Poincar� invariant differential equations for particles of arbitrary spin (original) (raw)
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On the equations of motion for particles with arbitrary spin in nonrelativistic mechanics
1975
It is well known that the electron motion in the external electromagnetic field is described by the relativistic Dirac equation. In this case, in the Foldy-Wouthuysen representation, the Hamiltonian includes the terms corresponding to the interaction of the particle magnetic moment with a magnetic field (∼ (1/m)(σH)) and the terms which are interpreted as a spin-orbit coupling (∼ (σ/m 2 ){(p − eA) × E)). Apart from these constituents the Hamiltonian includes the Darwin term (∼ (1/m 2 ) div E) .
International Journal of Quantum Chemistry, 2001
The relativistic dynamics of one spin-½ particle moving in a uniform magnetic field is described by the Hamiltonian . The discrete (and semidiscrete) eigenvalues and the corresponding eigenspinors are in principle known from the work of Dirac, Rabi, and Bloch. These are extensively reviewed here. Next, exact solutions are worked out for the recoil dynamics in relative coordinates, which involves the Hamiltonian . Exact solutions are also explicitly calculated in the case where the spin-½ particle has an anomalous magnetic moment such that its Hamiltonian is given by . Similar exact solutions are derived here when the recoiling particle has an anomalous magnetic moment, that is, the eigenvalues and eigenspinors of the Hamiltonian are explicitly obtained. The diagonalized and separable form of the Hamiltonian hD(π), written as , has exceedingly simple forms of eigenspinors. Similarly, the diagonalized and separable form of the operator hD(−k), written as , has very simple eigenspinors. The importance of these exact solutions is that the eigenspinors can be used as bases in a calculation involving many spin-½ particles placed in a uniform magnetic field. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 209–217, 2001
Equations of motion of a spinning relativistic particle in external fields
Journal of Experimental and Theoretical Physics, 1998
The motion of spinning relativistic particles in external electromagnetic and gravitational fields is considered. Covariant equations for this motion are demonstrated to possess pathological solutions, when treated nonperturbatively in spin. A self-consistent approach to the problem is formulated, based on the noncovariant description of spin and on the usual, "naïve" definition of the coordinate of a relativistic particle. A simple description of the gravitational interaction of first order in spin, is pointed out for a relativistic particle. The approach developed allows one to consider effects of higher order in spin. Explicit expression for the second-order Hamiltonian is presented. We discuss the gravimagnetic moment, which is a special spin effect in general relativity. 1 khriplovich@inp.nsk.su
On the Poincaré-invariant equations for particles with variable spin and mass
Reports on Mathematical Physics, 1975
The Poincaré-invariant equations without redundant components, describing the motion of a particle which can be in different spin and mass states are obtained. The quasirelativistic equation for a particle with arbitrary spin in external electromagnetic field is found. The group-theoretical analysis of these equations in carried out.
Dynamics of the relativistic electron spin in an electromagnetic field
Journal of Physics: Condensed Matter, 2020
A relativistic spin operator cannot be uniquely defined within relativistic quantum mechanics. Previously, different proper relativistic spin operators have been proposed, such as spin operators of the Foldy–Wouthuysen and Pryce type, that both commute with the free-particle Dirac Hamiltonian and represent constants of motion. Here we consider the dynamics of a relativistic electron spin in an external electromagnetic field. We use two different Hamiltonians to derive the corresponding spin dynamics. These two are: (a) the Dirac Hamiltonian in the presence of an external field, and (b) the semirelativistic expansion of the same. Considering the Foldy–Wouthuysen and Pryce spin operators we show that these lead to different spin dynamics in an external electromagnetic field, which offers possibilities to distinguish their action. We find that the dynamics of both spin operators involve spin-dependent and spin-independent terms, however, the Foldy–Wouthuysen spin dynamics additionally ...
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...
Advances in Mathematical Physics
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 coo...
Advances in Applied Clifford Algebras, 2017
Using Clifford and Spin-Clifford formalisms we prove that the classical relativistic Hamilton Jacobi equation for a charged massive (and spinning) particle interacting with an external electromagnetic field is equivalent to Dirac-Hestenes equation satisfied by a class of spinor fields that we call classical spinor fields. These spinor fields are characterized by having the Takabayashi angle function constant (equal to 0 or π). We also investigate a nonlinear Dirac-Hestenes like equation that comes from a class of generalized classical spinor fields. Finally, we show that a general Dirac-Hestenes equation (which is a representative in the Clifford bundle of the usual Dirac equation) gives a generalized Hamilton-Jacobi equation where the quantum potential satisfies a severe constraint and the "mass of the particle" becomes a variable. Our results can then eventually explain experimental discrepancies found between prediction for the de Broglie-Bohm theory and recent experiments. We briefly discuss de Broglie's double solution theory in view of our results showing that it can be realized, at least in the case of spinning free particles.The paper contains several Appendices where notation and proofs of some results of the text are presented.
Equations of motion for particles of arbitrary spin invariant under the Galileo group
Theoretical and Mathematical Physics, 1980
Systems of differential equations of first and second order are derived that are invariant under the Galileo group and describe the motion of a particle with arbitrary spin. These equations admit a Lagrangian formulation and describe the dipole, spin-orbit, and Darwin couplings of the particle to an external electromagnetic field; traditionally, these have been regarded as purely relativistic effects. Examples are given of infinite-component equations that are invariant under the Galileo group. The problem of the motion of a nonrelativistic particle with spin s = ~ in a homogeneous magnetic field is solved exactly.