teoria de tudo

teoria de tudo

(Parte 4 de 5)

Due to the exceptional structure of e8, the fermionic part of the curvature for the first generation is with D the covariant massive Dirac derivative in curved spacetime. The second and third fermionic generation parts of this curvature are similar.

3.2 Action

The most conservative approach to specifying the dynamics is to write down an action agreeing with the known standard model and gravitational action while satisfying our desire for minimalism. With these two motivations in mind, an action for everything can be economically expressed as a modified BF theory action over a four dimensional base manifold,

B is an e8 valued collection of 2-form and anti-

Grassmann 3-form Lagrange multiplier fields, =

B′ is the rest of the scalar part (the trace).

After varying = B and plugging it back in (3.7), this action — up to a boundary term — is

in which ∼e is the spacetime volume 4-form, R is the gravitational scalar curvature, and = F′ is the non-so(3,1) part of =F1 and =F2. This is recognizable as the action for the standard model and gravity, with a cosmological constant related to the Higgs vacuum expectation value,

The details of the action, and its agreement with the standard model and general relativity, can be worked out for each sector of the E8 Lie algebra.

3.2.1 Gravity

The modified BF action for gravity was discovered by MacDowell and Mansouri in 1977, [16] and revived by Smolin, Starodubtsev, and Freidel during their work on loop quantum gravity.[17, 18] The remarkable and surprising fact that gravity, described by the spin connection, ω, and frame, e, can be described purely in terms of a unified connection, ω +e, was the seed idea that led to the unification of all fields in a single connection.[2] The gravitational part of the action (3.7) is

FG + piG in which the gravitational part of the curvature (3.3) is

Extremizing the action under variation of the gravitational part of the Lagrange multiplier,

and plugging this back into the action gives

Multiplying this out gives three terms. The term quadratic in the Riemann curvature is the Chern-Simons boundary term,

Dropping this, the other two terms give the Palatini action for gravity,

equal to the Einstein-Hilbert action with cosmological constant, Λ = 34φ2. The magnitude of the Higgs, √ φ2, is a conformal factor that can be absorbed into the magnitude of the frame. The vacuum solution to Einstein’s equation with positive cosmological constant is de

Sitter spacetime (= R = Λ6ee and R = 4Λ), which should be considered the background vacuum spacetime for particle interactions in this theory. Since the symmetry of this spacetime is so(4,1) and not the Poincare group, the Coleman-Mandula theorem does not apply to restrict the unification of gravity within the larger group.

It should be emphasized that the connection (3.1) comprises all fields over the four dimensional base manifold. There are no other fields required to match the fields of the standard model and gravity. The gravitational metric and connection have been supplanted by the frame and spin connection parts of .A. The Riemannian geometry of general relativity has been subsumed by principal bundle geometry — a significant mathematical unification.

Devotees of geometry should not despair at this development, as principal bundle geometry is even more natural than Riemannian geometry. A principal bundle with connection can be described purely in terms of a mapping between tangent vector fields (diffeomorphisms) on a manifold, without the ab initio introduction of a metric.

3.2.2 Other bosons The part of the action (3.8) for the bosonic, non-so(3,1) parts of the connection is

in which the relevant parts of the curvature (3.2) are the mixed graviweak part (3.4), the electroweak part (3.5), and the so(8) part (3.6). The mixed graviweak part of the action is

which includes the kinetic Higgs term and gravitational torsion. The electroweak part of the action is

And the so(8) part of the action is

Fg +xΦxΦ)∗xΦxΦ〉 which includes the action for the gluons and a first guess at the action for the new fields. This action for the new fields is speculative at this stage and likely to change as our understanding of their role improves.

The use of the Hodge dual in this part of the action is required for general covariance but seems somewhat awkward from the viewpoint of this E8 theory. The Hodge star operator requires the frame part, e, to be extracted from the E8 connection, inverted to obtain the coframe, ⇀e, and contracted with the curvature. It would be better if there was a natural justification for this procedure, beyond the necessity to agree with known theory. An improved understanding will likely lead to a modification of this part of the action.

Choosing the anti-Grassmann Lagrange multiplier 3-form to be ≡· B = ∼e ·Ψ⇀e in the fermionic part of the action (3.8) gives the massive Dirac action in curved spacetime,

The coframe, ⇀e, in this action contracts with the frame part of the graviweak connection, to give the standard Higgs coupling term, ·

Ψφ .Ψ. The new, non-standard · Ψγµ .Ψwµ and

Ψγµ .ΨxµΦ terms are not yet well understood but seem promising for recovering the CKM matrix.

