Search results
Results From The WOW.Com Content Network
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics.
In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics , in particular to quantum field theory where they are an essential ingredient in ...
Strictly speaking, the spin group describes a fermion in a zero-dimensional space; however, space is not zero-dimensional, and so the spin group is used to define spin structures on (pseudo-)Riemannian manifolds: the spin group is the structure group of a spinor bundle.
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. 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 .
The existence of spinors in 3 dimensions follows from the isomorphism of the groups SU(2) ≅ Spin(3) that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as Pauli matrices. In 4 Euclidean dimensions, the corresponding isomorphism is Spin(4) ≅ SU(2) × SU(2).
In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold (,), one defines the spinor bundle to be the complex vector bundle: associated to the corresponding principal bundle: of spin frames over and the spin representation of its structure group on the space of spinors.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. [ 1 ] By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry ), and orientation ...
This general setting has multiple strengths: it clarifies their interpretation as fermions in physics, and it shows precisely how to define spin in General Relativity, or, indeed, for any Riemannian manifold or pseudo-Riemannian manifold. This is informally sketched as follows. The Weyl equation is invariant under the action of the Lorentz group.