Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 057, 13 pages      arXiv:1102.0065      https://doi.org/10.3842/SIGMA.2011.057
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Symmetry Operators and Separation of Variables for Dirac's Equation on Two-Dimensional Spin Manifolds

Alberto Carignano a, Lorenzo Fatibene b, Raymond G. McLenaghan c and Giovanni Rastelli d
a) Department of Engineering, University of Cambridge, United Kingdom
b) Dipartimento di Matematica, Università di Torino, Italy
c) Department of Applied Mathematics, University of Waterloo, Ontario, Canada
d) Formerly at Dipartimento di Matematica, Università di Torino, Italy

Received February 01, 2011, in final form June 02, 2011; Published online June 15, 2011

Abstract
A signature independent formalism is created and utilized to determine the general second-order symmetry operators for Dirac's equation on two-dimensional Lorentzian spin manifolds. The formalism is used to characterize the orthonormal frames and metrics that permit the solution of Dirac's equation by separation of variables in the case where a second-order symmetry operator underlies the separation. Separation of variables in complex variables on two-dimensional Minkowski space is also considered.

Key words: Dirac equation; symmetry operators; separation of variables.

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