This action works very well for one generation of fermions. The action for the other two generations should be similar, but is related by triality in a way that is not presently understood well enough to write down.

4. Summary

The “E8 theory” proposed in this work is an exceptionally simple unification of the standard model and gravity. All known fields are parts of an E8 principal bundle connection, in agreement with the Pati-Salam SU(2)L × SU(2)R × SU(4) grand unified theory, with a handfull of new fields suggested by the structure of E8. The interactions are described by the curvature of this connection, with particle quantum numbers corresponding to the vertices of the E8 polytope in eight dimensions. This structure suggests three fermionic generations related by triality,

T e = µ T µ = τ T τ = e

The action for everything, chosen by hand to be in agreement with the standard model, is concisely expressed as a modified BF theory action,

with gravity included via the MacDowell-Mansouri technique. The theory has no free parameters. The coupling constants are unified at high energy, and the cosmological constant and masses arise from the vacuum expectation values of the various Higgs fields,

In sum, everything is described by the pure geometry of an E8 principal bundle, perhaps the most beautiful structure in mathematics.

5. Discussion and Conclusion

There are a remarkable number of “coincidences” that work exactly right to allow all known fields to be unified as parts of one connection. The factors of 12 and 14 multiplying the spin connection and frame-Higgs result in the correct expressions for the gravitational Riemann curvature and the covariant Dirac derivative in curved spacetime. The fermions fit together perfectly in chiral representations under graviweak so(7,1), and the frame-Higgs has all the correct interactions. This frame-Higgs naturally gets a φ4 potential and produces a positive cosmological constant. Finally, and most impressively, the fit of all fields of the standard model and gravity to E8 is very tight. The structure of E8 determines exactly the spinor multiplet structure of the known fermions.

There are also aspects of this theory that are poorly understood. The relationship between fermion generations and triality is suggested by the structure of E8 but is not perfectly clear — a better description may follow from an improved understanding of the new w + xΦ fields and their relation to ω + eφ. This relationship may also shed light on how and why nature has chosen a non-compact form, E IX, of E8. Currently, the symmetry breaking and action for the theory are chosen by hand to match the standard model — this needs a mathematical justification.

Quantum E8 theory follows the methods of quantum field theory and loop quantum gravity — though the details await future work. One enticing possibility is that the gravitational and cosmological constants run from large values at an ultraviolet fixed point to the tiny values we encounter at low energies.[19, 20] At the foundational level, a quantum description of the standard model in E8 may be compatible with a spin foam description in terms of braided ribbon networks[21] through the identification of the corresponding finite groups. And there is a more speculative possibility: if the universe is described by an exceptional mathematical structure, this suggests quantum E8 theory may relate to an exceptional Kac-Moody algebra.[2]

The theory proposed in this paper represents a comprehensive unification program, describing all fields of the standard model and gravity as parts of a uniquely beautiful mathematical structure. The principal bundle connection and its curvature describe how the E8 manifold twists and turns over spacetime, reproducing all known fields and dynamics through pure geometry. Some aspects of this theory are not yet completely understood, and until they are it should be treated with appropriate skepticism. However, the current match to the standard model and gravity is very good. Future work will either strengthen the correlation to known physics and produce successful predictions for the LHC, or the theory will encounter a fatal contradiction with nature. The lack of extraneous structures and free parameters ensures testable predictions, so it will either succeed or fail spectacularly. If E8 theory is fully successful as a theory of everything, our universe is an exceptionally beautiful shape.

Acknowledgments

The author wishes to thank Peter Woit, Sergei Winitzki, Lee Smolin, Tony Smith, David Richter, Fabrizio Nesti, Sabine Hossenfelder, Laurent Freidel, David Finkelstein, Michael Edwards, James Bjorken, Sundance Bilson-Thompson, John Baez, and Stephon Alexander for valuable discussions and encouragement. Some of the work was carried out under the wonderful hospitality of the Perimeter Institute for Theoretical Physics. This research was supported by grant RFP1-06-07 from The Foundational Questions Institute (fqxi.org).

References

[1] M. Tegmark, “Is ‘the theory of everything’ merely the ultimate ensemble theory?” gr-qc/9704009.

[2] A.G. Lisi, “Clifford bundle formulation of BF gravity generalized to the standard model,” gr-qc/0511120.

[3] J.W. van Holten, “Aspects of BRST Quantization,” hep-th/0201124.

[4] J.F. Adams, Lectures On Exceptional Lie Groups, University of Chicago (1996).

[5] J.C. Pati and A. Salam, “Lepton number as the fourth color,” Phys. Rev. D 10 (1974) 275.

(Parte 4 de 5)

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