Spinor field
In differential geometry, given a spin structure on a n-dimensional Riemannian manifold (M, g) a section of the spinor bundle S is called a spinor field. The complex vector bundle
is associated to the corresponding principal bundle
of spin frames over M via the spin representation of its structure group Spin(n) on the space of spinors Δn.
In particle physics particles with spin s are described by 2s-dimensional spinor field, where s is an integer or a half-integer. Fermions are described by spinor field, while bosons by tensor field.
Formal definition
Let (P, FP) be a spin structure on a Riemannian manifold (M, g) that is, an equivariant lift of the oriented orthonormal frame bundle with respect to the double covering
One usually defines the spinor bundle[1] to be the complex vector bundle
associated to the spin structure P via the spin representation where U(W) denotes the group of unitary operators acting on a Hilbert space W.
A spinor field is defined to be a section of the spinor bundle S, i.e., a smooth mapping such that is the identity mapping idM of M.
See also
Notes
- ↑ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, p. 53
Books
- Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
- Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1