ANNALS OF PHYSICS 146, 262-288 (1983) Aspects of Supersymmetric Quantum Mechanics* FRED COOPER AND BARRY FREEDMAN+ Theoretical Division, Los Alamos National Laboratory, Mail Stop B 285, Los Alamos, New Mexico 87545 Received August 2, 1982 We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E, = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find P,(y) is identical to 1 YO(~)l’ in the quantum theory. I. INTRODUCTION There has been a renewed interest in supersymmetric field theories recently as a possible vehicle for solving the gauge hierarchy problem in unified theories. The gauge hierarchy problem is essentially the question of why the scalar Higgs meson masses are so light compared with the cutoff scale ,4 which might be the grand unified scale or Planck scale. In ordinary theories radiative corrections of meson masses are quadratic (A’) which requires extreme line tuning of bare masses, to obtain the correct physical masses. In supersymmetric theories, scalar masses can be kept zero because of the chiral invariance of the fermion sector plus the supersym- metry. Thus, if supersymmetry is dynamically broken at IO3 GeV, one would expect the masses that are produced would be at the level of the spontaneous breakdown rather than at A’. Of course, in any realistic theory of nature based on supersym- metry, supersymmetry must be broken because of the lack of degenerate boson partners for the light leptons. It is well known that if the classical approximation to a supersymmetric theory is supersymmetric, then spontaneous breakdown by perturbative radiative corrections is not possible. Thus, there are two possible scenarios for breakdown of supersymmetry. * Los Alamos National Laboratory is operated by the University of California for the U.S. Department of Energy under Contract W-7405-ENG-36. The US. Government’s right to retain a nonex- clusive royalty-free license in and to the copyright covering this paper, for governmental purposes, is acknowledged. ’ Present Address: University of Illinois, Champagne-Urbana, Illinois 61820. 262 0003-4916183 91.50 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form rwcrvcd.