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arXiv:hep-ph/0302136 · Physical Review A

Effect of the relativistic spin rotation on two-particle spin composition

Richard Lednicky, Vladimir Lvovich Lyuboshitz, Vladimir Lvovich Lyuboshitz

Published Mar 15, 2004• 2 citations
arXivHTMLPDF

Abstract

The effect of the relativistic spin rotation on two-particle spin states, conditioned by the setting of the spins of the particles in their rest frames and by the noncommutativity of the Lorentz transformations along noncollinear directions, is discussed. Particularly, the transition from the center-of-mass system (c.m.s.) of two spin-1∕2 particles to the laboratory is considered. When the vectors of the c.m.s. particle velocities are not collinear with the velocity vector of the c.m.s., the angles of the relativistic spin rotation for the two particles are different. As a result, the relative fractions of the singlet and triplet states in the relativistic system of two spin-1∕2 particles with a nonzero vector of relative momentum depend on the concrete frame in which the two-particle system is analyzed.

DOI
10.1103/physreva.69.034102

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The effect of the relativistic spin rotation on two-particle spin states, conditioned by the setting of the spins of the particles in their rest frames and by the noncommutativity of the Lorentz transformations along noncollinear directions, is discussed. Particularly, the transition from the center-of-mass system (c.m.s.) of two spin-1∕2 particles to the laboratory is considered. When the vectors of the c.m.s. particle velocities are not collinear with the velocity vector of the c.m.s., the angles of the relativistic spin rotation for the two particles are different. As a result, the relative fractions of the singlet and triplet states in the relativistic system of two spin-1∕2 particles with a nonzero vector of relative momentum depend on the concrete frame in which the two-particle system is analyzed.
The effect of the relativistic spin rotation, conditioned by the setting of the spin in the rest frame of a particle and by the noncommutativity of the Lorentz transformations along noncolinear directions, is discussed. In connection with this, the Thomas precession of the spin polarization vector at the motion of a particle along a curvilinear trajectory is considered. The transformations of the correlation tensor components for a system of two spin-1/2 particles at the transition from the c.m.s. of the particle pair to the laboratory frame are investigated. When the particle laboratory velocities are not colinear, the relativistic spin rotation angles for these particles are different. As a result, the relative fractions of the singlet and triplet states in the relativistic system of two free spin-1/2 particles with a nonzero vector of relative momentum depend on the concrete frame in which this two-particle system is analyzed.
This paper was concerned with the spin-momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigen states (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin-momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not observer-dependent quantity in inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and has shown that entanglement decreases with respect to the increasing of observer velocity. Finally, we argue that, Wigner rotation is induced by Lorentz transformations can be realized as controlling operator.
We study the orbits of two interacting particles described by a fully relativistic classical mechanical Hamiltonian. We use two sets of initial conditions. In the first set (dynamics 1) the system's center of mass is at rest. In the second set (dynamics 2) the center of mass evolves with velocity VVV. If dynamics 1 is observed from a reference frame moving with velocity $\ensuremath{-}V$, the principle of relativity requires that all observables must be identical to those of dynamics 2 seen from the laboratory frame. Our numerical simulations demonstrate that kinematic Lorentz space-time transformations fail to transform particle observables between the two frames. This is explained as a result of the inevitable interaction dependence of the boost generator in the instant form of relativistic dynamics. Despite general inaccuracies of the Lorentz formulas, the orbital periods are correctly predicted by the Einstein's time dilation factor for all interaction strengths.
We have studied a relativistically covariant Stern-Gerlach (SG) experiment for a relativistic spin and a Dirac spin. We have obtained the relativistic spin in an arbitrary frame by using the classical spin dipole tensor, which gives the covariant spin dipole interactions, and the relation between a spin and a spin magnetic dipole moment. The relativistic spin is shown to have problems to become a proper spin operator for a massive relativistic particle because of two reasons. First, the relativistic spin three-vector operators cannot satisfy the spin algebra. Second, the SG experiment for the relativistic particle provides a paradox between two observers in the particle rest frame and the laboratory frame, in which the particle is moving. We have shown that the paradox in the SG experiment is resolved by the Dirac spin, which is covariantly defined by a Lorentz transformation in the Dirac spinor representation. The Dirac spin three-vector operators satisfy the spin algebra. It is shown that the SG experiment for the Dirac spin in the inertial frame, where there is only magnetic field, can determine the spin without the information of the momentum of the particle. This shows that the reduced spin density matrix for the Dirac particle can be well-defined by integrating out the momentum degrees of freedom.
This paper is concerned with the spin–momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigenstates (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin and momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not an observer-dependent quantity in the inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and have shown that entanglement decreases with respect to the increase in observer velocity. Finally, we argue that Wigner rotation is induced by Lorentz transformations and can be realized as a controlling operator.
The spin is the prime example of a qubit. Encoding and decoding information in the spin qubit is operationally well defined through the Stern-Gerlach setup in the nonrelativistic (i.e., low velocity) limit. However, an operational definition of the spin in the relativistic regime is missing. The origin of this difficulty lies in the fact that, on the one hand, the spin gets entangled with the momentum in Lorentz-boosted reference frames, and on the other hand, for a particle moving in a superposition of velocities, it is impossible to "jump" to its rest frame, where spin is unambiguously defined. Here, we find a quantum reference frame transformation corresponding to a "superposition of Lorentz boosts," allowing us to transform to the rest frame of a particle that is in a superposition of relativistic momenta with respect to the laboratory frame. This enables us to first move to the particle's rest frame, define the spin measurements there (via the Stern-Gerlach experimental procedure), and then move back to the laboratory frame. In this way, we find a set of "relativistic Stern-Gerlach measurements" in the laboratory frame, and a set of observables satisfying the spin su(2) algebra. This operational procedure offers a concrete way of testing the relativistic features of the spin, and opens up the possibility of devising quantum information protocols for spin in the special-relativistic regime.