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Quantum Flatland and Monolayer Graphene from a Viewpoint of Geometric Algebra

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Quantum mechanical properties of the graphene are, as a rule, treated within the Hilbert space formalism. However a different approach is possible using the geometric algebra, where quantum mechanics is done in a real space rather than in the abstract Hilbert space. In this article the geometric algebra is applied to a simple quantum system, a single valley of monolayer graphene, to show the advantages and drawbacks of geometric algebra over the Hilbert space approach. In particular, 3D and 2D Euclidean space algebras Cl_{3, 0} and Cl_{2, 0} are applied to analyze relativistic properties of the graphene. It is shown that only three-dimensional Cl_{3, 0} rather than two-dimensional Cl_{2, 0} algebra is compatible with a relativistic flatland.
  • Center for Physical Sciences and Technology, Semiconductor Physics Institute, A. Goštauto 11, LT-01108 Vilnius, Lithuania
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