geometric phase

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Geometric phase

A unifying mathematical concept that describes the relation between the history of internal states of a system and the system's resulting orientation in space. Under various aspects, this concept occurs in geometry, astronomy, classical mechanics, and quantum theory. In geometry it is known as holonomy. In quantum theory it is known as Berry's phase, after M. Berry, who isolated the concept (which was already known in special cases) and explained its wide-ranging signi-ficance.

A system is envisioned whose possible states can be visualized as points in a suitable abstract space. At the same time, the system has some position or orientation in another space. A history of internal states can be represented by a curve in the first space; and the effect of this history on the disposition of the system, by a curve in the second space. The mapping between these two curves is described by the geometric phase. Especially interesting is the case when a closed curve (cycle) in the first space maps onto an open curve in the second, for then there is no net change in internal state, yet the disposition of the system with respect to the outside world is altered.

The power of the geometric phase ideas is that they make it possible, in complex dynamical problems, to find some simple universal regularities without having to solve the complete equations. Significant uses of these ideas include demonstrations of the fractional electric charge and quantum statistics of the quasiparticles in the quantum Hall effect, and of the occurrence of anomalies in quantum field theory. See Anyons, Hall effect, Quantum field theory

geometric phase

[‚jē·ə‚me·trik ′fāz]
(physics)
A unifying mathematical concept that describes the relation between the history of internal states of a system and the system's resulting orientation in space.
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References in periodicals archive ?
Since the discovery of a geometric effect by Berry [1] in the shape of an additional phase factor after an adiabatic and cyclic transport of a quantum system, Berry's phase has been intensively investigated and generalized: the extension to degenerate subspaces by Wilckzek [2], the removal of the adiabatic constraint by Aharonov and Anandan [3] and the cyclic condition by Samuel and Bhandari [4] using the early ideas of Pancharatnam [5] and the kinematic approach to geometric phases by Mukunda and Simon [6], to name a few.