Spin group

The spin group is used in physics to describe 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 spin structures 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), which in turn provides a footing for quantum gravity, as well as a formalization of Hawking radiation (where one of a pair of entangled, virtual fermions falls past the event horizon, and the other does not).

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.[3] 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

${\displaystyle \mathrm {T} V=\mathbb {R} \oplus V\oplus (V\otimes V)\oplus \cdots }$

The Clifford algebra Cl(V) is then the quotient algebra

${\displaystyle \operatorname {Cl} (V)=\mathrm {T} V/\left(v\otimes v-q(v)\right),}$

where ${\displaystyle q(v)}$ is the quadratic form applied to a vector ${\displaystyle v\in V}$. The resulting space is finite dimensional, naturally graded (as a vector space), and can be written as

${\displaystyle \operatorname {Cl} (V)=\operatorname {Cl} ^{0}\oplus \operatorname {Cl} ^{1}\oplus \operatorname {Cl} ^{2}\oplus \cdots \oplus \operatorname {Cl} ^{n}}$

where ${\displaystyle n}$ is the dimension of ${\displaystyle V}$, ${\displaystyle \operatorname {Cl} ^{0}=\mathbf {R} }$ and ${\displaystyle \operatorname {Cl} ^{1}=V}$. The spin algebra ${\displaystyle {\mathfrak {spin}}}$ is defined as

${\displaystyle \operatorname {Cl} ^{2}={\mathfrak {spin}}(V)={\mathfrak {spin}}(n),}$

where the last is a short-hand for V being a real vector space of real dimension n. It is a Lie algebra; it has a natural action on V, and in this way can be shown to be isomorphic to the Lie algebra ${\displaystyle {\mathfrak {so}}(n)}$ of the special orthogonal group.

The pin group ${\displaystyle \operatorname {Pin} (V)}$ is a subgroup of ${\displaystyle \operatorname {Cl} (V)}$'s Clifford group of all elements of the form

${\displaystyle v_{1}v_{2}\cdots v_{k},}$

where each ${\displaystyle v_{i}\in V}$ is of unit length: ${\displaystyle q(v_{i})=1.}$

The spin group is then defined as

${\displaystyle \operatorname {Spin} (V)=\operatorname {Pin} (V)\cap \operatorname {Cl} ^{\text{even}},}$

where ${\displaystyle \operatorname {Cl} ^{\text{even}}=\operatorname {Cl} ^{0}\oplus \operatorname {Cl} ^{2}\oplus \operatorname {Cl} ^{4}\oplus \cdots }$ 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.

If the set ${\displaystyle \{e_{i}\}}$ are an orthonormal basis of the (real) vector space V, then the quotient above endows the space with a natural anti-commuting structure:

${\displaystyle e_{i}e_{j}=-e_{j}e_{i}}$ for ${\displaystyle i\neq j,}$

which follows by considering ${\displaystyle v\otimes v}$ for ${\displaystyle v=e_{i}+e_{j}}$. This anti-commutation 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.

For a quadratic space V, a double covering of SO(V) by Spin(V) can be given explicitly, as follows. Let ${\displaystyle \{e_{i}\}}$ be an orthonormal basis for V. Define an antiautomorphism ${\displaystyle t:\operatorname {Cl} (V)\to \operatorname {Cl} (V)}$ by

${\displaystyle \left(e_{i}e_{j}\cdots e_{k}\right)^{t}=e_{k}\cdots e_{j}e_{i}.}$

This can be extended to all elements of ${\displaystyle a,b\in \operatorname {Cl} (V)}$ by linearity. It is an antihomomorphism since

${\displaystyle (ab)^{t}=b^{t}a^{t}.}$

Observe that Pin(V) can then be defined as all elements ${\displaystyle a\in \operatorname {Cl} (V)}$ for which

${\displaystyle aa^{t}=1.}$

Now define the automorphism ${\displaystyle \alpha \colon \operatorname {Cl} (V)\to \operatorname {Cl} (V)}$ which on degree 1 elements is given by

