Schwinger's magnetic model of matter: can it help us with Grand Unification

Paul J. Werbos

National Science Foundation

Arlington, Virginia, US 22230

The goal of this paper is to suggest how reconsideration of Schwinger's "magnetic model of matter" (MMM) can help us overcome some of the roadblocks towards a larger goal: the development of a mathematically well-posed and unified model of how all the forces of nature work together, capable of predicting the full spectrum of empirical data from the laboratory. MMM itself is not such a model - but neither is anything else available to us today. Rather, MMM can be useful as a kind of tool in coping with three major roadblocks which are limiting our progress towards that larger goal. In this introduction, I will start by discussing the larger goal, and mention MMM only as it connects to some of the key subgoals.

In all honesty - the goal of Grand Unification is not the only motivation here. I will argue that the most promising approach to building a finite unified field theory in 3+1 dimensions is to start from bosonic models which generate solitons; however, a bosonic unified theory would also have profound implications for the foundations of physics. Because of the many exact results which now exist for classical-quantum equivalence in the bosonic case, such a theory would seriously re-open the possibility of local realistic models powerful enough to address the complex empirical database of physics today. Section 3.4 will discuss this further, but this paper will mainly address the issue of Grand Unification in 3+1 dimensions, which is certainly a challenging enough starting point.

The three greatest roadblocks to the larger goal, in my view, are:

1        the theoretician/experimentalist divide, most notably the huge distance between true unified models like superstring theory and the practical phenomenological models used to make sense of the mid-to-low energy nuclear experiments which are the bread and butter of large nuclear laboratories today;

2        the physics/mathematics divide, the difficulty of formulating nontrivial quantum field theories which are truly well-posed according to the standards of mathematicians or even the more humble standards of rigorous engineers;

3        the mass prediction gap (not to be confused with the mass gap, the impossibility of really predicting the masses of quarks or leptons when using theories like quantum electrodynamics (QED) or quantum chromodynamics (QCD) which only become meaningful when we attach elaborate, nonphysical systems for regularization and renormalization as part of the definition of the theory.

The original motivation for this paper came from the theoretical side. Like the superstring people, I began by asking: "Can I come up with a well-defined quantum field theory which is finite, which reproduces all the tested predictions of the standard model of physics, but does not require renormalization and regularization as part of the definition of the theory?". However, unlike the superstring people, I asked: (1) can we do it without requiring additional, speculative dimensions; and (2) can we do it even without gravity, just to get started?

The biggest reason why QED requires renormalization is that the energy of self-repulsion of an electron will always be infinite, if we assume that the charge of an electron is all concentrated at a single point. The mass-energy predicted by a point-charge model will always be infinite, unless we adjust it in an ad hoc manner, through renormalization. Superstring theories can be finite, because they assume that the electron has a kind of nonzero radius - very small, as small as the Planck length, but that is enough. There is an easier way to achieve the same effect - by modeling the most elementary particles of nature as solitons, as compound systems whose charge is distributed over a finite region of space.

Of course, distributing the charge is not sufficient by itself to give us all that we need, but it is essentially a necessary condition; thus in order to get to the larger goal, this is the necessary starting point. Some physicists would worry whether there is any hope at all here; to create solitons, we need interaction terms which are not bilinear, and can any model of that sort be well-defined without renormalization? In fact, superstring theories have shown that this is possible, in principle; in any case, there is no mathematical result saying that models with third order nonlinearities cannot be well-defined without renormalization. In previous work, I have reviewed the extensive theoretical work and strong theorems for classical-quantum equivalence, which can provide both upper and lower bounds on energies and masses in bosonic field theories. One of the many important new opportunities ahead of us here is to exploit this equivalence, to prove that all of the Lagrangians discussed in section 3 do in fact yield finite well-defined theories.

Soliton models like the Skyrme model have in fact been very popular at times in empirical nuclear physics. They have been used to confront mid-to-low energy scattering data, in regimes where QCD could not be used directly. The argument has sometimes been that the Skyrme model provides a kind of approximation to the more fundamental model, QCD. But in my theoretical work, I was asking whether we might consider a more fundamental model than QCD, in which the quark itself is modeled as a soliton. In effect, I was asking whether we could overcome the mass prediction gap and the physics/mathematics gap for QED and QCD, by modeling leptons and quarks as solitons so small that we end up with the same actual empirical predictions. Instead of formulating QED and QCD as the limiting case of physically meaningless regularization models (like Pauli-Villars or fractional-dimension models), we could represent them as the limiting case of a family of finite field theories; any member of that family, of small enough radius, would be a legitimate theory of physics, consistent with all the empirical evidence supporting the standard model.

In pursuing this approach, I kept bumping into a fundamental obstacle - that the fields which yield solitons "want to be bosonic only." I also ran across important gaps in communication between different parts of the vast continent which physics has grown into.

One gap is that many orthodox physicists assume that a theory must be renormalizable in a certain sense, in order to be meaningful or useful as a theory of physics. More precisely, they require that the usual sort of perturbation theory - a kind of Taylor series about the zero or vacuum state - converges. This requires that the underlying coupling constants must be less than one. However, in the known soliton models for 3+1 dimensions, and in MMM, the coupling constants are larger. This entire family of models appears to fail the test.

