Shape Invariant Potential and Semi-Unitary Transformations (SUT) for Supersymmetry Harmonic Oscillator in T 4-Space

Abstract

Constructing the Semi-Unitary Transformations (SUT) to obtain the supersymmetric partner Hamiltonians for a one dimensional harmonic oscillator, it has been shown that under this transformation the supersymmetric partner loses its ground state in T 4 -space while its eigen functions constitute a complete orthonormal basis in a subspace of full Hilbert space. Keywords Supersymmetry · Superluminal Transformations · Semi Unitary TransformationsThere are a number of analytically solvable problems in non-relativistic quantum mechanics for which all the energy eigen values and eigen functions are explicably known. The question naturally arises as to why these potentials are solvable and what is the underlying symmetry property? No definite answer was known until 1983 when in a largely unnoticed paper, Grenden Shtein [1] pointed out that all these potentials have a property of shape invariance. Keeping in view the recent potential importance of tachyons [2-13] and the fact that these particles are not contradictory to special theory of relativity and are localized in time in view of second quantization and interaction of superluminal electromagnetic fields [14-17], we have constructed a Semi-Unitary Transformation (SUT) to obtain the supersymmetric partner Hamiltonians for one dimensional harmonic oscillator in T 4 -space (i.e the localization space for tachyons [18][19][20][21]) and it has been demonstrated that under this SUT the supersymmetric partner Hamiltonian H T + loses its ground state while its eigen functions constitute a complete orthonormal set in a subspace of full Hilbert space.In order to over come the various problems associated with superluminal Lorentz transformations (SLTs) [4][5][6][7][8][9][10][11][12][13], six-dimensional formalism [22][23][24][25][26] of space time is adopted with the symmetric structure of space and time having three space and three time components