(where one of a pair of entangled, virtual fermions falls past the event horizon, and the other does not).
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 formq.[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
The Clifford algebra Cl(V) is then the
quotient algebra
where is the quadratic form applied to a vector . The resulting space is finite dimensional, naturally
graded
(as a vector space), and can be written as
where is the dimension of , and . The
spin algebra
is defined as
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 of the
special orthogonal group
.
The pin group is a subgroup of 's Clifford group of all elements of the form
where each is of unit length:
The spin group is then defined as
where
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 are an orthonormal basis of the (real) vector space V, then the quotient above endows the space with a natural anti-commuting structure:
for
which follows by considering for . This anti-commutation turns out to be of importance in physics, as it captures the spirit of the
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, is the double cover of . Its multiplication law can be defined by lifting as follows. Call the
covering map
. Then is a set with two elements, and one can be chosen without loss of generality to be the identity. Call this . Then to define multiplication in , for choose paths satisfying , and . These define a path in defined satisfying . Since is a double cover, there is a unique lift of with . Then define the product as .
It can then be shown that this definition is independent of the paths , that the multiplication is continuous, and the group axioms are satisfied with inversion being continuous, making 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 be an
antiautomorphism
by
This can be extended to all elements of by linearity. It is an antihomomorphism since
Observe that Pin(V) can then be defined as all elements for which
Now define the automorphism which on degree 1 elements is given by
and let denote , which is an antiautomorphism of Cl(V). With this notation, an explicit double covering is the homomorphism given by
where . When a has degree 1 (i.e. ), 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 gives the same transformation as .
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 n = 2m an even number, its complexification
is . It can be written as the direct sum of a subspace of spinors and a subspace of anti-spinors:
The space is spanned by the spinors
for and the complex conjugate spinors span . 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. 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 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]
It is a multiplicative subgroup of the complexification 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
where the equivalence identifies (a, u) with (−a, −u).
This has important applications in 4-manifold theory and
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]
Cleven(1) = R the real numbers
Pin(1) = {+i, −i, +1, −1}
Spin(1) = O(1) = {+1, −1} the orthogonal group of dimension zero.
--
Cleven(2) = C the complex numbers
Spin(2) =
SO(2)
, which acts on z in R2 by double phase rotation z ↦ u2z. Corresponds to the abelian . dim = 1
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.
Indefinite signature
In
connected component of the identity of the indefinite orthogonal groupSO(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:
with Z(G′) the center of G′. This inclusion and the Lie algebra of G determine G entirely (note that it is not the case that 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:
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 can be more directly derived using results in homotopy theory. In particular we can find for as the three smallest have familiar underlying manifolds: is the point manifold, , and (shown using the
Then Theorem 4.41 in Hatcher tells us that there is a
long exact sequence
of homotopy groups
and we concentrate on a section at the end of the sequence:
Corollary 4.9 in Hatcher states for . So for , the exact sequence becomes
hence and are isomorphic as long as , so for , we have .
And since , we get .
The same argument can be used to show , by considering a fibration
where is the upper sheet of a two-sheeted
contractible
, and is the identity component of the proper Lorentz group (the proper orthochronous Lorentz group).
Center
The center of the spin groups, for n ≥ 3, (complex and real) are given as follows:[4]: 208
Quotient groups
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
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,
, 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
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
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
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 (since {±1} is central). As an example of these latter, given a cyclic group of odd order in SO(n), its preimage is a cyclic group of twice the order, and the subgroup Z2k+1 < Spin(n) maps isomorphically to Z2k+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⋅An → An, with the alternating group being the (rotational) symmetry group of the n-simplex.
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.