This page has 7 sections:
Conformal McDowell-Mansouri Gravity
Gravity in the D4-D5-E6 model begins with the Clifford algebra Cl(0,6) = R(8) with spin group Spin(0,6) = SU(4) = compact conformal group of 4-dimensional spacetime. The non-compact conformal group Spin(4,2) = SU(2,2) has the same Clifford algebra R(8) = Cl(4,2), but, to simplify the discussion, the compact group Spin(0,6) will be used in this page.
The conformal group Spin(0,6) = SU(4) is 15-dimensional, with a 10-dimensional subgroup Spin(0,5) = Sp(2) that is the de Sitter group.
The 15 infinitesimal generators of the conformal group are the 10 Poincare group generators of the de Sitter group plus one scale generator and 4 conformal generators.
Mohapatra (in section 14.6 of Unification and Supersymmetry, 2nd edition, Springer-Verlag 1992) shows that if the scale and conformal gauge degrees of freedom are fixed, then a Lagrangian with the conformal group as gauge group gives the usual Hilbert action for gravity.
After the scale and conformal gauges have been fixed, the conformal Lagrangian becomes a de Sitter Lagrangian. Einstein-Hilbert gravity can be derived from the de Sitter Lagrangian, as was first shown by MacDowell and Mansouri (Phys. Rev. Lett. 38 (1977) 739). For discussion of the MacDowell-Mansouri mechanism, see Freund (chapter 21 of Introduction to Supersymmetry, Cambridge 1986), or Ne'eman and Regge (Riv. Nuovo Cim. v. 1, n. 5 (1978) 1, at pages 25-28), or Nieto, Obregon, and Socorro.
What is the physical reason for fixing the scale and conformal degrees of freedom?
In the D4-D5-E6 model, the answer to that question comes from the answer to another question:
Since all rest mass comes from the Higgs mechanism, and since rest mass interacts through gravity, what is the relationship between gravity and the Higgs mechanism?
As remarked by Sardanashvily, Heisenberg and Ivanenko in the 1960s made the first atttempt to connect gravity with a symmetry breaking mechanism by proposing that the graviton might be a Goldstone boson resulting from breaking Lorentz symmetry in going from flat MInkowski spacetime to curved spacetime.
Sardanashvily (see also gr-qc/9405013, gr-qc/9407032, and gr-qc/9411013)
proposes that gravity be represented by a gauge theory with group GL(4), that GL(4) symmetry can be broken to either Lorentz SO(3,1) symmetry or SO(4) symmetry, and that the resulting Higgs fields can be interpreted as either the gravitational field (for breaking to SO(3,1) or the Riemannian metric (for breaking to SO(4). The identification of a pseudo-Riemannian metric with a Higgs field was made by Trautman (Czechoslovac Journal of Physics, B29 (1979) 107), by Sardanashvily (Phys. Lett. 75A (1980) 257), and by Ivanenko and Sardanashvily (Phys. Rep. 94 (1983) 1).
In the D4-D5-E6 model (using here the compact version) the conformal group Spin(0,6) = SU(4) is broken to the de Sitter group Spin(0,5) = Sp(2) by fixing the 1 scale and 4 conformal gauge degrees of freedom.
The resulting Higgs field is interpreted in the D4-D5-E6 model as the same Higgs field that gives mass to the SU(2) weak bosons and to the Dirac fermions by the Higgs mechanism.
The Higgs mechanism requires "spontaneous symmetry breaking" of a scalar field potential whose minima are not zero, but which form a 3-sphere SU(2). In particular, one real component of the complex Higgs scalar doublet is set to v / sqrt(2), where v is the modulus of the 3-sphere of minima, usually called the vacuum expectation value.
If the 3-sphere is taken to be the unit quaternions, then the "spontaneous symmetry breaking" requires choosing a (positive) real axis for the quaternion space.
In the standard model, it is assumed that a random vacuum fluctuation breaks the SU(2) symmetry and in effect chooses a real axis at random.
In the D4-D5-E6 model, the symmetry breaking from conformal Spin(0,6) to de Sitter Spin(0,5) by fixing the 1 scale and 4 conformal gauge degrees of freedom is a symmetry breaking mechanism that does not require perturbation by a random vacuum fluctuation.
