In mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:<math>1 \to \mathbb{Z}_2 \to \operatorname{Spin}(n) \to \operatorname{SO}(n) \to 1.</math>
The group multiplication law on the double cover is given by lifting the multiplication on <math>\operatorname{SO}(n)</math>.
As a Lie group, Spin(n) therefore shares its dimension, , and its Lie algebra with the special orthogonal group.
For , 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 −.
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 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
:<math>\mathrm{T}V= \mathbb {R} \oplus V \oplus (V\otimes V) \oplus \cdots </math>
The Clifford algebra Cl(V) is then the quotient algebra
:<math>\operatorname{Cl}(V) = \mathrm{T}V / \left( v \otimes v - q(v) \right) ,</math>
where <math>q(v)</math> is the quadratic form applied to a vector <math>v\in V</math>. The resulting space is finite dimensional, naturally graded (as a vector space), and can therefore be written as
:<math>\operatorname{Cl}(V) = \operatorname{Cl}^0 \oplus \operatorname{Cl}^1 \oplus \operatorname{Cl}^2 \oplus \cdots \oplus \operatorname{Cl}^n</math>
where <math>n</math> is the dimension of <math>V</math>, <math>\operatorname{Cl}^0 = \mathbb{R}</math> and <math>\operatorname{Cl}^1 = V</math>. The spin algebra <math>\mathfrak{spin}</math> is defined as the bivector subalgebra
:<math>\operatorname{Cl}^2 =\mathfrak{spin}(V) = \mathfrak{spin}(n) ,</math>
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 <math>\mathfrak{so}(n)</math> of the special orthogonal group: If the set <math>\{e_i\}</math> are an orthonormal basis of the (real) vector space V, then the quotient above endows the Clifford algebra with a natural anti-commuting structure:
:<math>e_i e_j = -e_j e_i</math> for <math>i \ne j ,</math>
which follows by considering <math>v\otimes v</math> for <math>v=e_i+e_j</math>. Then in <math>\mathfrak{spin}(n)</math> we have that the Lie commutator <math>[e_i \otimes e_j, e_j \otimes e_k ]=2 e_i \otimes e_k </math> and <math>[e_i \otimes e_j, e_k \otimes e_l ]=0</math>, so <math>e_i \otimes e_j \rightarrow 2e_i \otimes e_j -2e_j \otimes e_i</math> gives the isomorphism to <math>\mathfrak{so}(n)</math>. On the right hand side <math>\otimes </math> is the outer product. The multiplication by 2 explains why rotating a spinor by 360 degrees returns minus the spinor: in <math>e^{i \phi b}</math> dividing the basis element b by 2 gives half a rotation for 360 degrees.
The pin group <math>\operatorname{Pin}(V)</math> is a subgroup of <math>\operatorname{Cl}(V)</math>'s Clifford group of all elements of the form
:<math>v_1 v_2 \cdots v_k ,</math>
where each <math>v_i\in V</math> is of unit length: <math>q(v_i) = 1.</math>
The spin group is then defined as
:<math>\operatorname{Spin}(V) = \operatorname{Pin}(V) \cap \operatorname{Cl}^{\text{even ,</math>
where
<math>\operatorname{Cl}^\text{even}=\operatorname{Cl}^0 \oplus \operatorname{Cl}^2 \oplus \operatorname{Cl}^4 \oplus \cdots</math>
is the subspace generated by elements that are the product of an even number of vectors. That is, Spin(V) consists of all elements of Pin(V), given above, with the restriction to k being an even number. The restriction to the even subspace is key to the formation of two-component (Weyl) spinors, constructed below.
The anti-commutation of the Clifford algebra turns out to be of importance in physics, as it captures the spirit of the Pauli exclusion principle for fermions. A precise formulation is out of scope here, but it involves the creation of a spinor bundle on Minkowski spacetime; the resulting spinor fields can be seen to be anti-commuting as a by-product of the Clifford algebra construction. This anti-commutation property is also key to the formulation of supersymmetry. The Clifford algebra and the spin group have many interesting and curious properties, some of which are listed below.
