**Manual de Latex em Inglês**

Manual de Latex em Inglês

(Parte **3** de 5)

Although we are labeling triality partners as fermions of different generations, the exact relationship between triality and generations is more complicated and not yet clear to the author. One clue is that the triality partners of F4 (connected in the figure by pale blue and thin gray lines) may be collapsed to their midpoints to get a g2 subalgebra,

This triality collapse might relate to a description of graviweak interactions with a group smaller than F4.[9, 10] It also suggests physical fermions may be linear combinations of triality parners, such as

Table 6: The 8S− and 8V weights complete the D4 and 8S+ weight system of Table 5 to form the F4 root system. The 48 roots are projected from four dimensions to two and plotted, with lines shown between triality partners.

Guided by this triality symmetry, we will continue to label the triality partners with generation labels — though this should be understood as an idealization of a more complex and as yet unclear relationship between physical particles and triality partners.

If we wished, we could write the constituent particles of F4 as matrix elements of its smallest irreducible, 26 dimensional representation,[1] as we did for the gluons and quarks (2.3) in G2. We can also compute particle interactions by adding the roots in Tables 5 and 6, such as

between an electron, a frame-Higgs, and an electron neutrino. These graviweak interactions, described by the structure of F4, do not involve anti-fermions or color; to include all standard model interactions we will have to combine F4 and G2.

The coordinate axes chosen in Tables 5 and 6 are a good choice for expressing the quantum numbers for gravity and the electroweak fields, but they are not the standard axes for describing the F4 root system. We can rotate to our other coordinate system, using a pair of pi4 rotations (2.9) and thereby express the 48 roots of F4 in standard coordinates, shown in Table 7. These coordinate values are described by various permutations of

NMNMNM qII −1 all 3 HOHOHO qII 1 all 3 qIII 1 1 all 3

Table 7: Roots of F4 and weights of Table 2 described with allowed permutations of coordinate values.

To completely describe every field in the standard model and gravity we need to combine these two sets of quantum numbers. The graviweak F4 root system includes the two quantum numbers of so(3,1) gravity and the two of the su(2)L and su(2)R electroweak fields, with three generations of fermions related through so(7,1) triality (2.1). The G2 weight system includes the three quantum numbers of the su(3) strong fields and a u(1)B−L contributing to hypercharge, with fermions and anti-fermions related through duality (2.6). To match the quantum numbers of all known standard model and gravitational fields, the so(7,1) of F4 and su(3) and u(1)B−L of G2 + U(1) must act on three generations of 8 fermions for each of the 3 colors of quark, 1 uncolored lepton, and their anti-particles, as depicted in the periodic table, Figure 1. The weights of these 2 elements — corresponding to the quantum numbers of all gravitational and standard model fields — exactly match 2 roots out of the 240 of the largest simple exceptional Lie group, E8.

16S+ ±1/2 | even# > 0 128 |

Table 8: The 240 roots of E8.

“E8 is perhaps the most beautiful structure in all of mathematics, but it’s very complex.” — Hermann Nicolai

Just as we joined the weights of D2G and D2ew to form the F4 graviweak root system, the weights of F4 and G2 may be joined to form the roots of E8 — the vertices of the E8 polytope

— shown in Table 8. Combining these weights in eight dimensions requires the introduction of a new quantum number, w, with values determined by the F4 and G2 numbers. These quantum numbers uniquely identify each root of E8 as an elementary particle — Table 9.

NMNMNMNM νeL eL νeR eR ±1/2 | even#>0 −1/2 −1/2 −1/2 −1/2 8S+ l 8 |

HOHOHOHO νeL eL νeR eR ±1/2 | even#>0 1/2 1/2 1/2 1/2 8S+ l 8 |

NMNMNMNMNMNMNMNMNMNMNMNM uL dL uR dR ±1/2 | even#>0 −1/2 ±1/2... two>0 8S+ qI 24 |

HOHOHOHOHOHOHOHOHOHOHOHO uL dL uR dR ±1/2 | even#>0 1/2 ±1/2... one>0 8S+ qI 24 |

