An Exceptionally Simple Theory of Everything (part B)


(CONTINUED FROM 18/08/09)

2. The Standard Model Polytope
The structure of a simple Lie algebra is described by its root system. An N dimensional
Lie algebra, considered as a vector space, contains an R dimensional subspace, a Cartan
subalgebra, spanned by a maximal set of R inter-commuting generators, Ta,

(R is the rank of the Lie algebra) Every element of the Cartan subalgebra, C = CaTa, acts
linearly on the rest of the Lie algebra via the Lie bracket (the adjoint action). The Lie algebra
is spanned by the eigenvectors of this action, the root vectors, V , with each corresponding
to an eigenvalue,

Each of the (NR) non-zero eigenvalues, , (imaginary for real compact groups) is linearly
dependent on the coecients of C and corresponds to a point, a root, a , in the space dual
to the Cartan subalgebra. The pattern of roots in R dimensions uniquely characterizes the
algebra and is independent of the choice of Cartan subalgebra and rotations of the constituent
generators.
Since the root vectors, V , and Cartan subalgebra generators, Ta, span the Lie algebra,
they may be used as convenient generators | the Cartan-Weyl basis of the Lie algebra,

 

The Lie bracket between root vectors corresponds to vector addition between their roots, and
to interactions between particles,

Elements of the Lie algebra and Cartan subalgebra can also act on vectors in the various
representation spaces of the group. In these cases the eigenvectors of the Cartan subalgebra
(called weight vectors) have eigenvalues corresponding to the generalized roots (called
weights) describing the representation. From this more general point of view, the roots are
the weights of the Lie algebra elements in the adjoint representation space.
Each weight vector, V , corresponds to a type of elementary particle. The R coordinates
of each weight are the quantum numbers of the relevant particle with respect to the chosen
Cartan subalgebra generators.

 

2.1 Strong G2
The gluons, g 2 su(3), in the special unitary group of degree three may be represented using
the eight Gell-Mann matrices as generators,

The Cartan subalgebra, C = g3T3 + g8T8, is identi ed with the diagonal. This gives root
vectors | particle types | corresponding to the six non-zero roots, such as

 

for the green anti-blue gluon. (By an abuse of notation, the coecient, such as ggb , has the
same label as the particle eigenvector containing the coecient, and as the root | the usage
is clear from context.)
Since the Cartan subalgebra matrix in the standard representation acting on 3, and its
dual acting on 3, are diagonal, the weight vectors, V and V , satisfying

are the canonical unit vectors of the 3 and 3. The weights for these | the su(3) quantum
numbers of the quarks and anti-quarks | can be read o the diagonals of C and C = CT =
C.
The set of weights for su(3), the de ning 3, and its dual 3, are shown in Table 1. These
weights are precisely the 12 roots of the rank two simple exceptional Lie group, G2. The
weight vectors and weights of the 3 and 3 are identi ed as root vectors and roots of G2. The
G2 Lie algebra breaks up as

allowing a connection to be separated into the su(3) gluons, g, and the 3 and 3 quarks and
anti-quarks, : q and :q, related by Lie algebra duality. All interactions (2.1) between gluons and
quarks correspond to vector addition of the roots of G2, such as

We are including these quarks in a simple exceptional Lie algebra, g2, and not merely acting
on them with su(3) in some representation. The necessity of specifying a representation for
the quarks has been removed | a signi cant simpli cation of mathematical structure. And
we will see that this simpli cation does not occur only for the quarks in g2, but for all fermions
of the standard model.
Just as we represented the gluons in the (3 3) matrix representation (2.2) of su(3), we
may choose to represent the gluons and quarks using the smallest irreducible, (77), matrix
representation of g2,[6]

Squaring this matrix gives all interactions between gluons and quarks, equivalent to su(3)
acting on quarks and anti-quarks in the fundamental representation spaces.
The G2 root system may also be described in three dimensions as the 12 midpoints of
the edges of a cube | the vertices of a cuboctahedron. These roots are labeled g and qIII in
Table 2, with their (x; y; z) coordinates shown. These points may be rotated and scaled,

 

so that dropping the rst, B2, coordinate gives the projection to the roots in two dimensions.
In general, we can nd subalgebras by starting with the root system of a Lie algebra,
rotating it until multiple roots match up on parallel lines, and collapsing the root system along
these lines to an embedded space of lower dimension | a projection. Since the cuboctahedron
is the root system of so(6), we have obtained g2 by projecting along a u(1) in the Cartan
subalgebra of so(6),

This particular rotation and projection (2.4) generalizes to give the su(n) subalgebra of any
so(2n). We can also obtain g2 as a projected subalgebra of so(7) | the root system is the
so(6) root system plus 6 shorter roots, labeled qII , at the centers of the faces of the cube in
the gure of Table 1. The eight weights at the corners of a half-cube, labeled qI and l, also
project down to the roots of G2 and the origin, giving leptons and anti-leptons in addition
to quarks,

These three series of weights in three dimensions, and their rotations into su(3) coordinates,
are shown in Table 2. The action of su(3) on quarks and leptons corresponds to its action
on these sets of weights, while the u(1)BL quantum number, B2, is the baryon minus lepton
number, related to their hypercharge. The su(3) action does not move fermions between the
nine B2 grades in the table | each remains in its series, I, II, or III. Since this su(3) and   u(1)BL are commuting subalgebras, our grand uni cation of gauge elds follows the same path as the Pati-Salam 

(TO BE CONTINUED)

A. Garrett Lisi
SLRI, 722 Tyner Way, Incline Village, NV 89451
E-mail: alisi@hawaii.edu

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