Geometric construction
The spin groups can be constructed less explicitly but without appealing to Clifford algebras. As a manifold, <math>\operatorname{Spin}(n)</math> is the double cover of <math>\operatorname{SO}(n)</math>. Its multiplication law can be defined by lifting as follows. Call the covering map <math>p: \operatorname{Spin}(n) \rightarrow \operatorname{SO}(n)</math>. Then <math>p^{-1}(\{e\})</math> is a set with two elements, and one can be chosen without loss of generality to be the identity. Call this <math>\tilde e</math>. Then to define multiplication in <math>\operatorname{Spin}(n)</math>, for <math>a, b \in \operatorname{Spin}(n)</math> choose paths <math>\gamma_a, \gamma_b</math> satisfying <math>\gamma_a(0) = \gamma_b(0) = \tilde e</math>, and <math>\gamma_a(1) = a, \gamma_b(1) = b</math>. These define a path <math>\gamma</math> in <math>\operatorname{SO}(n)</math> defined <math>\gamma(t) = p(\gamma_a(t))\cdot p(\gamma_b(t))</math> satisfying <math>\gamma(0) = e</math>. Since <math>\operatorname{Spin}(n)</math> is a double cover, there is a unique lift <math>\tilde \gamma</math> of <math>\gamma</math> with <math>\tilde \gamma(0) = \tilde e</math>. Then define the product as <math>a \cdot b = \tilde \gamma (1)</math>.
It can then be shown that this definition is independent of the paths <math>\gamma_a, \gamma_b</math>, that the multiplication is continuous, and the group axioms are satisfied with inversion being continuous, making <math>\operatorname{Spin}(n)</math> a Lie group.
Double covering
For a quadratic space V, a double covering of SO(V) by Spin(V) can be given explicitly, as follows. Let <math>\{e_i\}</math> be an orthonormal basis for V. Define an antiautomorphism <math>t : \operatorname{Cl}(V) \to \operatorname{Cl}(V)</math> by
:<math>
\left(e_i e_j \cdots e_k\right)^t
= e_k\cdots e_j e_i. </math>
This can be extended to all elements of <math>a,b\in \operatorname{Cl}(V)</math> by linearity. It is an antihomomorphism since
:<math> (a b)^t = b^t a^t.</math>
Observe that <math>\operatorname{Pin}(V)</math> can then be defined as all elements <math>a \in \operatorname{Cl}(V)</math> for which
:<math>a a^t = 1.</math>
Now define the automorphism <math>\alpha\colon \operatorname{Cl}(V)\to\operatorname{Cl}(V)</math> which on degree 1 elements is given by
:<math>\alpha(v)=-v,\quad v\in V,</math>
and let <math>a^*</math> denote <math>\alpha(a)^t</math>, which is an antiautomorphism of <math>\operatorname{Cl}(V)</math>. With this notation, an explicit double covering is the homomorphism <math>\rho:\operatorname{Pin}(V)\to\operatorname O(V)</math> given by
:<math>\rho(a) v = a v a^* ,</math>
where <math>v \in V</math>. When <math>a</math> has degree 1 (i.e. <math>a\in V</math>), <math>\rho(a)</math> is the reflection across the hyperplane orthogonal to <math>a</math>; this follows from the anti-commuting property of the Clifford algebra.
This gives a double covering of both <math>\operatorname{O}(V)</math> by <math>\operatorname{Pin}(V)</math> and of <math>\operatorname{SO}(V)</math> by <math>\operatorname{Spin}(V)</math> because <math>a</math> gives the same transformation as <math>-a</math>.