${\displaystyle \alpha (v)=-v,\quad v\in V,}$

and let ${\displaystyle a^{*}}$ denote ${\displaystyle \alpha (a)^{t}}$, which is an antiautomorphism of Cl(V). With this notation, an explicit double covering is the homomorphism ${\displaystyle \operatorname {Pin} (V)\to \operatorname {O} (V)}$ given by

${\displaystyle \rho (a)v=ava^{*},}$

where ${\displaystyle v\in V}$. When a has degree 1 (i.e. ${\displaystyle a\in V}$), ${\displaystyle \rho (a)}$ corresponds a reflection across the hyperplane orthogonal to a; this follows from the anti-commuting property of the Clifford algebra.

This gives a double covering of both O(V) by Pin(V) and of SO(V) by Spin(V) because ${\displaystyle a}$ gives the same transformation as ${\displaystyle -a}$.

It is worth reviewing how spinor space and Weyl spinors are constructed, given this formalism. Given a real vector space V of dimension n = 2m an even number, its complexification is ${\displaystyle V\otimes \mathbf {C} }$. It can be written as the direct sum of a subspace ${\displaystyle W}$ of spinors and a subspace ${\displaystyle {\overline {W}}}$ of anti-spinors:

${\displaystyle V\otimes \mathbf {C} =W\oplus {\overline {W}}}$

The space ${\displaystyle W}$ is spanned by the spinors ${\displaystyle \eta _{k}=\left(e_{2k-1}-ie_{2k}\right)/{\sqrt {2}}}$ for ${\displaystyle 1\leq k\leq m}$ and the complex conjugate spinors span ${\displaystyle {\overline {W}}}$. 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 ${\displaystyle \textstyle {\bigwedge }W}$. 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 ${\displaystyle W}$ 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.[3]

The Spin^C group is defined by the exact sequence

${\displaystyle 1\to \mathrm {Z} _{2}\to \operatorname {Spin} ^{\mathbf {C} }(n)\to \operatorname {SO} (n)\times \operatorname {U} (1)\to 1.}$

It is a multiplicative subgroup of the complexification ${\displaystyle \operatorname {Cl} (V)\otimes \mathbf {C} }$ of the Clifford algebra, and specifically, it is the subgroup generated by Spin(V) and the unit circle in C. Alternately, it is the quotient

${\displaystyle \operatorname {Spin} ^{\mathbf {C} }(V)=\left(\operatorname {Spin} (V)\times S^{1}\right)/\sim }$

where the equivalence ${\displaystyle \sim }$ identifies (a, u) with (−a, −u).

This has important applications in 4-manifold theory and Seiberg–Witten theory. In physics, the Spin group is appropriate for describing uncharged fermions, while the Spin^C group is used to describe electrically charged fermions. In this case, the U(1) symmetry is specifically the gauge group of electromagnetism.

In low dimensions, there are isomorphisms among the classical Lie groups called exceptional isomorphisms. For instance, there are isomorphisms between low-dimensional spin groups and certain classical Lie groups, owing to low-dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Writing R for the reals, C for the complex numbers, H for the quaternions and the general understanding that Cl(n) is a short-hand for Cl(Rⁿ) and that Spin(n) is a short-hand for Spin(Rⁿ) and so on, one then has that[3]

Cl^even(1) = R the real numbers

Pin(1) = {+i, −i, +1, −1}

Spin(1) = O(1) = {+1, −1}     the orthogonal group of dimension zero.

