychronakos A. P., “On the electromagnetic interactions of anyons”, Phys. Lett. B, 304 (1993), 105–110, arXiv: ; Ghosh S., “Spinning particles in 2+1 dimensions”, Phys. Lett. B, 338 (1994), 235–240, arXiv: ; Erratum Phys. Lett. B, 347 (1995), 468, arXiv: hep-th/9301037hep-th/9406089hep-th/9406089 | DOI | MR | DOI | MR | DOI | MR | DOI | MR | DOI

[24] Djemaï A. E. F., Smail H., “On noncommtative classical mechanics”, Commun. Theor. Phys., 41 (2004), 837–844, arXiv: hep-th/0309034 | MR | Zbl

[25] Djemaï A. E. F., “Introduction to Dubois-Violette's noncommutative differential geometry”, Internat. J. Theoret. Phys., 34 (1995), 801–887 | DOI | MR | Zbl

[26] Doplicher S., Fredenhagen K., Roberts J. E., “Spacetime quantization induced by classical gravity”, Phys. Lett. B, 331 (1994), 39–44 ; Doplicher S., Fredenhagen K., Roberts J. E., “The quantum structure of space-time at the Planck scale and quantum fields”, Comm. Math. Phys., 172 (1995), 187–220 , arXiv: http://projecteuclid.org/getRecord?id=euclid.cmp/1104273963hep-th/0303037 | DOI | MR | DOI | MR | Zbl

[27] Amorim R., “Tensor operators in noncommutative quantum mechanics”, Phys. Rev. Lett., 101 (2008), 081602, 4 pp., arXiv: ; Amorim R., “Dynamical symmetries in noncommutative theories”, Phys. Rev. D, 78 (2008), 105003, 7 pp., arXiv: ; Amorim R., “Fermions and noncommutative theories”, J. Math. Phys., 50 (2009), 022303, 7 pp., arXiv: ; Amorim R., “Tensor coordinates in noncommutative mechanics”, J. Math. Phys., 50 (2009), 052103, 7 pp., arXiv: 0804.44000808.30620808.39030804.4405 | DOI | MR | DOI | MR | DOI | MR | DOI | MR

[28] Abreu E. M. C., Mendes A. C. R., Oliveira W., The noncommutative Doplicher–Fredenhagen–Roberts–Amorim space, arXiv: 1003.5322

[29] Dirac P. A. M., “A reformulation of the Born–Infeld electrodynamics”, Proc. Roy. Soc. London. Ser. A, 257 (1960), 32–43 ; Hanson A., Regge T., Teitelboim C., Constrained Hamiltonian systems, Academia Nazionale dei Lincei, Roma, 1976; Sundermeyer K., Constrained dynamics. With applications to Yang–Mills theory, general relativity, classical spin, dual string model, Lectures Notes in Physics, 169, Springer, Berlin, New York, 1982 ; Bergmann P. G., “Non-linear field theories”, Phys. Rev., 75 (1949), 680–685 | DOI | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[30] Banerjee N., Banerjee R., Ghosh S., “Quantization of second class systems in the Batalin–Tyutin formalism”, Ann. Physics, 241 (1995), 237–257, arXiv: (and references therein) hep-th/9403069 | DOI | MR | Zbl

[31] Rajaraman R., “Hamiltonian formulation of the anomalous chiral Schwinger model”, Phys. Lett. B, 154 (1985), 305–309 | DOI | MR

[32] Faddeev L., Shatashivilli S. L., “Realization of the Schwinger term in the Gauss law and the possibility of correct quantization of a theory with anomalies”, Phys. Lett. B, 167 (1986), 225–228 | DOI

[33] Neves C., Oliveira W., “Clebsch parameterization from the symplectic point of view”, Phys. Lett. A, 321 (2004), 267–272, arXiv: hep-th/0310064 | DOI | MR | Zbl

[34] Wotzasek C., “On the Wess–Zumino term for a general anomalous gauge theory with second class constraints”, Internat. J. Modern Phys. A, 5 (1990), 1123–1133 ; Wotzasek C., “Derivation of the Wess–Zumino term for the chiral Schwinger model using Dirac bracket formalism”, J. Math. Phys., 32 (1991), 540–543 | DOI | DOI | MR | Zbl

[35] Batalin I. A., Fradkin E. S., “Operator quantization of dynamical systems with irreducible first- and second-class constraints”, Phys. Lett. B, 180 (1987), 157–162 ; Erratum Phys. Lett. B, 236 (1990), 528–529 ; Batalin I. A., Tyutin I. V., “Existence theorem for the effective gauge algebra in the generalized canonical formalism with Abelian conversion of second-class constraints”, Internat. J. Modern Phys. A, 6 (1991), 3255–3282 ; Batalin I. A., Fradkin E. S., Fradkina T. E., “Another version for operatorial quantization of dynamical systems with irreducible constraints”, Nuclear Phys. B, 314 (1989), 158–174 ; Erratum Nuclear Phys. B, 323 (1989), 734–735 | DOI | MR | DOI | DOI | MR | Zbl | DOI | MR | DOI | MR

[36] Ananias Neto J., Neves C., Oliveira W., “Gauging the SU(2) Skyrme model”, Phys. Rev. D, 63 (2001), 085018, 9 pp., arXiv: ; Abreu E. M. C., Ananias Neto J., Mendes A. C. R., Neves C., Oliveira W., Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism, arXiv: hep-th/00080701001.2254 | DOI | MR

[37] Hong S.-T., Kim Y.-W., Park Y.-J., Rothe K. D., “Symplectic embedding and Hamilton–Jacobi analysis of Proca model”, Modern Phys. Lett. A, 17 (2002), 435–451, arXiv: hep-th/0112170 | DOI | MR | Zbl

