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arXiv:quant-ph/9607025 · Physical Review A

Kinematics and hydrodynamics of spinning particles

Erasmo Recami, Giovanni Salesi

Published Jan 1, 1998• 68 citations
arXivHTMLPDF

Abstract

In the first part (Secs. I and II) of this paper, starting from the Pauli current, we obtain the decomposition of the nonrelativistic field velocity into two orthogonal parts: (i) the ``classical'' part, that is, the velocity $\mathbf{w}=\mathbf{p}/m$ in the center of mass (c.m.), and (ii) the ``quantum'' part, that is, the velocity V of the motion of the c.m. frame (namely, the internal ``spin motion'' or Zitterbewegung). By inserting such a complete, composite expression of the velocity into the kinetic-energy term of the nonrelativistic classical (i.e., Newtonian) Lagrangian, we straightforwardly get the appearance of the so-called quantum potential associated, as it is known, with the Madelung fluid. This result provides further evidence of the possibility that the quantum behavior of microsystems is a direct consequence of the fundamental existence of spin. In the second part (Secs. III and IV), we fix our attention on the total velocity $\mathbf{v}=\mathbf{w}+\mathbf{V},$ now necessarily considering relativistic (classical) physics. We show that the proper time entering the definition of the four-velocity ${v}^{\ensuremath{\mu}}$ for spinning particles has to be the proper time \ensuremath{\tau} of the c.m. frame. Inserting the correct Lorentz factor into the definition of ${v}^{\ensuremath{\mu}}$ leads to completely different kinematical properties for ${v}^{2}.$ The important constraint ${p}_{\ensuremath{\mu}}{v}^{\ensuremath{\mu}}=m,$ identically true for scalar particles but just assumed a priori in all previous spinning-particle theories, is herein derived in a self-consistent way.

