%0 Journal Article
%T Classical Analog of Extended Phase Space SUSY and Its Breaking
%A Gagik Ter-Kazarian
%J ISRN Mathematical Physics
%D 2013
%R 10.1155/2013/639424
%X 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¨C12]). 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
%U http://www.hindawi.com/journals/isrn.mathematical.physics/2013/639424/