A unified field theory is a concept that goes back to the early twentieth century. It is the idea of producing a single field that encompasses both electromagnetism and gravity. In pursuit of this ideal, geometry has been scoured and there have been many attempts in this direction by many people including Einstein, Weyl, and many others and whose contributions are being evaluated in the historical research. I am not yet certain that I believe that a unified field theory is the perfect description of nature. Unlike these early workers on this problem for whom empirical reality consisted of three spatial dimensions, I am in a situation where regardless of detailed dynamics we know that the universe has four macroscopic spatial dimensions and is compact and the best candidate for the shape of the universe is a scaled 4-sphere. Rather than a unified field theory on this entire space (which is an ‘Einstein space’ in the sense that the Ricci curvature is a constant multiple of the metric) more natural seems to assume that an electromagnetic potential is global while gravity is equivalent to choice of a three-dimensional hypersurface. A hypersurface could be determined by critical point of an Einstein-Hilbert action modified by a ‘matter Lagrangian’ determined by the electromagnetic potential. It’s best to understand this picture with frozen time rather than analysis on a ‘spacetime’ manifold. At any given moment, our physical universe is a hypersurface of a scaled 4-sphere and the induced metric of the hypersurface is a critical point of the Einstein-Hilbert action. When there are no electromagnetic terms in the Lagrangian then the solution will be the equatorial 3-sphere which is a sensible solution for the gravitational field equations. In the presence of a global electromagnetic potential, a 1-form A, the Euler-Lagrange equations will be similar in form to Einstein’s gravitational field equations (without time components). Suppose M->S4(1/h) is such a critical point. On one hand, the Einstein tensor for M is equal to the stress-energy tensor determined by A but on the other hand the Ricci curvature formula for submanifolds applies and shows that the Einstein tensor can also be written in terms of the traceless part of a 2-tensor determined by the second fundamental form. Therefore these two symmetric two-tensors can be identified on M. This identification gives the relation between geometry of M and the electromagnetic field rather than a single field uniting gravity and electromagnetism. Assuming that the four-dimensional universe has a static fixed shape of S4(1/h), gravity is being determined by electromagnetism.

The construction above I believe is closer to the truth regarding nature rather than a unified field theory of the type produced, say by Kaluza, where beyond the metric on a lower dimensional space, an additional dimension allows insertion of electromagnetic potentials using extra dimensions. I also base my ideas on unification of electromagnetism and gravity based on a different type of support than considerations of mathematical elegance: I have shown by a sharp match of the slope of redshift using a scaled 4-sphere model of the universe that this geometry has empirical grounds. Quite a bit is known about geometry of a 4-sphere such as the fact that the Dirac spectrum are integer multiples of h, that the eigenspinors are in one-to-one correspondence with spin groups. The fact that a Dirac equation can be written for photons (in addition to fermions) implies that photons can be understood on a scaled 4-sphere naturally. I suspect that all quantum phenomena can be recovered as a consequence of this geometry. On the other hand, there is a perfectly good geometric theory of gravity based on selection of appropriate hypersurfaces which in the case of no ‘matter field’ reverts correctly to a equatorial 3-sphere. Schematically, 4-sphere geometry is producing the foundations for quantum phenomena as well as gravitational equations with the right cosmological constant for every critical hypersurface. Electromagnetism operates globally.

Many people would probably complain that my empirical evidence for four macroscopic spatial dimensions is somehow forced: that 5, 8, 10 and 12 fold rotational symmetry observed in quasicrystals is genuinely different from crystals but strict adherence to parsimony would suggest otherwise. This is absolutely crucial to seriously move away from the Kaluza-Klein type assumptions about ‘compactified’ versus noncompactified dimensions. In an S4 model of the universe there are no strongly preferred directions a priori; it is the solution of the gravitational variational equations that choose a hypersurface. Regardless of which hypersurface is thus chosen, the normal lines on these submanifolds all have closed large radius 1/hbar.

ADVANTAGES OF HINDSIGHT

The strands of historical thought on basic issues in physics and mathematics is quite revealing and fruitful. One juncture of history that I find fascinating is 1900 when two simultaneous events: Planck’s resolution of the blackbody radiation problem by introducing quanta and the solution of the problem of discrete spectra for planar bounded domains. More than a century later, it is not a difficult problem to find the link, with the spectrum of the Dirac operator on a 4-sphere scaled by 1/h will be integral multiples of h. On a 4-sphere Dirac equations are automatically ‘geometrized’. We know now that Weyl’s introduction to gauge invariance had been tremendously fruitful for particle theory via non-abelian gauge theories and these can easily be considered on a scaled 4-sphere universe. Without the tremendous efforts made by the pioneers for unified field theories and sharpening and simplification of a century we can use a differential 1-form to represent the electromagnetic potential and produce gravitational field equations by a well-worn action principle based on the Einstein-Hilbert action. There is here an extremely simple unity in this setup but it is not quite a unified field theory of electromagnetism and gravity without losing sight of quantum phenomena.

Of course there are grave problems with contemporary physics as well, even from an unstudied viewpoint: expansionary big bang cosmology is just wrong. Scaling the radius of a 4-sphere by 1/h links microscopic with macroscopic physics which is not compatible with nonstationary four dimensional universe, and empirical result I have obtained matching the slope of the redshift reinforces this. Given that we accept the evidence for a compact universe, such a link is crucial for understanding effects of global geometry on microscopic physics. We can consider the characterization of eigenspinors of the Dirac equation as indication that this link between macroscopic and microscopic leads immediately to the infrastructure required for quantum mechanics.

I admit ignorance regarding string theories but if a unification theory without contradictions is achievable on a scaled 4-sphere (which has empirical evidence) then it is a better candidate than a highly complex theory.

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