Amir D. Aczel (amirdaczel.com) writes about mathematics and physics and has published 18 books, numerous newspaper and magazine articles, as well as professional research papers.

A Higgs candidate event from the ATLAS detector of the LHC. Courtesy of CERN

What made me fall in love with theoretical physics many years ago (in 1972, when I first met Werner Heisenberg) was its stunningly powerful relationship---far beyond any reasonable expectation---with pure mathematics. Many great minds have pondered this mysteriously deep connection between something as abstract as mathematics, based on theorems and proofs that seem to have little to do with anything "real," and the physical universe around us. In addition to Heisenberg, who brilliantly applied abstract matrix theory to quantum physics, Roger Penrose has explored the deep relation between the two fields---and also, to a degree, between them and the human mind---in his book The Road to Reality

. And in 1960, the renowned quantum physicist and Nobel Laureate Eugene Wigner of Princeton wrote a fascinating article

that tried to address the mysterious nature of this surprising relationship. Wigner marveled at the sheer mystery of why mathematics works so well in situations where there seems to be no obvious reason why it does. And yet, it works. Wigner had contributed much to physics using very advanced mathematics: he was one of the pioneers of using mathematical groups to model physical phenomena. Group theory is the mathematical branch dealing with concepts of symmetry. As Wigner helped us see, symmetries can reveal the deepest secrets of physical reality. Such symmetries, for example, allowed Steven Weinberg to actually predict both the existence of the Z boson and the actual masses of the Z and the two W bosons, which act inside nuclei of matter to produce radioactive decay. In doing so, Weinberg exploited what he called "the Higgs mechanism," which he hypothesized to break a primeval symmetry of the early universe and thus impart masses to these three particles---and presumably also to all other matter in the cosmos. When the discovery of the Higgs boson was announced at CERN on the 4th of July, this immensely important scientific triumph of our time also lent further support for Weinberg's theory. The fact that the “pure” mathematics of group theory can help produce such accurate physical predictions as Weinberg's seems like nothing less than a miracle. But in fact, the connection between mathematical groups and physics was established eight decades ago by the brilliant German-Jewish mathematician Emmy Noether

, who barely escaped Nazism only to die from an abdominal tumor in the United States shortly after obtaining a professorship at Bryn Mawr, which had allowed her to leave Europe. Noether devised and proved two key theorems in mathematics, called Noether's Theorems

. These powerful mathematical results established the relationship between the symmetries of group theory and the all-important conservation laws in physics---such as the conservation of energy, momentum, and electric charge. Continuous groups

, the work of the Norwegian mathematician Sophus Lie

(pronounced "lee"), have played a key role in physics, and these come into play through Noether's Theorems. The technical explanations of how symmetry works in theoretical physics are beyond the scope of this article, but the point is that finding a pure-math kind of symmetry allows a physicist to do a lot: such as discover an entire new theory! As a very quick example: symmetry through time is what gives physics the key concept of the conservation of energy---the paramount property that energy can only change form (for example, from mass to sheer energy, as per Einstein's famous formula), but never be created or destroyed

.* In the 1960s, symmetry went wild! Murray Gell-Mann, sitting at an international conference at CERN in Geneva one day in 1962, was able to look at symmetries described by formulas written on the board, run down the aisle and write excitedly on the board his prediction of the existence of a new particle

, called the Omega Minus! (later confirmed in particle accelerator work

).

A digram illustrating the symmetry of Gell-Mann's "Eightfold Way" (which preceded his Omega-minus discovery and resulting new symmetry).*** Courtesy of CERN

The electroweak symmetry of the photon, the Z, and the two Ws was broken when these four photon-like particles turned into 3 massive particles (hence mass was "imparted" to them through the Higgs mechanism) and one remaining massless photon (see my previous post

).** The deep theoretical idea of a broken symmetry producing mass thus led to the theoretical birth of the Higgs boson: the then-hypothesized, and now tentatively confirmed, existence of the field associated with the Higgs particle, which through an interaction with the electroweak field gave us mass. Peter Higgs and his colleagues (Brout, Englert, Hagen, Kibble, and Guralnik, all of whom had the same idea in 1964) were able to exploit the idea of a symmetry to predict the existence of a particle-wave-field that gave mass to other particles and to itself when the universe was very young. The Higgs itself was thus born from the pure mathematical idea of symmetry, captured through the theory of continuous groups. To be sure, this idea was already "in the air" before these papers were published, and other theoretical physicists had understood it well and published papers about it from 1954 to 1961. But a deep trap quickly materialized. A physicist now at MIT, Jeffrey Goldstone, had (with the help of Steven Weinberg and Abdus Salam) proved a theorem that showed that

, under certain circumstances, "bad" particles---massless like the photon---somehow appear when the primeval symmetry of the universe (in which the Higgs supposedly did its mass-giving magic) breaks down. This was devastating news: the physicists didn't want these massless bosons there, since they ruined the theory about how mass can be imparted to particles. Something had to be done about it! So then came our "gang of 6" (Higgs, Brout, Englert, Hagen, Kibble, Guralnik) and, technically, what they all did was to show that the offending Goldstone theorem did not apply to the particular symmetry relevant to the early universe.**** And so, with the hurdle of the nasty Goldstone-Weinberg-Salam theorem finally removed, the road was finally wide open for the greatest symmetry of all time to break down dramatically through the interaction of the electroweak field with the Higgs field, resulting in mass being given to the Ws and the Z, leaving only our lonely photon as massless. This purely theoretical advance in 1964, culminating in Weinberg's Nobel paper of 1967, thus made the Higgs mechanism emerge triumphant, and mass was shown to be conferred to particles---allowing us to come into being and contemplate the birth of the universe we live in now. It all happened because of the mathematical idea of symmetry and the uncannily powerful relationship between pure mathematics and theoretical physics! * Two more quick examples of symmetries in physics: Einstein's general theory of relativity enjoys an important symmetry called "general covariance

"---and it gives the theory its power and validity. Maxwell's theory of electromagnetism has a particular Lie-group structure that allows the theory to remain valid even when "rotated" in an abstract mathematical space through the action of a mathematical group. The group involved here is the group of all possible rotations of a circle (rotations by any given angle). This Lie group is called U(1). ** This symmetry is modeled by the continuous group SU(2)xU(1)

, a product of the "circle rotations" group U(1) and the group of special unitary 2 by 2 matrices, SU(2). This composite group is believed to have governed the electroweak symmetry that existed shortly after the Big Bang and which broke down through its field interacting with the Higgs field. It was explained by Steven Weinberg, Sheldon Glashow, and Abdus Salam,

The full standard model is represented today by the composite group SU(3)xSU(2)xU(1), which adds the quarks (with their 3 color charges) to the picture. *** It shows the proton, neutron, and xi, sigma, and lambda mesons (intermediate-sized composite particles). This is one representation of the Lie group SU(3), the group of special unitary 3 by 3 matrices. **** It's called a Yang-Mills gauge symmetry: a particular kind of continuous, Lie group symmetry.