Abstract

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Abstract and Introduction

An attempt is made to address a stylized question posed to Ernst Strauss by Albert Einstein regarding the amount of freedom present in the construction of our field theoretic universe.

Let X4 be a 4-dimensional C∞ manifold with a chosen orientation and unique spin structure.

It is otherwise considered to be without a geometry in that no metric, symplectic, complex, volume, quaternionic or other structure is yet imposed upon it.

This is to consider to what extent the opening epigrammatic question of Einstein could be considered a scientific program, rather than a philosophical one, by asking whether the observed world could be extracted from little more than such initial data as above.

Einstein did not specify what he meant exactly. So I have reformulated his question as follows:

“Starting from X4, as the topological structure underlying the SpaceTime construction, to what extent can the observed universe together with stylized contents and laws mirroring its own be generated without further assumptions?”

so that what we are effectively asking is whether there is a plausible map:

…

This recovers the fundamental or seemingly fundamental stylized aspects of the observed universe from which we appear to have emerged.

This does not answer “Why is there something rather than nothing?”

Instead I answer: “Why might we expect a world of the richness we have found through science and observation, to arise out of something which is minimally determined beyond being a low dimensional arena for the rules of calculus to collide with those of linear algebra.”

From our perspective, a fundamental theory is simply a theory that effectively discourages further scientific search for a more foundational layer. Were the world recoverable from a manifold X4 (with little more than minor additional data), it would not bring either physics or theology to an end by any means.

It would, however move the technically minded at last away from the search for fundamental law and focus them instead upon the consequences of the rules encoded leaving the search for an explanation illuminating the initial input as a purely philosophical or, perhaps, religious question.

1.1 Strategy: Ideas over Instantiation (following Dirac)

This work arises from a particular orientation which should be shared with the reader up front. At its heart, it’s belief is that most advances that are held up for years or even decades are blocked because of confounding factors particular to instantiations of the needed idea confused for being problems with the idea itself.

This perspective animates Dirac’s 1963 Scientific American article [3] where he argues that experiment can only be used to check agreement with the instantiation of an idea, rather than the idea itself. Dirac references Schrodinger’s failure to take spin into account leading to a superficial failure to agree with experiment.

He could just as easily, however, been writing about his own superficial mistake in viewing the electron and proton as anti-particles to each other despite the obvious mass asymmetry as pointed out by Heisenberg.

In all such cases, the initial instantiations of radical physical ideas were either flawed in a way the underlying ideas were not, or the presentation was such that it caused the wrong pictures to form in the minds of those who heard it. Thus our belief is that we should be following Dirac at this juncture and looking for natural theories and not over-indexing on their initial instantiations.

Further, we have noticed something exceedingly interesting and no less odd. The tiny minority of theorists who have contributed directly to physical law all appear to share a common quixotic focus on beauty and internal coherence rather than an immediate emphasis on formulae, instantiation and experiment. We take from this that Einstein, Dirac and Yang were not giving general advice as to how to do physics but rather very specific advice as to how to seek new physical law to the almost negligible subset of working theorists who might follow.

As such it is our contention that one should search for a theory that is geometrically and algebraically natural and quite close to our world at a stylistic level (e.g. chiral, three generations, etc…). If such a theory can be found, then, given the seemingly idiosyncratic nature of the various peculiarities of the Standard Model, it is our (historically well motivated but partially unjustified) belief that it will quite likely be that initial instantiations will be confounded by difficulties that are likely to prove inessential and thus surmountable.

1.2 About the Present Work

This work was begun while I was finishing a combined Bachelor’s and Master’s degree program at the University of Pennsylvania which ended in 1985.

At the time neutrinos were not yet claimed to posses mass and there were quite possibly 15 particles in a generation of matter under grand unified ideas like versions of SU(5).

The present theory began on the narrow hope or ‘joke’ that in some sense if:

as the dimension of internal Fermionic quantum numbers, then we would be in the unique case where peculiar spinorial methods could unify the auxiliary fiber bundle Geometry of Ehresmann with the intrinsic geometry of Riemann.