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1. Introduction 
2. The Standard Model Polytope
3. Dynamics
4. Summary
5. Discussion and Conclusion
We exist in a universe described by mathematics. But which math? Although it is interesting to consider that the universe may be the physical instantiation of all mathematics, [1] there is a classic principle for restricting the possibilities:
The mathematics of the universe should be beautiful. A successful description of nature should be a concise, elegant, unified mathematical structure consistent with experience.
Hundreds of years of theoretical and experimental work have produced an extremely successful pair of mathematical theories describing our world. The standard model of particles and interactions described by quantum field theory is a paragon of predictive excellence.
General relativity, a theory of gravity built from pure geometry, is exceedingly elegant and effective in its domain of applicability. Any attempt to describe nature at the foundational level must reproduce these successful theories, and the most sensible course towards
unification is to extend them with as little new mathematical machinery as necessary. The further we drift from these experimentally verified foundations, the less likely our mathematics is to correspond with reality.
In the absence of new experimental data, we should be very
careful, accepting sophisticated mathematical constructions only when they provide a clear simplification. And we should pare and unite existing structures whenever possible.
careful, accepting sophisticated mathematical constructions only when they provide a clear simplification. And we should pare and unite existing structures whenever possible.
The standard model and general relativity are the best mathematical descriptions we have of our universe. By considering these two theories and following our guiding principles, we will be led to a beautiful unification.
1.1 A connection with everything
The building blocks of the standard model and gravity are fields over a
four dimensional base manifold. The electroweak and strong gauge fields
are described by Lie algebra valued connection 1-forms,
four dimensional base manifold. The electroweak and strong gauge fields
are described by Lie algebra valued connection 1-forms,
while the gravitational fields are described by the spin connection,
a Clifford bivector valued 1-form, and the frame
a Clifford vector valued 1-form.
The frame may be combined with a multiplet of Higgs scalar fields, φ, to interact with the electroweak gauge fields and fermions to give them masses. The fermions are represented as Grass-mann valued spinor fields, { ν . e , e,
. u,. . . .}, with the spin connection and
. u,. . . .}, with the spin connection and
gauge fields acting on them in fundamental representations. The electroweak W acts on doublets of left chiral fermions, {[ ν . eL , e . L ], . . .}; the strong g acts on triplets of red, green, and blue colored quarks, {[ u . r , u . g , u . b ], . . .}; and the electroweak B
acts on all with an interesting pattern of hypercharges. The left and right chiral parts of the gravitational spin connection, ω, act on the frame and on the left and right chiral fermions.
This structure, depicted in Figure 1, is repeated over three generations of fermions with different masses. This diverse collection of fields in various algebras and representations is, inarguably, a mess.
It is difficult at first to believe they can be unified as aspects of a unique mathematical structure — but they can. The gauge fields are known to combine naturally as the connection
of a grand unified theory with a larger Lie group, and we continue with unification in this spirit. The spin connection, frame, and Higgs may be viewed as Lie algebra elements and included as parts of a “graviweak” connection.
Relying on the algebraic structure of the exceptional Lie groups, the fermions may also be recast as Lie algebra elements and included naturally as parts of a BRST extended connection.[2, 3] The result of this program is a single
principal bundle connection with everything,
principal bundle connection with everything,
In this connection the bosonic fields, such as the strong g = dx i g i A T A , are Lie algebra valued 1-forms, and the fermionic fields, such as u . = u . A T A , are Lie algebra valued Grassmann numbers.
(These Grassmann fields may be considered ghosts of former gauge fields, or accepted a priori as parts of this superconnection.) The dynamics are described by the curvature,
with interactions between particles given by their Lie bracket. For example, the interaction between two quarks and a gluon is specified by the Lie bracket between their generators, with a corresponding Feynman vertex,
It is a remarkable property of the exceptional Lie groups that some of their Lie brackets are equivalent to the action of a subgroup on vectors in fundamental representation spaces, just as they occur in the standard model.[4]
For example, the bracket between the gluons and a set of colored quarks in A . can give the su(3) action on the defining 3,
When all standard model particles and interactions are identified this way, the entire ensemble corresponds to a uniquely beautiful Lie group — the largest simple exceptional group, E8.
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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, T a ,
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Abstract: All fields of the standard model and gravity are unified as an E8 principal bundle connection. A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs,
and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E8 superconnection are described by the curvature and action over a four dimensional base manifold.
Keywords: ToE.
Keywords: ToE.
(R is the rank of the Lie algebra) Every element of the Cartan subalgebra, C = C a T a , 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 β , with each corresponding to an eigenvalue,
Each of the (N −R) non-zero eigenvalues, α β , (imaginary for real compact groups) is linearly dependent on the coefficients of C and corresponds to a point, a root, α aβ , 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 β , and Cartan subalgebra generators, T a , 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 β , 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
Table 1: The su(3) weight vectors and weight coordinates of the gluon, quark, and anti-quark weights form the G2 root system.
