A Stranger in Gaugeland...
Once upon a time, there was a clockmaker who woke up one morning and found himself in a strange foreign land. I could go on about how he was forced to come to terms with his situation and build a new life in this place, but that is another story and shall be told at another time. For now, I will tell you about just one aspect of this land to which he had to adjust.
In our world, we take for granted the fact that it is easy to know the time of day in any part of the Earth -- to relate a time in a different location to that in our own, we need only look up its time zone from a table that is available anywhere that the internet is. In Gaugeland, however, life is not so simple. There is no universal table telling you the shift in time at all locations; rather, in each and every time zone there is a marker (which they called a “gauge”) which tells you how to adjust your watch when entering or leaving the zone. Worse, the divisions of their time zones are sporadic, sometimes being as small as a single house, or even a single patch of road. So for example, if you were returning from enjoying a cup of tea with a friend and wanted to make a trip to the baker on your way home, you might have to:
- check the gauge in front of your friend’s house to find out how to adjust your watch when entering the street
- read the gauge at the corner to see how the time zone changed when turning right on Main St
- look for the gauge at the baker to see how the time changed when entering his house (as opposed to walking past it, which would be yet a different time)
For the people of Gaugeland, this was just an ordinary part of life; one simply did not expect to be able to know the time at a different place until you got there. To the clockmaker, however, this was anathema. Part of the pleasure of his job had been knowing that he was making people’s lives more orderly by helping them all know the correct time, and it drove him insane to see even this last vestige of his old life torn away from him.
So the clockmaker went to the King of Gaugeland and said, “Good king, surely you see that the time zones are madness! In my old land there were only 24 time zones, but here I was able to count over 3000 separate zones before I grew sick of keeping track. At the very least, perhaps we could make a table that tells people which time zone is where so that they do not have to change their watches every five steps.”
The King gave the clockmaker an amused smile, and replied, “Very well. If you can assemble such a table, then I will have it printed and distributed everywhere in the kingdom.”
The clockmaker beamed, taking this as a declaration of full support. He thanked the King and immediately started his task. As he left the castle he looked at the great clock on the front and saw that it read “12:00”. He realized that the castle would be the perfect reference point for his time zone table, so he opened his notebook and wrote, “Castle is at T=0.” He smiled, feeling like he was already making progress.
He then journeyed north of the castle, and it only took him five minutes to find a gauge which read “Travelers from the South, set your clocks forward 15 minutes.” In his notebook he recorded, “North of castle is at T=+15 minutes.”
He then walked east, and as before it was only a few minutes he found another gauge, which read: “Travelers from the West, set your clocks forward 15 minutes.” Into his notebook he wrote, “Northeast of castle is at T=+30 minutes.”
South he travelled, and as before it was only a few minutes before he found a gauge which read: “Travelers from the North, set your clocks forward 15 minutes.” He wrote, “East of castle is at T=+45 minutes,” and grinned. True, he had only covered the tiniest fraction of the land, but it was now only a matter of time before all time zones were tabulated, and a semblance of sanity was brought to this strange place.
He walked west, returning to the castle. At the castle he saw a gauge -- this one, of course, much grander than all of the ones he had encountered so far -- which read “WELCOME, SUBJECTS, TO THE CASTLE OF THE KING OF GAUGELAND! GOOD TRAVELERS FROM THE EAST, PLEASE SET YOUR CLOCKS 15 MINUTES FORWARD.”
He paused, and read the gauge again. This could not be! He had returned to where he had started, so he must be back at T=0. The gauge must surely be wrong -- the correct setting should be “45 minutes backward.” But when he looked at the great castle clock, he saw that it read “1:20”, which was consistent with T=+1 hour and 20 minutes of walking.
Thinking that he must have made a mistake, the clockmaker walked the circuit again, but experienced the same results. He then walked in a different circle taking him northwest of the castle... and this time found that he was an hour behind.
Well, he thought, if this is the way that this place is, then he may as well at least write down what is on the gauge signs so that there could be a table of something. But when he repeated his original northeast circuit (to double check his recordings) he found that the gauge signs had changed -- this time when walking in the very same circle, he was only 30 minutes ahead.
Now he knew why the king had smiled: in the land of Gaugeland, it was physically impossible to know the time of distant places! That night he dreamed of Escher’s staircase -- a circle of stairs in which you had to climb continuously upward just to return where you started -- but with the additional madness that the stairs themselves kept moving around so that you never knew how far up each step would take you from one moment to the next.
