Spinor bundle

Short description: Geometric structure

In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M,g), one defines the spinor bundle to be the complex vector bundle π𝐒:𝐒->M associated to the corresponding principal bundle π𝐏:𝐏->M of spin frames over M and the spin representation of its structure group Spin(n) on the space of spinors Δn.

A section of the spinor bundle 𝐒 is called a spinor field.

## Contents

* 1 Formal definition
* 2 See also
* 3 Notes
* 4 Further reading

## 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 FSO(M)->M with respect to the double covering ρ:Spin(n)->SO(n) of the special orthogonal group by the spin group.

The spinor bundle 𝐒 is defined [1] to be the complex vector bundle 𝐒=𝐏×κΔn associated to the spin structure 𝐏 via the spin representation κ:Spin(n)->U(Δn), 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 Spin(n).[2]

## See also

* Clifford bundle
* Clifford module bundle
* Orthonormal frame bundle
* Spin geometry
* Spinor
* Spinor representation

## Notes

1. ↑ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 53
2. ↑ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 pages 20 and 24

## Further reading

* 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

* v
* t
* e

Tensors

Glossary of tensor theory

Scope|

| Mathematics|

* coordinate system
* multilinear algebra
* Euclidean geometry
* tensor algebra
* dyadic algebra
* differential geometry
* exterior calculus
* tensor calculus

|

* Physics
* Engineering

* continuum mechanics
* electromagnetism
* transport phenomena
* general relativity
* computer vision

Notation|

* index notation
* multi-index notation
* Einstein notation
* Ricci calculus
* Penrose graphical notation
* Voigt notation
* abstract index notation
* tetrad (index notation)
* Van der Waerden notation

Tensor
definitions|

* tensor (intrinsic definition)
* tensor field
* tensor density
* tensors in curvilinear coordinates
* mixed tensor
* antisymmetric tensor
* symmetric tensor
* tensor operator
* tensor bundle
* two-point tensor

Operations|

* tensor product
* exterior product
* tensor contraction
* transpose (2nd-order tensors)
* raising and lowering indices
* Hodge star operator
* covariant derivative
* exterior derivative
* exterior covariant derivative
* Lie derivative

Related
abstractions|

* dimension
* basis
* vector, vector space
* multivector
* covariance and contravariance of vectors
* linear transformation
* matrix
* spinor
* Cartan formalism
* differential form
* exterior form
* connection form
* geodesic
* manifold
* fiber bundle
* Levi-Civita connection
* affine connection

Notable tensors|

| Mathematics|

* Kronecker delta
* Levi-Civita symbol
* metric tensor
* nonmetricity tensor
* Christoffel symbols
* Ricci curvature
* Riemann curvature tensor
* Weyl tensor
* torsion tensor

|

Physics|

* moment of inertia
* angular momentum tensor
* spin tensor
* Cauchy stress tensor
* stress–energy tensor
* EM tensor
* gluon field strength tensor
* Einstein tensor
* metric tensor (GR)

Mathematicians|

* Leonhard Euler
* Carl Friedrich Gauss
* Augustin-Louis Cauchy
* Hermann Grassmann
* Gregorio Ricci-Curbastro
* Tullio Levi-Civita
* Jan Arnoldus Schouten
* Bernhard Riemann
* Elwin Bruno Christoffel
* Woldemar Voigt
* Élie Cartan
* Hermann Weyl
* Albert Einstein

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