Spinor space
It is worth reviewing how spinor space and Weyl spinors are constructed, given this formalism. Given a real vector space V of dimension an even number, its complexification is <math>V \otimes \mathbf{C}</math>. It can be written as the direct sum of a subspace <math>W</math> of spinors and a subspace <math>\overline{W}</math> of anti-spinors:
:<math>V \otimes \mathbf{C} = W \oplus \overline{W}</math>
The space <math>W</math> is spanned by the spinors
<math>\eta_k = \left( e_{2k-1} - ie_{2k} \right) / \sqrt 2</math>
for <math>1\le k\le m</math> and the complex conjugate spinors span <math>\overline{W}</math>. It is straightforward to see that the spinors anti-commute, and that the product of a spinor and anti-spinor is a scalar.
The spinor space is defined as the exterior algebra <math>\textstyle{\bigwedge} W</math>. The (complexified) Clifford algebra acts naturally on this space; the (complexified) spin group corresponds to the length-preserving endomorphisms. There is a natural grading on the exterior algebra: the product of an odd number of copies of <math>W</math> correspond to the physics notion of fermions; the even subspace corresponds to the bosons. The representations of the action of the spin group on the spinor space can be built in a relatively straightforward fashion.
As in definite signature, there are some accidental isomorphisms in low dimensions:
{| class="wikitable"
|+ Accidental Isomorphisms
! <math>\text{Spin}(p, q)</math> !! 1 !! 2 !! 3
|-
! 1
| <math>\text{GL}(1, \mathbb{R})</math> || ||
|-
! 2
| <math>\text{SL}(2, \mathbb{R})</math> || <math>\text{SL}(2, \mathbb{R}) \times \text{SL}(2, \mathbb{R})</math> ||
|-
! 3
| <math>\text{SL}(2, \mathbb{C})</math> || <math>\text{Sp}(4, \mathbb{R}) </math>|| <math>\text{SL}(4, \mathbb{R}) </math>
|-
! 4
| <math>\text{Sp}(1,1)</math> || <math>\text{SU}(2,2)</math> ||
|-
! 5
| <math>\text{SL}(2, \mathbb{H})</math> || ||
|-
! 6
| || <math>\text{SU}(2,2, \mathbb{H})</math> ||
|}
Note that <math>\text{Spin}(p, q) = \text{Spin}(q, p)</math>.
Topological considerations
Connected and simply connected Lie groups are classified by their Lie algebra. So if G is a connected Lie group with a simple Lie algebra, with G′ the universal cover of G, there is an inclusion
:<math> \pi_1 (G) \subset \operatorname{Z}(G'), </math>
with Z(G′) the center of G′. This inclusion and the Lie algebra <math>\mathfrak{g}</math> of G determine G entirely (note that it is not the case that <math>\mathfrak{g}</math> and π<sub>1</sub>(G) determine G entirely; for instance SL(2, R) and PSL(2, R) have the same Lie algebra and same fundamental group Z, but are not isomorphic).
The definite signature Spin(n) are all simply connected for n > 2, so they are the universal coverings of SO(n).
In indefinite signature, Spin(p, q) is not necessarily connected, and in general the identity component, Spin<sub>0</sub>(p, q), is not simply connected, thus it is not a universal cover. The fundamental group is most easily understood by considering the maximal compact subgroup of SO(p, q), which is SO(p) × SO(q), and noting that rather than being the product of the 2-fold covers (hence a 4-fold cover), Spin(p, q) is the "diagonal" 2-fold cover – it is a 2-fold quotient of the 4-fold cover. Explicitly, the maximal compact connected subgroup of Spin(p, q) is
:Spin(p) × Spin(q)/{(1, 1), (−1, −1)}.
This allows us to calculate the fundamental groups of SO(p, q), taking p ≥ q:
:<math>\pi_1(\mbox{SO}(p,q)) = \begin{cases}
0 & (p,q)=(1,1) \mbox{ or } (1,0) \\
\mathbb{Z}_2 & p > 2, q = 0,1 \\
\mathbb{Z} & (p,q)=(2,0) \mbox{ or } (2,1) \\
\mathbb{Z} \times \mathbb{Z} & (p,q) = (2,2) \\
\mathbb{Z} \times \mathbb{Z}_2 & p > 2, q=2 \\
\mathbb{Z}_2 \times \mathbb{Z}_2 & p, q >2\\
\end{cases}</math>
Thus once the fundamental group is Z<sub>2</sub>, as it is a 2-fold quotient of a product of two universal covers.