NMNMNMNM νµL µL νµR µR ±1/2 | odd#>0 −1/2 1/2 1/2 1/2 8S− l 8 |

HOHOHOHO νµL µL νµR µR ±1/2 | odd#>0 1/2 −1/2 −1/2 −1/2 8S− l 8 |

NMNMNMNMNMNMNMNMNMNMNMNM cL sL cR sR ±1/2 | odd#>0 1/2 ±1/2 ... two>0 8S− qI 24 |

HOHOHOHOHOHOHOHOHOHOHOHO cL sL cR sR ±1/2 | odd#>0 −1/2 ±1/2 ... one>0 8S− qI 24 |

NMNMNMNMNMNMNMNMNMNMNMNM tL bL tR bR ±1 0 −1 8V qII 24

Table 9: The 240 roots of E8 assigned elementary particle labels according to F4 and G2 subgroups.

The E8 root system was first described as a polytope by Thorold Gosset in 1900,[12] and the triacontagonal projection plotted by hand in 1964. This plot,[13] now with elementary particle symbols assigned to their associated roots according to Table 9, is shown in Figure 2, with lines drawn between triality partners.

Figure 2: The E8 root system, with each root assigned to an elementary particle field. – 17 –

The interactions between all standard model and gravitational fields correspond to the

Lie brackets between elements of the E8 Lie algebra, and thus to the addition of E8 roots. The Lie algebra breaks into the standard model (2.12) as

The 26 is the the traceless exceptional Jordan algebra — the smallest irreducible representation space of F4 — and the 7 is the smallest irreducible representation space of G2. Each 8

not in the standard model,

The new xΦ field carries weak hypercharge and color, has three generations, and couples leptons to quarks. This breakdown of E8 is possible because F4 is the centralizer of G2 in E8,

To display this subalgebra structure, the E8 root system may be rotated in eight dimensions, projected to two, and plotted, as shown in Figures 3 and 4.1 In these plots, the root coordinates have been transformed by a rotation,

V 3 w x y z equivalent to the redefinition of the Cartan subalgebra generators according to (2.4) and (2.9). Since the spaces containing the F4 and G2 root systems are orthogonal in E8, these plots of E8 showing a rotation between the two are especially pretty and convenient for identifying interactions between particles. Also, the central cluster of 72 roots in Figure 4 is the E6 root system, which acts on each of the three colored and anti-colored 27 element clusters of the exceptional Jordan algebra.

1An animation of this rotation is available at http://deferentialgeometry.org/anim/e8rotation.mov

Figure 3: The E8 root system, rotated a little from F4 towards G2. – 19 –

Inspecting the e6 subalgebra of e8 reveals how the fermions and anti-fermions — up to now described as living in real representations — are combined in complex representations.[14] The e6 decomposes to graviweak so(7,1) acting on three complex generations of fermions as

in which the final u(1) is the complex structure, i, related to the w quantum number, and the 16SC is a complex spinor acted on by the so(9,1). Although considering its e6, f4, and g2 subalgebras is useful, the E8 Lie algebra may be broken down to the standard model via a more direct route,[8]

This decomposition is directly visible

Figure 5: A periodic table of E8.

in Table 9, in which the first four coordinate axes are of so(7,1) and the last four are of so(8). The so(7,1) decomposes into the graviweak fields, and the so(8) decomposes into strong su(3), u(1)B−L, and new fields via the embedding of su(4) in so(8). A matched triality rotation of so(7,1) and so(8) relates the three generations of fermions.

The Lie algebra structure of E8, and its relation to the structure of the standard model, is depicted in Figure 5 — a periodic table of E8. A comparison of this structure with Figure 1 shows the extremely close fit to the standard model, with only a handful of new particles suggested by the structure of E8.

2.4.1 New particles

After all algebraic elements of the standard model have been fit to the E8 Lie algebra there are a few e8 elements remaining, representing new, non-standard particles. There are two new quantum numbers, X and w, representing the Pati-Salam partner to weak hypercharge and a new quantum number related to generations. Each of these corresponds to new u(1) valued fields, X and w, which presumably have large masses impeding their measurement.

