Mathematics and the Higgs
In general, I tend to resist talking about the thing that everyone is talking about, but I find reason to make an exception today. I do want to say something about yesterday’s announcement from physicists at the LHC that they saw the Higgs particle. Frank Wilczek describes the significance of this observation (particularly nicely) in a blog post that appears on both the FQXi Blog and NOVA’s Nature of Reality Blog. He says:
It confirms, as it completes, the Standard Model of fundamental physics. It hints at the splendid new prospect of supersymmetry while debunking rival speculations. Most fundamentally, it reaffirms our scientific faith that nature works according to precise yet humanly comprehensible laws—and, importantly, rewards our moral commitment to testing that faith rigorously.
Those “precise yet humanly comprehensible laws” are interpreted mathematics. In this context, mathematically formulated is the meaning of “humanly comprehensible.” I’m not suggesting that the observation is an achievement in mathematics. It is not. I only want to take note of the fact that mathematics is the reasoning, as well as the strategy, that brings this quantum mechanical world into view. The extent to which the effort rests on mathematics is as important as the extent to which it rests on the accelerator. Yet the point is rarely made.
In an earlier post, however, one that anticipated the July 4th announcement, Frank Wilczek points, more than once, to mathematics. He explains that modern physics proposes a way to simplify the laws, or the equations, that describe nature if we’re willing to see the empty space of our everyday perception as a medium “whose influence complicates how matter is observed to move.”
He refers to this as “a time-honored, successful strategy.” Classical mechanics, for example, which postulates complete symmetry among the three dimensions of space, wouldn’t account for the actually observed motions (that are not symmetric in all directions) without the idea of “a pervasive gravitational field.”
A much more modern example occurs in quantum chromodynamics (QCD), our fundamental theory of the strong force between quarks and gluons. There we discover that the universe is filled with a medium, the sigma (σ) field, that forms a sort of cosmic molasses for protons and neutrons. The σ field slows protons and neutrons down. Allowing a bit of poetic license, we can say that the σ field gives protons and neutrons mass. Many consequences of the σ field have been calculated and successfully observed, so that to modern physicists it is now every bit as real as Earth’s gravity field. But the σ field exists everywhere and everywhen; it is not tied to Earth.
In the theory of the weak force, we need to do a similar trick for less familiar particles, the W and Z bosons. We could have beautiful equations for those particles if their masses were zero; but their masses are observed not to be zero. So we postulate the existence of a new all-pervasive field, the so-called Higgs condensate, which slows them down. This proposal, which here I’ve described only loosely and in words, comes embodied in specific equations and leads to many testable predictions. This proposal has been resoundingly successful. (emphasis my own)
Since no known matter had the properties necessary for the Higgs condensate, the search was on for the one that did. To answer the question, what is the Higgs particle specifically? Wilczek says the following
“There’s a quotation I love from Heinrich Hertz, about Maxwell’s equations, that’s relevant here.
To the question: “What is Maxwell’s theory?” I know of no shorter or more definite answer than the following: “Maxwell’s theory is Maxwell’s system of equations.”
Similarly, Higgs particles are the entities that obey the equations of Higgs particle theory. Those equations prescribe everything about how Higgs particles move, interact with other particles, and decay—with just one, albeit glaring, exception: The equations do not determine the mass of the Higgs particle. The theory can accommodate a wide range of values for that mass.
And so a tremendous amount of analysis has been done to narrow the range within which the Higgs particle mass is expected to be. Wilczek also explains:
Physicists will have used intricate equations and difficult calculations to predict not only the mere existence of the Higgs particle, but also (given its mass) its rate of production in the complex, extreme conditions of ultra high energy proton-proton collisions. Those equations will also have accurately rendered the relative rates at which the Higgs particle decays in different ways. Yet the most challenging task of all may be computing the much larger, competing background “noise” from known processes, in order to successfully contrast the Higgs’ “signal.” Virtually every aspect of our current understanding of fundamental physics comes into play, and gets a stringent workout, in crafting these predictions.
Because the equations in the Standard Model stipulate four different forces (the strong, the weak, electromagnetic and gravitational) and six different materials that they act on, the search is always on for the simpler, prettier, unified theory. There are proposals for such a theory, where there is only one kind of material and one force. To make them work quantitatively, the equations of the Standard Model have to be expanded to accommodate a concept called supersymmetry (one of a number of proposals) and supersymmetry predicts the existence of many additional new fundamental particles that are likely to be accessible to the LHC.
It is mathematics that gives the various models of the universe their structure and that determines the specifications for what has come to be called the Higgs particle. In another way, it is mathematics that provides the way to decipher the data produced when theories are tested. And the testing is a herculean effort. Wilczek says it right, “…detection requires cunning.” The search for the Higgs has relied on a history of inventive and probing analyses, clever programming, and endless calculations. In his June blog Wilczek also takes note of something many Higgs stories ignore:
Finding the Higgs boson depends on assuming that the Standard Model is reliable, so we can work around the “background noise”. Here years of hard bread-and-butter work at earlier accelerators—especially the Large Electron-Positron Collider (LEP), which previously occupied the same CERN tunnel in which the LHC resides today, and the Tevatron at Fermilab, as well as at the LHC itself—pays off big. Over the years, many thousands of quantitative predictions of the Standard Model have been tested and verified. Its record is impeccable; it has earned our trust.
I care about the absence of mathematics in physics discussions only because it gives every non-scientist yet another opportunity to ignore its living presence and fail to see how it functions, not only to describe, but to perceive.