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3.1. Force Strength Formula.

 After dimensional reduction to 4-dimensional spacetime, the F4 model has four gauge groups: U(1) for electromagnetism; SU(2) for the weak force; SU(3) for the color force; and Spin(5) for gravity.

The Laplace-Beltrami operators for the four gauge groups G are defined on the following compact rank one symmetric spaces G/K of

dimension < 4:

Spin(2) = U(1) = S1 (abelian)      G/K = S1 = T1;

Spin(3) = SU(2) = Sp(1) = S3   G/K = S3 and G/K = S2 = SU(2)/U(1);

SU(3)             G/K = SU(3)/S(U(2) x U(1)) = CP2; and

Spin(5) = Sp(2)           G/K = Spin(5)/Spin(4) = S4.

Wolf28 has shown that T4, S2 x S2, CP2, and S4 are representatives of the four equivalence classes of complete simply connected Riemannian manifolds with quaternionic structure.

For G/K, with Riemannian structure given by the Killing form

B(Xi,Xj) = gij of G, the Laplace-Beltrami operator


where Ls is the Laplace-Beltrami operator on the sphere S(r,p0) of radius r at point p‚ of G/K and A(r) is the area of S(r,p). Let V(r) be the volume of S(r,p), L be the diameter of G/K, and p be antipodal to p0.

is the solution to Poisson's equation L(G/K) u(p) = (p,p0) - (p,p) such that u(q) dq = 0. The term - (p,p) is necessary because compactness of G/K requires that [(p,p0) - (p,p)] dq = 0 for Poisson's equation to have a solution (Helgason27, Ch.II, §§4,5).

g(p,p0) - g(p,p) is the sum of Green's functions for the propagator at p0 of a gauge boson particle of gauge group G and the propagator at p of a gauge boson antiparticle. Physically, the gauge boson antiparticle at p is ignored and

is the Green's function for the propagator of the gauge bosons of the gauge group G. V(L) is the volume of G/K.

The factor 1/V(L) normalizes the delta function (p,p0) to 1.

The F4 model compares the strengths of two different forces, insofar as pure gauge boson terms are concerned, by the inverse ratio of the factors 1/V(L) (or n times 1/V(L) if n copies of G/K are needed to make a 4-dimensional manifold), because the stronger force has to be reduced more to normalize the delta function (p,p0) to 1.

The 4-dimensional manifold made up of Cartesian products of a symmetric space G/K is considered to be a natural compactified base manifold spacetime M for the force G. Those spacetimes are:

T4 for U(1); S2 x S2 for SU(2); CP2 for SU(3); and S4 for Spin(5).

The force strength is proportional to the volume V(M). That is consistent with the lattice structure of the 4-dimensional F4 model, as the force is propagated by gauge bosons and the gauge bosons are identified with links between points in the spacetime base manifold.

In the F4 model, the 8 first generation spinor fermion particles are characterized by a compact parallelizable manifold Q8+ = RP1 x S7.

Q8+ is the Silov boundary of a bounded domain D8+ which is isomorphic to the symmetric space D8+ = Spin(10)/Spin(8) x U(1), so that Q8+ has a local symmetry Spin(8) and another local U(1) symmetry that gives the complex phase of the propagator amplitude.

The geometry of Silov boundaries, bounded domains, and symmetric spaces is discussed in Helagason12 and in Hua 11. Let

[D] denotes the Silov boundary of the bounded domain isomorphic to D.

To see how a given force acts on the basis fermions

{n, e, ru, gu, bu, rd, gd, bd} of Q8+ = RP1 x S7, consider Sn in S7, where S7 has basis fermions {e, ru, gu, bu, rd, gd, bd}, as a sphere Sn in the tangent space R7 of S7. Then there is a subspace Rn+1 of R7 such that

Sn in Rn+1, so that the tangent bundle of Sn includes n+1 independent linear combinations of the basis fermions {e, ru, gu, bu, rd, gd, bd} of S7.

SU(3) acts on S5 = [SU(4)/S(U(3) x U(1))]

S5 in R6 <-> {ru, gu, rd, gd, bu, bd}

SU(2) acts on RP1 x S2 = [Spin(5)/SU(2) x U(1)]

RP1 x S2 in RP1 x R3 <-> {n, e, (r+g+b)u, (r+g+b)d}

U(1) acts on S1

S1 in R2 <-> {e, (r+g+b)(u+d)}

Spin(5) acts on RP1 x S4 = [Spin(7)/Spin(5) x U(1)]

RP1 x S4 in RP1 x R5 <-> {n, e, (r+g+b)u, (r+g+b)d, ru, gu} and

{n, e, (r+g+b)u, (r+g+b)d, ru, gu} generates {n,e,ru,gu,bu,rd,gd,bd} =

= RP1 x S7 by bu = (r+g+b)u - ru - gu and the individual d quarks by transforming (r+g+b)d to (r+g+b)u, then producing the individual u quark, and then transforming back to the d quark space. Therefore Spin(5) gravity effectively acts on all the fermions.

The action of each of the four forces is such that Q has a local symmetry of the gauge group of the force and another local U(1) propagator phase symmetry.

Let M be the irreducible m-dimensional manifold on which the gauge group acts naturally as a part of 4-dimensional spacetime = M^4/m,

Q be the spinor manifold with natural local action by the gauge group,

D be the bounded complex domain of which Q is the Silov boundary, and µ be the mass scale for effective theories below symmetry breaking at µ.

The force strength constants of the four forces are proportional to three factors, taking into account the fundamental F4 model lattice point of view that fermions and charged gauge bosons constitute lattice vertices and gauge bosons constitute the links joining the vertices:

The force strength is proportional to V(M).

The force strength is proportional to V(Q), the volume of the part of the half-spinor fermion manifold Q8+ = RP1 x S7 on which the force acts naturally locally, because charged gauge bosons can emit or absorb gauge bosons in the same way as charged fermions do.

When emitting or absorbing other gauge bosons, a charged gauge boson corresponds to a vertex, just as a fermion would, and therefore corresponds to a part of the spinor fermion manifold that is associated with a vertex at a point in the spacetime base manifold.

The V(Q) factor does not apply to any force whose gauge group

does not include charged gauge bosons, that is, it does not apply to the abelian gauge group of electromagnetism since photons do not carry charge of any kind.

The third proportionality factor for force strengths is 1/(V(D)^1/m), where D is the bounded domain of which Q is the Silov boundary. It is a normalization factor to be used if the dimension of the spinor fermion manifold Q is different from the dimension m of the base manifold M to normalize the radius of the half-spinor manifold Q to be consistent with the unit radius of the base manifold M. For the abelian U(1) gauge group of electromagnetism there is no factor 1/(V(D)1/m) because photons carry no charge of any kind, so for U(1), 1/(V(D)^1/m)=1.

In addition to the product V(M) V(Q) / (V(D)^1/m) of the three proportionality factors, the relative strengths of the four forces are affected by two other processes:

effective field theory mass factors 1/µ^2 must be included for gravitation in the effective field theory below the Planck mass and for the weak force in the effective field theory below the weak boson mass range; and

renormalization group effects (running coupling constant) must be taken into account in evaluating the force strength away from the characteristic energy of the force.

The resulting formula for calculating the force strength is then:

force strength = [1/µ^2] [V(M)] [V(Q)/V(D)^1/m] at the

characteristic energy level of each force, with renormalization effects accounting for running the force strength to other energy levels.

The plan of this paper is first to calculate the force strengths at characteristic energy levels, and then to estimate roughly the renormalization effects.



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