Gauge-fixing the 1 scale degree of freedom fixes a length scale. It can be chosen to be the magnitude of the vacuum expectation value, or radius of the SU(2) 3-sphere.
Gauge-fixing the 4 conformal degrees of freedom fixes the (positive) real axis of the SU(2) 3-sphere consistently throughout 4-dimensional spacetime.
Therefore, the D4-D5-E6 model Higgs field comes from the breaking of Spin(0,6) conformal symmetry to Spin(0,5) de Sitter gauge symmetry, from which Einstein-Hilbert gravity can be constructed by the MacDowell-Mansouri mechanism.
The Feynman spin-2 flat spacetime construction of Einstein-Hilbert gravity allows the D4-D5-E6 model to be based on a fundamental D4 lattice 4-dimensional spacetime.
Donoghue has shown a way to formulate gravity as an effective field theory at low energies. He has also written a shorter survey article.
Renormalizable quantum theories are well known for the three forces of the Standard Model (electromagnetism, the weak force, and the color force), and they are part of the D4-D5-E6 model.
The standard model quantum path integral sum over histories breaks the gauge group invariance of the Lagrangian, because the path integral must not overcount paths by including more than one representative of each gauge-equivalence class of paths. The remaining quantum symmetry is the symmetry of BRST cohomology classes. Knowledge of the BRST symmetry tells you which ghosts must be used in quantum calculations, so the BRST cohomology can be taken to be the basis for the quantum theory.
A good description of BRST cohomology is in the paper of Garcia-Compean, Lopez-Romero, Rodriguez-Segura, and Socolovsky. As they state, the only force for which a renormalizable quantum theory is not well known is gravity.
They discuss two current approaches to quantum gravity:
string theory, which abandons point particles even at the classical level; and
redefinition of classical general relativity in terms of new variables, the Ashtekar variables, and trying to use the new variables to construct a quantum theory of gravity.
Nieto, Obregon, and Socorro have shown that the MacDowell-Mansouri Spin(0,5) = Sp(2) de SitterLagrangian for gravity used in the D4-D5-E6 model is
equal to the Lagrangian for gravity in terms of the Ashtekar variables plus
a cosmological constant term - which may decay to zero during inflation,
an Euler topological term - which for 4-dim spacetime is a 4-form that is proportional to the square root of the determinant of the 4x4 matrix representing the curvature 2-form (see sec. 11.4 of Nakahara, Geometry, Toplolgy, and Physics, Adam Hilger 1990), and
a Pontrjagin topological term - which for 4-dim spacetime is proportional to the trace of the square of the 4x4 matrix representing the curvature 2-form (see sec. 11.4 of Nakahara, Geometry, Toplolgy, and Physics, Adam Hilger 1990) and which may correspond to black holes that decay during inflation to produce particles and residual Planck-mass black holes. Medina and Nieto have shown that the Pontrjagin term may be related to Chern-Simons theory for gauge group Spin(2,3), which in turn may be related to conformal field theory in 1+1 dimensions. Smolin has surveyed the Crane ladder of dimensions going from 3+1 dimensions to 2+1 (Chern-Simons topological theory with massive gravitons) to 1+1 (WZW conformal field theory). Smolin says that the ladder is related to the holographic hypothesis of 't Hooft and Susskind that a quantum field theory on the interior of a black hole is best described by a quantum field theory on its boundary. By using the Crane ladder, you can study black holes with the methods of topological quantum field theory, knot theory, and category theory that are described by Baez in his series This Week's Finds in Mathematical Physics as well as in his books and papers.
Therefore, although the quantum gravity methods of string theory cannot be used in the D4-D5-E6 model because the D4-D5-E6 model uses fundamental point particles at the classical level, methods based on Ashtekar variables are available. Two such approaches are:
a topological approach based on loop groups and spin networks; and
an algebraic approach based on getting BRST transformations from Maurer-Cartan horizontality conditions.
Spin Networks are defined by Baez as "...graphs embedded in a manifold S (the space of spacetime) with edges labelled by representations of a Lie group G and with vertices labelled by intertwining operators. Spin networks define gauge-invariant functions on the space A of connections of any G-bundle over S...".
The intertwining operators are related to the fermion creation and annihilation operators in the 3x3 Octonion Nilpotent Heisenberg Algebra Matrix Model.
