Spinor field

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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

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

is 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 Δ_n.

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

One usually defines the spinor bundle^[1] $\pi _{{{\mathbf S}}}:{{\mathbf S}}\to M\,$ 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 id_M of M.

Notes

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

Books

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

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Spinor field

Formal definition

See also

Notes

Books