# 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

${\displaystyle \pi _{\mathbf {S} }:{\mathbf {S} }\to M\,}$

is associated to the corresponding principal bundle

${\displaystyle \pi _{\mathbf {P} }:{\mathbf {P} }\to M\,}$

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 ${\displaystyle \mathrm {F} _{SO}(M)\to M}$ with respect to the double covering ${\displaystyle \rho :{\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)\,.}$

One usually defines the spinor bundle[1] ${\displaystyle \pi _{\mathbf {S} }:{\mathbf {S} }\to M\,}$ to be the complex vector bundle

${\displaystyle {\mathbf {S} }={\mathbf {P} }\times _{\kappa }\Delta _{n}\,}$

associated to the spin structure P via the spin representation ${\displaystyle \kappa :{\mathrm {Spin} }(n)\to {\mathrm {U} }(\Delta _{n}),\,}$ 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 ${\displaystyle \psi :M\to {\mathbf {S} }\,}$ such that ${\displaystyle \pi _{\mathbf {S} }\circ \psi :M\to M\,}$ is the identity mapping idM of M.

## Notes

1. ^ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, p. 53