Journal of Applied Mathematics and Physics, 2013, 1, 105109 Published Online November 2013 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2013.15016 Open Access JAMP Path Integral Quantization of Superparticle with 1/4 Supersymmetry Breaking Nasser Ismail Farahat, Hanaa Abdulkareem Elegla Physics Department, Islamic University of Gaza, Gaza, Palestine Email: nfarahat@iugaza.edu.ps, helegla@iugaza.edu.ps Received September 28, 2013; revised October 28, 2013; accepted November 5, 2013 Copyright © 2013 Nasser Ismail Farahat, Hanaa Abdulkareem Elegla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We present path integral quantization of a massive superparticle in 4d which preserves 1 4 of the target space su persymmetry with eight supercharges, and so corresponds to the partial breaking 8NN2 . Its worldline action contains a WessZumino term, explicitly breaks 4d Lorentz symmetry and exhibits one complex fermionic symmetry. We perform the HamiltonJacobi 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 m ethod. Keywords: HamiltonJacobi Formalism; Singular Lagrangian; Supersymmetry 1. Introduction The theory of constrained systems is a basis of modern physics: gauge field theories, quantum gravity, supergra vity, string and superstring models are examples of sys tems with constraints. For such theories, one should sp ec i fy not only an evolution equation but also additional re quirements (constraints) being imposed on initial condi tions [1]. A standard consistent way of dealing with singular systems was first formulated by Dirac [2]. In Dirac for mulism, when a singular Lagrangian in configuration space is transformed into a singular Lagrangian in phase space, the inherent constraints would be generated, which are called Dirac primary constraints [35]. Through the consistency conditions, step by step, using these primary constraints may generate more new inherent constraints, which are called Dirac secondary constraints. Such a way to calculate different constraints in Dirac formalism is named as DiracBergmann algorithm, which was first propos ed by Bergmann [6,7]. Canonical path integral method is a kind of quantiza tion method [8,9], which depends on HamiltonJacobi formalism shown by Güler [10,11]. This method has some very useful properties of obviating the need to dis tinguish primary and secondary constraints and the first and the second types of constraints. The method is sim pler, and does not have such a hypothesis of Diracs con jecture, thus it has evoked much attention [1220]. Partial breaking of global supersymmetr y (PBGS) [21 24] is widely understood to be an inborn feature of su persymmetric extended objects. The concept of PBGS provides a manifestly offshell supersymmetric world volume description of various superbranes in terms of Goldstone sup erfields , ii x . The physical wo r l d  volume multiplets of the given superbrane are interpreted as Goldstone superfields realizing the spontaneous brea king of the full brane supersymmetry group down to its unbroken worldvolume subgroup [25,26]. The technical tools here are the method of nonlinear realizations [2729]. Recently, there has been a growing interest in PBGS options other than the 1/2 breaking [3034]. This is essentially due to the discovery of the super gravity solutions preserving 1/4 or 1/8 of the 11d11d supersymmetry [30] and their subsequent interpretation in terms of intersecting branes. Since branelike world volume effective actions which would be capable of de scribing those solutions are still unknown, it seems inter esting to study pointlike models that mimic the exotic supersymmetry breaking options inherent in the inter secting branes. Such models could share some character
