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# Spinor field

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### Spinor field

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 Δn.

In particle physics particles with spin s are described by 2s-dimensional spinor field, where s is a integer or a half-integer. Fermions are described by spinor field, while bosons by tensor field.

## Contents

• Formal definition 1
• Notes 3
• Books 4

## 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[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 idM of M.

## Notes

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

## Books

• Lawson, H. Blaine;
• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry,
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