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APPENDIX 2: The Spin(8) Clifford Algebra.

General references for Clifford algebras and spin structures are Porteous43, Benn and Tucker44, Freund7, Penrose and Rindler45, Bröcker and tom Dieck8, Choquet-Bruhat and DeWitt-Morrette46, Harvey21, and Lawson and Michelsohn20. Much of the following is from Harvey21.

Spin groups are usually defined by Clifford algebras. The Spin(8) Clifford algebra Cl(8), is the Clifford algebra of euclidean 8-dimensional space.

Cl(8) is a graded algebra of dimension 28 = 256 = 16 x 16 =

= 1+8+28+56+70+56+28+8+1. Cl(8) is isomorphic to R(16), the 16 x 16 matrix algebra over the real numbers R.

The first subspace of Cl(8) is the 1-dimensional scalar (grade 0) space of the trivial scalar representation of Spin(8).

The second subspace is the 8-dimensional vector (grade 1) space of the vector representation of Spin(8).

The third subspace is the 28-dimensional bivector (grade 2) space of the adjoint representation of Spin(8). The bivector space closes under the Lie algebra product of Spin(8), and is isomorphic to the Lie algebra of Spin(8).

The eighth subspace is the 1-dimensional pseudovector (grade 8) space of the pseudovector representation of Spin(8). The pseudovector V(8) determines the unit volume element of the underlying Euclidean space R8 of Cl(8). Clifford multiplication by V(8) splits Cl(8) into two parts:

the 128-dimensional self-dual part Cl(8)+ for which V(8)a = a; and the 128-dimensional anti-self-dual part Cl(8)- for which V(8)a = -a.

The even subspaces of Cl(8) form a subalgebra of dimension 128 =

= 1+28+70+28+1 = 64+64 = 82 + 82, called the even subalgebra Cl(8)e.

Cl(8)e is a subalgebra because the Clifford product of even elements of Cl(8) are also even elements. Cl(8)e can be represented as two diagonal 8 x 8 real submatrices of the 16 x 16 matrix algebra Cl(8).

One of the diagonal 8 x 8 real submatrices, Cl(8)e+, is self-dual.

The other, Cl(8)e-, is anti-self-dual.

Since the bivector 28-dimensional even subspace is isomorphic to the Lie algebra of Spin(8) it is clear that Spin(8) is a subalgebra of Cl(8)e.

The odd subspaces of Cl(8) also have dimension 128 =

= 8+56+56+8 = = 64+64 = 82 + 82, and are denoted Cl(8)o.

Clifford multiplication by Cl(8)o takes Cl(8)o into Cl(8)e and takes Cl(8)e into Cl(8)o. Cl(8)o can be represented as two off-diagonal 8 x 8 real submatrices of the 16 x 16 matrix algebra Cl(8).

One, Cl(8)o+, is self-dual. The other, Cl(8)o-, is anti-self-dual.

The space on which the 16 x 16 real matrix algebra Cl(8) acts by matrix multiplication is the spinor space SPINOR(8). SPINOR(8) is also a 16 x 16 real matrix. If Cl(8) is considered to be acting on SPINOR(8) by right-multiplication, then a minimal right-ideal of SPINOR(8) is a 1 x 16 real row vector. The 16 independent 1 x 16 minimal right-ideals make up the 16-dimensional space of spacetime Spin(8) spinors, reducible to 2 mirror image 8-dimensional irreducible half-spinor spaces.

In the F4 model, the minimal right-ideal of SPINOR(8) acts as the g-matrices of 8-dimensional spacetime in a conventional way.

If Cl(8) is considered to be acting on SPINOR(8) by left-multiplication, the a minimal left-ideal of SPINOR(8) is a 16x1 real column vector. In the F4 model, the minimal left-ideal of SPINOR(8) acts as the space of spinor fermion particles and antiparticles. Since this is an unconventional interpretation of spinor space, the following discussion is in terms of left-ideals. Of course, similar results apply to right-ideals.

