Path Integral Quantization of Superparticle with 1/4 Supersymmetry Breaking

ABSTRACT

We present path integral quantization of a massive
superparticle in *d *=4 which preserves 1/4
of the target space supersymmetry with eight supercharges, and so corresponds
to the partial breaking *N* = 8 to *N* = 2. Its worldline action contains a Wess-Zumino term,
explicitly breaks *d *=4 Lorentz symmetry
and exhibits one complex fermionic *k*-symmetry.
We perform the Hamilton-Jacobi formalism of constrained systems, to obtain the
equations of motion of the model as total differential equations in many
variables. These equations of motion are in exact agreement with those obtained
by Dirac’s method.

Cite this paper

Farahat, N. and Elegla, H. (2013) Path Integral Quantization of Superparticle with 1/4 Supersymmetry Breaking.*Journal of Applied Mathematics and Physics*, **1**, 105-109. doi: 10.4236/jamp.2013.15016.

Farahat, N. and Elegla, H. (2013) Path Integral Quantization of Superparticle with 1/4 Supersymmetry Breaking.

References

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[3] K. Sundermeyer, “Lecture Notes in Physics,” Spring-Verlag, Berlin, 1982.

[4] D. M. Gitman and I. V. Tyutin, “Quantization of Fields with Constraints,” Springer-Verlag, Berlin, 1990.

http://dx.doi.org/10.1007/978-3-642-83938-2

[5] J. Govaerts, “Hamiltonian Quantisation and Constrsined Dynamics,” Vol. 4, Leuven University Press, 1991.

[6] J. L. Anderson and P. G. Bergmann, “Constraints in Covariant Field Theories,” Physical Review, Vol. 83, No. 5, 1951, pp. 1018-1025.

http://dx.doi.org/10.1103/PhysRev.83.1018

[7] P. G. Bergmann and J. Goldberg, “Dirac Bracket Transformations in Phase Space,” Physical Review, Vol. 98, No. 2, 1955, pp. 531-538.

http://dx.doi.org/10.1103/PhysRev.98.531

[8] S. I. Muslih, “Path Integral Quantization of Electromagnetic Theory,” Nuovo Cimento B, Vol. 115, No. 1, 2000, p. 7.

[9] S. I. Muslih, “Quantization of Parametrization Invariant Theories,” Nuovo Cimento B, Vol. 115, 2002, p. 1.

[10] Y. Güler, “Integration of Singular Systems,” Nuovo Cimento B, Vol. 107, No. 10, 1992, pp. 1143-1149.

http://dx.doi.org/10.1007/BF02727199

[11] Y. Güler, “Canonical Formulation of Constrained Systems,” Nuovo Cimento B, Vol. 107, No. 12, 1992, pp. 1389-1395. http://dx.doi.org/10.1007/BF02722849

[12] N. I. Farahat and Y. Güler, “Treatment of a Relativistic Particle in External Electromagnetic Field as a Singular System,” Nuovo Cimento B, Vol. 111, No. 4, 1996, pp. 513-520. http://dx.doi.org/10.1007/BF02724560

[13] E. M. Rabei and S. Tawfiq, “Hamilton-Jacobi Treatment of QED and Yang-Mills Theory as Constrained Systems,” Hadronic Journal, Vol. 20, 1997, p. 232.

[14] S. I. Muslih, “Canonical Path Integral Quantization of Einstein’s Gravitational Field,” General Relativity and Gravitation, Vol. 34, No. 7, 2002, pp. 1059-1065.

http://dx.doi.org/10.1023/A:1016561904569

[15] N. I. Farahat and Z. Nassar, “Relativistic Classical Spinning Particle as Singular System of Second Order,” Islamic University Journal, Vol. 13, 2005, p. 239.

[16] N. I. Farahat and Z. Nassar, “Treatment of a Spinning Particle or Super-gravity in One Dimension Singular System,” Hadronic Journal, Vol. 25, 2002, p. 239.

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[18] S. I. Muslih and Y. Güler, “The Feynman Path Integral Quantization of Constrained Systems,” Nuovo Cimento B, Vol. 112, 1997, p. 97.

[19] S. I. Muslih, “Reduced Phase-Space Quantization of Constrained Systems,” Nuovo Cimento B, Vol. 117, No. 4, 2002, p. 383.

[20] N. I. Farahat and H. A. Elegla, “Hamilton-Jacobi Formulation of Siegle Superparticle,” Turkish Journal of Physics, Vol. 30, No. 6, 2006, pp. 473-478.

