Lab Note 1: My 1984 Unpublished Paper on Geometrodynamics with Sources

The following is a PDF version of my 1984 unpublished paper:

An Extension of Reinich’s Already Unified Theory to Electromagnetic Sources, In a Simply-Connected Spacetime Topology

This is a good starting point for my blog, first, for historical reasons, and secondly, because this paper is still of interest to me today, and still influences my work heavily in at least two ways:

First: writing this paper was the first time that I because attuned to the importance of duality and duality symmetry.  Reinich showed that gravitation and electromagnetism are, classically, “already unified” if one considers “source-free” electrodynamics, which, implicitly, also thereby introduces a duality symmetry between electric and magnetic fields.  This led me to consider whether and under what circumstances Wheeler / Reinich’s approach might be extended to include sources.  The fly in the ointment, of course, is the absence of an observed symmetry between electric and magnetic sources.  We definitely observe electric charges.  We definitely do not observe magnetic charges, or, if we do, we don’t know that they are magnetic charges and we do not experience them in the same way as we experience electric charges.  Therefore, to extend Reinich, which is the cornerstone of Wheeler’s “Geometrodynamics,” to include sources, one must of necessity grapple with the absence of observed duality between electric and magnetic sources.  I have therefore come to regard the question of WHY we do not observe a duality symmetry between electric and magnetic courses, and, in modern parlance, the question of HOW one might start with such a symmetry and then “break” or “hide” this symmetry, to be the oldest unresolved question in theoretical physics, dating all the way back to Maxwell’s time.  Of course, with the quantum revolution which started right in 1900, this question got lost in the shuffle, but it still subsists as a fundamentally-important, unresolved physics question.

Secondly, in this 1984 paper, I first became aware of that there exist — at least formally speaking — third rank antisymmetric sources, in addition to the first rank sources that we regard classically as electric currents.  You will see these in the 1984 paper.  MTW’s Gravitation also talks about first/third rank duality, but only in a formal way, and without connecting the third rank antisymmetric tensors to anything physical beyond their first-rank duals.  For twenty years, I was aware of these third rank antisymmetric sources, but thought of them as some unexplained, unphysical oddity with merely formal utility.  Then, in early 2005, it occurred to me that these third rank antisymmetric sources, in non-abelian gauge theory, may, as a formal matter, come to be regarded as baryons.  Those have followed my work are aware that since that time, I have become absolutely convinced that it will one day be recognized that baryon sources are, in fact, to be regarded in spacetime as third-rank antisymmetric tensor sources.  If the establishment ever catches on to this, you know where you first heard it.

Let me now tiptoe into some of the mathematics which I think is important to keep in mind.   I will start with the known mathematical identity:

between any two antisymmetric tensors A and B in spacetime, where * denotes “duality” employing the Levi-Civita tensor in the usual way.  This identity applies even where A and B are non-Abelian, and it is one of the most important, yet least-utilized identities in (classical) physics.    Especially, when we associate A and B with the field strength tensors F of QED and QCD, then the third-rank antisymmetric term can be thought of as a third-rank antisymmetric source (and also as a magnetic charge) while  is of course related to first-rank sources, namely, current densities J.  Further, equation (1), in terms of its differential order, is in the nature of a dynamical equation between the first and third rank sources, and it is at the same differential order as the Bianchi identity of gravitational theory.  It is as a result of all these connections that the fundamental relationship of the 1984 paper,   emerges as an alternative statement for Maxwell’s equations with sources, so long as there is a duality symmetry between electric and magnetic sources.  Note, particularly, the discussion in section 6 of this 1984 paper, which is based on one of Einstein’s final papers on the Relativistic Theory of the Non-Symmetric Field, and which contains an explicit calculation showing the strength of the electromagentic and gravitational field equations to be identical.  I believe that in this last paper, Einstein was laying out for future generations, an important guidepost for achieving unfication between (classical) gravitational and electromagentic theories.   So, the identity (1) is important to keep in mind, and one ought to be inquiring into the physical content of this identity, and not just its mathematical existence.

The second piece of mathematics to keep in mind is what happens to third-rank antisymmetric tensors in the above, when Yang-Mills (non-Abelian) gauge theory is considered.  t’Hooft and Polyakov were the first to recognize that magnetic sources become non-vanishing in non-Abelian gauge theories.  For Abelian gauge theory this term is identical to zero, but for Yang-Mills, if one defines a field strength tensor F in the usual way, then, with group indexes suppressed:

which in the language of forms is:

This object is non-zero, it is a source, and it ought to exist somewhere in the observable universe.  Also, in light of how such sources enter into equation (1), on a par with electric sources, this is, in my view, the start of the trail that leads to understanding baryons.  Again, the seeds of these two lines of inquiry, are already contained in the 1984 paper.   I will have more to say about baryons in a later lab note.

Finally, one should also keep in mind that the electric current, in the language of forms, is specified by:
 
This too, is a three-form, based on a third-rank, antisymmetric tensor.  We need to understand these third rank antisymmetric tensors on their own terms, and not just as a formalism in relation to their first-rank duals.  If I have not made myself clear, let me do so one more time: these third-rank antisymmetric tensor sources underlie the very existence of baryons, which in turn underlie the vast bulk of our material universe.  One day, everyone will come to recognize this.

In the spirit of the blog, this is my first lab note.

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