--

Cl^even(2) = C the complex numbers

Spin(2) = U(1) = SO(2), which acts on z in R² by double phase rotation z ↦ u²z. Corresponds to the abelian ${\displaystyle D_{1}}$.     dim = 1

--

Cl^even(3) = H the quaternions

Spin(3) = Sp(1) = SU(2), corresponding to ${\displaystyle B_{1}\cong C_{1}\cong A_{1}}$.     dim = 3

--

Cl^even(4) = H ⊕ H

Spin(4) = SU(2) × SU(2), corresponding to ${\displaystyle D_{2}\cong A_{1}\times A_{1}}$.     dim = 6

--

Cl^even(5)= M(2, H) the two-by-two matrices with quaternionic coefficients

Spin(5) = Sp(2), corresponding to ${\displaystyle B_{2}\cong C_{2}}$.     dim = 10

--

Cl^even(6)= M(4, C) the four-by-four matrices with complex coefficients

Spin(6) = SU(4), corresponding to ${\displaystyle D_{3}\cong A_{3}}$.     dim = 15

There are certain vestiges of these isomorphisms left over for n = 7, 8 (see Spin(8) for more details). For higher n, these isomorphisms disappear entirely.

In indefinite signature, the spin group Spin(p, q) is constructed through Clifford algebras in a similar way to standard spin groups. It is a double cover of SO₀(p, q), the connected component of the identity of the indefinite orthogonal group SO(p, q). For p + q > 2, Spin(p, q) is connected; for (p, q) = (1, 1) there are two connected components.[4]^: 193 As in definite signature, there are some accidental isomorphisms in low dimensions:

Spin(1, 1) = GL(1, R)

Spin(2, 1) = SL(2, R)

Spin(3, 1) = SL(2, C)

Spin(2, 2) = SL(2, R) × SL(2, R)

Spin(4, 1) = Sp(1, 1)

Spin(3, 2) = Sp(4, R)

Spin(5, 1) = SL(2, H)

Spin(4, 2) = SU(2, 2)

Spin(3, 3) = SL(4, R)

Spin(6, 2) = SU(2, 2, H)

Note that Spin(p, q) = Spin(q, p).

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

${\displaystyle \pi _{1}(G)\subset \operatorname {Z} (G'),}$

with Z(G′) the center of G′. This inclusion and the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of G determine G entirely (note that it is not the case that ${\displaystyle {\mathfrak {g}}}$ and π₁(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, Spin₀(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:

${\displaystyle \pi _{1}({\mbox{SO}}(p,q))={\begin{cases}0&(p,q)=(1,1){\mbox{ or }}(1,0)\\\mathbb {Z} _{2}&p>2,q=0,1\\\mathbb {Z} &(p,q)=(2,0){\mbox{ or }}(2,1)\\\mathbb {Z} \times \mathbb {Z} &(p,q)=(2,2)\\\mathbb {Z} &p>2,q=2\\\mathbb {Z} _{2}&p,q>2\\\end{cases}}}$

Thus once p, q > 2 the fundamental group is Z₂, as it is a 2-fold quotient of a product of two universal covers.

The maps on fundamental groups are given as follows. For p, q > 2, this implies that the map π₁(Spin(p, q)) → π₁(SO(p, q)) is given by 1 ∈ Z₂ going to (1, 1) ∈ Z₂ × Z₂. For p = 2, q > 2, this map is given by 1 ∈ Z → (1,1) ∈ Z × Z₂. And finally, for p = q = 2, (1, 0) ∈ Z × Z is sent to (1,1) ∈ Z × Z and (0, 1) is sent to (1, −1).

Fundamental groups of SO(n)

The fundamental groups ${\displaystyle \pi _{1}(\operatorname {SO} (n))}$ can be more directly derived using results in homotopy theory. In particular we can find ${\displaystyle \pi _{1}(\operatorname {SO} (n))}$ for ${\displaystyle n>3}$ as the three smallest have familiar underlying manifolds: ${\displaystyle SO(1)}$ is the point manifold, ${\displaystyle SO(2)\cong S^{1}}$, and ${\displaystyle SO(3)\cong \mathbb {RP} ^{3}}$ (shown using the axis-angle representation).

