Supersymmetry and the Dirac Equation
1988
28 pages
Published in:
- Annals Phys. 187 (1988) 1-28
Report number:
- (10)-LA-UR-88-0169
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Abstract: (Elsevier)
We discuss in detail two supersymmetries of the 4-dimensional Dirac operator / kD 2 where / kD = ∂ − ieA , namely the usual chiral supersymmetry and a separate complex supersymmetry. Using SUSY methods developed to categorize solvable potentials in 1-dimensional quantum mechanics we systematically study the cases where the spectrum, eigenfunctions, and S -matrix of / kD 2 can be obtained analytically. We relate these solutions to the solutions of the ordinary massive Dirac equation in external fields. We show that whenever a Schrödinger equation for a potential V ( x ) is exactly solvable, then there always exists a corresponding static scalar field ϕ ( x ) for which the Jackiw-Rebbi type (1 + 1)-dimensional Dirac equation is exactly solvable with V ( x ) and ϕ ( x ) being related by V ( x ) = ϕ 2 ( x ) + ϕ ′( x ). We also discuss and exploit the supersymmetry of the path integral representation for the fermion propagator in an external field.- OPERATOR: DIRAC
- ELECTROMAGNETIC FIELD: EXTERNAL FIELD
- QUANTUM ELECTRODYNAMICS
- SUPERSYMMETRY: WITTEN INDEX
- FERMION: PROPAGATOR
- FIELD THEORY: PATH INTEGRAL
- SCHROEDINGER EQUATION
- POTENTIAL
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