This madness drove him to obsession. He studied the gauges of Gaugeland, and learned that it was not everywhere that had this property. In remote parts of the wilderness, walking in a circle did not take you into a different time. It was only where people lived that this strangeness occurred, and as the people moved from place to place the distortion changed with them. Furthermore, where it was present it strongly impacted peoples’ comings and goings, which in turn fed back into a fluctuation of the gauges. The people and the gauges of Gaugeland were tied together by a deep, mysterious connection...
Seeking order in the madness, the clockmaker found that although he could not figure out why Gaugeland was this way, he could at least through experimentation determine the rules that it obeyed. He learned that although the behaviors of the gauges and the people were complicated, the rule that gave rise to them was actually quite simple. I will not tell you the details, but suffice it to say that magic of the kingdom arranged matters so that the gauges and the population density were just what they needed to be in order to minimize a quantity called the “action”, which was a mathematical function of the time history of the two quantities.
With this, he was now satisfied. For although there was still madness in his new home, at least there was an order to be found within it.
I was not fully honest with you in telling the above fable, for it turns out that Gaugeland is not really a distant, magical place but rather it is the Universe in which we live.
I say that, however, with a caveat: I am not referring to how matter bends spacetime, which is a quite different subject. Instead, it turns out that at every point in space and time each particle has an associated angle called its “phase”. (More precisely, all the particles of a particular kind together form a single field which has a phase at every point in spacetime; you can think of the individual particles themselves as being like waves of an ocean -- their identity is an illusion, for the true thing is the field and they are just visible excitations of it.) This phase is similar to the time of day of our story in that it is a cyclic quantity -- advancing a phase by 360 degrees obtains an identical phase, just as advancing the time by 24 hours gets you the same time of day.
As you travel from one place to another, you pass through different “phase zones” (my own invented term), so you have to update your “phase” so that it matches with the locals. If you did not, then you wouldn’t be able to have a meaningful conversation with them about the phase, just like here you would not be able to talk correctly about the time if you travelled a thousand miles away without changing your watch. What makes life complicated (and interesting!) is the fact that when you travel in a circle, your phase “watch” might not be reset to where it was when you started -- whenever this happens, you have discovered the presence of an electromagnetic field somewhere inside your circle! And just as the time zones and the people of Gaugeland are linked by magic, so too is it that the phase zones in our world are bound together with the presence of charged particles; the former influence how the latter move, and the latter dictate the “madness” in the former.
Even stranger, it turns out that at each point in space there is not only a phase, but there is also a member of “SU(2)” and “SU(3)” -- which you could almost think of respectively as rotations in 2- and 3-dimensional space, except that since there are complex numbers thrown into the mix the full story is a little more complicated. Particles interact with these just as they interact with U(1) (the proper name for the group of “phases” that we were discussing earlier), and likewise the curls (the limit of what happens when you walk in an infinitely small circle) of these gauge potentials (the proper name for the the “gauges” of Gaugeland) influence how the particles move.
Now, most treatments of gauge fields begin where I shall end, which is to mention that the exact “phase zone” at each point is actually not important; what is important is that we know how to compare the phase at one point in space to the phase at another. (This is important because the faster the phase changes over space, the more momentum a particle has.) Thus, if you wanted, you could “set the clock forward” at some point in space and then update all of the local gauges, and everything would be fine. This is referred to as the gauge invariant nature of a theory -- you don’t care about exactly what the phase zones are as long as you keep proper track of how to update your “watch” when you move from one point to another. This is a local symmetry because it gives you something that you could change at every point in space and time without altering the behavior dictated by the theory; no matter what you do, you will not be able to get rid of the fact that walking in a circle doesn’t get you where you started.
This might seem horrendously complicated, but understand that I have told you everything that we know about the fundamental nature of the laws of the universe. That is, there is nothing more complicated going on than a set of particles, a set of numbers (one from U(1), SU(2), and SU(3)) telling you how to update your three local “clocks” as you move through space and time, and a rule (the “action”) which tells you everything you need to know about how distortions in the latter affect the former and vice versa. This is what is meant when we say that the Standard Model of the Universe is U(1) x SU(2) x SU(3).
Obligatory Warning: The above is a “fairy tale introduction” to gauge field theory, so it should not be taken as a mathematically rigorous description. For example, since we treat spacetime as being continuous (except in so-called “lattice gauge theories”) the gauge potentials are not actually members of U(1), SU(2), and SU(3), but rather are members of their associated Lie algebras. But that is another story, and shall be told at another time