2.1 Strong G2
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The gluons, g ∈ su(3), in the special unitary group of degree three may be represented using the eight Gell-Mann matrices as generators,
@threadreaderapp unroll
The Cartan subalgebra, C = g 3 T 3 + g 8 T 8 , is identified 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 coefficient, such as g gb̄ , has the same label as the particle eigenvector containing the coefficient, 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 3̄, are diagonal, the weight vectors, V β and V̄ β , satisfying
are the canonical unit vectors of the 3 and 3̄. The weights for these — the su(3) quantum numbers of the quarks and anti-quarks — can be read off the diagonals of C and C̄ = −C T = −C.
The set of weights for su(3), the defining 3, and its dual 3̄, 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 3̄ are identified 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 3̄ quarks and anti-quarks, q . and q̄, 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 significant simplification of mathematical structure.
And we will see that this simplification 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, (7 × 7), 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 q III in Table 2, with their (x, y, z) coordinates shown. These points may be rotated and scaled,
Table 2: Weights of gluons, three series of quarks and anti-quarks, and leptons, in three dimensions, projecting down to the G2 root system in the last two coordinates.
so that dropping the first, B 2 , coordinate gives the projection to the roots in two dimensions. In general, we can find 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 q II , at the centers of the faces of the cube
in the figure of Table 1. The eight weights at the corners of a half-cube, labeled q I 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) B−L quantum number, B 2 , is the
baryon minus lepton number, related to their hypercharge. The su(3) action does not move fermions between the nine B 2 grades in the table — each remains in its series, I, II, or III. Since this su(3) and
u(1) B−L are commuting subalgebras, our grand unification of gauge fields follows the same path as the Pati-Salam SU (2) L × SU (2) R × SU (4) GUT.[5]
2.2 Graviweak 𝘍4
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The interactions between other gauge fields are more involved and separate from the strong gluons. Most importantly, the weak W acts only on left-chiral fermions, as determined by their gravitational so(3, 1) quantum numbers.
Also, the Higgs, φ, needs to be combined with the gravitational frame, e, to make a 1-form interacting correctly with the electroweak gauge fields and the fermions.
These interactions imply that the spin connection, which acts on the frame, and the electroweak gauge fields, which act on the Higgs, must be combined in a graviweak gauge group. The best candidate for this unification is so(7, 1), which breaks up as
and has the desired balance of gravity and left-right symmetric electroweak gauge fields acting on the frame-Higgs.
2.2.1 Gravitational 𝘋2
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For its action on spinors, gravity is best described using the spacetime Clifford algebra, Cl(3, 1) — a Lie algebra with a symmetric product. The four orthonormal Clifford vector generators,
are written here as (4 × 4) Dirac matrices in a chiral representation, built using the Kronecker product of Pauli matrices,
These may be used to write the gravitational frame as
with left and right chiral parts, e L/R = i(e 4 ±e ε σ ε ), and the coefficients,
The d2 = so(3, 1) = Cl 2 (3, 1) valued gravitational spin connection is written using the six Clifford bivector generators, γ μν = 12 [γ μ , γ ν ], as
with six real coefficients redefined into the spatial rotation and temporal boost parts,
These relate to the left and right-chiral (selfdual and anti-
selfdual) parts of the spin connection,
selfdual) parts of the spin connection,
which are sl(2, C) valued but not independent, ω τR = ω τ L ∗ . The Cartan subalgebra of gravity, in several different
coordinates, is
coordinates, is
Taking the Lie bracket with C gives root vectors and roots
for the spin connection, such as
for the spin connection, such as
for ω L ∧ , and weight vectors and weights for the frame, such as
for e ∧ . L , are in the 4 of the spinor T . The fermions, such as the left-chiral spin-up up quark, u representation space (2.8) with weight vectors, such as [1, 0, 0, 0], equal to the canonical unit vectors, and weights read off the diagonal of C.
The collection of fields and their weights are shown in Table 3. The two coordinate systems in the table are related by a π 4 rotation and scaling,
Table 3: Gravitational 𝘋2 weights for the spin connection,
frame, and fermions, in two coordinate systems.
frame, and fermions, in two coordinate systems.
Unlike other standard model roots, the roots of so(3, 1) are not all imaginary — the coordinates along the ω T 3 axis are real. The Spin + (3, 1) Lie group of gravity, with Lie algebra so(3, 1), is neither simple nor compact
— it is somorphic to SL(2, C) = SL(2, R) × SL(2, R). According to the ADE classification of Lie groups it is still labeled D2 — the same as Spin(4) = SU (2) × SU (2) — since it has the same root system, albeit with one real axis.
2.2.2 Electroweak 𝘋2
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