Spinor bundle - Alchetron, The Free Social Encyclopedia

In differential geometry, given a spin structure on a n -dimensional Riemannian manifold ( M , g ) , one defines the spinor bundle to be the complex vector bundle π S : S → M associated to the corresponding principal bundle π P : P → M of spin frames over M and the spin representation of its structure group S p i n ( n ) on the space of spinors Δ n . .

A section of the spinor bundle S is called a spinor 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 F S O ( M ) → M with respect to the double covering ρ : S p i n ( n ) → S O ( n ) of the special orthogonal group by the spin group.

The spinor bundle S is defined to be the complex vector bundle

S = P × κ Δ n

associated to the spin structure P via the spin representation κ : S p i n ( n ) → U ( Δ n ) , where U ( W ) denotes the group of unitary operators acting on a Hilbert space W . It is worth noting that the spin representation κ is a faithful and unitary representation of the group S p i n ( n ) .

References

Spinor bundle Wikipedia

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