Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

Abstract

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a (2 + 1)-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a (2 + 1)-dimensional Minkowski (flat) space. For each of the sets we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.

Algebra of Symmetry Operators for Klein-Gordon-Fock Equation

“…In particular, in [31][32][33], a complete classification of spaces admitting a simply transitive action of the motions groups G 4 was obtained, provided that the Klein-Gordon-Fock equation is exactly solved by non-commutative integration methods. In [34][35][36][37][38], a similar problem was solved for Dirac-Fock equation.…”

Algebra of symmetry operators for Klein-Gordon-Fock equation

“…In particular, in [31]- [33], a complete classification of spaces admitting a simply transitive action of the motions groups G 4 was obtained, provided that the Klein-Gordon-Fock equation is exactly solved by non-commutative integration methods. In [34] - [38], a similar problem was solved for Dirac-Fock equation.…”

Dirac equation in the curved spacetime and generalized uncertainty principle: A fundamental quantum mechanical approach with energy-dependent potentials

“…The Dirac equation in low dimensions (2 + 1) has been intensely studied for the past three decades by many authors in theoretical physics [9], [10], [11], [12]. The low dimensional systems have always been attracting attention due to theoretical and experimental interesting results such as high energy particle theory, condensed matter physics (monolayer structures), topological field theory, and string theory.…”