In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g), a section of the spinor bundle S is called a spinor field. A spinor bundle is the complex vector bundle 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 ?n.
In particle physics, particles with spin s are described by a 2s-dimensional spinor field, where s is an integer or a half-integer. Fermions are described by spinor field, while bosons by tensor field.
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 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.
- ^ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, p. 53