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 associated to the corresponding principal bundle 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.

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 with respect to the double covering

One usually defines the spinor bundle[1] to be the complex vector bundle

associated to the spin structure P via the spin representation 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 such that is the identity mapping idM of M.

See also

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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