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arXiv:1504.07432 · International Journal of Quantum Information

Wigner rotation via Fermi–Walker transport and relativistic EPR correlations in the Schwarzschild spacetime

K. Bakke, Claudio Furtado, Alexandre M. de M. Carvalho

Published Mar 1, 2015• 7 citations
arXivHTMLPDF

Abstract

The Wigner rotation angle for a particle in a circular motion in the Schwarzschild spacetime is obtained via the Fermi–Walker transport of spinors. Then, by applying the Wentzel, Kramers, Brillouin (WKB) approximation, a possible application of the Fermi–Walker transport of spinors in relativistic Einstein–Podolsky–Rosen (EPR) correlations is discussed, where it is shown that the spins of the correlated particle undergo a precession in an analogous way to that obtained by Terashima and Ueda [H. Terashima and M. Ueda, Phys. Rev. A 69, 032113 (2004)] via the application of successive infinitesimal Lorentz transformations. Moreover, from the WKB approach, it is also shown that the degree of violation of the Bell inequality depends on the Wigner rotation angle obtained via the Fermi–Walker transport. Finally, the relativistic effects from the geometry of the spacetime and the accelerated motion of the correlated particles is discussed in the nonrelativistic limit.

DOI
10.1142/s0219749915500203

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The Wigner rotation angle for a particle in a circular motion in the Schwarzschild spacetime is obtained via the Fermi–Walker transport of spinors. Then, by applying the Wentzel, Kramers, Brillouin (WKB) approximation, a possible application of the Fermi–Walker transport of spinors in relativistic Einstein–Podolsky–Rosen (EPR) correlations is discussed, where it is shown that the spins of the correlated particle undergo a precession in an analogous way to that obtained by Terashima and Ueda [H. Terashima and M. Ueda, Phys. Rev. A 69, 032113 (2004)] via the application of successive infinitesimal Lorentz transformations. Moreover, from the WKB approach, it is also shown that the degree of violation of the Bell inequality depends on the Wigner rotation angle obtained via the Fermi–Walker transport. Finally, the relativistic effects from the geometry of the spacetime and the accelerated motion of the correlated particles is discussed in the nonrelativistic limit.
We present a geometric description to study the relativistic EPR correlations in curved spacetime background given by the application of the Fermi-Walker transport in the relativistic EPR states and we show that its result has the same effect as the applications of successive infinitesimal Lorentz boosts in the relativistic EPR states. We also show that the expression for the Bell inequality due to the Fermi-Walker transport is equivalent to the expression demonstrated by Terashima and Ueda \cite{TU2}, where the degree of violation of the Bell inequality is dependent of the angle of the Wigner rotation. This geometrical approach to study the relativistic EPR correlations is a promissing formulation to investigate the EPR correlations in the general relativity background.
We present a geometric approach to study the relativistic EPR correlations in curved space–time background given by the application of the Fermi–Walker transport in the relativistic EPR states and show that its result has the same effect as the applications of successive infinitesimal Lorentz boosts in the relativistic EPR states. We also show that the expression for the Bell inequality due to the Fermi–Walker transport is equivalent to the expression demonstrated by Terashima and Ueda,20 where the degree of violation of the Bell inequality depends on the angle of the Wigner rotation. This geometrical approach for study of the relativistic EPR correlations is a promising formulation to investigate the EPR correlations in the general relativity background.
We formulate the Einstein-Podolsky-Rosen (EPR) gedankenexperiment within the framework of relativistic quantum theory to analyze a situation in which measurements are performed by moving observers. We point out that under certain conditions the perfect anti-correlation of an EPR pair of spins in the same direction deteriorates in the moving observers' frame due to the Wigner rotation, and show that the degree of the violation of Bell's inequality prima facie decreases with increasing velocity of the observers if the directions of the measurement are fixed. However, this does not imply a breakdown of non-local correlation since the perfect anti-correlation is maintained in appropriately chosen different directions. When considering moving frames we must take account of this relativistic effect on the EPR correlation and on the violation of Bell's inequality for quantum communication.
We investigate in a covariant manner the spin-induced non-geodesic motion of massive spin 1/2 particles in an arbitrary gravitational field for trajectories that are initially geodesic when spin is ignored. Using the WKB approximation for the wave function in curved spacetime, we compute the O(hbar) correction to the Wigner rotation of the spin 1/2 particle, whose O(1) contribution is zero on timelike geodesics. We develop conditions for the motion of observers in which the Wigner rotation is null. For the spherically symmetric Schwarzschild metric, we consider specific examples of particle motion in the equatorial plane for (i) circular orbits and (ii) radially infalling trajectories. For the former case we consider the entanglement for a perfectly anti-correlated EPR entangled pair of spins as the separate qubits traverse the circular orbit in same direction.
In this thesis, entanglement under fully relativistic settings are discussed. The thesis starts with a brief review of the relativistic quantum mechanics. In order to describe the effects of Lorentz transformations on the entangled states, quantum mechanics and special relativity are merged by construction of the unitary irreducible representations of Poincar{é} group on the infinite dimensional Hilbert space of state vectors. In this framework, the issue of finding the unitary irreducible representations of Poincar{é} group is reduced to that of the little group. Wigner rotation for the massive particles plays a crucial role due to its effect on the spin polarization directions. Furthermore, the physical requirements for constructing the correct relativistic spin operator is also studied. Then, the entanglement and Bell type inequalities are reviewed. Special attention has been devoted to two historical papers, by EPR in 1935 and by J.S. Bell in 1964. The main part of the thesis is based on the Lorentz transformation of the Bell states and the Bell inequalities on these transformed states. It is shown that entanglement is a Lorentz invariant quantity. That is, no inertial observer can see the entangled state as a separable one. However, it was shown that the Bell inequality may be satisfied for the Wigner angle dependent transformed entangled states. Since the Wigner rotation changes the spin polarization direction with the increased velocity, initial dichotomous operators can satisfy the Bell inequality for those states. By choosing the dichotomous operators taking into consideration the Wigner angle, it is always possible to show that Bell type inequalities can be violated for the transformed entangled states.