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In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties.
In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. [1]
It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian
Non-abelian gauge theories are now handled by a variety of means. Methods for quantization are covered in the article on quantization. The main point to quantization is to be able to compute quantum amplitudes for various processes allowed by the theory. Technically, they reduce to the computations of certain correlation functions in the vacuum ...
[16] [17] [18] New quantum numbers, such as ground state degeneracy [15] (which can be defined on a closed space or an open space with gapped boundaries, including both Abelian topological orders [19] [20] and non-Abelian topological orders [21] [22]) and the non-Abelian geometric phase of degenerate ground states, [1] were introduced to ...
A Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)).
Trigonal bipyramidal molecular shape ax = axial ligands (on unique axis) eq = equatorial ligand (in plane perpendicular to unique axis). The Berry mechanism, or Berry pseudorotation mechanism, is a type of vibration causing molecules of certain geometries to isomerize by exchanging the two axial ligands (see the figure) for two of the equatorial ones.
(This statement generalizes to any sheaf of groups G, not necessarily abelian, using the non-abelian cohomology set H 1 (X,G).) By definition, an E-torsor over X is a sheaf S of sets together with an action of E on X such that every point in X has an open neighborhood on which S is isomorphic to E, with E acting on itself by translation.