S4 Physics Basics

LEGAL DISCLAIMERS

Although the following material appears in a public forum, the ideas and content of the material is protected by the Uniform Trade Secrets Act of 1985 for the sole commercial use by S4 Group, to register as a Limited Liability Company in New York State. The Company reserves the right to take legal action against any entity who use any novel unpublished idea or their mathematical consequencs for commercial use.

PHILOSOPHY OF SCIENCE

Modern empirical science has relied heavily on statistical hypothesis testing. A scientific statement or hypothesis under empiricist scientific philosophy is accepted by convention if the p-value of the hypothesis is lower than 0.01 or 0.05. A stronger philosophy of science is to reject a hypothesis if there are any exceptions to the hypothesis.

Following such a stringent criterion for acceptance of a scientific statement leads to a good science by mathematical completeness. Thus instead of testing simple local questions, one tests a mathematically coherent theory as a whole as ONE hypothesis on a variety of experimental data. Mathematical coherence allows the "hypothesis" to be the full theory. This is useful because divisions between physics, chemistry, and biology can be removed by this exercise.

PROPOSED "HYPOTHESIS"

There exists a radius R (measured in meters) such that the shape of the universe is a round 4-sphere of radius R. Fields/particles in the universe are fully described by an SU(2)xSU(2)xSU(3) gauge theory and follow classical mechanics rather than "quantum mechanics". Electromagnetism is an SU(2) gauge theory rather than U(1) gauge theory. The physical universe is a smooth 3-manifold that evolves in S4(R) according to joint classical movement of all particles/fields contained within it.

EVIDENCE FOR 4+1 DIMENSIONS RATHER THAN 3+1 DIMENSIONS

In 1984 Schechtman et. al. discovered crystal structures with 5-fold symmetry that came to be known as "quasi-crystals". The crystallographic group for 3 dimensions do not allow 5-fold symmetry but that of 4 dimensions does. By Ockham's razor, the existence of a fourth macroscopic dimension, and 4D crystals provides a better theory for this phenomenon.

So this provides evidence for 4 MACROSCOPIC dimensions. On the other hand, no quasi-crystal found thus far has symmetries not possible in 4 dimensions (I need better search for this). Therefore, a 4-dimensional theory is consistent with evidence from crystallography.

EVIDENCE FOR BOUNDEDNESS OF THE UNIVERSE

If we assume that all EM particles of the universe was concentrated at a single point ~20 billion years ago, and microwave particles had diffused since then, then the cosmic background radiation with a lower bound of ~2.7 K is impossible unless the universe is bounded. This is so because diffusions starting at a point in an unbounded universe has gaussian decay and hence cannot have a lower bound in density in finite time.

WHY QUANTUM MECHANICS IS UNNECESSARY

Quantum mechanics was developed to address two fundamental problems. First, the description of the blackbody intensity distribution which classical physics on 3-dimensional flat space could not explain. Second, to explain the stability of the electron-proton system, which in flat space would lead to acceleration and energy emission, and collapse.

Both of these problems are easily overcome by classical physics on S4(R). For the first problem, all that Planck required was quantization of electron energy. This energy quantization is a consequence of COMPACTNESS of S4 in the S4 scientific theory rather than a different behavior "in the small" versus "in the large". Planck's energy quantum is related to the radius of the universe.

An electron-proton system on S4(R) has no acceleration and hence matter is stable.

WHY GAUGE GROUP SU(2) RATHER THAN U(1) FOR EM?

The topology of SU(2) is the same as that of a three-sphere. SU(2) is also isomorphic to unit quaternions. For static EM particles on the round 4-sphere, Atiyah, Hitchin and Singer have a full characterization of magnetic monopoles on S4, the so-called instantons. For this, see their 1974 or their other work.

Since S4 has no 2-dimensional cohomology, there is no obstruction to classical mechanics for any particle/field, EM, weak nuclear, or strong nuclear. Furthermore, the Maxwell's equations reduce to the anti-self-duality equation for the curvature of a connection on an SU(2) bundle on S4.

CALCULATING THE RADIUS OF THE UNIVERSE VIA PLANCK CONSTANT

The Planck constant is the smallest quantum of energy possible. For a universe that is a round 4-sphere of radius R, the spectral gap must equal Planck constant. The Laplacian scales as 1/R^2 and the eigenvalues of the Laplacian on the 4-sphere are (n+2)(n+3)/12? in particular

R = h^{-1/2}.

Interestingly, S4(R) is an Einstein manifold as well, and the Ricci curvature is a simple function of R. This is a nontrivial predicted equality of a quantity from general relativity and quantum mechanics, by a theory simpler than both.

General relativity has been tested only on binary pulsars and the solar system, neither of which actually require the complex energy-momentum tensor quantity.

THE STANDARD MODEL

The standard model is an U(1)xSU(2)xSU(3) gauge theory on 3+1 dimensional universe. The standard model for S4 physics is simply an SU(2)xSU(2)xSU(3) gauge theory on S4(R). The compactness of S4(R) provides a spectral gap for any smooth function, without need for a special "quantum" Yang-Mills theory.

CLASSICAL MECHANICS

Let Q be quaternions. Classical Hamiltonian mechanics on QxQ can be elegantly described in terms of the natural symplectic form on QxQ and a Hamiltonian function. Although S4 is not a symplectic manifold (it has no 2 dimensional cohomology) Hamiltonian mechanics is well-defined for a subset of Hamiltonian functions that can be "lifted" to QxQ. That is to say, S4 is the quaternionic projective space with cover QxQ. Hamiltonian functions on S4 correspond to those Hamiltonian functions on QxQ which are invariant under joint multiplication by unit quaternions. There is no difficulty in describing classical mechanics of such Hamiltonians on QxQ, thus classical mechanics is well-defined using symplectic methods for S4(R).

IMPLICATIONS

The Universe is jointly DETERMINISTIC, although it may be highly non-deterministic locally.