Spinor field

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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

is 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 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[edit]

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[edit]


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