N. I. FARAHAT, H. A. ELEGLA 106 istic features of the systems of intersecting branes, much like the ordinary superparticle bears a similarity to the GreenSchwarz superstring. Superparticle models exhib iting 3/4 or 1/4 PBGS have been constructed [3538]. The 1/4 breaking of the original supersymmetry down to manifests itself in the presence of only one complexsymmetry in the corresponding worldline action. This is achieved at the cost of the explicit break ing of the target space Lorentz symmetry down to SO(3) symmetry (in the fermionic sector). 8N 8N 2N 48 t In the present paper we study the canonical path in tegral quantization for model as a ty pical example of massive superparticles with 1/4 PBGS in an ordinary four dimensional Minkowski spacetime (with as the target superspace and explicitly bro ken Lorentz symmetry) [39]. Prior to quantization, Ham iltonian analysis is accomplished in full detail, by ob taining the set of inherent constraints, and the equations of motion as total differential equations. Our paper is organized as follows. HamiltonJacobi formulation is presented in Section 2. In Section 3, the canonical path integral qu antizatio n of our model is investigated. In Sec tion 4, the conclusion is given. 2 n N R 2. HamiltonJacobi Formalism of Constrained Systems The system that is described by the Lagrangian ,, ii Lqq , , is constrained system if the Hessian matrix 1, ,i 2,1,,, ij ij L ij n qq (1) has a rank , . In this case we have momenta are dependent of each other. The generalized momenta i corresponding to the generalized coor dinates are defined as, n p i q rrnr 1, ,, aa L pan q r (2) 1,,.n L pnr q (3) Since the rank of the Hessian matrix is nr , one may solve (2) for as a q ,, aai qq a qp . (4) Substituting (4) into (3), we obtain relations in ,, ia qpq and t in the form ,, ,,. aa qiaa L pHqqq q a pt (5) The canonical Hamiltonian 0 is defined as 0,, ,. iaa aapH HLqqqtp qp The set of HamiltonJacobi partial differential equa tions (HJPDE) is ex p re ss ed as ;; ;0, ,0, 1,,, aa a SS Hqqpp qq nr n (7) where 000 ; pH (8) . pH (9) with 0 qt and being the action. The equations of motion are obtained as total differential equations in many variables such as, S d aa H q p d,t (10) d H p q d,t (11) dd aa H. Hp t p (12) where ,a St q .These equation are integrable if and only if 0 dH0, (13) d0, 1,,. nr n (14) If the conditions (13) and (14) are not satisfied iden tically, we consider them as new constraints and we ex amine their variations. Thus repeating this procedure, one may obtain a set of constraints such that all the variations vanish, then we may solve the equations of motion (10) and (11) to get the canonical phasespace coordinates as ,,,, 1,, aaaa qq ttpp ttr . (15) In this case the path integral representation may be written as 1 Out In dd expd, nr t aa a t aa S H qpiHpt p (16) 1, ,,0,1, ,.anr nrn We should notice that the integral (16) is an integra tion over the canonical phase space coordinates , aa qp. 3. HamiltonJacobi Formulation of Superparticle with 1/4 Supersymmetry Breaking (6) The action functional of a massive superparticle model exhibiting 1/4 PBGS, is written as [39] Open Access JAMP
N. I. FARAHAT, H. A. ELEGLA 107 00 2 11 d. (17) 22 ii ii Semim e where 00 , 222 2 ,1,2,3. ii ii ii ii iii i x xi ii and ,i are four complex fermions parameterizing the odd sector of the model. The Lagrangian is 00 2 11 , 22 ii ii Lemim e (18) The singularity of the the Lagrangian follows from the fact that the rank of the Hessian matrix ij is two. The canonical momenta defined in (2) and (3) read as 00 0 1, 222 2 ii ii Liiii Px e x (19) 1, iii i L Pxii e x i (20) 0, 2 ii Li Pm iPH (21) 0, 2 ii Li Pm iPH (22) 0, 2i i i Li Pm H i (23) 0, 2i i i Li Pm H i (24) and 0. (25) ee L PH e Now the velocities 0 and i i P can be expressed in terms of the momenta and respectively as 0 P 0. 222 2 oii iii i xeP ii (26) and . iiii xePii (27) The canonical Hamiltonian is obtained as 00 2 1. 