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Gauge Theories

From time to time some aspect of science or technology crops up in the daily news. Whenever this happens, scientists and engineers are expected by their friends and acquaintances to be able to explain that aspect on short notice and without notes. As a result of this expectation I have, on occasion, found myself in somebody’s living room, cocktail in hand, trying to sum up everything a contemporary American should know about Higgs bosons, the standard model, or Apple’s new Yosemite operating system. Some of these subjects are ripe for cavalier generalizations,while othrs are virtually impossible to get across in fewer than five or ten semester hours. One good example of the latter, which crops up whenever some physics experiment does something, is “what is a gauge theory?”

If you bring up Wikipedia’s entry for “gauge theory” you get:

“A gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.” There are, of course, links to Wikipedia entries for “field theory,” Lagrangian,” “invariant,” “continuous group,” and “transformations.” Inside each of the discussions so linked are a bunch of further terms, also linked, etc, etc.. This obviously is not a subject that lends itself to satisfactory answers satisfying idle curiosity.

The Encyclopaedia Britannica entry for “gauge theory”, being professionally written and edited, is a bit more to the point for the average reader:

A gauge theory is “A mathematical theory involving both quantum mechanics and Einstein’s theory of relativity that is commonly used to describe subatomic particles and their associated wave fields.” Several percent of the population remembers from high school physics that Albert Einstein was not a fan of quantum mechanics (God doesn’t throw dice) so that anything that combines his theory and quantum mechanics is going to be doing something wonderful. Also we are warned that it’s a “mathematical theory,” and so any adequate answer is going to be in terms of higher mathematics. There is, however, obviously more to the subject that begs for further explanation, and both Wikipedia and Britannica labor on for many pages. The challenge is in providing that explanation in simple prose without the use of mathematics., however, which neither is quite able to pull off.

The problem that those two explanations have is that a gauge theory is any physics description of how the universe works that is contained in mathematical equations that have a certain property. That mathematical property is difficult to describe in so many words, while what most curious people would want to find out about is what the physical theories themselves are about. So perhaps we can sneak up on a simplified explanation by ignoring the mathematics and just worrying about the physics.

The first gauge theory was invented in the nineteenth century by James Clerk Maxwell; several decades before quantum mechanics, and before the discovery of the electron or any other elemental particles. Maxwell had no idea that he had produced a gauge theory for the simple reason that the term wouldn’t be invented for another century. What Maxwell did was to take a large number of algebraic relationships among electrical and magnetic properties that had been recently discovered by Michael Faraday and others and combine them all into four equations. Two of these equations were couched in differential calculus and the other two were in vector algebra, and it would never have occurred to Maxwell to worry about whether or not they had the mathematical property needed to be a gauge theory.

Einstein had a very high opinion of Maxwell, and kept a portrait of Maxwell hanging in his office at Princeton’s Institute for Advanced Study. Maxwell’s equations constitute a very powerful handle on the physics of electricity and magnetism, and are in constant use today in designing everything from power plants to cell phones. Beyond their practical present uses, however, their appearance in nineteenth century science was earthshaking, and affected not only science but also philosophy. Isaac Newton’s equations didn’t constitute a gauge theory, but they had the exciting property that if the mass, position and velocity of every particle in the universe were known at a particular time, then it would be theoretically possible to use those equations to calculate everything that had ever happened before that time and everything that would happen in the future. In short, Newton had destroyed free will. In addition, his view of gravity wasn’t a field theory, i.e., a massive object did not generate a gravitational force field that propagated into the space surrounding it; instead, gravity’s attractive force just magically flew at infinite speed between any two masses, creating the dreaded “action at a distance” problem.

Newton only worried about gravity, and he didn’t know about or deal with electricity or magnetism (he lived in the seventeenth century). None-the-less, the nineteenth century theory of electricity that Maxwell had access to used exactly the same form of equation to describe electrical force that Newton used for gravity – the product of either two masses or of two electrical charges divided by the square of the distance between them. Maxwell’s equations, however, are field equations. The remarkable thing about them was that they coupled the electrical and magnetic fields such that electric fields depend upon the rate of change of the magnetic field and vice versa. Adding these four new equations to Newton’s three equations really messed up the prospects for calculating the past and future from just the mass, position and velocity of every particle. If any of the particles were charged, or if there were any electric of magnetic fields involved, then all bets were off. But by describing fields, Maxwell’s equations had the possibility of possessing the weird mathematical property that would make them gauge fields, and as luck would have it, they did.

Maxwell’s equations also contained a velocity with which fields propagated. They said quite clearly that if you drove electrons back and forth along a copper wire, then the electrons would radiate an electromagnetic wave out into the surrounding space at the speed of light. Of course, electrons wouldn’t be discovered for another few decades…. Maxwell was way ahead of his time and missed out on a lot of potentially nifty patents.

What the solutions to Maxwell’s equations and the wave functions of subatomic particles calculated from quantum mechanics have in common is that they deal with the amplitudes, phases and polarizations of waves; electrical and magnetic waves in the first instance, and probability waves the second (well, actually, a quantum mechanical wave function is the complex square root of a probability, but we agreed not to use mathematics). In neither case, however, are the amplitudes, phases or polarizations really directly observable. You can concoct an experiment whereby you can measure a photon (a quantum of electromagnetic wave) or a subatomic particle (a quantum of that particle’s wave function) from which you may be able to deduce information about one or another of those quantities, but they really aren’t directly measurable. In short, both Maxwell’a and quantum mechanics theories are systems of equations dealing in hidden quantities that can’t be measured, but from which truly measurable quantities can be calculated.

By now you should be comfortable imagining photons as ripples of electrical and magnetic fields moving along at the speed of light, and can almost believe that other subatomic particles move around as similar ripples in some other kind of field. Photons may also be, of course, perfectly good subatomic particles, so there is a wave function for them as well, which also ripples along at the speed of light and contains all the information needed to describe the electrical and magnetics fields.

Wave functions come in a great variety of shapes and sizes. Chemists, for example, work with standing waves…. Wave functions that don’t go anywhere but just bounce back and forth inside a molecule. Something called a Higgs field permeates all space, consisting of Higgs bosons that could exist anywhere if they wanted to but don’t exist unless prompted, the field being composed of that possibility. If a wave function can interact with the Higgs field it has mass and can’t ever go as fast as light, while if it can’t interact it can’t slow down or stop and must go at exactly the speed of light. One property they can have is that they can look the same no matter how the observer is moving with respect to them. When this occurs, the particle associated with that wave function can be part of a gauge theory.

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