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Physics Essays

Supersymmetric algebra of the negative-kinetic-energy supersymmetric quantum mechanics

Lu Huan, Wan Buyong, Liu Yao, Luo Guang

Published Sep 24, 2026• 0 citations
Publisher

Abstract

The issues of the negative energy and the negative kinetic energy remain among the most profound and actively debated topics in physics. Based on the negative-kinetic-energy supersymmetric quantum mechanics (NKE-SUSYQM), this paper explores the supersymmetric algebra in depth. First, we begin with a concise overview of how the negative-kinetic-energy Schrödinger equation is derived and how NKE-SUSYQM is thereby established. Second, we analyze the fundamental quantum operators and elucidate the complete structure of the supersymmetric algebra within NKE-SUSYQM. Finally, we provide a detailed comparative study that contrasts the positive- and the negative-kinetic-energy SUSYQM, highlighting their key similarities and differences.

DOI
10.4006/0836-1398-39.3.308

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The issues of the negative energy and the negative kinetic energy remain among the most profound and actively debated topics in physics. Based on the negative-kinetic-energy supersymmetric quantum mechanics (NKE-SUSYQM), this paper explores the supersymmetric algebra in depth. First, we begin with a concise overview of how the negative-kinetic-energy Schrödinger equation is derived and how NKE-SUSYQM is thereby established. Second, we analyze the fundamental quantum operators and elucidate the complete structure of the supersymmetric algebra within NKE-SUSYQM. Finally, we provide a detailed comparative study that contrasts the positive- and the negative-kinetic-energy SUSYQM, highlighting their key similarities and differences.
The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’s equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac’s equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrödinger equation, and the other is the NKE Schrödinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author’s opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.
Abstract Liénard-type nonlinear one dimensional oscillator is quantized using van Roos symmetric ordering recipe for the kinetic-like part of the new derived Hamiltonian. The corresponding Schrödinger equation is exactly solved in momentum space via the approach of supersymmetric quantum mechanics (SUSYQM). The bound-states energy spectra and corresponding wave functions are given explicitly in terms of the ambiguity parameters. The limiting case of no deformation agrees exactly with the eigenenergies and eigenfunctions of the ordinary quantum harmonic oscillator.
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.