2 ii HePPPPm (28) The set of HJPDE’s are 00 2 10, 2 ii HP ePPPPm 00, 2 ii i HPmiP (30) 00, 2 ii i HPmiP (31) 00, 2 ii i HPm i (32) 00, 2 ii i HPm i (33) and 0. ee HP (34) Therefore, the total differential equations for the cha racteristics read as 0 dddddd 222 2 oii iii i xeP , ii (35) dddd ii ii xeP ii , (36) 0 dP0, (37) d0 i P, (38) 0 d 2 iPm d, (39) 0 d 2 iPm d, (40) 0 dd d 2 iii iPP m , i (41) 0 dd d 2 ii i iPP m , i (42) and 002 1 dd 2 ii e PPPPPm . (43) To check whether the set of Equations (35)(43) are integrable or not, let us consider the total variations of the set of (HJPDE)’s. The variati on of dH0, (44) dH 0, (45) dH 0, (46) d i H 0, (47) d i H 0, (48) are identically zero, whereas 00 2 1 dd 2 ii ee d. PPPPmH t (49) (29) where Open Access JAMP
N. I. FARAHAT, H. A. ELEGLA 108 00 2 10. 2 ii e HPPPPm (50) is a new constraint. Thus the equations of motion (35) (43) and the new constraint (50) represent an integrable system. Now to obtain the path integral quantization of this system, we can use (12) to obtain the canonical action integral as 00 2 11 d. 22 ii ii SePPPPemim (51) By using (51) and (16) the path integral for the system is expressed as 00 00 00 2 ,,;, , 1 dddd exp2 1d 2 ii ii i ii xxxx i xPPie PPPP em im (52) 4. Conclusion The path integral qantization of constrained systems is obtained for using the canonical path integral method, which based on the constrained Hamilton theory. The equations of motion are obtained as total differential equations in many variables, and the integrability con ditions were shown to be equivalent to the vanishing of the variation of each , i.e. ’s, then the sys tem is integrable. In this paper, we examined Hamilto nian treatment of a massive superparticle model with 1/4 partial breaking of global supersymmetry which propa gates in four dimensional flat spacetime. We obtain con straints in phase space, which contains all kinds of con straints (primary and secondary, first and second class ones). This example is very illustrative, since it allows a comparison between all features of Diracs and Hamil tonJacobi formalisms. In Dirac’s formalism, we must reduce any constrained singular system to one with first class constraints only, and we must call attention to the presence of arbitrary variables in some of the Hamil tonian equations of motion due to the fact that we have gauge dependent variables, therefore we have made a gauge fixing. This does not occur in HamiltonJacobi formalism since it provides a gaugeindependent descrip tion of the systems evolution due to the fact that the HamiltonJacobi functio n S contains all the solu tions that are related by gauge transformations. Our results are in agreement with those given in Dirac’s method [39]. dH 0 REFERENCES [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,” SpringVer lag, Berlin, 1982. [4] D. M. Gitman and I. V. Tyutin, “Quantization of Fields with Constraints,” SpringerVerlag, Berlin, 1990. http://dx.doi.org/10.1007/9783642839382 [5] J. Govaerts, “Hamiltonian Quantisation and Constrsined Dynamics,” Vol. 4, Leuven University Press, 1991. [6] J. L. Anderson and P. G. Bergmann, “Constraints in Co variant Field Theories,” Physical Review, Vol. 83, No. 5, 1951, pp. 10181025. http://dx.doi.org/10.1103/PhysRev.83.1018 [7] P. G. Bergmann and J. Goldberg, “Dirac Bracket Trans formations in Phase Space,” Physical Review, Vol. 98, No. 2, 1955, pp. 531538. http://dx.doi.org/10.1103/PhysRev.98.531 [8] S. I. Muslih, “Path Integral Quantization of Electromag netic 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 Ci mento B, Vol. 107, No. 10, 1992, pp. 11431149. http://dx.doi.org/10.1007/BF02727199 [11] Y. Güler, “Canonical Formulation of Constrained Sys tems,” Nu ovo Ci men to 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. 