DOI
10.1103/physreva.57.98

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In the first part (Secs. I and II) of this paper, starting from the Pauli current, we obtain the decomposition of the nonrelativistic field velocity into two orthogonal parts: (i) the classical'' part, that is, the velocity $\mathbf{w}=\mathbf{p}/m$ in the center of mass (c.m.), and (ii) the quantum'' part, that is, the velocity V of the motion of the c.m. frame (namely, the internal ``spin motion'' or Zitterbewegung). By inserting such a complete, composite expression of the velocity into the kinetic-energy term of the nonrelativistic classical (i.e., Newtonian) Lagrangian, we straightforwardly get the appearance of the so-called quantum potential associated, as it is known, with the Madelung fluid. This result provides further evidence of the possibility that the quantum behavior of microsystems is a direct consequence of the fundamental existence of spin. In the second part (Secs. III and IV), we fix our attention on the total velocity v=w+V,\mathbf{v}=\mathbf{w}+\mathbf{V},v=w+V, now necessarily considering relativistic (classical) physics. We show that the proper time entering the definition of the four-velocity v\ensuremathμ{v}^{\ensuremath{\mu}}v\ensuremathμ for spinning particles has to be the proper time \ensuremath{\tau} of the c.m. frame. Inserting the correct Lorentz factor into the definition of v\ensuremathμ{v}^{\ensuremath{\mu}}v\ensuremathμ leads to completely different kinematical properties for v2.{v}^{2}.v2. The important constraint p\ensuremathμv\ensuremathμ=m,{p}_{\ensuremath{\mu}}{v}^{\ensuremath{\mu}}=m,p\ensuremathμ​v\ensuremathμ=m, identically true for scalar particles but just assumed a priori in all previous spinning-particle theories, is herein derived in a self-consistent way.
In this note, we first obtain the decomposition of the non-relativistic field velocity into the classical part (i.e., the velocity w=p/m OF the center-of-mass (CM), and the so-called quantum part (i.e., the velocity V of the motion IN the CM frame (namely, the internal spin-motion or Zitterbewegung), these two parts being orthogonal. Our starting point is the Pauli current. Then, by inserting such a composite expression of the velocity into the kinetic energy term of the non-relativistic newtonian lagrangian, we get the appearance of the so-called "quantum potential" (which makes the difference between classical and quantum behaviour) as a pure consequence of the internal motion. Such a result carries further evidence about the possibility that the quantum behaviour of micro-systems be a direct consequence of the fundamental existence of spin.
Starting from the Pauli current we obtain the decomposition of the nonrelativistic local velocity in two parts: one parallel and the other orthogonal to the impulse. The former is recognized as the “classical” part, that is, the velocity of the center-of-mass, and the latter the “quantum” one, that is, the velocity of the motion in the center-of-mass frame (namely, the internal “spin motion” or zitterbewegung), Inserting the complete expression of the velocity into the kinetic energy term of the classical nonrelativistic (i.e. Newtonian) Lagrangian, we straightforwardly get the appearance of the so-called quantum potential associated, as it is known, to the Madelung fluid. In such a way, the quantum mechanical behavior of particles appears to be strictly correlated to the existence of spin and zitterbewegung.
Starting from the Pauli current we obtain the decomposition of the non-relativistic local velocity in two parts: one parallel and the other orthogonal to the momentum. The former is recognized to be the ``classical'' part, that is the velocity of the center-of-mass, and the latter the ``quantum'' one, that is the velocity of the motion in the center-of-mass frame (namely, the internal ``spin motion'' or {\em Zitterbewegung}). Inserting the complete expression of the velocity into the kinetic energy term of the classical non-relativistic (i.e., Newtonian) Lagrangian, we straightforwardly derive the so-called ``quantum potential'' associated to the Madelung fluid. In such a way, the quantum mechanical behaviour of particles appears to be strictly correlated to the existence of spin and Zitterbewegung.
Classical mechanics of a free particle is based on Newtonian mechanics and Lagrangian/Hamilton theories. A single frame of reference is chosen (the lab) and particles with the same mass mo (for example) moving at different velocities are considered in the free particle scenario (i.e in the absence of force (potentials)). We argue that the above is identical to considering a single particle with mass mo at time=to and x=0 viewed by various frames moving at different constant velocities -v. From the point of view of each frame, the particle moves at a velocity v. Thus the Newtonian picture of an mo particle moving at different speeds may be considered as a problem of special relativity. We argue (as in (1)) that (x,t)=(0,t) transforms to (x’,t’) such that x’/t’=v. If one uses a metric (1 0) (first row) (0 -1) second row, one may derive special relativistic relationships. In particular, one may obtain the Lorentz invariant -Et + px if (p,E) and (x,t) transform in the same manner. We argue that the relationships obtained from special relativity pertain to classical mechanics and so one may mathematically obtain a function L(v) such that p=dL/dv and E=pv - L for a free particle. In other words, special relativity governs Lagrangian results in the free particle case and is not really needed (in this special case). More importantly, it is special relativity which links E with t and p with x. As noted above, if one considers Netwonian mechanics in a single frame, the notion of special relativity does not arise, but the scaling of time by E and of x by p does i.e. A= -Et+px is the classical action both relativistically and nonrelativistically. It is just not written in this way usually (i.e. A = -mo t sqrt(1-vv) (c=1) relatiivstically and A =t .5mo vv nonrelativistically) We argued in (2) that scaling must manifest itself physically and that it accounts for the physical wavelength in quantum mechanics. We argue that the quantum mechanics of a free particle ultimately follows from special relativity. As a result, ideas of quantum mechanics would not have appeared in Newtonian mechanics nor Lagrangian or Hamiltonian theories. When special relativity was developed it focused on transformations related to different frames, but these are equivalent to describing one frame with many different velocities (same mo). Thus the physics in one frame is governed by special relativity even if it is nonrelativistic. We argue that the notion of the wavelength which appears even in the Schrodinger formalism follows from scaling of x by p which ultimately originates from special relativity. In other words, both the Lagrangian formalism and quantum mechanics follow from special relativity in the free particle case. We argue that free particle quantum mechanics is the starting point of overall quantum mechanics.
A quantum-mechanical non-relativistic spinnig particle is represented equivalently with a non-linear vector field which defines a new kind of hydrodynamics of a spinning fluid. In this hydrodynamics, quantum effects are separated as non-linear terms which mean the occurrence of “internal potential” and “internal magnetic field”. Mathematically, the method is based upon the replacement of the original calculus in terms of a spinor ψ by the calculus in terms of tensors formed as the bilinear expressions in ψ*. This article (hereafter called I) and also the major part of the subsequent one (hereafter called II) which will appear in succession, are the translation of the paper which appeared in Soryusiron-Kenkyu 7, pp. 600–640, (January 1955) in Japanese, while a brief account of these works was reported in Prog. Theor. Phys. 12 (1954), 810. Very recently articles by Bohm, Schiller, and Tiomno appeared (Nuovo Cimento 1, Suppl. (1955), 48, 67), in which they investigated, independently of us, the similar hydrodynamical method for the purpose of a “ causal interpretation ” of quantum theory in the case of a non-relativistic spinning particle. However, their method stands upon Euler angle representation of the spinor (cf II), not directly concerned with the self-contained tensor formulation of the spinor equation and ψ. This replacement is ensured by setting up the identities which should hold among those tensor quantities.
A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.