Abstract
We review the recent results on development of vector models of spin and apply them to study the influence of spin-field interaction on the trajectory and precession of a spinning particle in external gravitational and electromagnetic fields. The formalism is developed starting from the Lagrangian variational problem, which implies both equations of motion and constraints which should be presented in a model of spinning particle. We present a detailed analysis of the resulting theory and show that it has reasonable properties on both classical and quantum level. We describe a number of applications and show how the vector model clarifies some issues presented in theoretical description of a relativistic spin: (A) one-particle relativistic quantum mechanics with positive energies and its relation with the Dirac equation and with relativistic Zitterbewegung; (B) spin-induced noncommutativity and the problem of covariant formalism; (C) three-dimensional acceleration consistent with coordinate-independence of the speed of light in general relativity and rainbow geometry seen by spinning particle; (D) paradoxical behavior of the Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body in ultrarelativistic limit, and equations with improved behavior.
Highlights
Basic notions of Special and General Relativity have been formulated before the discovery of spin, so they describe the properties of space and time as they are seen by spinless testparticle
We describe a number of applications and show how the vector model clarifies some issues presented in theoretical description of a relativistic spin: (A) one-particle relativistic quantum mechanics with positive energies and its relation with the Dirac equation and with relativistic Zitterbewegung; (B) spin-induced noncommutativity and the problem of covariant formalism; (C) three-dimensional acceleration consistent with coordinate-independence of the speed of light in general relativity and rainbow geometry seen by spinning particle; (D) paradoxical behavior of the Mathisson-PapapetrouTulczyjew-Dixon equations of a rotating body in ultrarelativistic limit, and equations with improved behavior
In [25] we presented the Lagrangian minimally interacting with gravitational field, while in [26, 27] it has been extended to the case of nonminimal interaction through the gravimagnetic moment
Summary
Basic notions of Special and General Relativity have been formulated before the discovery of spin, so they describe the properties of space and time as they are seen by spinless testparticle. Since the only constant of motion in arbitrary background is S2, we write (we could start with P2 + (mc)2 + f(S2, P2) = 0; assuming that this equation can be resolved with respect to P2, we arrive essentially at the same expression) With this value of P2, we can exclude P휇 from MPTD equations, obtaining closed system with second-order equation for x휇 (so we refer to the resulting equations as Lagrangian form of MPTD equations). The singularity determines behavior of the particle in ultrarelativistic limit To clarify this point, consider the standard equations of a spinless particle interacting with electromagnetic field in the physical time parametrization x휇(t) factor.
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