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

Abstract

We construct relativistic-invariant spinning-particle Lagrangian without auxiliary variables. Spin is considered as a composed quantity constructed on the base of non-Grassmann vector-like variable. The variational problem guarantees both fixed value of spin and Frenkel condition on spin-tensor. Taking into account the Frenkel condition, we obtain, inevitably, relativistic corrections to the algebra of position variables: their classical brackets became noncommutative, with the "parameter of non-commutativity" proportional to the spin-tensor. This leads to a number of interesting consequences in quantum theory. We construct the relativistic quantum mechanics in canonical formalism (in physical-time parametrization) and in covariant formalism (in arbitrary parametrization). We show how state-vectors and operators of covariant formulation can be used to compute mean values of physical operators of position and spin. This proves relativistic covariance of canonical formalism. Various candidates for position and spin operators of an electron acquire clear meaning and interpretation in the Lagrangian model of Frenkel electron. We also establish the relation between Frenkel electron and positive-energy sector of Dirac equation, this allowed us to turn to the long-standing problem on spin and position operators of Dirac theory. Contrary to widely assumed opinion, our results argue in favor of Pryce's (d)-type operators. This implies that effects of non-commutativity could be presented at the Compton wave length, in contrast to conventional expectations at the Planck length. At last, we present the manifestly covariant form of spin and position operators of Dirac equation.

25 pages, matches with published version, references added

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Cited by in corpus (27)

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• Frenkel electron on an arbitrary electromagnetic background and magnetic Zitterbewegung
• Relativistic corrections to the algebra of position variables and spin-orbital interaction
• Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations in ultra-relativistic regime and gravimagnetic moment
• Frenkel electron and a spinning body in a curved background
• Ultra-relativistic spinning particle and a rotating body in external fields
• Relativistic electron spin dynamics in a strong unipolar laser field
• Lagrangian for the Frenkel electron
• Spin and localization of relativistic fermions and uncertainty relations
• Relativistic effects due to gravimagnetic moment of a rotating body
• General relativity effects in precision spin experimental tests of fundamental symmetries
• On the relationship between the modifications to the Raychaudhuri equation and the canonical Hamiltonian structures
• World-line geometry probed by fast spinning particle
• Probing deformed quantum commutators
• Geometric Constructions Underlying Relativistic Description of Spin on the Base of Non-Grassmann Vector-Like Variable
• Reparametrization Invariance and Some of the Key Properties of Physical Systems
• Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle
• Nonminimal spin-field interaction of the classical electron and quantization of spin
• System of interacting harmonic oscillators in rotationally invariant noncommutative phase space
• Canonical Structure of Noncommutative Quantum Mechanics as Constraint System
• Minimal momentum estimation in noncommutative phase space of canonical type with preserved rotational and time reversal symmetries
• Pryce's mass-center operators and the anomalous velocity of a spinning electron
• A Generalized Model for the Classical Relativistic Spinning Particle
• New symmetries, conserved quantities and gauge nature of a free Dirac field
• Acceleration of particles in Schwarzschild and Kerr geometries