We derive the classical analog of the extended phase space quantum mechanics of the particle with odd degrees of freedom which gives rise to ( )-realization of supersymmetry (SUSY) algebra. By means of an iterative procedure, we find the approximate ground state solutions to the extended Schr？dinger-like equation and use these solutions further to calculate the parameters which measure the breaking of extended SUSY such as the ground state energy. Consequently, we calculate a more practical measure for the SUSY breaking which is the expectation value of an auxiliary field. We analyze nonperturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of the theory. Particular attention is given to the algebraic properties of shape invariance and spectrum generating algebra. 1. Introduction An interesting question of keeping the symmetry between canonical coordinates and momenta in the process of quantization deserves an investigation. From its historical development, this aspect of statistical quantum mechanics, unfortunately, has attracted little attention. However, much use has been made of the technique of ordering of canonical coordinates ( ) and momenta ( ) in quantum mechanics [1, 2]. It was observed that the concept of an extended Lagrangian in phase space allows a subsequent extension of Hamilton’s principle to actions minimum along the actual trajectories in -, rather than in -space. This leads to the phase space formulation of quantum mechanics. Consequently this formalism was developed further in [3] by addressing the extended phase space stochastic quantization of Hamiltonian systems with first class holonomic constraints. This in a natural way results in the Faddeev-Popov conventional path-integral measure for gauge systems. Continuing along this line in the present paper we address the classical analog of the extended phase space ( )-SUSY quantum mechanics [4] of the particles which have both bosonic and fermionic degrees of freedom, that is, the quantum field theory in -dimensions in -space, exhibiting supersymmetry (for conventional SUSY quantum mechanics see [5–12]). We analyze in detail the non-perturbative mechanism for supersymmetry breaking in the instanton picture ([13]). This paper has been organized as follows. In the first part (Sections 2 and 3), we derive the classical analog of the extended phase space SUSY quantum mechanics and obtain the integrals of motion. Consequently, we describe the extended phase space ( )-SUSY algebra. In
References
[1]
Y. Sobouti and S. Nasiri, “A phase space formulation of quantum state functions,” International Journal of Modern Physics B, vol. 7, no. 18, pp. 3255–3272, 1993.
[2]
S. Nasiri, Y. Sobouti, and F. Taati, “Phase space quantum mechanics—direct,” Journal of Mathematical Physics, vol. 47, no. 9, Article ID 092106, 15 pages, 2006.
[3]
G. T. Ter-Kazarian and Y. Sobouti, “An extended phase-space stochastic quantization of constrained Hamiltonian systems,” Journal of Physics A, vol. 41, no. 31, Article ID 315303, 8 pages, 2008.
[4]
G. Ter-Kazarian, “An extended phase-space SUSY quantum mechanics,” Journal of Physics A, vol. 42, no. 5, Article ID 055302, 12 pages, 2009.
[5]
H. Nicolai, “Supersymmetry and spin systems,” Journal of Physics A, vol. 9, no. 9, pp. 1497–1506, 1976.
[6]
E. Witten, “Dynamical breaking of supersymmetry,” Nuclear Physics B, vol. 188, no. 3, pp. 513–554, 1981.
[7]
E. Witten, “Constraints on supersymmetry breaking,” Nuclear Physics B, vol. 202, no. 2, pp. 253–316, 1982.
[8]
A. Inomata and G. Junker, “Quasiclassical path-integral approach to supersymmetric quantum mechanics,” Physical Review A, vol. 50, no. 5, pp. 3638–3649, 1994.
[9]
A. Inomata, H. Kuratsuji, and C. C. Gerry, Path Integrals and Coherent States of SU(2) and SU(1,1), Singapore, 1992.
[10]
J. Gamboa and M. Plyushchay, “Classical anomalies for spinning particles,” Nuclear Physics B, vol. 512, no. 1-2, pp. 485–504, 1998.
[11]
G. Grignani, M. Plyushchay, and P. Sodano, “A pseudoclassical model for -invariant planar fermions,” Nuclear Physics B, vol. 464, no. 1-2, pp. 189–212, 1996.
[12]
J. L. Cortés, M. S. Plyushchay, and L. Velázquez, “A pseudoclassical model for the massive Dirac particle in dimensions,” Physics Letters B, vol. 306, no. 1-2, pp. 34–40, 1993.
[13]
P. Salomonson and J. W. van Holten, “Fermionic coordinates and supersymmetry in quantum mechanics,” Nuclear Physics B, vol. 196, no. 3, pp. 509–531, 1982.
[14]
A. M. Polyakov, “Quark confinement and topology of gauge theories,” Nuclear Physics B, vol. 120, no. 3, pp. 429–458, 1977.
[15]
L. Gendenshtein, “Derivation of exact spectra of the Schrdinger equation by means of supersymmetry,” Pisma Zhurnal Eksperimental noi i Teoreticheskoi Fiziki, vol. 38, pp. 299–302, 1983.
[16]
A. Gangopadhyaya, J. Mallow, and U. P. Sukhatme, “Shape invariance and its connection to potential algebra,” in Supersymmetry and Integrable Models, vol. 502 of Lecture Notes in Physics, pp. 341–350, Springer, 1998.
[17]
A. B. Balantekin, “Algebraic approach to shape invariance,” Physical Review A, vol. 57, no. 6, pp. 4188–4191, 1998.
[18]
F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Physics Reports, vol. 251, no. 5-6, pp. 267–385, 1995.
