By Barry A. Cipra
In a fitting homage to “an extraordinary man,” Alan Newell of the University of Arizona gave the inaugural Martin D. Kruskal Prize Lecture at this year’s SIAM Conference on Nonlinear Waves and Coherent Structures, held June 1316 at the University of Washington, Seattle. Newell described some of his ideas for an alternative to the Standard Model in particle physics, based on phase transitions in patternforming systems, taking seriously the slogan on a Tshirt his longtime friend and colleague was known to wear: “Subvert the Dominant Paradigm.”
“Martin Kruskal was one of kind,” Newell says. “He would always encourage doing slightly unusual things.” Dogged determination was another of his attributes, Newell recalls. “Once a problem gripped him, he never let go.”
Both traits were key to Kruskal’s signature achievement: the discovery of solitons (see this supplementary article). The particlelike behavior he and Norman Zabusky observed in numerical solutions to a classic wave equation inspired a paradigm shift in mathematical physics. Fifty years ago, integrable systems of differential equations were virtually synonymous with linearity; within a decade, integrable systems of nonlinear equations had taken center stage, a position the nonlinear theory has occupied ever since. “Linear theory,” Newell quips, “is a rest home for applied mathematicians.”
Newell makes no claim that his own observations will shift any paradigms. But if they do, it will be due in part to the example of a man who was a role model for subversive thinking.
Whence Symmetry
The Standard Model is the outrageously successful theory that accounts for threefourths of physics. (It unifies the strong, weak, and electromagnetic forces; only gravity escapes its embrace.) In the broadest of outlines, it posits quantum fields based on certain unitary and special unitary groups, from which spring the bosons and fermions of the observed (and sometimes unobserved) universe. Its most recent tour de force is the apparent spotting of the long elusive Higgs boson, which purports to explain why some particles have inertia or mass—the ratio of momentum to speed, or energy to speed squared.
A key word in the Standard Model is symmetry. In accord with the principle discovered by Emmy Noether nearly a hundred years ago, conserved properties stem from symmetries (put very loosely: Things that don’t change depend on changes that effect no change). The Standard Model sports a global spacetime playground based on translational and rotational symmetries, dotted with the swings, seesaws, and monkey bars of the symmetry groups SU(3), SU(2), and U(1). These symmetries (along with a batch of “accidental” ones that seemingly come for free) enforce a physics of conserved energy and momenta (both straight and angled), spin and charge (electric, color, and weak hyper). In particular, the unitary groups give rise to “fractional” spin and charge: quantities that require a full two or three turns to remain invariant.
To Newell, those groups seem somewhat jerryrigged: They’re posited precisely to give the results the theory needs. Accordingly, he set out to see if it would be possible to start with nothing more than the symmetries of translation and rotation and “stress” them into producing objects with fractional invariants. The Standard Model already makes use of symmetry breaking, of course. (It’s part of how the Higgs boson accomplishes its massive undertaking.) But Newell’s aim is to squeeze the local gauge symmetries out of the global symmetry of spacetime.
It doesn’t take a degree in theoretical physics to imagine that it might be possible. Evidence for spatial symmetry breaking is as plain as the ridges on your fingertips. “We have such systems all over the place,” Newell says. “They’re called patternforming systems.”
The “granddaddy” of patternforming systems, Newell notes, is Rayleigh–Bénard convection, with its roiling regularity. A thin layer of fluid, heated from below, becomes a rhythmic sea of stripes, with hot fluid forcing its way up along one edge of each stripe and denser, cool fluid diving down along the other edge. The width of the stripes (or honeycombs, also a frequently observed pattern) is determined by the physics, but the orientation is a more or less random choice. Indeed, different portions of the fluid may opt for different orientations. The upshot is that, in striped patterns, “phase grain boundaries” are widely seen, along with point defects known as concave and convex disclinations (see Figure 1). On your fingertips, these point defects are known as triradii and loops.
Figure 1. What goes around comes around. Could point defects known as concave (left) and convex (right) disclinations, which arise in patternforming systems, account for fractional charges and spin in the Standard Model of particle physics?

For these twodimensional disclinations, an imaginary twoheaded arrow perpendicular to a stripe, when transported continuously in a circle around the point defect, turns only halfway around, a natural analog to the fermion spin of 1/2. Newell and colleagues have worked out an extensive 2D theory of “phase diffusion” to account for the stable patterns observed in phase grain boundaries. When the theory is taken into higher dimensions, the analog of concave disclinations leads to loops that twist by multiples of \(2\pi/3\). In one case, the result is a defect with index \(±2/3\), which Newell calls a “pattern up quark”; in another case, the result has index \(±1/3\), for a “pattern down quark.” In the convex case, the 3D analog is a “pattern lepton” with index \(±1\).
Despite the suggestive names and results, “this is not an attempt to replace the Standard Model,” Newell says. It’s “just a little game” that’s “more than likely destined for the dustbins of history.” Nonetheless, it’s interesting to see what happens to the simplest symmetries when a system is stressed far from equilibrium," he adds. “There’s a lot of interesting geometry. And there are so many open questions.”
Barry A. Cipra is a mathematician and writer based in Northfield, Minnesota.