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The pair (P, η) defines the structure of an affine geometry on M, making it into an affine manifold. The affine Lie algebra aff(n) splits as a semidirect product of R n and gl(n) and so η may be written as a pair (θ, ω) where θ takes values in R n and ω takes values in gl(n).
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Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J 1 Y of the jet bundle J 1 Y → Y of Y is an affine bundle morphism over X. In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X. (That is, the ...
Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold.For instance, these are gauge theory of dislocations in continuous media when =, the generalization of metric-affine gravitation theory when is a world manifold and, in particular, gauge theory of the fifth force.
It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations . In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations .
Malvin Ruderman and Charles Kittel of the University of California, Berkeley first proposed the model to explain unusually broad nuclear spin resonance lines in natural metallic silver. The theory is an indirect exchange coupling : the hyperfine interaction couples the nuclear spin of one atom to a conduction electron also coupled to the spin ...
1: Direct J-coupling in hydrogen gas. +½ spin 1 H polarizes electron close to it to -½ spin state. Antiparallel spins can exist within the same orbital as described by Pauli exclusion principle. Antiparallel spins can exist within the same orbital as described by Pauli exclusion principle.
A Yukawa interaction is also used in the Standard Model to describe the coupling between the Higgs field and massless quark and lepton fields (i.e., the fundamental fermion particles). Through spontaneous symmetry breaking , these fermions acquire a mass proportional to the vacuum expectation value of the Higgs field.