[21] J. Bagger and J. Wess, “Partial Breaking of Extended Supersymmetry,” Physics Letters B, Vol. 138, No. 1-3, 1984, pp. 105-110.

http://dx.doi.org/10.1016/0370-2693(84)91882-3

[22] J. Hughes, J. Liu and J. Polchinski, “Supermembranes,” Physics Letters B, Vol. 180, No. 4, 1986, pp. 370-374.

http://dx.doi.org/10.1016/0370-2693(86)91204-9

[23] S. Bellucci, E. Ivanov and S. Krivonos, “Superbranes and Super Born-Infeld Theories from Nonlinear Realizations,” Nuclear Physics B—Proceedings Supplements, Vol. 102-103, 2001, pp. 26-41.

http://dx.doi.org/10.1016/S0920-5632(01)01533-X

[24] J. Hughes and J. Polchinski, “Partially Broken Global Supersymmetry And The Superstring,” Nuclear Physics B, Vol. 278, No. 1, 1986, pp. 147-169.

http://dx.doi.org/10.1016/0550-3213(86)90111-2

[25] E. Ivanov and S. Krivonos, “N = 1, D = 2 Supermem-Brane In The Coset Approach,” Physics Letters B, Vol. 453, No. 3-4, 1999, pp. 237-244.

http://dx.doi.org/10.1016/S0370-2693(99)00314-7

[26] S. Bellucci, E. Ivanov and S. Krivonos, “Superworldvolume Dynamics of Superbranes From Nonlinear Realizations,” Physics Letters B, Vol. 482, No. 1-3, 2000, pp. 233-240. http://dx.doi.org/10.1016/S0370-2693(00)00529-3

[27] S. Coleman, J. Wess and B. Zumino, “Structure of Phenomenological Lagrangians I,” Physical Review, Vol. 177, No. 5, 1969, pp. 2239-2247.

http://dx.doi.org/10.1103/PhysRev.177.2239

[28] C. Callan, S. Coleman, J. Wess and B. Zumino, “Structure of Phenomenological Lagrangians II,” Physical Review, Vol. 177, No. 5, 1969, pp. 2247-2250.

http://dx.doi.org/10.1103/PhysRev.177.2247

[29] D. V. Volkov and J. Sov, “Phenomenological Lagrangians,” Soviet Journal of Particles & Nuclei, Vol. 4, 1973, p. 3.

[30] G. Papadopoulos and P. K. Townsend, “Intersecting MBranes,” Physics Letters B, Vol. 380, No. 3-4, 1996, pp. 273-279. http://dx.doi.org/10.1016/0370-2693(96)00506-0

[31] M. Berkooz, M. Douglas and R. Leigh, “Branes Inter-Secting At Angles,” Nuclear Physics B, Vol. 480, No. 1-2, 1996, pp. 265-278.

http://dx.doi.org/10.1016/S0550-3213(96)00452-X

[32] N. Ohta and P. K. Townsend, “Supersymmetry of MBranes at Angles,” Physics Letters B, Vol. 418, No. 1-2, 1998, pp. 77-84.

http://dx.doi.org/10.1016/S0370-2693(97)01396-8

[33] J. P. Gauntlett and C. M. Hull, “BPS States with Extra Supersymmetry,” Journal of High Energy Physics, Vol. 2000, 2000.

http://dx.doi.org/10.1088/1126-6708/2000/01/004

[34] J. P. Gauntlett, G. W. Gibbons, C. M. Hull and P. K. Townsend, “BPS States of D=4, N=1 Supersymmetry,” Communications in Mathematical Physics, Vol. 216, No. 2, 2001, pp. 431-459.

http://dx.doi.org/10.1007/s002200000341

[35] I. Bandos and J. Lukierski, “Tensorial Central Charges and New Superparticle Models with Fundamental Spinor Coordinates,” Modern Physics Letters A, Vol. 14, No. 19, 1999, p. 1257.

http://dx.doi.org/10.1142/S0217732399001358

[36] I. Bandos, J. Lukierski and D. Sorokin, “Superparticle Models with Tensorial Central Charges,” Physical Review D, Vol. 61, No. 4, 2000, Article ID: 045002.

http://dx.doi.org/10.1103/PhysRevD.61.045002

[37] F. Delduc, E. Ivanov and S. Krivonos, “1/4 Partial Break-Ing Of Global Supersymmetry And New Superparticle Actions,” Nuclear Physics B, Vol. 576, No. 1-3, 2000, pp. 196-218.

http://dx.doi.org/10.1016/S0550-3213(00)00106-1

[38] S. Fedoruk and V. Zima, “Massive Superparticle with Tensorial Central Charges,” Modern Physics Letters A, Vol. 15, No. 37, 2000, p. 2281.

http://dx.doi.org/10.1142/S0217732300002875

[39] S. Bellucci, A. Galajinsky, E. Ivanov and S. Krivonos, “Quantum Mechanics of a Superparticle with 1/4 Supersymmetry Breaking,” Physical Review D, Vol. 65, No. 10, 2002, Article ID: 104023.

http://dx.doi.org/10.1103/PhysRevD.65.104023

[1] M. Henneaux and C. Teitelboim, “Quantization of Gauge Systems,” Princeton University Press, New Jersey, 1992.

