Why Does E=mc2_ - Brian Cox [77]
Returning to the task at hand, since quantum theory defines the rules of the game, we are obliged to talk of electron fields. But having specified our field and laid out the landscape, we are not quite done. The mathematics of quantum fields has a surprise lurking. There is some redundancy. For every point on the landscape, be it hill or valley, the mathematics says that we must specify not only the value of the field at a particular point (say, the height above sea level in our real-field analogy), corresponding to the probability that a particle will be found there, but we need also to specify something called the “phase” of the field. The simplest picture of a phase is to imagine a clock face or a dial (or a gauge) with only one clock hand. If the hand points to 12 o’clock, then that is one possible phase, or if it points to half-past, then that would be a different phase. We have to imagine placing a tiny clock face at each and every point on our landscape, with each one telling us the phase of the field at that point. Of course, these are not real clocks (and they certainly do not measure time). The existence of the phase is something that was familiar to quantum physicists well before Glashow, Weinberg, and Salam came along. More than that, everyone knew that although the relative phase between different points of the field matters, the actual value does not. For example, you could wind all of the tiny clocks forward by ten minutes and nothing would change. The key is that you must wind every clock by the same amount. If you forget to wind one of them, then you will be describing a different electron field. So there appears to be some redundancy in the mathematical description of the world.
Back in 1954, several years before Glashow, Weinberg, and Salam constructed the Standard Model, two physicists sharing an office at the Brookhaven Laboratory, Chen Ning Yang and Robert Mills, pondered the possible significance associated with the redundancy in setting the phase. Physics often proceeds when people play around with ideas without any good reason, and Yang and Mills did just that. They wondered what would happen if nature actually did not care about the phase at all. In other words, they played around with the mathematical equations while messing up all the phases, and tried to work out what the consequences might be. This might sound weird, but if you sit a couple of physicists in an office and allow them some freedom, this is the sort of thing they get up to. Returning to the landscape analogy, you might imagine walking over the field, haphazardly changing the little dials by different amounts. What happens is at first sight simple—you are not allowed to do it. It is not a symmetry of nature.
To be more specific, let’s go back and look at only the second line of the master equation. Now strike out all of the W, B, and G bits. What we have is then the simplest possible theory of particles that we could imagine: The particles just sit around and never interact with each other. That little portion of the master equation very definitely does not stay the same if we suddenly go and redial all the little clocks (that isn’t something that you are supposed to be able to see by just looking at the equation). Yang and Mills knew this, but they were more persistent. They asked