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 ${\displaystyle \pi _{\mathbf {S} }:{\mathbf {S} }\to M\,}$ 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 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 ${\displaystyle \mathrm {F} _{SO}(M)\to M}$ 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 id_M of M.

See also

• Orthonormal frame bundle
• Spinor
• Spin manifold
• Spinor representation
• Spin geometry

Notes

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

References

• 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