Fig. 17 illlustrates the structure of Cl(8):

Fig. 17

If the minimal left-ideal 16x1 column vector space of SPINOR(8) is taken to include an 8-dimensional 8 x 1 self-dual column vector of the self-dual part Cl(8)e+ of Cl(8)e, then it describes an 8-dimensional self-dual half-spinor representation S8+ of Spin(8).

If the minimal left-ideal 16 x 1 column vector space of SPINOR(8) is taken to include an 8-dimensional 8 x 1 anti-self-dual column vector of the anti-self-dual part Cl(8)e- of Cl(8)e, then it describes an 8-dimensional anti-self-dual half-spinor representation S8- of Spin(8).

S8+ and S8- are sometimes called the left-handed and right-handed mirror image half-spinor representations of Spin(8).

In the F4 model, S8+ represents the 8 first generation fermion particles and S8-represents the 8 first generation fermion antiparticles.

In the F4 model, the 16-dimensional 16 x 1 column left-ideal vector space containing S8+ can be represented by a compact homogeneous symmetric space D8+.

The 16-dimensional 16 x 1 column left-ideal vector space containing S8- can similarly be represented by an isomorphic compact homogeneous symmetric space D8- having 16 real dimensions.

8 dimensions of D8+ lie in Cl(8)e and 8 dimensions in Cl(8)o.

Since Clifford multiplication by Cl(8)o interchanges Cl(8)o and Cl(8)e,

the 8 dimensions in Cl(8)o can be considered the imaginary part and

the 8 dimensions in Cl(8)e the real part of D8+, with D8+ having 8 complex dimensions. D8+ should be an 8-complex-dimensional symmetric space with a local U(1) symmetry for its complex structure and a local Spin(8) symmetry for its Spin(8) structure.

So, D8+ = Spin(10)/(Spin(8) x U(1)).

D8+ is isomorphic to a bounded complex homogeneous domain D8+.

I conjecture, but do not prove, that the Silov boundary ¶[D8+] of D8+ is isomorphic to the real part S8+ of D8+.

Identifying the real part S8+ of D8+ with the Silov boundary ¶[D8+] gives S8+ = RP1 x S7. Similarly, S8- = RP1 x S7.

In the Lagrangian approach used in the F4 model, the Lagrangian action gives, by the Euler-Lagrange equations, the (classical) differential equation that is a generalization of the Dirac equation. The solutions to the generalized Dirac equation are given by Green's operators. The Green's operators give propagator amplitudes.

Since the Silov boundary is the part of the bounded homogeneous domain that is necessary and sufficient to define the harmonic functions with respect to the Laplacian, using the Poisson kernel, a Silov boundary manifold is a natural manifold to use to define manifold used to construct a Green's operator for a propagator amplitude.

Triality between Base Manifold and Spinor Manifold:

The 8-dimensional half-spinor manifolds Q8+ = RP1 x S7 of the F4 model are related by triality to the 8-dimensional base manifold V of the vector representation of Spin(8).

The Spin(8) Clifford algebra is the lowest dimensional Clifford algebra in which all spinors are not pure spinors. A pure spinor in n (n even) dimensions can be represented, up to proportionality, by a (1/2)n plane through the origin of the vector space V. The (1/2)n plane is (anti-) self-dual. Pure 8-spinors determine (as 8 is even) a projective (1/2)(n-2) plane (called the a-plane or the b-plane in the left-handed and right-handed half-spinor spaces) on q, the non-singular quadric (n-2) surface on the (n-1)-dimensional (complex) projective space PV associated with the vector space V, defined by the non-zero null vectors of V. Since the projective pure spinors in even dimension have dimension (1/8)n(n-2), the space of even-dimensional pure spinors has dimension (1/8)n(n-2)+1, or 7 in the 8-dimensional case.

