In mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)
1 → Z 2 → Spin ( n ) → SO ( n ) → 1. {\displaystyle 1\to \mathbb {Z} _{2}\to \operatorname {Spin} (n)\to \operatorname {SO} (n)\to 1.}
The group multiplication law on the double cover is given by lifting the multiplication on SO ( n ) {\displaystyle \operatorname {SO} (n)} . As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −I. Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n). A distinct article discusses the spin representations.
Use for physics models The spin group is used in physics when describing the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the electron. Strictly speaking, the spin group describes a fermion in a zero-dimensional space; however, space is not zero-dimensional, and so the spin group is used to define (non-existent) spin structures as a calculation tool on (pseudo-)Riemannian manifolds: the spin group is the structure group of a spinor bundle. The affine connection on a spinor bundle is the spin connection; the spin connection can simplify calculations in general relativity. The spin connection in turn enables the Dirac equation to be written in curved spacetime (effectively in the tetrad coordinates).
Construction Construction of the Spin group often starts with the construction of a Clifford algebra over a real vector space V with a definite quadratic form q. The Clifford algebra is the quotient of the tensor algebra TV of V by a two-sided ideal. The tensor algebra (over the reals) may be written as
T V = R ⊕ V ⊕ ( V ⊗ V ) ⊕ ⋯ {\displaystyle \mathrm {T} V=\mathbb {R} \oplus V\oplus (V\otimes V)\oplus \cdots }
The Clifford algebra Cl(V) is then the quotient algebra
Cl ( V ) = T V / ( v ⊗ v − q ( v ) ) , {\displaystyle \operatorname {Cl} (V)=\mathrm {T} V/\left(v\otimes v-q(v)\right),}
where q ( v ) {\displaystyle q(v)} is the quadratic form applied to a vector v ∈ V {\displaystyle v\in V} . The resulting space is finite dimensional, naturally graded (as a vector space), and can therefore be written as
Cl ( V ) = Cl 0 ⊕ Cl 1 ⊕ Cl 2 ⊕ ⋯ ⊕ Cl n {\displaystyle \operatorname {Cl} (V)=\operatorname {Cl} ^{0}\oplus \operatorname {Cl} ^{1}\oplus \operatorname {Cl} ^{2}\oplus \cdots \oplus \operatorname {Cl} ^{n}}
where n {\displaystyle n} is the dimension of V {\displaystyle V} , Cl 0 = R {\displaystyle \operatorname {Cl} ^{0}=\mathbb {R} } and Cl 1 = V {\displaystyle \operatorname {Cl} ^{1}=V} . The spin algebra s p i n {\displaystyle {\mathfrak {spin}}} is defined as the bivector subalgebra
Cl 2 = s p i n ( V ) = s p i n ( n ) , {\displaystyle \operatorname {Cl} ^{2}={\mathfrak {spin}}(V)={\mathfrak {spin}}(n),}
where the last is a short-hand for V being a real vector space of real dimension n. It is a Lie algebra with the commutator as multiplication; it has a natural action on V, and is isomorphic to the Lie algebra s o ( n ) {\displaystyle {\mathfrak {so}}(n)} of the special orthogonal group: If the set { e i } {\displaystyle \{e_{i}\}} are an orthonormal basis of the (real) vector space V, then the quotient above endows the Clifford algebra with a natural anti-commuting structure:
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