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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 algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.
The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the geometric phase. This mechanics analogue of the geometric phase is known as the Hannay angle . Berry phase in quantum mechanics
Consider a generic (possibly non-Abelian) gauge transformation acting on a component field = =.The main examples in field theory have a compact gauge group and we write the symmetry operator as () = where () is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. up to a ...
This is a complication of the strong force making it inherently non-linear, contrary to the linear theory of the electromagnetic force. QCD is a non-abelian gauge theory. The word non-abelian in group-theoretical language means that the group operation is not commutative, making the corresponding Lie algebra non-trivial.
[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 ...
Mathematically the term within the trace is known as the holonomy, which describes a mapping of the fiber into itself upon horizontal lift along a closed loop. The set of all holonomies itself forms a group, which for principal bundles must be a subgroup of the gauge group.
It is generally argued that the Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, Φ and A, in quantum mechanics.Classically it was possible to argue that only the electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructs, that due to gauge freedom are not even unique for a given electromagnetic field.