[38] Batalin I. A., Fradkin E. S., “Operational quantization of dynamical systems subject to second class constraints”, Nuclear Phys. B, 279 (1987), 514–528 | DOI | MR

[39] Kin Y.-W., Park Y.-J., Rothe K. D., “Hamiltonian embedding of SU(2) Higgs model in the unitary gauge”, J. Phys. G: Nucl. Part. Phys., 24 (1998), 953–961, arXiv: hep-th/9711092 | DOI

[40] Wotzasek C., “The Wess–Zumino term for chiral bosons”, Phys. Rev. Lett., 66 (1991), 129–132 | DOI | MR | Zbl

[41] Vytheeswaran A. S., “Gauge invariances in the Proca model”, Internat. J. Modern Phys. A, 13 (1998), 765–778, arXiv: hep-th/9701050 | DOI | MR | Zbl

[42] Mitra P., Rajaraman R., “New results on systems with second class constraints”, Ann. Physics, 203 (1990), 137–156 | DOI | MR | Zbl

[43] Neves C., Oliveira W., Rodrigues D. C., Wotzasek C., “Hamiltonian symplectic embedding of the massive noncommutative U(1) theory”, J. Phys. A: Math. Gen., 37 (2004), 9303–9315, arXiv: ; Abreu E. M. C., Neves C., Oliveira W., “Noncommutativity from the symplectic point of view”, Internat. J. Modern Phys. A, 21 (2006), 5359–5369, arXiv: ; Neves C., Oliveira W., Rodrigues D. C., Wotzasek C., “Embedding commutative and noncommutative theories in the symplectic framework”, Phys. Rev. D, 69 (2004), 045016, 13 pp., arXiv: hep-th/0310082hep-th/0411108hep-th/0311209 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR

[44] Dirac P. A. M., Lectures on quantum mechanics, Dover, 2001

[45] Faddeev L., Jackiw R., “Hamiltonian reduction of unconstrained and constrained systems”, Phys. Rev. Lett., 60 (1988), 1692–1694 ; Woodhouse N. M. J., Geometric quantization, Oxford Mathematical Monographs, The Clarendon Press, New York; Oxford University Press, 1980 | DOI | MR | Zbl | MR

[46] Barcelos-Neto J., Wotzasek C., “Symplectic quantization of constrained systems”, Modern Phys. Lett. A, 7 (1992), 1737–1747 | DOI | MR | Zbl

[47] Gotay M. J., Nester J. M., Hinds G., “Presymplectic manifolds and the Dirac–Bergmann theory of constraints”, J. Math. Phys., 19 (1978), 2388–2399, (and references therein) | DOI | MR | Zbl

[48] Barcelos-Neto J., “BFFT quantization with nonlinear constraints”, Phys. Rev. D, 55 (1997), 2265–2273, arXiv: hep-th/9701072 | DOI | MR

[49] Abreu E. M. C., Mendes A. C. R., Neves C., Oliveira W., Wotzasek C., Xavier L. M. V., “New considerations about the Maxwell–Podolsky-like theory through the symplectic embedding formalism”, Modern Phys. Lett. A, 25 (2010), 1115–1127, arXiv: 0812.0950 | DOI | Zbl

[50] Montani H., “Symplectic analysis of constrained systems”, Internat. J. Modern Phys. A, 8 (1993), 4319–4337 | DOI | MR | Zbl

[51] Connes A., Noncommutative geometry year 2000, arXiv: math.QA/0011193 | MR

[52] Ghosh S., “Bosonization in the noncommutative plane”, Phys. Lett. B, 563 (2003), 112–116, arXiv: hep-th/0303022 | DOI | MR | Zbl

[53] Moyal J. E., “Quantum mechanics as a statistical theory”, Proc. Cambridge Philos. Soc., 45 (1949), 99–124 ; Vey J., “Déformation du crochet de poisson sur une variété symplectique”, Comment. Math. Helv., 50 (1975), 421–454 ; Flato M., Lichnerowicz A., Sternheimer D., “Déformations 1-différentiables des algèbres de Lie attachés à une variété symplectique ou de contact”, Compositio Math., 31 (1975), 47–82 ; Flato M., Lichnerowicz A., Sternheimer D., “Deformations of Poisson brackets, Dirac brackets and applications”, J. Math. Phys., 17 (1976), 1754–1762 http://www.numdam.org/item?id=CM_1975__31_1_47_0 | DOI | MR | Zbl | DOI | MR | Zbl | MR | Zbl | DOI | MR

[54] Horváthy P. A., “The non-commutative Landau problem”, Ann. Physics, 299 (2002), 128–140, arXiv: hep-th/0201007 | DOI | MR | Zbl

[55] Floreanini R., Jackiw R., “Selfdual fields as charge density solitons”, Phys. Rev. Lett., 59 (1987), 1873–1876 | DOI

[56] Bazeia D., “A quantum-mechanical anomaly re-examined”, Modern Phys. Lett. A, 6 (1991), 1147–1153 | DOI

[57] Banerjee R., Ghosh S., “The chiral oscillator and its applications in quantum theory”, J. Phys. A: Math. Gen., 31 (1998), L603–L608, arXiv: quant-ph/9805009 | DOI | MR | Zbl

[58] Banerjee R., Mukherjee P., “A canonical approach to the quantization of the damped harmonic oscillator”, J. Phys. A: Math. Gen., 35 (2002), 5591–5598, arXiv: quant-ph/0108055 | DOI | MR | Zbl