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arXiv:hep-th/0503037 · Journal of Physics A Mathematical and General

Superevolution

N. S. Manton

Published Jun 15, 2005• 2 citations
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Abstract

Usually, in supersymmetric theories, it is assumed that the time evolution of states is determined by the Hamiltonian, through the Schrödinger equation. Here we explore the superevolution of states in superspace, in which the supercharges are the principal operators. The superevolution equation is consistent with the Schrödinger equation, but it avoids the usual degeneracy between bosonic and fermionic states. We discuss superevolution in supersymmetric quantum mechanics and in a simple supersymmetric field theory.

DOI
10.1088/0305-4470/38/26/015

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Usually, in supersymmetric theories, it is assumed that the time evolution of states is determined by the Hamiltonian, through the Schrödinger equation. Here we explore the superevolution of states in superspace, in which the supercharges are the principal operators. The superevolution equation is consistent with the Schrödinger equation, but it avoids the usual degeneracy between bosonic and fermionic states. We discuss superevolution in supersymmetric quantum mechanics and in a simple supersymmetric field theory.
An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail.
Supersymmetry (SUSY) postulates that for every fermion there is boson of equal mass (i.e. energy). In quantum mechanics this comes about because for every quantum Hamiltonian there is a partner Hamiltonian that has the same energy spectrum above the ground state of the original system. In particle physics, one “sector” is populated by bosons and the other sector by fermions and SUSY predicts that the lowest lying fermion state is energetically degenerate with the first excited boson state. Evidence for SUSY has proven to be elusive and it is now believed that SUSY is a broken symmetry. In January 2009 at a conference dedicated to Bob Wyatt, we began to look at SUSY as a way to develop new computational methods and approaches. Up until now, SUSY has been more of a mathematical technique that has been used more or less as a way to obtain stationary solutions to the Schrödinger equation for a variety of T&F Cat # K11224, Chapter 4, Page 54, 15-9-2010 one-dimensional potential systems. In this chapter, we will discuss some of the work we have been doing in developing “SUSY” inspired methods for performing quantum many-body calculations and quantum scattering calculations. We begin with a brief overview of the SUSY theory and some of its elementary results. We then discuss how we have used the approach to develop both analytical and numerical solutions of the stationary Schrödinger equation. Finally, we conclude by discussing our recent extension of SUSY to higher dimensions and for scattering theory.
The authors present the supercoherent states. They are introduced as the eigenstates of the supersymmetric annihilation operator of the quantum mechanical supersymmetric harmonic oscillator. They have a compact expression in terms of the standard (bosonic) coherent states. For each value of the complex parameter z there are two linearly independent orthogonal supercoherent states. One of them has pure fermionic character and is the more classical, while the other is fully supersymmetric (i.e. the mean value of the Klein operator when the system is set in this state vanishes). There are no purely bosonic supercoherent states. The supercoherent fermionic states saturate both the Heisenberg uncertainty relation and the new entropic uncertainty recently introduced by Deutsch (1983), while the supersymmetric ones almost (but not exactly) make it. They give a classical (or almost classical) mean value for the energy of the system and do not spread along their time evolution.