Supersymmetry and the Dirac equation

Cooper, Fred; Khare, Avinash; Musto, R.; Wipf, Andreas

doi:10.1016/0003-4916(88)90279-5

Cited by 106 publications

(84 citation statements)

References 30 publications

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“…First of all, we consider the Dirac equation in 1+1 dimensions with Lorentz scalar potential φ(x). We show that whenever the one-dimensional Schrödinger equation is analytically solvable for a potential V (x), then there always exists a corresponding Dirac scalar potential problem which is also analytically solvable [101]. It turns out that, on the one hand, φ(x) is essentially the superpotential of the Schrödinger problem and on the other hand, it can be looked upon as the kink solution of a scalar field theory in 1+1 dimensions.…”

Section: Supersymmetry and The Dirac Equationmentioning

confidence: 91%

“…We also discuss the problem of the Dirac equation in an external magnetic field in two dimensions and show that there is always a supersymmetry in the problem in the massless case. We also classify a number of magnetic field problems whose solutions can be algebraically obtained by using the concepts of SUSY and shape invariance [101]. In addition, we show that the Euclidean Dirac operator in four dimensions, in the background of gauge fields, can always be cast in the language of SUSY QM.…”

Section: Supersymmetry and The Dirac Equationmentioning

confidence: 97%

“…In addition, we show that the Euclidean Dirac operator in four dimensions, in the background of gauge fields, can always be cast in the language of SUSY QM. Finally, we discuss the path integral formulation of the fermion propagator in an external field and show how the previously known results for the constant external field can be very easily obtained by using the ideas of SUSY [101].…”

Section: Supersymmetry and The Dirac Equationmentioning

confidence: 97%

“…Thus we have reduced the problem to that of SUSY in one dimension with superpotential W (y) + k where W (y) must be independent of k. This constraint on W (y) strongly restricts the allowed forms of shape invariant W (y) for which the spectrum can be written down algebraically. In particular from Table 4.1 we find that the only allowed forms are (i) W (y) = ω c y + c 1 (ii) W (y) = a tanh y + c 1 (iii) W (y) = a tan y + c 1 (iv) W (y) = c 1 − c 2 exp (−y) for which W (y) can b e written in terms of simple functions and for which the spectrum can be written down algebraically [202,101]. In particular when…”

Section: Pauli Equation and Supersymmetrymentioning

confidence: 99%

“…Several aspects of the Dirac equation have also been studied within SUSY QM formalism [100,101,102]. In particular, it has been shown using the results of SUSY QM and shape invariance that whenever there is an analytically solvable Schrödinger problem in 1-dimensional QM then there always exists a corresponding Dirac problem with scalar interaction in 1+1 dimensions which is also analytically solvable.…”

Section: Introductionmentioning

confidence: 99%

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Parasupersymmetry in quantum mechanics

Khare

1993

Journal of Mathematical Physics

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In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all 1 have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

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Section: Supersymmetry and The Dirac Equationmentioning

confidence: 91%

Section: Supersymmetry and The Dirac Equationmentioning

confidence: 97%

Section: Supersymmetry and The Dirac Equationmentioning

confidence: 97%

Section: Pauli Equation and Supersymmetrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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