Mathisson’s helical motions for a spinning particle: Are they unphysical?

Abstract

It has been asserted in the literature that Mathisson's helical motions are unphysical, with the argument that their radius can be arbitrarily large. We revisit Mathisson's helical motions of a free spinning particle, and observe that such statement is unfounded. Their radius is finite and confined to the disk of centroids. We argue that the helical motions are perfectly valid and physically equivalent descriptions of the motion of a spinning body, the difference between them being the choice of the representative point of the particle, thus a gauge choice. We discuss the kinematical explanation of these motions, and we dynamically interpret them through the concept of hidden momentum. We also show that, contrary to previous claims, the frequency of the helical motions coincides, even in the relativistic limit, with the zitterbewegung frequency of the Dirac equation for the electron.

Center of Mass, Spin Supplementary Conditions, and the Momentum of Spinning Particles

“…The rules for transition between spin conditions, and the quantities that are fixed (for different solutions corresponding to the same physical body), were established in [13], where the numerical solutions were compared in the Kerr spacetime, and it was shown that, within the limit of validity of the pole-dipole approximation, the different solutions are contained within a minimal worldtube, formed by all the possible positions of the center of mass, which lies inside the convex hull of the body's worldtube. These rules were further discussed in [16], and used to show that the helical motions are fully consistent solutions, always contained within the minimal worldtube (and to clarify the misunderstanding that led to the contrary claims in the literature).…”

“…The different solutions given by the different spin conditions corresponding to the same physical motion are compared in simple examples, and their differences dissected. Building on the works in [13,16] (where the equivalence was shown for free particles in flat spacetime), we prove the equivalence of the solutions to dipole order in curved spacetime; in particular, we clarify the dependence of the spin-curvature force on the spin condition, as being precisely what ensures the equivalence, and the connection of that with the geodesic deviation equation. 3.…”

“…A significant step towards its understanding was taken in [17], where a generalized concept of "hidden momentum" (first discovered in the context of classical electrodynamics [18][19][20][21][22]) was introduced in general relativity, and applied to the study of the TD and CP conditions (the latter designated therein by a different name, the "laboratory frame centroid"). These ideas were further worked out, with emphasis on the FMP condition, in recent works by the authors [16,24].…”

“…Later treatments, most notably the works by Möller [6,7], dealing with extended bodies, shed some light on the interpretation of the spin condition, as it being a choice of representative point in the body; more precisely, choosing it as the center of mass ("centroid") as measured in the rest frame of an observer of 4-velocity u α -since in relativity, the center of mass of a spinning body is an observer-dependent point. Different choices have been proposed; the best known ones are the Frenkel-Mathisson-Pirani (FMP) condition [8,16], which chooses the centroid as measured in a frame comoving with it; the Corinaldesi-Papapetrou (CP) condition [9], which chooses the centroid measured by the observers of zero 3-velocity (u i = 0) in a given coordinate system; and the Tulczyjew-Dixon (TD) condition [10,11], which chooses the centroid measured in the zero 3-momentum frame (u α ∝ P α ). A more recent condition, proposed in [12,13], dubbed herein the "Ohashi-Kyrian-Semerák (OKS) condition" (which, as we shall see, seems to be favored in many applications), chooses the centroid measured with respect to some u α parallel-transported along its worldline.…”

A Detailed Analysis of the Quantum Relativistic Motion of an Electron

“…In our view, the waves are electromagnetic waves in B⊂M [2] ; they are naturally entangled by virtue of the geometry of quotient spaces in B. Furthermore, if our geometry of a particle at (t, x, y, z) in [1] carrying its wave energy around (it, iz, ix, iy) in B⊂ [2] is correct, then the misspecified geometries will likely hamper the development of the technologies sought for (Costa et al, 2012, for a shared interest in www.ccsenet.org/apr Applied Physics Research Vol. 5, No.…”