NationMaster - Encyclopedia: Spinor bundle

In mathematics and theoretical physics, spinors are certain geometric entities bound up with physical theories of 'spin', and the mathematics of Clifford algebras, that in a sense are kinds of twisted tensors. From a geometric point of view, spinors are organised into spinor bundles. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

Given a differentiable manifold M with a metric of signature (p,q) over it, a spinor bundle over M is a vector bundle over M such that its fiber is a spinor representation of This article or section is in need of attention from an expert on the subject. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ...

Spin(p,q),

a double cover of the identity component of the special orthogonal group SO(p,q). In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

Spinor bundles inherit a connection from a connection on the vector bundle V (see spin connection). In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... For the theory, see Cartan connection applications For the chromosomal formation, see meiosis. ...

When

p + q ≤ 3

there are some further possibilities for covering groups of the orthogonal group, so other bundles (anyonic bundles). In mathematics, specifically topology, a covering map is a continuous surjective map p : C â†’ X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics and physics, an anyon is a type of projective representation of a Lie group. ...

From associated bundles

The language of associated bundles is helpful in expressing the meaning of spinor bundles. The existence of a spin structure is extra information on a real vector bundle. In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... A spin structure on a smooth orientable Riemannian manifold is a lift of the principal O(n) bundle associated to its tangent bundle to a principal Spin(n) bundle. ...

Here the two groups SO and Spin are involved (for a fixed choice of signature $(p, q)$ ), the former having a faithful matrix representation of dimension $n = p + q$ , but the latter acting (in general) only faithfully in a higher dimension, on a space of spinors. Spin is a double cover of the identity component of SO, so that the latter is a quotient of the former. (If p and q are both non-zero, then the special orthogonal group has 2 components, while the spin group has only one.) That does mean that transition data with values in Spin give rise to transition data for SO, automatically: passing to a quotient group simply loses information. The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...

Therefore a Spin-bundle always gives rise to an associated bundle with fibers $mathbb{R}^n$ , since Spin acts on $mathbb{R}^n$ , via its quotient SO. Conversely, there is a lifting problem for SO-bundles: there is a consistency question on the transition data, in passing to a Spin-bundle. The obstruction to the lifting is known to be the second Stiefel-Whitney class. Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles . ...

See also