On a model of a classical relativistic particle of constant and universal mass and spin

Abstract

The deformation of the classical action for a point-like particle recently suggested by Staruszkiewicz gives rise to a spin structure which constrains the values of the invariant mass and the invariant spin to be the same for any solution of the equations of motion. Both these Casimir invariants, the square of the four-momentum vector and the square of the Pauli-Lubański pseudo-vector, are shown to preserve the same fixed values also in the presence of an arbitrary external electromagnetic field. In the "free" case, in the centre-of-mass reference frame, the particle moves along a circle of fixed radius with arbitrary varying frequency. In a homogeneous magnetic field, a number of rotational "states" is possible, with frequencies slightly different from the cyclotron frequency, and "phase-like" transitions with spin flops occure at some critical value of the particle's three-momentum. In the last section, the article of Kuzenko, Lyakhovich and Segal (1994) in which, in fact, an equivalent model had been earlier proposed and elaborated, is briefly reviewed and compared with Staruszkiewicz's approach and our results. PACS numbers: 03.30.+p; 03.50.De; 41.60.Ap 1 Introduction. The Staruszkiewicz model From the times of Frenkel [1] and Mathisson [2], or from later works [3, 4], a lot of attempts have been undertaken to unambiguously formulate the dynamics of a classical spinning particle [5]-[10].Most of them deal with generalizations of the classical point-mass Lagrangian (−mc √ẋẋ ) through the introduction of terms with higher derivatives or "inner" variables, and then try to restrict the undesirable freedom by making use of some geometrical [11,12] or symmetry [13,14] considerations. A thorough analysis of extra variables responsible for the spin structure has been carried out by Hanson and Regge [15]. Rivas [16] proposed to make use of a complete set of parameters of the kinematical symmetry group and obtained considerable restrictions on the spin dynamics (so-called "atomic hypothesis").However, a lot of problems arising within the classical description of spin still have to be solved. Corresponding Lagrangians, especially in their interaction part, are rather ambiguous, cumbersome and have no correspondence with generally accepted gauge theories and with the structure of the Lorentz force in the spinless limit in particular [17]. The Schrödinger zitterbewegung motion, being a common feature of different spin models, results in the problem of radiation of electromagnetic waves, and the invariant spin is not bound to preserve its value in the presence of external fields (as one expects for an innate characteristic of elementary particle). Thus, the "zitterbewegung" radius and the magnitude of spin are, as a rule, to be fixed "by hands". Generally, it is not quite clear which properties of the spin could arise in the framework of a successively classical relativistic model.Meanwhile, in a short note [18] Staruszkiewicz has offered a fairly simple relativistic model for a classical particle with spin 1 . Specifica...

Spin-one-half particles in strong electromagnetic fields: Spin effects and radiation reaction

“…The spin is also conserved in the quantum mechanical description of this setup. Although the effect of the spin-dependent effective mass [55,85,87] including the possibility of a negative mass [86] has been considered in the literature, this behavior is in contrast to the predictions of the classical Foldy-Wouthuysen model as well as the quantum mechanical Dirac theory. In the classical Foldy-Wouthuysen model the force on the electron does not depend on its spin orientation in the case of homogeneous electromagnetic fields.…”

“…Note that it renders the modified Lorentz force of the Frenkel model spin dependent via the additional prefactor m/M in Eq. (31a) [55,85,86]. The effective mass (22) has been explained by a magnetic-energy contribution to the electron's total relativistic energy [87].…”

World sheets of spinning particles

“…The model without third invariant 7 describes the irreducible massive spinning particle, though it admits inclusion of interaction only with the constant curvature gravity 8 and constant electromagnetic field. The minimal models of 6,7 , were rederived for various reasons, reformulated, and reinterpreted in various ways, see [9][10][11][12][13][14][15] and references therein. The most frequent common feature of the reformulations is that some linear space L of higher dimension is considered as the phase space of spinning particle instead of T * M 6 , while some extra constraints are imposed on linear coordinates.…”

A generalized model for the classical relativistic spinning particle

“…In Ref. [13] the dynamics of this special case is analyzed and the authors show that a nontrivial oscillatory solution can be proposed for the system. Hence, we expect to find a more general oscillatory solution for the equations of motion (29) and (30).…”

“…[14], a special gauge is proposed, and it is claimed that it gives the well-established results of Ref. [13] in the Lagrangian approach.…”