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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]]
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| :<math>\pi_{\mathbf S}:{\mathbf S}\to M\,</math>
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| is associated to the corresponding [[principal bundle]] | |
| :<math>\pi_{\mathbf P}:{\mathbf P}\to M\,</math>
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| of spin frames over ''M'' via the [[spin representation]] of its structure group Spin(''n'') on the space of [[spinor]]s Δ<sub>''n''</sub>.
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| ==Formal definition==
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| Let ('''P''', ''F''<sub>'''P'''</sub>) be a [[spin structure]] on a [[Riemannian manifold]] (''M, g'') that is, an [[equivariant]] lift of the oriented [[orthonormal frame bundle]] <math>\mathrm F_{SO}(M)\to M</math> with respect to the double covering <math>\rho: {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n)\,.</math>
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| One usually defines the [[spinor bundle]]<ref>{{citation | last1=Friedrich|first1=Thomas|title = Dirac Operators in Riemannian Geometry| year=2000|page=53}}</ref> <math>\pi_{\mathbf S}:{\mathbf S}\to M\,</math> to be the [[complex vector bundle]]
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| :<math>{\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\,</math> | |
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| associated to the [[spin structure]] '''P''' via the [[spin representation]] <math>\kappa: {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\,</math> where U('''W''') denotes the [[Group (mathematics)|group]] of [[unitary operator]]s acting on a [[Hilbert space]] '''W'''.
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| A '''spinor field''' is defined to be a section of the [[spinor bundle]] '''S''', i.e., a smooth mapping <math>\psi : M \to {\mathbf S}\,</math> such that <math>\pi_{\mathbf S}\circ\psi: M\to M\,</math>
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| is the identity mapping id<sub>''M''</sub> of ''M''.
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| ==See also==
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| * [[Orthonormal frame bundle]]
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| * [[Spinor]]
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| * [[Spin manifold]]
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| * [[Spinor representation]]
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| * [[Spin geometry]]
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| ==Notes==
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| {{Reflist}}
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| ==Books==
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| * {{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise |author2-link=Marie-Louise Michelsohn| title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 | postscript=<!--None-->}}
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| * {{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}}
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| [[Category:Riemannian geometry]]
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| [[Category:Structures on manifolds]]
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| [[Category:Algebraic topology]]
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| [[Category:Quantum field theory]]
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| [[Category:Spinors]]
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| {{differential-geometry-stub}}
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| [[ca:Fibrat d'espinors]]
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| [[es:Fibrado de espinores]]
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| [[zh:旋量丛]]
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