The maps on fundamental groups are given as follows. For , this implies that the map is given by going to . For , this map is given by . And finally, for , is sent to and is sent to .
Fundamental groups of SO(n)
The fundamental groups <math>\pi_1(\operatorname{\text{SO(n))</math> can be more directly derived using results in homotopy theory. In particular we can find <math>\pi_1(\operatorname{\text{SO(n))</math> for <math>n > 3</math> as the three smallest have familiar underlying manifolds: <math>\operatorname{\text{SO(1)</math> is the point manifold, <math>\operatorname{\text{SO(2) \cong S^1</math>, and <math>\operatorname{\text{SO(3) \cong \mathbb{RP}^3</math> (shown using the axis-angle representation).
The proof uses known results in algebraic topology.
{| class="wikitable collapsible collapsed"
! Proof
|-
|
First consider the action of <math>\operatorname{SO}(n)</math> on <math>\mathbb{R}^n</math>, in particular on the vector <math>v = (1, 0, \cdots, 0)</math>. The orbit of this vector is <math>\text{Orbit}_{\text{SO}(n)}(v) = S^{n-1}</math>, while the stabilizer is <math>\text{Stab}_{\text{SO}(n)}(v) = \text{SO}(n-1)</math>. Thus from the orbit-stabilizer theorem one obtains an isomorphism
<math display = block> \text{SO}(n)/\text{SO}(n-1) \cong S^{n-1}.</math>
Geometrically, this provides a fibration
<math display = block> \text{SO}(n-1) \rightarrow \text{SO}(n) \rightarrow S^{n-1}.</math>
Then Theorem 4.41 in Hatcher tells us that there is a long exact sequence of homotopy groups
<math display = block>\cdots \rightarrow \pi_k(\text{SO}(n-1)) \rightarrow \pi_k(\text{SO}(n)) \rightarrow \pi_k(S^{n-1}) \rightarrow \pi_{k-1}(\text{SO}(n-1)) \rightarrow \cdots </math>
and we concentrate on a section at the end of the sequence:
<math display = block>\pi_2(S^{n-1}) \rightarrow \pi_1(\text{SO}(n-1))\rightarrow \pi_1(\text{SO}(n)) \rightarrow \pi_1(S^{n-1}).</math>
Corollary 4.9 in Hatcher states <math>\pi_k(S^n) = 0</math> for <math>k < n</math>. So for <math>n > 3</math>, the exact sequence becomes
<math display = block> 0 \rightarrow \pi_1(\text{SO}(n-1))\rightarrow \pi_1(\text{SO}(n)) \rightarrow 0,</math>
hence <math>\pi_1(\text{SO}(n))</math> and <math>\pi_1(\text{SO}(n-1))</math> are isomorphic as long as <math>n > 3</math>, so for <math>n > 3</math>, we have <math>\pi_1(\text{SO}(n)) \cong \pi_1(\text{SO}(3))</math>.
And since <math>\text{SO}(3) \cong \mathbb{RP}^3 \cong S^3 / \{\pm 1\}</math>, we get <math>\pi_1(\text{SO}(3)) \cong \mathbb{Z}_2</math>.
|}
The same argument can be used to show <math>\pi(\text{SO}(1,n)^\uparrow) \cong \pi(\text{SO}(n))</math>, by considering a fibration
<math display = block> \text{SO}(n) \rightarrow \text{SO}(1,n)^\uparrow \rightarrow H^n,</math>
where <math>H^n</math> is the upper sheet of a two-sheeted hyperboloid, which is contractible, and <math>\text{SO}(1,n)^\uparrow</math> is the identity component of the proper Lorentz group (the proper orthochronous Lorentz group).
Center
The center of the spin groups, for , (complex and real) are given as follows: