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 [math]\displaystyle{ \pi_{\mathbf S}:{\mathbf S}\to M\, }[/math] associated to the corresponding principal bundle [math]\displaystyle{ \pi_{\mathbf P}:{\mathbf P}\to M\, }[/math] 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 [math]\displaystyle{ \mathrm F_{SO}(M)\to M }[/math] with respect to the double covering [math]\displaystyle{ \rho: {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n)\,. }[/math]

One usually defines the spinor bundle [math]\displaystyle{ \pi_{\mathbf S}:{\mathbf S}\to M\, }[/math] to be the complex vector bundle

[math]\displaystyle{ {\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\, }[/math]

associated to the spin structure P via the spin representation [math]\displaystyle{ \kappa: {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\, }[/math] 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 [math]\displaystyle{ \psi : M \to {\mathbf S}\, }[/math] such that [math]\displaystyle{ \pi_{\mathbf S}\circ\psi: M\to M\, }[/math] 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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