teoria de tudo

teoria de tudo

(Parte 2 de 5)

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, 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, I, or II. 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

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.

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, eL/R = i(e4±eεσε), and the coefficients,

with six real coefficients redefined into the spatial rotation

Table 3: Gravitational D2 weights for the spin connection, frame, and fermions, in two coordinate systems.

and temporal boost parts,

These relate to the left and right-chiral (selfdual and antiselfdual) parts of the spin connection, ωL/R = (ωτL/R)iστ = ωS ∓ iωT which are sl(2,C) valued but not independent, ωτR = ωτ∗L . The Cartan subalgebra of gravity, in several different coordinates, is

Taking the Lie bracket with C gives root vectors and roots for the spin connection, such as[

for ω∧L, and weight vectors and weights for the frame, such as[

for e∧T. The fermions, such as the left-chiral spin-up up quark, .u∧L, are in the 4 of the spinor 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 pi4 rotation andscaling, [

Unlike other standard model roots, the roots of so(3,1) are not all imaginary — the

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

The electroweak gauge field, W ∈ su(2)L, acts on left-chiral doublets, such as [.uL, .dL]. The Pati-Salam GUT introduces a partner to this field, B1 ∈ su(2)R, acting on all right-chiral fermion doublets. Part of this field, B31 i2σ3 ∈ u(1)R, joins with the u(1)B−L complement, B2, of the strong su(3) to give the electroweak B ∈ u(1)Y . The left-right electroweak partner fields may be joined in a d2 partner to gravity,

Since both W and B1 act on the Higgs doublet, [φ+,φ0], it is sensible to consider the 4 real fields of this Higgs doublet to be components of a vector acted on by the so(4). This suggests we proceed as we did for gravity, using a complementary chiral matrix representation for the four orthonormal basis vectors of Cl(4),

with coefficients equal to those of the Higgs doublet,

The d2 = so(4) = Cl2(4) valued electroweak connection breaks up into two su(2) parts,

The U and V fields are analogous to the 1iωT and ωS of gravity, and are related to the electroweak W and B1, analogous to the ωL and ωR, by the same pi4 rotation and scaling (2.9). The Cartan subalgebra,

gives root vectors and roots for the electroweak fields,

Table 4: Weights for electroweak D2, for B2 from Table 2, and electroweak hypercharge and charge.

such as W±, and weight vectors and weights for the

Higgs, such as[ for φ0. The fermions are acted on in the standard

4, equivalent to the independent su(2)L and su(2)R action on left and right-chiral Weyl doublets, such as

[uL,dL] and [uR,dR]. The electroweak D2 weights for various fields are shown in Table 4.

The two right-chiral gauge fields, B±1 , are not part of the standard model. They are a necessary part of the Pati-Salam GUT, and presumably have large masses or some other mechanism breaking left-right symmetry and impeding their detection. As in the

Pati-Salam GUT, the B2 weights from Table 2 and the B31 weights may be scaled and rotated ((2.4) and (2.9)) into two new coordinates, including the weak hypercharge,

This scaling implies a weak hypercharge coupling constant of g1 = √ 3/5 and Weinberg angle satisfying sin2 θW = 3/8, typical of almost all grand unified theories. There is also a new quantum number partner to the hypercharge, X, corresponding to the positive combination of quantum numbers B31 and B2. The hypercharge may be scaled and rotated with W3 to give the electric charge,

These weights, shown in Table 4, are in agreement with the known standard model quantum numbers, and justify our use of the corresponding particle labels.

2.2.3 Graviweak D4

The electroweak d2 = so(4) and gravitational d2 = so(3,1) combine as commuting parts of a graviweak d4 = so(7,1). The 4 Higgs fields, φ, a vector of the electroweak so(4), combine with the 4 gravitational so(3,1) vectors of the frame, e, into 16 bivector valued fields, eφ, of the graviweak D4 gauge group. This combination is achieved by adding the weights of Table 3 with those of Table 4 to obtain the weights of D4 in four dimensions, as shown in Table 5. The weights of the fermions also add to give their D4 weights.

The fermion weights correspond to the fundamental positive-chiral spinor representation space, 8S+, of D4. To construct this explicitly, we use Trayling’s model,[7] and combine our Cl(3,1) and Cl(4) basis generators into eight Clifford basis vector elements of Cl(7,1), represented as (16 × 16) matrices,

These allow us to build the spin connection, ω = 12ωµνΓµν,

Table 5: Graviweak D4 roots for 24 bosons and weights for 8S+ fermions.

the electroweak connection, wew = 12ωµνewΓ′µν, the frame, e = eµΓµ, and the Higgs, φ = φµΓ′µ, as Cl(7,1) valued fields, with the same coefficients as before. The frame and

Higgs multiply to give the frame-Higgs, eφ = eµφνΓµΓ′ν, a Clifford bivector valued 1-form. Together, these fields

Since our chosen Cl(7,1) representation is chiral, H1 may be represented by its positive-chiral part, the (8×8) first quadrant of the (16 × 16) rep, shown here acting on a νeL eL νeR eR

The fractions, 12 and 14, multiplying fields in H1 are necessary for fitting gravity and the electroweak connection together in D4, and for obtaining the correct dynamics from curvature.

The first 24 weights in Table 5 are the roots of D4. This particular root system has a uniquely beautiful set of symmetries called triality,[8] rotations of the root system by 2pi 3

that leave it invariant. A triality rotation matrix, T, can permute the coordinates of the rootsystem,

taking each root to its first triality partner, then to its second, and back — satisfying T3 = 1. As an example, the above triality rotation gives

showing the equivalence of these roots under this triality rotation. Six of the roots,

are their own triality partners — they lie in the plane orthogonal to this triality rotation.

The last 8 weights of Table 5, representing one generation of leptons as 8S+, are rotated by triality into the other fundamental representation spaces of D4: the negative-chiral spinor and the vector,

These two new sets of weights are equivalent to the 8S+ under this triality rotation — they carry the same quantum numbers and have the same interactions with the triality rotated roots of D4. Given this relationship, we tentatively consider these three triality partners of

8S+ as the three generations of fermions, such as

The complete set of weights, D4 + (8S++ 8S−+ 8V ), including these new triality partners, is the root system of the rank four simple exceptional group, F4.

The 48 roots of F4 are shown in Tables 5 and 6. These roots, in four dimensions, are the vertices of the 24-cell polytope and its dual. Using the breakdown of F4 into D4 and the three triality-equivalent fundamental representation spaces, the graviweak bosons (2.10) and three generations of leptons (or quarks) may be written as parts of a F4 connection,

(Parte 2 de 5)

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