In
differential geometry, given a
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
on an
-dimensional orientable
Riemannian manifold one defines the spinor bundle to be the
complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.
Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
associated to the corresponding
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
of spin frames over
and the
spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
of its
structure group on the space of
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s
.
A section of the spinor bundle
is called a spinor field.
Formal definition
Let
be a
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
on a
Riemannian manifold that is, an
equivariant lift of the oriented
orthonormal frame bundle with respect to the double covering
of the
special orthogonal group by the
spin group.
The spinor bundle
is defined to be the
complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.
Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
associated to the
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
via the
spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
where
denotes the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
s acting on a
Hilbert space It is worth noting that the spin representation
is a faithful and
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
of the group
[ pages 20 and 24]
See also
*
Clifford bundle
*
Clifford module bundle
*
Orthonormal frame bundle
*
Spin geometry
*
Spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
*
Spinor representation
Notes
Further reading
*
*
,
Algebraic topology
Riemannian geometry
Structures on manifolds
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