The proof uses known results in algebraic topology.[5]

Proof

First consider the action of ${\displaystyle \operatorname {SO} (n)}$ on ${\displaystyle \mathbb {R} ^{n}}$, in particular on the vector ${\displaystyle v=(1,0,\cdots ,0)}$. The orbit of this vector is ${\displaystyle {\text{Orbit}}_{{\text{SO}}(n)}(v)=S^{n-1}}$, while the stabilizer is ${\displaystyle {\text{Stab}}_{{\text{SO}}(n)}(v)={\text{SO}}(n-1)}$. Thus from the orbit-stabilizer theorem one obtains an isomorphism $${\displaystyle {\text{SO}}(n)/{\text{SO}}(n-1)\cong S^{n-1}.}$$ Geometrically, this provides a fibration $${\displaystyle {\text{SO}}(n-1)\rightarrow {\text{SO}}(n)\rightarrow S^{n-1}.}$$ Then Theorem 4.41 in Hatcher tells us that there is a long exact sequence of homotopy groups $${\displaystyle \cdots \rightarrow \pi _{k}({\text{SO}}(n-1))\rightarrow \pi _{k}({\text{SO}}(n))\rightarrow \pi _{k}(S^{n-1})\rightarrow \pi _{k-1}({\text{SO}}(n-1))\rightarrow \cdots }$$ and we concentrate on a section at the end of the sequence: $${\displaystyle \pi _{2}(S^{n-1})\rightarrow \pi _{1}({\text{SO}}(n-1))\rightarrow \pi _{1}({\text{SO}}(n))\rightarrow \pi _{1}(S^{n-1}).}$$ Corollary 4.9 in Hatcher states ${\displaystyle \pi _{k}(S^{n})=0}$ for ${\displaystyle k<n}$. So for ${\displaystyle n>3}$, the exact sequence becomes $${\displaystyle 0\rightarrow \pi _{1}({\text{SO}}(n-1))\rightarrow \pi _{1}({\text{SO}}(n))\rightarrow 0,}$$ hence ${\displaystyle \pi _{1}({\text{SO}}(n))}$ and ${\displaystyle \pi _{1}({\text{SO}}(n-1))}$ are isomorphic as long as ${\displaystyle n>3}$, so for ${\displaystyle n>3}$, we have ${\displaystyle \pi _{1}({\text{SO}}(n))\cong \pi _{1}({\text{SO}}(3))}$. And since ${\displaystyle {\text{SO}}(3)\cong \mathbb {RP} ^{3}\cong S^{3}/\{\pm 1\}}$, we get ${\displaystyle \pi _{1}({\text{SO}}(3))\cong \mathbb {Z} _{2}}$.

The same argument can be used to show ${\displaystyle \pi ({\text{SO}}(1,n)^{\uparrow })\cong \pi ({\text{SO}}(n))}$, by considering a fibration

$${\displaystyle {\text{SO}}(n)\rightarrow {\text{SO}}(1,n)^{\uparrow }\rightarrow H^{n},}$$

where ${\displaystyle H^{n}}$ is the upper sheet of a two-sheeted hyperboloid, which is contractible, and ${\displaystyle {\text{SO}}(1,n)^{\uparrow }}$ is the identity component of the proper Lorentz group (the proper orthochronous Lorentz group).

The center of the spin groups, for n ≥ 3, (complex and real) are given as follows:[4]^: 208

${\displaystyle {\begin{aligned}\operatorname {Z} (\operatorname {Spin} (n,\mathbf {C} ))&={\begin{cases}\mathrm {Z} _{2}&n=2k+1\\\mathrm {Z} _{4}&n=4k+2\\\mathrm {Z} _{2}\oplus \mathrm {Z} _{2}&n=4k\\\end{cases}}\\\operatorname {Z} (\operatorname {Spin} (p,q))&={\begin{cases}\mathrm {Z} _{2}&p{\text{ or }}q{\text{ odd}}\\\mathrm {Z} _{4}&n=4k+2,{\text{ and }}p,q{\text{ even}}\\\mathrm {Z} _{2}\oplus \mathrm {Z} _{2}&n=4k,{\text{ and }}p,q{\text{ even}}\\\end{cases}}\end{aligned}}}$

Quotient groups can be obtained from a spin group by quotienting out by a subgroup of the center, with the spin group then being a covering group of the resulting quotient, and both groups having the same Lie algebra.