[2] P. A. M. Dirac, “Lectures on Quantum Mechanics (Belfer Graduate School of Science),” Yeshiva University, New York, 1964.

[3] K. Sundermeyer, “Lecture Notes in Physics,” Spring-Verlag, Berlin, 1982.

[4] D. M. Gitman and I. V. Tyutin, “Quantization of Fields with Constraints,” Springer-Verlag, Berlin, 1990.

http://dx.doi.org/10.1007/978-3-642-83938-2

[5] J. Govaerts, “Hamiltonian Quantisation and Constrsined Dynamics,” Vol. 4, Leuven University Press, 1991.

[6] J. L. Anderson and P. G. Bergmann, “Constraints in Covariant Field Theories,” Physical Review, Vol. 83, No. 5, 1951, pp. 1018-1025.

http://dx.doi.org/10.1103/PhysRev.83.1018

[7] P. G. Bergmann and J. Goldberg, “Dirac Bracket Transformations in Phase Space,” Physical Review, Vol. 98, No. 2, 1955, pp. 531-538.

http://dx.doi.org/10.1103/PhysRev.98.531

[8] S. I. Muslih, “Path Integral Quantization of Electromagnetic Theory,” Nuovo Cimento B, Vol. 115, No. 1, 2000, p. 7.

[9] S. I. Muslih, “Quantization of Parametrization Invariant Theories,” Nuovo Cimento B, Vol. 115, 2002, p. 1.

[10] Y. Güler, “Integration of Singular Systems,” Nuovo Cimento B, Vol. 107, No. 10, 1992, pp. 1143-1149.

http://dx.doi.org/10.1007/BF02727199

[11] Y. Güler, “Canonical Formulation of Constrained Systems,” Nuovo Cimento B, Vol. 107, No. 12, 1992, pp. 1389-1395. http://dx.doi.org/10.1007/BF02722849

[12] N. I. Farahat and Y. Güler, “Treatment of a Relativistic Particle in External Electromagnetic Field as a Singular System,” Nuovo Cimento B, Vol. 111, No. 4, 1996, pp. 513-520. http://dx.doi.org/10.1007/BF02724560

[13] E. M. Rabei and S. Tawfiq, “Hamilton-Jacobi Treatment of QED and Yang-Mills Theory as Constrained Systems,” Hadronic Journal, Vol. 20, 1997, p. 232.

[14] S. I. Muslih, “Canonical Path Integral Quantization of Einstein’s Gravitational Field,” General Relativity and Gravitation, Vol. 34, No. 7, 2002, pp. 1059-1065.

http://dx.doi.org/10.1023/A:1016561904569

[15] N. I. Farahat and Z. Nassar, “Relativistic Classical Spinning Particle as Singular System of Second Order,” Islamic University Journal, Vol. 13, 2005, p. 239.

[16] N. I. Farahat and Z. Nassar, “Treatment of a Spinning Particle or Super-gravity in One Dimension Singular System,” Hadronic Journal, Vol. 25, 2002, p. 239.

[17] S. I. Muslih and Y. Güler, “Is Gauge Fixing of Constrained Systems Necessary?” Nuovo Cimento B, Vol. 113, 1998, p. 277.

[18] S. I. Muslih and Y. Güler, “The Feynman Path Integral Quantization of Constrained Systems,” Nuovo Cimento B, Vol. 112, 1997, p. 97.

[19] S. I. Muslih, “Reduced Phase-Space Quantization of Constrained Systems,” Nuovo Cimento B, Vol. 117, No. 4, 2002, p. 383.

[20] N. I. Farahat and H. A. Elegla, “Hamilton-Jacobi Formulation of Siegle Superparticle,” Turkish Journal of Physics, Vol. 30, No. 6, 2006, pp. 473-478.

