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4.1. The F4 Model Ansatz and the MacDowell-Mansouri Mechanism.

 The F4 model of physics is an 8-dimensional theory that breaks down to a 4-dimensional theory at the Planck mass. At the Planck mass, the F4 model gravity is a 4-dimensional Yang-Mills theory with gauge group Spin(5). It has an action curvature term of the form

- Rab /\ *Rab

(where a,b = 1,2,3,4,5), is renormalizable (see Collins18), and has 10 gauge bosons, or Spin(5) gravitons, corresponding to the 10 infinitesimal generators of Spin(5)= Sp(2). 8 of the 10 gauge bosons carry charges.

6 of them carry color charge, and may therefore be considered as corresponding to quarks. 2 carry no color charge, but can carry electric charge and so may be considered as corresponding to leptons. The remaining 2 of the 10 are neutral and correspond to the Cartan subalgebra of Spin(5). It does not look at all like gravity.

However, with the MacDowell-Mansouri mechanism5,6,7, gravity can be derived from an action curvature term of the form

Rab /\ Rcd eabcd with gauge group Spin(5).

The key ansatz in the F4 model that produces gravity from the Spin(5) Yang-Mills action Rab /\ *Rab is:

the Hodge operator * acts on the Spin(4) = S3 x S3 subgroup

of Spin(5) by interchanging the self-dual S3's.

 

To go from the Spin(5) Yang-Mills action - Rab /\ *Rab to the Spin(5) MacDowell-Mansouri action Rab /\ Rcd eabcd, use the F4 model ansatz that the Spin(5) Rab transform under the spacetime duality * operation as follows:

R12 -> R34 ; R13 -> R24 ; R14 -> -R23 ;

R23 -> -R14 ; R24 -> R13 ; R34 -> R12

for the 6 of the Rab that correspond to the Spin(4) subgroup of Spin(5);

and R15 -> R15 ; R25 -> R25 ; R35 -> R35 ; and R45 -> R45

for the 4 of the Rab that correspond to S4 = Spin(5)/Spin(4).

The result of applying the ansatz to the Yang-Mills Spin(5) action

- Rab /\ *Rab (where a,b = 1,2,3,4,5) is the following action:

- Rab/\ Rcd eabcd + - Ra5/\Ra5 , where a,b = 1,2,3,4 .

The term - Ra5/\Ra5 (which was in effect discarded by

MacDowell and Mansouri5,6,7) looks like it has 4 vector bosons

R15, R25, R35, R45. They can be considered to be the translations of parallel transport on the 4-dimensional spacetime base manifold that are needed for formulation of quantum field theory on the its tangent space,

as required because the diffeomorphisms Diff(V4) of spacetime are not analytic and so do not define an exponential map that gives spacetime canonical coordinates in a neighborhood of an identity (Mayer23).

The term - Rab/\ Rcd eabcd gives Einstein gravity by the MacDowell-Mansouri procedure.

The MacDowell-Mansouri procedure begins, following Ne'eman and Regge6 and Freund7, with the action

Rab/\ Rcd eabcd , where a,b,c,d = 1,2,3,4,

Rab = drab + rac/\rcb - 8 l^2 ra/\rb , with 1-forms rab where l is

the Wigner-Inönü contraction scaling factor of the Spin(5)

structure constant being contracted to Spin(4) and with ra and rb defined

by ra5/\r5b = -8 l^2 ra/\rb.

The factor 8 is due to: a5,b5; a5,5b; 5a,b5; and 5a,5b all give the same ra/\rb , but all 4 correspond to the same Spin(4) structure constant f[cd]ab so that f[cd]ab is rescaled by -4 l^2; and

[ra5,r5b] = 2 ra5/\r5b so that the structure is further doubled to -8 l^2 by the rescaling.

Define, for a,b,c,d = 1,2,3,4, `Rab = drab + rac/\rcb , to get:

`Rab /\ `Rcd eabcd - 16 l^2 `Rab /\rc/\rd eabcd +

+ 64 l^4 ra/\rb/\rc/\rd eabcd .

Then, rescale the entire action by 1 / 1024G l^2 to get

(1/16G) (1/64) (1/ l^2) `Rab /\ `Rcd eabcd -

- (1/16G) (1/4) `Rab /\rc/\rd eabcd +

+ (1/16G) l^2 ra/\rb/\rc/\rd eabcd .

 

For the second term, note that there are 4 ways to sum

`Rab /\rc/\rd eabcd , and define `Rab /\rc/\rd eabcd = 4 R.

The second term, - (1/16G) (1/4) `Rab /\rc/\rd eabcd ,

then becomes the classical Einstein Lagrangian - (1/16G) R .

 

For the third term, note that there are 4 x 3 x 2 x 1 = 24 ways to sum ra/\rb/\rc/\rd eabcd , and define -12 l2 = L to be a cosmological constant. The third term, + (1/16G) l2 ra/\rb/\rc/\rd eabcd , then becomes a cosmological term - (1/16G) 2L .

Note that in the l -> 0 limit corresponding to the Inönü-Wigner contraction of Spin(5) into the Poincaré group the L = 0 theory is obtained that and l^-1 is the characteristic distance of anti-de Sitter spacetime. However, in the F4 model the cosmological term is natural and the simplest assumption is that it takes the critical value with respect to the full Spin(5) gauge group.

Assuming that the cosmological term begins at the critical value based on the full Spin(5) gauge group, and that full Spin(5) value is converted to an equivalent value with respect to the rescaled Spin(4) gauge group by the S4 in R5 to S3 in R4 unit sphere radius ratio of (5/4), the universe would begin to evolve as an inflationary universe with vacuum curvature density of (5/4) times the critical density of a standard 4-dimensional universe.

 

The first term, (1/16G) (1/64) (1/l2) `Rab /\ `Rcd eabcd (a,b,c,d=1,2,3,4) effectively has nonabelian gauge group

Spin(4) = SU(2) x SU(2). It is a surface term, needed in the quantum theory to take care of second derivatives in the Einstein-Hilbert R term if the base manifold M is compact with boundary M and metric variations are allowed that vanish on M but whose normal derivatives do not

(Kolb and Turner56; Narlikar and Padmanabhan57). This leads to the Wheeler-DeWitt equation for the wave function Y of the universe. With Vilenkin boundary condition of a purely expanding solution

i Y-1 Y / R > 0 creating the universe from R=0 (nothing), in terms of Airy functions: Y(R) [proportional to] Ai(z(R)) Ai(z(R0)) + i Bi(z(R)) Bi(z(R0)).56

 


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