Regular Article – Theoretical Physics
Generalization of Geyer’s commutation relations with respect to the orthogonal group in even dimensions
Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russia
Accepted: 23 October 2020
Published online: 14 December 2020
A connection between the deformed Duffin–Kemmer–Petiau (DKP) algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators and for an additional operator obeying para-Fermi statistics of order 2 based on the Lie algebra is established. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The representation for the operator in terms of generators of the orthogonal group SO(2M) correctly reproducing action of this operator on the state vectors of Fock space is obtained. A connection of the Geyer operator with the operator of so-called G-parity and with the CPT- operator of the DKP-theory is established. In a para-Grassmann algebra a noncommutative, associative star product (the Moyal product) as a direct generalization of the star product in the algebra of Grassmann numbers is introduced. Two independent approaches to the calculation of the Moyal product are considered. It is shown that in calculating the matrix elements in the basis of parafermion coherent states of various operator expressions it should be taken into account constantly that we work in the so-called Ohnuki and Kamefuchi’s generalized state-vector space , whose state vectors include para-Grassmann numbers in their definition, instead of the standard state-vector space (the Fock space).
© The Author(s) 2020
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