[21] J. Bagger and J. Wess, “Partial Breaking of Extended Supersymmetry,” Physics Letters B, Vol. 138, No. 1-3, 1984, pp. 105-110.

http://dx.doi.org/10.1016/0370-2693(84)91882-3

[22] J. Hughes, J. Liu and J. Polchinski, “Supermembranes,” Physics Letters B, Vol. 180, No. 4, 1986, pp. 370-374.

http://dx.doi.org/10.1016/0370-2693(86)91204-9

[23] S. Bellucci, E. Ivanov and S. Krivonos, “Superbranes and Super Born-Infeld Theories from Nonlinear Realizations,” Nuclear Physics B—Proceedings Supplements, Vol. 102-103, 2001, pp. 26-41.

http://dx.doi.org/10.1016/S0920-5632(01)01533-X

[24] J. Hughes and J. Polchinski, “Partially Broken Global Supersymmetry And The Superstring,” Nuclear Physics B, Vol. 278, No. 1, 1986, pp. 147-169.

http://dx.doi.org/10.1016/0550-3213(86)90111-2

[25] E. Ivanov and S. Krivonos, “N = 1, D = 2 Supermem-Brane In The Coset Approach,” Physics Letters B, Vol. 453, No. 3-4, 1999, pp. 237-244.

http://dx.doi.org/10.1016/S0370-2693(99)00314-7

[26] S. Bellucci, E. Ivanov and S. Krivonos, “Superworldvolume Dynamics of Superbranes From Nonlinear Realizations,” Physics Letters B, Vol. 482, No. 1-3, 2000, pp. 233-240. http://dx.doi.org/10.1016/S0370-2693(00)00529-3

[27] S. Coleman, J. Wess and B. Zumino, “Structure of Phenomenological Lagrangians I,” Physical Review, Vol. 177, No. 5, 1969, pp. 2239-2247.

http://dx.doi.org/10.1103/PhysRev.177.2239

[28] C. Callan, S. Coleman, J. Wess and B. Zumino, “Structure of Phenomenological Lagrangians II,” Physical Review, Vol. 177, No. 5, 1969, pp. 2247-2250.

http://dx.doi.org/10.1103/PhysRev.177.2247

[29] D. V. Volkov and J. Sov, “Phenomenological Lagrangians,” Soviet Journal of Particles & Nuclei, Vol. 4, 1973, p. 3.

[30] G. Papadopoulos and P. K. Townsend, “Intersecting MBranes,” Physics Letters B, Vol. 380, No. 3-4, 1996, pp. 273-279. http://dx.doi.org/10.1016/0370-2693(96)00506-0

[31] M. Berkooz, M. Douglas and R. Leigh, “Branes Inter-Secting At Angles,” Nuclear Physics B, Vol. 480, No. 1-2, 1996, pp. 265-278.

http://dx.doi.org/10.1016/S0550-3213(96)00452-X

[32] N. Ohta and P. K. Townsend, “Supersymmetry of MBranes at Angles,” Physics Letters B, Vol. 418, No. 1-2, 1998, pp. 77-84.

http://dx.doi.org/10.1016/S0370-2693(97)01396-8

[33] J. P. Gauntlett and C. M. Hull, “BPS States with Extra Supersymmetry,” Journal of High Energy Physics, Vol. 2000, 2000.

http://dx.doi.org/10.1088/1126-6708/2000/01/004

[34] J. P. Gauntlett, G. W. Gibbons, C. M. Hull and P. K. Townsend, “BPS States of D=4, N=1 Supersymmetry,” Communications in Mathematical Physics, Vol. 216, No. 2, 2001, pp. 431-459.

http://dx.doi.org/10.1007/s002200000341

[35] I. Bandos and J. Lukierski, “Tensorial Central Charges and New Superparticle Models with Fundamental Spinor Coordinates,” Modern Physics Letters A, Vol. 14, No. 19, 1999, p. 1257.

http://dx.doi.org/10.1142/S0217732399001358

[36] I. Bandos, J. Lukierski and D. Sorokin, “Superparticle Models with Tensorial Central Charges,” Physical Review D, Vol. 61, No. 4, 2000, Article ID: 045002.

http://dx.doi.org/10.1103/PhysRevD.61.045002

[37] F. Delduc, E. Ivanov and S. Krivonos, “1/4 Partial Break-Ing Of Global Supersymmetry And New Superparticle Actions,” Nuclear Physics B, Vol. 576, No. 1-3, 2000, pp. 196-218.

http://dx.doi.org/10.1016/S0550-3213(00)00106-1

[38] S. Fedoruk and V. Zima, “Massive Superparticle with Tensorial Central Charges,” Modern Physics Letters A, Vol. 15, No. 37, 2000, p. 2281.

http://dx.doi.org/10.1142/S0217732300002875

[39] S. Bellucci, A. Galajinsky, E. Ivanov and S. Krivonos, “Quantum Mechanics of a Superparticle with 1/4 Supersymmetry Breaking,” Physical Review D, Vol. 65, No. 10, 2002, Article ID: 104023.

http://dx.doi.org/10.1103/PhysRevD.65.104023