Harman Patil (Editor)

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

Contents

π S : S → M

is associated to the corresponding principal bundle

π P : 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 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 F S O ( M ) → M with respect to the double covering ρ : S p i n ( n ) → S O ( n ) .

One usually defines the spinor bundle π S : S → M 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.

A spinor field is defined to be a section of the spinor bundle S, i.e., a smooth mapping ψ : M → S such that π S ∘ ψ : M → M is the identity mapping idM of M.

Books

  • 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 

    • References

    Spinor field Wikipedia
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