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 π𝐒:𝐒->M associated to the corresponding principal bundle π𝐏:𝐏->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 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.

## Contents

* 1 Formal definition
* 2 See also
* 3 Notes
* 4 References

## 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 FSO(M)->M with respect to the double covering ρ:Spin(n)->SO(n).

One usually defines the spinor bundle[1] π𝐒:𝐒->M to be the complex vector bundle

𝐒=𝐏×κΔn

associated to the spin structure P via the spin representation κ:Spin(n)->U(Δn), 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 ψ:M->𝐒 such that π𝐒∘ψ:M->M is the identity mapping idM 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

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