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arXiv:hep-th/9403076 · Physical Review Letters

Thomas Precession, Berry Potential, and the Meron

R. Shankar, Harsh Mathur

Published Sep 19, 1994• 26 citations
arXivHTMLPDF

Abstract

The Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a non-Abelian Berry vector potential. We ask what object produces the non-Abelian potential in parameter space, in the same sense that the Abelian vector potential arising in the adiabatic transport of a nondegenerate level is produced by a monopole (centered at the point where the level becomes degenerate with another) as shown by Berry. We find that it is a meron, an object in four Euclidean dimensions with instanton number \textonehalf{}, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate.

DOI
10.1103/physrevlett.73.1565

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The Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a non-Abelian Berry vector potential. We ask what object produces the non-Abelian potential in parameter space, in the same sense that the Abelian vector potential arising in the adiabatic transport of a nondegenerate level is produced by a monopole (centered at the point where the level becomes degenerate with another) as shown by Berry. We find that it is a meron, an object in four Euclidean dimensions with instanton number \textonehalf{}, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate.
We discuss the algebra and the interpretation of the anomalous Zeeman effect and the spin-orbit coupling within the Dirac theory. Whereas the algebra for the anomalous Zeeman effect is impeccable and therefore in excellent agreement with experiment, the physical interpretation of that algebra uses images that are based on macroscopic intuition but do not correspond to the meaning of this algebra. The interpretation violates the Lorentz symmetry. We give an alternative intuitive description of the meaning of this effect, which respects the symmetry and is exact. It can be summarized by stating that a magnetic field makes any charged particle spin. We show also that the traditional discussion about magnetic monopoles confuses two issues, viz. the symmetry of the Maxwell equations and the quantization of charge. These two issues define each a different concept of magnetic monopole. They cannot be merged together into a unique all-encompassing issue. We also generalize the minimal substitution for a charged particle, and provide some intuition for the magnetic vector potential. We finally explore the algebra of the spin-orbit coupling, which turns out to be badly wrong. The traditional theory that is claimed to reproduce the Thomas half is based on a number of errors. An error-free application of the Dirac theory cannot account for the Thomas precession, because it only accounts for the instantaneous local boosts, not for the rotational component of the Lorentz transformation. This runs contrary to established beliefs, but Thomas precession can be understood in terms of the Berry phase on a path through the velocity space of the Lorentz group manifold. These results clearly reveal the limitations of the prevailing working philosophy to "shut up and calculate".
Abstract This work revisit on the intrinsic connection between Dirac monopole theory and Berry geometric phases by extending Dirac's theory to the parameter space. Using the simplest two-mode Hamiltonian model, we explicitly visualize Dirac strings with endpoints in the parameter space, demonstrating that these endpoints correspond to accidental degeneracy points of energy eigenvalues in Hermitian systems. We show that non-integrable phase factors, induced by such Dirac strings, directly give rise to the well-known Berry connection and curvature, which can be re-derived by following the spirit of Dirac's monopole theory with the aid of the quantum adiabatic theorem. Our results indicate that the Berry geometric phase is essentially the non-integrable phase factor induced by Dirac strings with endpoints in the parameter space. These results further deepen our understanding on monopole and geometric phases, and would inspire more studies on them especially when Berry's framework is not applicable.
The non-Abelian Berry’s phase effect for a family of operators H0+k⋅V is considered, where H0 is rotationally invariant, V is a vector operator, and k varies over the unit vectors in R3. The parameter space is the two-sphere. The time evolution in the adiabatic limit is given by a connnection on a fiber bundle over the two-sphere. All connections consistent with the rotational symmetry are classified, and the time evolution is explicitly calculated for a nondegenerate Hamiltonian, as well as for a Hamiltonian with a double degeneracy. In the nondegenerate case, the connection is uniquely determined by the symmetry. In the doubly degenerate case, the connection is in some instances not determined by the symmetry. The case of approximate degeneracy is also discussed. A possible experimental test of this effect using optical pumping is described elsewhere.
Abstract Quantum states can acquire a geometric phase called the Berry phase after adiabatically traversing a closed loop, which depends on the path not the rate of motion. The Berry phase is analogous to the Aharonov–Bohm phase derived from the electromagnetic vector potential, and can be expressed in terms of an Abelian gauge potential called the Berry connection. Wilczek and Zee extended this concept to include non-Abelian phases—characterized by the gauge-independent Wilson loop—resulting from non-Abelian gauge potentials. Using an atomic Bose–Einstein condensate, we quantum-engineered a non-Abelian SU(2) gauge field, generated by a Yang monopole located at the origin of a 5-dimensional parameter space. By slowly encircling the monopole, we characterized the Wilczek–Zee phase in terms of the Wilson loop, that depended on the solid-angle subtended by the encircling path: a generalization of Stokes’ theorem. This observation marks the observation of the Wilson loop resulting from a non-Abelian point source.
Berry's phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model. This topology change of Berry's phase is visualized as a result of the lensing effect; the monopole supposed to be located at the level crossing point appears at the displaced point when the variables of the model deviate from the precisely adiabatic movement. The effective magnetic field generated by Berry's phase is determined by a simple geometrical consideration of the magnetic flux coming from the displaced Dirac monopole.