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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