Physics: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 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 ${\mathrm{F}}_{SO}(M)\to M$ with respect to the double covering $\rho :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{S}\mathrm{O}(n).$

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

$\mathbf{𝐒}=\mathbf{𝐏}{\times }_{\kappa }{\mathrm{\Delta }}_n$

associated to the spin structure P via the spin representation $\kappa :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{U}({\mathrm{\Delta }}_n),$ 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.

See also

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

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 

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