Spinor field

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 formula_1 associated to the corresponding principal bundle formula_2 of spin frames over "M" via the spin representation of its structure group Spin("n") on the space of spinors Δ"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.
Formal definition.
Let (P, "F"P) be a spin structure on a Riemannian manifold ("M, g") that is, an equivariant lift of the oriented orthonormal frame bundle formula_3 with respect to the double covering formula_4
One usually defines the spinor bundle formula_1 to be the complex vector bundle
associated to the spin structure P via the spin representation formula_7 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 formula_8 such that formula_9 
is the identity mapping id"M" of "M".