The Spin(8) triality isomorphisms V = S8+ = S8- among the vector and half-spinor representations of Spin(8) map the 7-dimensional space of null vectors in V into the 7-dimensional spaces of pure spinors in S8+ and S8-. The a-planes of q, the b-planes of q, and the points of q all correspond to each other. (Penrose and Rindler45)

The F4 triality correspondence is not a conventional supersymmetry correspondence between gauge bosons and fermions.

S8+ = RP1 x S7 and Helicity States of Fermions:

Since RP1 is not simply-connected, RP1 x RP7 admits two spinor structures, left-handed and right-handed. Reflection in RP1 leaves the two spinor structures invariant, while reflection in RP7 interchanges the two spinor structures (Penrose and Rindler45).

Given a left-handed spinor field S8±L over RP1 x RP7,

RP1 reflection takes S8±L into S8±L but RP7 reflection takes S8±L into S8±L over RP¡ and into S8±R , a right-handed spinor field, over RP7.

Therefore, the double cover S7 of RP7 should be used to describe S8+ so that both left-handed and right-handed spinor fields exist for the Dirac fermions, to be reflected into each other by RP7 reflection.

If the northern hemisphere of S7 is taken to correspond to one RP7, then the southern hemisphere of S7 corresponds to its equatorial reflection.

Therefore, if the northern hemisphere of the S7 carries a left-handed spinor structure, then the southern hemisphere of the S7 carries a right-handed spinor structure (Penrose and Rindler45).

Conformal Structure:

The global Spin(10) symmetry of D8 is a euclideanized version of Spin(8,2), the 4-fold covering group of the 8-dimensional conformal group Cnf(8) (Penrose and Rindler45). The conformal nature of the transformation is important because the conformal group not only preserves the causal light-cone structure of spacetime but also leaves invariant the spinor bundle on an n-dimensional manifold (and is, in a sense, the largest group that does so) (Lawson and Michelsohn20).

The local Spin(8) x Spin(2) = Spin(8) x U(1) symmetry of D8+ means that D8 has a natural Spin(8) gauge group with complex U(1) phase for gauge boson propagator amplitudes.

The F4 model 8-dimensional spacetime S8 = Spin(9)/Spin(8) does not have built-in full conformal structure, but it can be induced by the F4 triality automorphism between S8, S8+, and S8-, or, equivalently, by looking at the octonionic structure of S8 = OP1.

The F4 model is massless in 8-dimensional spacetime, but has massive particles after dimensional reduction to 4-dimensional spacetime. Although conformal symmetry is not preserved in conventional massive field theories, Iwo Bialynicki-Birula47 has pointed out that if the mass m of a fermion (m2 for bosons) is the amplitude for intermediate stops in a Feynman sum over paths, and if all paths are made up of Planck-length F4 lattice light-cone segments as in the F4 model or a Feynman checkerboard, then conformal symmetry would be preserved because all motion, viewed on the Planck length scale, would be on a light-cone.

Dirac Operators and Spherical Space:

Spin(8) coincides with SU(2,L7) as a multiplicative group in M(2,Cl(7)), where L7 is the Clifford semigroup consisting of all a in Cl(7) such that for each x in R8 there is a x in R8 such that ax = xa', where a' is the image of a under the principal automorphism of Cl(7).

L7 = G7 plus {0}, where G7 is the Clifford group of Cl(7), a multiplicative group of all invertible finite products in Cl(7) of elements of R = R7.

G7 in Cl(7) is the analogue of GL(2,R) in Cl(2).

Spin(8) is a 2-fold covering of the group of all transformations of

R7 plus infinity = S7 preserving the spherical metric.

is a linear fractional transformation in SU(2,L7), and therefore in Spin(8), then it is well defined and bijective on R7 » {°} = S7, and the only Spin(8) orbit is R7 plus infinity = S7.

Similar results hold for hyperbolic space, with the spin group Spin0(8,1) coinciding with SL(2,L6).

The above results are from Gilbert and Murray49, which contains further details.

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