Abstract
We develop Hamiltonian formalism and quantize supersymmetric non-Abelian multiwave system (nAmW) in D=3 spacetime constructed as a simple counterpart of 11D multiple M-wave system. Its action can be obtained from massless superparticle one by putting on its worldline 1d dimensional reduction of the 3d SYM model in such a way that the new system still possesses local fermionic kappa-symmetry.
The quantization results in a set of equation of supersymmetric field theory in an unusual space with su(N)-valued matrix coordinates. Their superpartners, the fermionic su(N)-valued matrices, cannot be split on coordinates and momenta in a covariant manner and hence are included as abstract operators acting on the state vector in the generic form of our D=3 Matrix model field equations. We discuss the Clifford superfield representation for the quantum state vector and in the simplest case of N = 2 elaborate it in a bit more detail. As a check of consistency, we show that the bosonic Matrix model field equations obtained by quantization of the purely bosonic limit of our D=3 nAmW system have a nontrivial solution.
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References
P. West, Introduction to strings and branes, Cambridge University Press, Cambridge U.K. (2012).
I.A. Bandos and A.A. Zheltukhin, Null super p-branes quantum theory in four-dimensional space-time, Fortsch. Phys. 41 (1993) 619 [INSPIRE].
F.A. Berezin and M.S. Marinov, Particle spin dynamics as the Grassmann variant of classical mechanics, Annals Phys. 104 (1977) 336 [INSPIRE].
L. Brink, P. Di Vecchia and P.S. Howe, A locally supersymmetric and reparametrization invariant action for the spinning string, Phys. Lett. 65B (1976) 471.
L. Brink, P. Di Vecchia and P.S. Howe, A lagrangian formulation of the classical and quantum dynamics of spinning particles, Nucl. Phys. B 118 (1977) 76 [INSPIRE].
V.D. Gershun and V.I. Tkach, Classical and quantum dynamics of particles with arbitrary spin, JETP Lett. 29 (1979) 288 [Pisma Zh. Eksp. Teor. Fiz. 29 (1979) 320] [INSPIRE].
P.S. Howe, S. Penati, M. Pernici and P.K. Townsend, Wave equations for arbitrary spin from quantization of the extended supersymmetric spinning particle, Phys. Lett. B 215 (1988) 555 [INSPIRE].
F. Bastianelli, P. Benincasa and S. Giombi, Worldline approach to vector and antisymmetric tensor fields, JHEP 04 (2005) 010 [hep-th/0503155] [INSPIRE].
F. Bastianelli, P. Benincasa and S. Giombi, Worldline approach to vector and antisymmetric tensor fields. II., JHEP 10 (2005) 114 [hep-th/0510010] [INSPIRE].
F. Bastianelli, O. Corradini and E. Latini, Higher spin fields from a worldline perspective, JHEP 02 (2007) 072 [hep-th/0701055] [INSPIRE].
R. Bonezzi, A. Meyer and I. Sachs, Einstein gravity from the N = 4 spinning particle, arXiv:1807.07989 [INSPIRE].
L. Brink and J.H. Schwarz, Quantum superspace, Phys. Lett. 100B (1981) 310 [INSPIRE].
J.A. de Azcarraga and J. Lukierski, Supersymmetric particles with internal symmetries and central charges, Phys. Lett. B 113 (1982) 170 [INSPIRE].
J.A. de Azcarraga and J. Lukierski, Supersymmetric particles in N = 2 superspace: phase space variables and hamiltonian dynamics, Phys. Rev. D 28 (1983) 1337 [INSPIRE].
W. Siegel, Hidden local supersymmetry in the supersymmetric particle action, Phys. Lett. 128B (1983) 397 [INSPIRE].
M.B. Green, M. Gutperle and H.H. Kwon, Light cone quantum mechanics of the eleven-dimensional superparticle, JHEP 08 (1999) 012 [hep-th/9907155] [INSPIRE].
I.A. Bandos, Spinor moving frame, M0-brane covariant BRST quantization and intrinsic complexity of the pure spinor approach, Phys. Lett. B 659 (2008) 388 [arXiv:0707.2336] [INSPIRE].
I.A. Bandos, D = 11 massless superparticle covariant quantization, pure spinor BRST charge and hidden symmetries, Nucl. Phys. B 796 (2008) 360 [arXiv:0710.4342] [INSPIRE].
E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335 [hep-th/9510135] [INSPIRE].
A.A. Tseytlin, On non-Abelian generalization of Born-Infeld action in string theory, Nucl. Phys. B 501 (1997) 41 [hep-th/9701125] [INSPIRE].
R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].
Y. Lozano and D. Rodriguez-Gomez, Fuzzy 5-spheres and pp-wave matrix actions, JHEP 08 (2005) 044 [hep-th/0505073] [INSPIRE].
B. Janssen and Y. Lozano, On the dielectric effect for gravitational waves, Nucl. Phys. B 643 (2002) 399 [hep-th/0205254] [INSPIRE].
B. Janssen and Y. Lozano, A microscopical description of giant gravitons, Nucl. Phys. B 658 (2003) 281 [hep-th/0207199] [INSPIRE].
