### 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 language | English |
---|---|

Article number | 3101 |

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | European Physical Journal C |

Volume | 74 |

Issue number | 10 |

DOIs | |

Publication status | Published - 21 Oct 2014 |

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### ASJC Scopus subject areas

- Engineering (miscellaneous)
- Physics and Astronomy (miscellaneous)

### Cite this

*European Physical Journal C*,

*74*(10), 1-18. [3101]. https://doi.org/10.1140/epjc/s10052-014-3101-2

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

Research output: Contribution to journal › Article

*European Physical Journal C*, vol. 74, no. 10, 3101, pp. 1-18. https://doi.org/10.1140/epjc/s10052-014-3101-2

}

TY - JOUR

T1 - Lagrangian for Frenkel electron and position’s non-commutativity due to spin

AU - Deriglazov, Alexei A.

AU - Pupasov-Maksimov, Andrey M.

PY - 2014/10/21

Y1 - 2014/10/21

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84907382143&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907382143&partnerID=8YFLogxK

U2 - 10.1140/epjc/s10052-014-3101-2

DO - 10.1140/epjc/s10052-014-3101-2

M3 - Article

VL - 74

SP - 1

EP - 18

JO - European Physical Journal C

JF - European Physical Journal C

SN - 1434-6044

IS - 10

M1 - 3101

ER -