513520. http://dx.doi.org/10.1007/BF02724560 [13] E. M. Rabei and S. Tawfiq, “HamiltonJacobi Treatment of QED and YangMills 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. 10591065. http://dx.doi.org/10.1023/A:1016561904569 [15] N. I. Farahat and Z. Nassar, “Relativistic Classical Spin ning Particle as Singular System of Second Order,” Is lamic University Journal, Vol. 13, 2005, p. 239. [16] N. I. Farahat and Z. Nassar, “Treatment of a Spinning Particle or Supergravity in One Dimension Singular Sys tem,” Hadronic Journal, Vol. 25, 2002, p. 239. [17] S. I. Muslih and Y. Güler, “Is Gauge Fixing of Con strained 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 PhaseSpace Quantization of Con strained Systems,” Nuovo Cimento B, Vol. 117, No. 4, 2002, p. 383. Open Access JAMP
N. I. FARAHAT, H. A. ELEGLA Open Access JAMP 109 [20] N. I. Farahat and H. A. Elegla, “HamiltonJacobi Formu lation of Siegle Superparticle,” Turkish Journal of Phys ics, Vol. 30, No. 6, 2006, pp. 473478. [21] J. Bagger and J. Wess, “Partial Breaking of Extended Supersymmetry,” Physics Letters B, Vol. 138, No. 13, 1984, pp. 105110. http://dx.doi.org/10.1016/03702693(84)918823 [22] J. Hughes, J. Liu and J. Polchinski, “Supermembranes,” Physics Letters B, Vol. 180, No. 4, 1986, pp. 370374. http://dx.doi.org/10.1016/03702693(86)912049 [23] S. Bellucci, E. Ivanov and S. Krivonos, “Superbranes and Super BornInfeld Theories from Nonlinear Realiza tions,” Nuclear Physics B—Proceedings Supplements, Vol. 102103, 2001, pp. 2641. http://dx.doi.org/10.1016/S09205632(01)01533X [24] J. Hughes and J. Polchinski, “Partially Broken Global Supersymmetry And The Superstring,” Nuclear Physics B, Vol. 278, No. 1, 1986, pp. 147169. http://dx.doi.org/10.1016/05503213(86)901112 [25] E. Ivanov and S. Krivonos, “N = 1, D = 2 Supermem Brane In The Coset Approach,” Physics Letters B, Vol. 453, No. 34, 1999, pp. 237244. http://dx.doi.org/10.1016/S03702693(99)003147 [26] S. Bellucci, E. Ivanov and S. Krivonos, “Superworldvo lume Dynamics of Superbranes From Nonlinear Realiza tions,” Physic s Letters B, Vol. 482, No. 13, 2000, pp. 233 240. http://dx.doi.org/10.1016/S03702693(00)005293 [27] S. Coleman, J. Wess and B. Zumino, “Structure of Phe nomenological Lagrangians I,” Physical Review, Vol. 177, No. 5, 1969, pp. 22392247. http://dx.doi.org/10.1103/PhysRev.177.2239 [28] C. Callan, S. Coleman, J. Wess and B. Zumino, “Struc ture of Phenomenological Lagrangians II,” Physical Re view, Vol. 177, No. 5, 1969, pp. 22472250. http://dx.doi.org/10.1103/PhysRev.177.2247 [29] D. V. Volkov and J. Sov, “Phenomenological Lagran gians,” Soviet Journal of Particles & Nuclei, Vol. 4, 1973, p. 3. [30] G. Papadopoulos and P. K. Townsend, “Intersecting M Branes,” Physics Letters B, Vol. 380, No. 34, 1996, pp. 273279. http://dx.doi.org/10.1016/03702693(96)005060 [31] M. Berkooz, M. Douglas and R. Leigh, “Branes Inter Secting At Angles,” Nuclear Physics B, Vol. 480, No. 12, 1996, pp. 265278. http://dx.doi.org/10.1016/S05503213(96)00452X [32] N. Ohta and P. K. Townsend, “Supersymmetry of M Branes at Angles,” Physics Letters B, Vol. 418, No. 12, 1998, pp. 7784. http://dx.doi.org/10.1016/S03702693(97)013968 [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/10.1088/11266708/2000/01/004 [34] J. P. Gauntlett, G. W. Gibbons, C. M. Hull and P. K. To w n send, “BPS States of D=4, N=1 Supersymmetry,” Com munications in Mathematical Physics, Vol. 216, No. 2, 2001, pp. 431459. 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 Physic s 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 Re view 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. 13, 2000, pp. 196218. http://dx.doi.org/10.1016/S05503213(00)001061 [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 Super symmet ry Breaking ,” Physical Review D, Vol. 65, No. 10, 2002, Article ID: 104023. http://dx.doi.org/10.1103/PhysRevD.65.104023