The use of the Pati-Salam model also implies a non-standard pair of fields, B±1 , interacting with right-chiral fermions. In addition, there is a new field, xΦ, interacting with leptons and quarks. This field factors into three generations, x1/2/3, corresponding to different w quantum numbers, and a new Higgs scalar, Φ, for each color and anti-color. The new field, xΦ, is a joining of x and Φ in the same way eφ is a joining of the gravitational frame, e, and the Higgs, φ. Since the frame-Higgs is a composite field — a simple bivector — its degrees of freedom do not exhaust the algebraic sector it inhabits. Specifically, eφ = eµφνΓµΓ′ν uses 16 algebraic elements but, because it is simple, has only 4 (for e = eµΓµ) plus 4 (for φ = φνΓ′ν) equals 8 algebraic field degrees of freedom. How or why these 16 algebraic elements are restricted is not understood — but this restriction is necessary to recover the standard model and gravity. Because the 18 algebraic degrees of freedom inhabited by xΦ appear amenable to the same sort of factorization as eφ (see Table 9), it is natural to factor it into three x fields and three colored and three anti-colored Higgs fields, Φ. It could be possible that this new xΦ gives different masses to the different generations of quarks and leptons, producing the CKM and PMNS matrices. Also, since it mixes leptons and quarks, the existence of this field predicts proton decay, as does any grand unified theory.

The interactions between the new fields, w and xΦ, are analogous to the interactions between the gravitational spin connection and the frame-Higgs, ω and eφ. This suggests that a better understanding of the triality relationship between generations will involve how these two sets of fields may be more intimately related.

This is a somewhat arbitrary choice, selected for leaving W3 and color invariant. Once the first generation of fermions, with correct charges and spins, are assigned to elements of e8, this T rotates them to the second and third generations. The second and third generations only have the correct spins and charges when considered as equivalent under this T. When considered as independent fields with E8 quantum numbers, irrespective of this triality relationship, the second and third generation of fields do not have correct charges and spins. The W3 and color charges are invariant under our choice of T but the spins and hypercharges are only correct through triality equivalence. This relationship between fermion generations and triality is the least understood aspect of this theory.

It is conceivable that there is a more complicated way of assigning three generations of fermions to the E8 roots to get standard model quantum numbers for all three generations without triality equivalence. There is such an assignment known to the author that gives the correct hypercharges for all three generations, but it is not a triality rotation and it produces unusual spins. A correct description of the relationship between triality and generations, if it exists, awaits a better understanding.

3. Dynamics

The dynamics of a connection is specified by the action functional, S[ .A]. Classically, extrem- izing this action, constrained by boundary data, determines the value of the connection, .A(x), over a region of the base manifold. The value of the connection may also be used to infer topological properties of the base manifold. Quantum mechanically, the action of a connection over the base manifold determines the probability of experiencing that connection.[15] Since quantum mechanics is fundamental to our universe, it may be more direct to describe a set of quantum connections as a spin foam, with states described as a spin network. Under more conventional circumstances, the extensive methods of quantum field theory for a non-abelian gauge field may be employed, with propagators and interactions determined by the action. In any case, the dynamics depends on the action, and the action depends on the curvature of the connection.

3.1 Curvature The connection with everything, an e8 valued collection of 1-forms and Grassmann fields,

The curvature of this connection, an e8 valued collection of 2-forms and Grassmann 1-forms,

may be computed and broken up into standard model parts. The so(7,1) part of the curvature,

Few includes the gravitational so(3,1) part, the mixed graviweak 4 × (2 + 2) part, and the electroweak su(2)L + su(2)R part. The gravitational so(3,1) part of the curvature is in which = R is the Riemann curvature 2-form, e is the spacetime area bivector, and φ2 is the amplitude of the Higgs squared. The mixed graviweak 4 × (2 + 2) part is

Tφ − eDφ (3.4) in which = T is the gravitational torsion and D is the covariant derivative. And the electroweak su(2)L + su(2)R part of the curvature is

The so(8) part of the curvature,

Fg + xΦxΦ (3.6)

The last term does not easily separate — xΦxΦ contributes to all three parts of = F2. The u(1) and u(1)B−L parts are

And the strong su(3) part is

(Parte **3** de 5)