 Spinor Field
Get Spinor Field essential facts below. View Videos or join the Spinor Field discussion. Add Spinor Field to your PopFlock.com topic list for future reference or share this resource on social media.
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 $\pi _{\mathbf {S} }:{\mathbf {S} }\to M\,$ associated to the corresponding principal bundle $\pi _{\mathbf {P} }:{\mathbf {P} }\to M\,$ of spin frames over M via the spin representation of its structure group Spin(n) on the space of spinors ?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, FP) be a spin structure on a Riemannian manifold (M, g) that is, an equivariant lift of the oriented orthonormal frame bundle $\mathrm {F} _{SO}(M)\to M$ with respect to the double covering $\rho :{\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)\,.$ One usually defines the spinor bundle $\pi _{\mathbf {S} }:{\mathbf {S} }\to M\,$ to be the complex vector bundle

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