The answer to this gap is that we do not really need to meet this very narrow requirement in order to be well-posed or useful. In fact, some leaders of axiomatic quantum field theory have argued⁷ that we will need to move towards non-perturbative theories and methods in order to have any hope of overcoming the physics/mathematics gap. Soliton models can indeed be well-defined and finite, at least as well-defined as the standard model of physics, as shown by the classic, authoritative work of Rajaraman. Rajaraman still uses "Taylor series" types of perturbation, but he uses a Taylor series, in effect, about a nonzero starting point (the classical soliton solution). More creative forms of perturbation analysis have also proven essential to efficient calculation and convergence in the challenging domain of many-body practical QED calculations.

But how do we handle the remaining difficulty - the fact that the known well-defined soliton models are built up from bosonic fields? How could we reproduce the predictions of the standard model of physics, which combines both bosonic fields and fermionic fields, when we only seem to have bosonic fields available?

On occasion, major physicists have argued that we could construct well-defined fermionic soliton models by somehow assuming "classical anti-commuting fields". I have yet to find or construct a reasonably well-defined mathematical way of doing this which really works.

Bosonization has been the mainstream, most promising approach to bridging the gap between soliton models and the standard model. Bosonization studies how quantum field theories which are purely bosonic in nature may actually result in solitons (or point particles) which behave like fermions.

The literature on bosonization was already huge and diverse, when Makhankov et al wrote their review of the subject. More recently, Vachaspati has attempted to construct a "bosonic standard model" based on bosonization, but encountered certain difficulties which have not yet been resolved. Makhankov et al also noted that the evidence for bosonization in 3+1 dimensions is very persuasive, but still not worked out so completely as the 1+1-D case.

In order to address this roadblock and open the door to new approaches to the mathematical formulation of field theory, I have proposed that we should revisit a model field theory (HtJR) independently proposed by Hasenfratz and 'tHooft and by Jackiw and Rebbi.

My argument here is not that HtJR is "true" or that it is more plausible, in the end, than the skyrmion model or others as a theory of elementary particles. My argument is three-fold: (1) because of the work reviewed by Rajaraman, HtJR (and the simpler Georgi-Glashow model which it is based on) are extremely promising as nontrivial targets for axiomatic field theory, far more interesting than the φ⁴ field theories, suitable both for nonperturbative and novel perturbative approaches; (2) because of the work of Hasenfratz, 'tHooft, Jackiw and Rebbi, and because of the properties discussed in Section III and the Appendix, we are closer to being able to work out the details of bosonization with this model than with other models in this class; and (3) because bosonization and axiomatic properties may actually be similar for other soliton models, in the limit as the radius goes to zero, work in this direction may be an important starting point, even if we may shift in the end to other soliton models.

I will not say more about these theoretical considerations in this Introduction. A major goal of this paper - in section 3 and in the Appendix - is to elaborate on these points, to review MMM, and to suggest additional model field theories for mathematical investigation. Until we begin to investigate at least some of the soliton-generating bosonic field theories capable of "bosonization", we will not have any way to try to predict or explain the masses of the most elementary particles known to physics (currently leptons and quarks).

But then there is the empirical side. Even before I started to study some of the properties of the HtJR model, I immediately noted a major difficulty in trying to use it in a "bosonic standard model": the spin-½ particles which it predicts are dyons - they possess both electric and magnetic charge.

Aside from what Vachaspati has already attempted, there are three obvious ways to try to deal with this problem:

1        to construct a modified model of the quark, based on Schwinger's MMM, where he proposes, in effect, that the quark is a dyon;

2        to reinterpret the topological charge in the HtJR model as a mixture of ordinary electrical and magnetic charge, such that the soliton only possesses electrical charge;

3        or work out the mathematics, as proposed here, but use it only as a steppingstone to understanding the properties of alternative soliton models, such as the more difficult Skyrme model, which may or may not have fewer difficulties.

We do not yet know which of these three approaches will work best, in the end, in getting to the larger goal here. Thus to get to the larger goal, I would propose that the community of physicists work as vigorously as possible in all three directions, in parallel, without becoming overly committed to one or the other.

When I first encountered this three-fold choice, I asked myself: "What is the empirical evidence right now that the quark does not possess magnetic charge, as Schwinger proposed?" I looked for definitive papers, similar to the classic three tests of general relativity versus Newtonian gravity, which would rule out Schwinger's model or define what is needed to rule it out. To my great surprise, after a very thorough search, I found very little work addressing this head-to-head comparison. So far as I know, Sawada of Japan is the only person who has gone directly to the empirical data to try to find out what they say about the QCD-versus-Schwinger issue. So far as I know, Feinberg is the only living person who has directly addressed the theoretical implications of Sawada's calculations. To my greatest surprise, these preliminary indications appear to support Schwinger's model over QCD as we now understand it (with a mass gap). Today's best understanding of atomic nuclei (Section 2.4) also suggests that the strong nuclear force may be longer-range than we would have thought form QCD.

I am not suggesting that this preliminary evidence disproves QCD. On the contrary, I am suggesting that we need to change this situation, by getting better evidence. Because of the great importance and great difficulty of the larger goal here, I am also proposing that we should at least work hard to do full justice to the possibility that approach (1) might work out in the end.

Since I have started getting a bit deeper into these issues (see Appendix), I see more and more hope for approach (2). Perhaps this will reduce the real need for approach (2). Nevertheless, I do hope that someone will be able to follow up on approach (1) as well, at least by conducting more decisive experiments. Because I am really coming at this from the mathematical side, and because the empirical literature here is huge, my review in section 2 is incomplete; however, I hope it will provide at least a starting point for the empirical nuclear (or electromagnetic) physicist, by giving some sense of what is available in the more complete and detailed papers I cite. For more details, of course, it would be easy enough to look at the electronic versions of these papers at arXiv.org and at the APS web site.