## The Triumph of the Accountants, or, How I Learned to Stop Worrying and Love the Higgs Boson

Trust the math.

No matter how crazy it seems.

If you are at all hooked into the world of science–and if you are not, you probably are looking for a different website–you must be aware of the hoopla surrounding the discovery of the Higgs boson. The news was splashed across the papers, filled the covers of *Nature* and *Science* and *Physical Review*. Peter Higgs, he of the eponymous particles, a gentle and shy man, became an instant celebrity, which ramped up afterwards with the awarding of the Nobel Prize to him and Brout. Physicists argued, spittle flying, over who truly deserved the prize.

But what was the fuss about the Higgs? Why was it so important?

It short, it was all about the math. It was, in the end, making sure the microscopic books added up. It was about *accounting*.

For many people math is about as much fun as a tooth cleaning; important, they assume, but not something you’d *want* to do. Barbie dolls complain “Math is hard!” sending the wrong message to generations of girls. (Though just recently we saw the first woman recipient of the Fields prize, the math equivalent of the Nobel.) They see mathematical formulas as an arbitrary jumble of symbols.

But math has its own grammar, its own narrative; even its own surprising plot twists. This is true not only in the field of abstract mathematics, where one can discover true statements–the endless infinitude of primes, the murderous fact that square root of 2 cannot be represented as a fraction, the shock at understanding that not only are there are infinite number of numbers, there are different *kinds* of infinity, each infinitely bigger than the previous one–but in the application of mathematics to physics.

Some of the early applications were useful. Newton showed that if the strength of gravity fell off as square of the distance, then not only could he explain Kepler’s elliptical orbits for planets, he could also predict the parabolic paths of comets, and of artillery shells. Later astronomers used Newton’s equations to deduce, from tiny deviations their orbits, the existence of new planets. Heinrich Hertz, studying Maxwell’s equations describing light as an electromagnetic wave, realized that there must be electromagnetic waves with very different wavelengths, and then went and discovered radio.

Still, Herschel had discovered the planet Uranus simply by using a telescope, so using math to find one was clever but not shocking. Radio is nothing more than a “color” far beyond red and infrared.

But as mathematical physics became more and more sophisticated, on occasion physicists painted themselves into a corner.

While Maxwell’s equations described light and radio as electromagnetic waves, physicists were puzzled. Most waves moved in a medium. What about light? Shouldn’t there be a “luminiferous ether”? Isn’t the Earth moving through that ether? But Michaelson and Morley’s experiment to detect the ether failed. And Maxwell’s equations lacked any term for the medium.

In 1905, Albert Einstein said, Let’s take Maxwell’s equations seriously. Assume there is *no* medium dependence for light and radio and x-rays and all electromagnetic waves. From that, he deduced the equations for special relativity, which in turn imply that observers moving at high relative velocities disagree on their measurements of space and time. He also derived *E=mc ^{2}*, the cosmic exchange rate between energy and mass. Totally crazy ideas, but they were

*confirmed experimentally*.

Quantum mechanics arose in the same way. In 1900, Max Planck imagined a box full of electromagnetic waves, heated by an outside source. Standard calculations suggested more and more energy in the shorter wavelengths, building up to infinity in the so-called “ultraviolet catastrophe,” something which was *not* observed experimentally. Out of desperation Planck limited the energy of an electromagnetic wave by making it proportion to the frequency. This accounting trick solved the problem and matched experiment, and a few years later Einstein borrowed the same idea to explain the photoelectric effect, envisioning light as little packets (“photons,” a name not coined until 1926) with a wavelength below a certain limit can knock electrons out of a metal. This won him the Nobel Prize in 1921.

If light is a wave but also comes in little particle-light packets, then why can’t particles like electrons also behave like waves? Niels Bohr and then Louis de Broglie developed this idea, which sounded totally crazy, until Davisson and Germer in 1927 demonstrated electrons diffracting off a crystal just like light.

But the corresponding theory of quantum mechanics also leads to entanglement, which Einstein objected to, calling it “spooky action at a distance” although it is really a *correlation* and not a *causation*. Entanglement has been demonstrated experimentally and you can even buy commercial quantum cryptography equipment using entanglement.

It gets weirder. In 1927 Paul Dirac wanted to combine special relativity and quantum mechanics, the two Big Theories of the early 20th century. He came up with a nice equation, but rather strangely, it predicted particles no one had seen, particles with the opposite properties of those expected. It wasn’t until 1931, when Carl Anderson discovered *positrons* in a cosmic ray shower, that it was realized Dirac’s equation predicted the existence of *anti-matter*.

In all these cases the mathematics derived to explain known physical phenomena also predicted results which seemed not just counter-intuitive but downright ridiculous… until they were experimentally confirmed.

And it’s the same way with the Higgs boson.

In the postwar era not only had quantum mechanics, special relativity, Dirac’s equation, antimatter, etc., had been accepted by the physics community, we had become skilled in dealing with its crazier aspects. Similar to how the ultraviolet catastrophe arose from a theoretical buildup of an infinite amount of energy, calculations in *quantum field theory* routinely also led to infinities. This is because Dirac’s equation not only allowed for the existence of antimatter, it positively demanded pairs of particles and antiparticles be created and absorbed in a blink of an eye. Fortunately physicists found a way to tame the infinities, essentially a way to subtract infinity from infinity to get a finite answer. Not only was the answer finite, it agreed with experiment! A true triumph of mathematical ingenuity.

Not only that, but Dutch physicist Gerard ‘t Hooft proved a class of quantum field theories known as *gauge field theories* could always be tamed, or *renormalized*, in this way.

That was the good news. The bad news was, an essential ingredient, *mass*, turned out to *not* be renormalizable under the recipes of gauge quantum field theories. Put mass in your Lagrangian (the descriptive formula for a theory) and it blew up. Who knew something so basic could be so much trouble?

This is where Peter Higgs and others (Anderson, Brout, Englert, Guralnik, Hagen, and Kibble, to name the major ones) came in. They devised an admitted Rube Goldbergian mechanism for creating particles with mass that didn’t cause any additional infinities, satisfying the account books. But nothing comes for free, and the price here was the existence of an additional, undiscovered particle which permeates all of space and time. The particle coupled to all other fundamental fermions like an anchor, which is a good description for mass.

It’s a crazy idea, but without the Higgs boson the rest of the “Standard Model” of particle physics did not add up. And the agreement of the Standard Model with experiment was breathtakingly precise.

So the multibillion dollar/euro search for the Higgs at the Large Hadron Collider was actually a fairly safe bet. Like radio waves from Maxwell’s equations, like antimatter from Dirac’s equation, the Higgs boson was almost an unescapeable conclusion.

Almost. Despite the luxurious mathematical gilding, physics is at bottom an empirical science. And it is truism among experimental physicists that the only thing they enjoy more than proving a theory right is proving a theory *wrong*.

In the end, the accountants were right. The books added up, the Higgs boson, avoiding uncomfortable infinities from here to Ursa Major and beyond, was discovered last year, and Higgs, and Peter Higgs and Francois Englert were awarded the Nobel Prize in Physics in 2013.

And I repeat the message:

No matter how crazy it seems,

trust the math.

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