The configuration space of gravity using the Ashtekar variables is not the space of metrics, but the space of connections on an SL(2,C) bundle (with compact real form SU(2) = Spin(3)) over S.
The constraints of canonical quantum gravity are then polynomial, and the diffeomorphism constraint is invariance under diffeomorphisms of S. Since loops are invariant under diffeomorphisms of S, and since Wilson loops can represent gauge theories, Rovelli and Smolin proposed a loop representation of quantum gravity.
John Baez, in week 55 of This Week's Finds in Mathematical Physics, discusses work of Rovelli and Smolin and of Loll in which Rovelli and Smolin have shown that the loop representation of quantum gravity is equivalent to an SU(2) Spin Network representation and
Loll has calculated that spatial volume eigenstates of the Spin Network space of states are discrete and nonzero, on the scale of the Planck length, for Spin Networks with at least 4 edges at each vertex.
(Baez says that Rovelli and Smolin made a sign error that led them, in an earlier paper, to conclude that trivalent Spin Networks could have nonzero spatial volume, and that Loll showed at a 1995 Warsaw workshop that trivalent Spin Networks had zero spatial volume, but 4-valent Spin Networks had nonzero Planck-scale spatial volume.)
This leads to the 4-dimensional HyperDiamond Feynman Checkerboard based on the 4-dimensional lattice structure of Michael Gibbs and David Finkelstein.
Gibbs views the 4-dimensional HyperDiamond Feynman Checkerboard as being a lattice in a 4-dimensional spacetime, while Finkelstein developed the Quantum Graphs used in construction of the lattice as a theory of abstract Quantum Graphs not embedded in spacetime, but from which spacetime should be derived.
The D4-D5-E6 model has the discrete structure of the Finkelstein-Gibbs HyperDiamond lattice, and has a gravity sector based on the conformal MacDowell-Mansouri mechanism, which has been shown to be closely related to to the Ashtekar variable picture by Nieto, Obregon, and Socorro.
Baez says that Spin Networks "...were invented in the early 1970s by Penrose ...", and that while the Baez-Rovelli-Smolin-Loll Spin Networks "...involve graphs embedded in a pre-existing manifold that represents space, his [Penrose's] spin networks were abstract graphs ... intended as a purely combinatorial substitute for a spacetime manifold."
Since the Finkelstein-Gibbs 4-dimensional HyperDiamond lattice has spatial structure of the 3-dimensional Diamond lattice, whose natural Spin Network has 4-valent tetrahedral structure, Loll's results support the Finkelstein-Gibbs HyperDiamond Spin Network and the D4-D5-E6 model.
As Michael Gibbs says:
a 3-valent Spin Network is fundamentally 2-dim like graphite;
a 4-valent Spin Network is fundamentally 3-dim like diamond.
Garcia-Compean, Lopez-Romero, Rodriguez-Segura, and Socolovsky review the mathematical structure of BRST symmetry.
H. S. Yang and B. H. Lee describe the relationship between the cohomology of the compact Lie algebra of a Lagrangian gauge theory and the BRST cohomology.
Blaga, Moritsch, Schweda, Sommer, Tataru, and Zerrouki describe the BRST cohomology for gravity using the Ashtekar variables.
Nieto, Obregon, and Socorro have shown that Lagrangian action of the Ashtekar variables is a Chern-Simons action if the Killing metric of the de Sitter group is used instead of the Levi-Civita tensor.
Smolin and Soo have shown that the Chern-Simons invariant of the Ashtekar-Sen connection is a natural candidate for the internal time coordinate for classical and quantum cosmology, so that the D4-D5-E6 model uses Chern-Simons time.
The D4-D5-E6 model has
a quantum theory of the standard model forces (electromagnetism, the weak force, and the color force), and
a quantum theory of MacDowell-Mansouri-Ashtekar gravity.
The two quantum theories must be combined in order to calculate how standard model particles and fields interact in the presence of gravity.
Moritsch, Schweda, Sommer, Tataru, and Zerrouki have done this by using Maurer-Cartan horizontality conditions to get BRST transformations for Yang-Mills gauge fields in the presence of gravity.
This gives a complete quantum structure for the D4-D5-E6 model.
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