Lagrangian for Frenkel electron and position’s non-commutativity due to spin

Alexei A. Deriglazov, Andrey M. Pupasov-Maksimov

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

We construct a relativistic spinning-particle Lagrangian where spin is considered as a composite quantity constructed on the base of a non-Grassmann vector-like variable. The variational problem guarantees both a fixed value of the spin and the Frenkel condition on the spin-tensor. The Frenkel condition inevitably leads to relativistic corrections of the Poisson algebra of the position variables: their classical brackets became noncommutative. We construct the relativistic quantum mechanics in the canonical formalism (in the physical-time parametrization) and in the covariant formalism (in an arbitrary parametrization). We show how state vectors and operators of the covariant formulation can be used to compute the mean values of physical operators in the canonical formalism, thus proving its relativistic covariance. We establish relations between the Frenkel electron and positive-energy sector of the Dirac equation. Various candidates for the position and spin operators of an electron acquire clear meaning and interpretation in the Lagrangian model of the Frenkel electron. Our results argue in favor of Pryce’s (d)-type operators as the spin and position operators of Dirac theory. This implies that the effects of non-commutativity could be expected already at the Compton wavelength. We also present the manifestly covariant form of the spin and position operators of the Dirac equation.

Original languageEnglish
Article number3101
Pages (from-to)1-18
Number of pages18
JournalEuropean Physical Journal C
Volume74
Issue number10
DOIs
Publication statusPublished - 21 Oct 2014

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operators
Electrons
electrons
Quantum theory
Dirac equation
formalism
Algebra
Tensors
Wavelength
state vectors
Composite materials
brackets
metal spinning
quantum mechanics
algebra
sectors
tensors
formulations
composite materials
wavelengths

ASJC Scopus subject areas

  • Engineering (miscellaneous)
  • Physics and Astronomy (miscellaneous)

Cite this

Lagrangian for Frenkel electron and position’s non-commutativity due to spin. / Deriglazov, Alexei A.; Pupasov-Maksimov, Andrey M.

In: European Physical Journal C, Vol. 74, No. 10, 3101, 21.10.2014, p. 1-18.

Research output: Contribution to journalArticle

Deriglazov, Alexei A. ; Pupasov-Maksimov, Andrey M. / Lagrangian for Frenkel electron and position’s non-commutativity due to spin. In: European Physical Journal C. 2014 ; Vol. 74, No. 10. pp. 1-18.
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