Quotienting out by the entire center yields the minimal such group, the projective special orthogonal group, which is centerless, while quotienting out by {±1} yields the special orthogonal group – if the center equals {±1} (namely in odd dimension), these two quotient groups agree. If the spin group is simply connected (as Spin(n) is for n > 2), then Spin is the maximal group in the sequence, and one has a sequence of three groups,

Spin(n) → SO(n) → PSO(n),

splitting by parity yields:

Spin(2n) → SO(2n) → PSO(2n),

Spin(2n+1) → SO(2n+1) = PSO(2n+1),

which are the three compact real forms (or two, if SO = PSO) of the compact Lie algebra ${\displaystyle {\mathfrak {so}}(n,\mathbf {R} ).}$

The homotopy groups of the cover and the quotient are related by the long exact sequence of a fibration, with discrete fiber (the fiber being the kernel) – thus all homotopy groups for k > 1 are equal, but π₀ and π₁ may differ.

For n > 2, Spin(n) is simply connected (π₀ = π₁ = Z₁ is trivial), so SO(n) is connected and has fundamental group Z₂ while PSO(n) is connected and has fundamental group equal to the center of Spin(n).

In indefinite signature the covers and homotopy groups are more complicated – Spin(p, q) is not simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact SO(p) × SO(q) ⊂ SO(p, q) and the component group of Spin(p, q).

The spin group appears in a Whitehead tower anchored by the orthogonal group:

${\displaystyle \ldots \rightarrow {\text{Fivebrane}}(n)\rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)}$

The tower is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed. Killing the π₃ homotopy group in Spin(n), one obtains the infinite-dimensional string group String(n).

Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational point groups).

Given the double cover Spin(n) → SO(n), by the lattice theorem, there is a Galois connection between subgroups of Spin(n) and subgroups of SO(n) (rotational point groups): the image of a subgroup of Spin(n) is a rotational point group, and the preimage of a point group is a subgroup of Spin(n), and the closure operator on subgroups of Spin(n) is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as binary polyhedral groups.

Concretely, every binary point group is either the preimage of a point group (hence denoted 2G, for the point group G), or is an index 2 subgroup of the preimage of a point group which maps (isomorphically) onto the point group; in the latter case the full binary group is abstractly ${\displaystyle \mathrm {C} _{2}\times G}$ (since {±1} is central). As an example of these latter, given a cyclic group of odd order ${\displaystyle \mathrm {Z} _{2k+1}}$ in SO(n), its preimage is a cyclic group of twice the order, ${\displaystyle \mathrm {C} _{4k+2}\cong \mathrm {Z} _{2k+1}\times \mathrm {Z} _{2},}$ and the subgroup Z_2k+1 < Spin(n) maps isomorphically to Z_2k+1 < SO(n).

Of particular note are two series:

• higher binary tetrahedral groups, corresponding to the 2-fold cover of symmetries of the n-simplex; this group can also be considered as the double cover of the symmetric group, 2⋅A_n → A_n, with the alternating group being the (rotational) symmetry group of the n-simplex.
• higher binary octahedral groups, corresponding to the 2-fold covers of the hyperoctahedral group (symmetries of the hypercube, or equivalently of its dual, the cross-polytope).

For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group.

• The essential dimension of spin groups is OEIS:A280191.
• Grothendieck's "torsion index" is OEIS:A096336.

• Karoubi, Max (2008). K-Theory. Springer. pp. 210–214. ISBN 978-3-540-79889-7.