Hydrodynamics of Spinning Particles (original) (raw)
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Kinematics and hydrodynamics of spinning particles
Physical Review A, 1998
In the first part ͑Secs. I and II͒ of this paper, starting from the Pauli current, we obtain the decomposition of the nonrelativistic field velocity into two orthogonal parts: ͑i͒ the ''classical'' part, that is, the velocity w ϭp/m in the center of mass ͑c.m.͒, and ͑ii͒ the ''quantum'' part, that is, the velocity V of the motion of the c.m. frame ͑namely, the internal ''spin motion'' or Zitterbewegung͒. By inserting such a complete, composite expression of the velocity into the kinetic-energy term of the nonrelativistic classical ͑i.e., Newtonian͒ 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 IV͒, we fix our attention on the total velocity vϭwϩV, now necessarily considering relativistic ͑classical͒ physics. We show that the proper time entering the definition of the four-velocity v for spinning particles has to be the proper time of the c.m. frame. Inserting the correct Lorentz factor into the definition of v leads to completely different kinematical properties for v 2 . The important constraint p v ϭm, identically true for scalar particles but just assumed a priori in all previous spinning-particle theories, is herein derived in a self-consistent way.
About kinematics and hydrodynamics of spinning particles: some simple considerations
In the first part (Sections 1 and 2) of this paper -starting from the Pauli current, in the ordinary tensorial language-we obtain the decomposition of the non-relativistic field velocity into two orthogonal parts: (i) the "classical" part, that is, the velocity w = p/m of the centerof-mass (CM), and (ii) the so-called "quantum" part, that is, the velocity V of the motion in the CM 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 non-relativistic 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 carries further evidence that the quantum behaviour of micro-systems can be a direct consequence of the fundamental existence of spin. In the second part (Sections 3 and 4), we fix our attention on the total velocity v = w + V , it being now necessary to pass to relativistic (classical) physics; and we show that the proper time entering the definition of the four-velocity v µ for spinning particles has to be the proper time τ of the CM frame. Inserting the correct Lorentz factor into the definition of v µ leads to completely new kinematical properties for v 2 . The important constraint pµv µ = 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.
1998
Starting from the formal expressions of the hydrodynamical (or “local”) quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce —in terms of the ordinary tensorial language— a new definition for the field of any quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin 1/2 particle. This operator appears as the sum of the ordinary part describing the mean motion (the motion OF the center-of-mass), and of a second part associated with the so-called Zitterbewegung, which is the spin "internal'' motion observed IN the center-of-mass frame (CMF). This spin component of the velocity operator is non-zero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a non-constant spin function, but also in the Schroedinger case, when the wave-function is a spin eigenstate. Thus, one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts: which constitutes the non-relativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the Hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore, that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a non-uniform spin vector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schroedinger theory), also a component which is parallel to the rotor of the spin vector.
A Velocity Field and Operator for Spinning Particles in (Nonrelativistic) Quantum Mechanics
Foundations of Physics, 1998
Starting from the formal expressions of the hydrodynamical (or “local”) quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce—in terms of the ordinary tensorial language—a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin- \({\frac{1}{2}}\) particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the so-called Zitterbewegung, which is the spin “internal” motion observed in the center-of-mass frame (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a nonconstant spin function, but also in the Schrödinger case, when the wavefunction is a spin eigenstate. Thus, one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore, that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a nonuniform spin vector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schrödinger theory), also a component which is parallel to the curl of the spin vector.
Molecular Physics, 2018
Michael Baer is known for his many achievements in the field of non-adiabatic quantum dynamics and the theory of reaction dynamics. Among his main tools of inquiry is the study of two-state quantum systems. I was fortunate enough to collaborate with him on such models in the first part of the previous decade and the last years of the previous century. Here, I return to the original two-state system, that is Pauli's electron with a spin and show how this system can be interpreted as a vortical fluid. The similarities and difference between spin flows and classical ideal flows are elucidated.
About the velocity operator for spinning particles in quantum mechanics
Starting from the formal expressions of the hydrodynamical (or "local") quantities employed in the applications of Clifford Algebras to quantum mechanics, we introduce -in terms of the ordinary tensorial framework-a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (non-relativistic) velocity operator for a spin 1 2 particle. This operator is the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the so-called zitterbewegung, which is the spin "internal" motion observed in the center-of-mass frame. This spin component of the velocity operator is non-zero not only in the Pauli theoretical framework, i.e. in presence of external magnetic fields and spin precession, but also in the Schrödinger case, when the wave-function is a spin eigenstate. In the latter case, one gets a decomposition of the velocity field for the Madelung fluid into two distinct parts: which constitutes the non-relativistic analogue of the Gordon decomposition for the Dirac current. We find furthermore that the zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In presence of a non-constant spin vector (Pauli case) we have, besides the component normal to spin present even in the Schrödinger theory, also a component of the local velocity which is parallel to the rotor of the spin vector. †
A Classical and Spinorial Description of the Relativistic Spinning Particle
arXiv: High Energy Physics - Theory, 2016
In a previous work we showed that spin can be envisioned as living in a phase space that is dual to the standard phase space of position and momentum. In this work we demonstrate that the second class constraints inherent in this "Dual Phase Space" picture can be solved by introducing a spinorial parameterization of the spinning degrees of freedom. This allows for a purely first class formulation that generalizes the usual relativistic description of spinless particles and provides several insights into the nature of spin and its relationship with spacetime and locality. In particular, we find that the spin motion acts as a Lorentz contraction on the four-velocity and that, in addition to proper time, spinning particles posses a second gauge invariant observable which we call proper angle. Heuristically, this proper angle represents the amount of Zitterbewegung necessary for a spin transition to occur. Additionally, we show that the spin velocity satisfies a causality cons...
VELOCITY FIELD AND OPERATOR FOR SPINNING PARTICLES IN (NONRELATIVISTIC) QUANTUM MECHANICS
1998
Starting from the formal expressions of the hydrodynamical (or``local '') quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce± ± in terms of the ordinary tensorial language± ± a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin-1 2 particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass ), and of a second part associated with the socalled Zitterbewegung, which is the spin``internal '' motion observed in the centerof-mass from (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework , i.e., in the presence of external electromagnetic fields with a nonconstant spin function , but also in the Schro È dinger case, when the wavefunction is a spin eigenstate. Thus , one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore , that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a nonuniform spin vector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schro È dinger theory), also a component which is parallel to the curl of the spin vector.
Chaos (Woodbury, N.Y.), 2018
We present the results of a theoretical investigation of hydrodynamic spin states, wherein a droplet walking on a vertically vibrating fluid bath executes orbital motion despite the absence of an applied external field. In this regime, the walker's self-generated wave force is sufficiently strong to confine the walker to a circular orbit. We use an integro-differential trajectory equation for the droplet's horizontal motion to specify the parameter regimes for which the innermost spin state can be stabilized. Stable spin states are shown to exhibit an analog of the Zeeman effect from quantum mechanics when they are placed in a rotating frame.
Quantization of generalized spinning particles: New derivation of Dirac’s equation
Journal of Mathematical Physics, 1994
Quantization of generalized Lagrangian systems suggests that wave functions for elementary particles must be defined on the kinematical space rather than on configuration space. For spinning particles the center of mass and center of charge are different points. Their separation is of the order of the Compton wavelength. Spin-1/2 particles arise if the classical model rotates but no half integer spins are obtained for systems with spin of orbital nature. Dirac’s equation is obtained when quantizing the classical relativistic spinning particles whose center of charge is circling around its center of mass at the speed c. Internal orientation of the electron completely characterizes its Dirac’s algebra.