P.S. Howe, U. Lindström and L. Wulff, Superstrings with boundary fermions, JHEP 08 (2005) 041 [hep-th/0505067] [INSPIRE].
On the covariance of the Dirac-Born-Infeld-Myers action, JHEP 02 (2007) 070 [hep-th/0607156] [INSPIRE].
D.P. Sorokin, Coincident (super)Dp-branes of codimension one, JHEP 08 (2001) 022 [hep-th/0106212] [INSPIRE].
S. Panda and D. Sorokin, Supersymmetric and kappa invariant coincident D0-branes, JHEP 02 (2003) 055 [hep-th/0301065] [INSPIRE].
I.A. Bandos, Action for the eleven dimensional multiple M-wave system, JHEP 01 (2013) 074 [arXiv:1207.0728] [INSPIRE].
I.A. Bandos and C. Meliveo, Covariant action and equations of motion for the eleven dimensional multiple M0-brane system, Phys. Rev. D 87 (2013) 126011 [arXiv:1304.0382] [INSPIRE].
J.M. Drummond, P.S. Howe and U. Lindström, Kappa symmetric non-Abelian Born-Infeld actions in three-dimensions, Class. Quant. Grav. 19 (2002) 6477 [hep-th/0206148] [INSPIRE].
I.A. Bandos, Supersymmetric non-Abelian multiwaves in D = 3 AdS superspace, JHEP 11 (2013) 143 [arXiv:1309.0512] [INSPIRE].
J. Bagger and N. Lambert, Modeling multiple M 2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108] [INSPIRE].
A. Gustavsson, Algebraic structures on parallel M 2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
M.R. Douglas, On D = 5 super Yang-Mills theory and (2, 0) theory, JHEP 02 (2011) 011 [arXiv:1012.2880] [INSPIRE].
N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, M 5-branes, D4-branes and quantum 5D super-Yang-Mills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE].
B. de Wit, J. Hoppe and H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
H.S. Snyder, Quantized space-time, Phys. Rev. 71 (1947) 38 [INSPIRE].
A. Connes, Noncommutative geometry, Academic Press, Noew York U.S.A. (1995).
A. Connes, M.R. Douglas and A.S. Schwarz, Noncommutative geometry and matrix theory: Compactification on tori, JHEP 02 (1998) 003 [hep-th/9711162] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
L. Brink and J.H. Schwarz, Clifford algebra superspace, CALT-68-813 (1980) [INSPIRE].
A. Ferber, Supertwistors and conformal supersymmetry, Nucl. Phys. B 132 (1978) 55 [INSPIRE].
T. Shirafuji, Lagrangian mechanics of massless particles with spin, Prog. Theor. Phys. 70 (1983) 18 [INSPIRE].
A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky and E.S. Sokatchev, Harmonic superspace, Cambridge University Press, Cambridge U.K. (2001).
I.A. Bandos, Superparticle in Lorentz harmonic superspace. (In Russian), Sov. J. Nucl. Phys. 51 (1990) 906 [INSPIRE].
I.A. Batalin and E.S. Fradkin, Operatorial Quantization of Dynamical Systems Subject to Second Class Constraints, Nucl. Phys. B 279 (1987) 514 [INSPIRE].
P.A.M. Dirac, Lectures on quantum mechanics, Academic Press, New York U.S.A. (1967).
I.A. Batalin, E.S. Fradkin and T.E. Fradkina, Another version for operatorial quantization of dynamical systems with irreducible constraints, Nucl. Phys. B 314 (1989) 158 [Erratum ibid. B 323 (1989) 734] [INSPIRE].
E.S. Egorian and R.P. Manvelyan, Quantization of dynamical systems with first and second class constraints, Theor. Math. Phys. 94 (1993) 173 [Teor. Mat. Fiz. 94 (1993) 241] [INSPIRE].
I.A. Bandos, J. Lukierski and D.P. Sorokin, Superparticle models with tensorial central charges, Phys. Rev. D 61 (2000) 045002 [hep-th/9904109] [INSPIRE].
D.P. Sorokin, Double supersymmetric particle theories, Fortsch. Phys. 38 (1990) 923 [INSPIRE].
F. Delduc, S. Kalitsyn and E. Sokatchev, Learning the ABC of light cone harmonic space, Class. Quant. Grav. 6 (1989) 1561 [INSPIRE].
S. Fedoruk and V.G. Zima, Covariant quantization of d = 4 Brink-Schwarz superparticle with Lorentz harmonics, Theor. Math. Phys. 102 (1995) 305 [Teor. Mat. Fiz. 102 (1995) 420] [hep-th/9409117] [INSPIRE].
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Bandos, I., Sabido, M. Hamiltonian approach and quantization of D = 3,\( \mathcal{N}=1 \) supersymmetric non-Abelian multiwave system. J. High Energ. Phys. 2018, 112 (2018). https://doi.org/10.1007/JHEP09(2018)112
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DOI: https://doi.org/10.1007/JHEP09(2018)112