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a great question: How can we change the equation so that it does stay the same? The answer is fantastic. We need to add back precisely the missing bits of the master equation that we just struck out, and nothing else will do. In so doing we conjure into existence the force mediators and suddenly we go from a world without any interactions to a theory that has the potential to describe our real world. The fact that the master equation does not care about the values on the clock faces (or gauges) is what we mean by gauge symmetry. The remarkable thing is that demanding gauge symmetry leaves us no choice in what to write down: Gauge symmetry leads inexorably to the master equation. To put it another way, the forces that make our world interesting exist as a consequence of the fact that gauge symmetry is a symmetry of nature. As a postscript, we should add that Yang and Mills set the ball rolling, but their work was primarily of mathematical interest and it came well before particle physicists even knew which particles the fundamental theory ought to describe. It was Glashow, Weinberg, and Salam who had the wit to take their ideas and apply them to a description of the real world.

So we have seen how the first two lines of the master equation that underpins the Standard Model of particle physics can be written, and we hope to have given some flavor as to its scope and content. Moreover, we have seen that it is not ad hoc; instead we are led inexorably to it by the draw of gauge symmetry. Now that we have a better feel for this most important of equations, we can get back to the task that originally motivated us. We were trying to understand to what extent nature’s rules allow for the possibility that mass can actually be converted into energy, and vice versa. The answer lies, of course, within the master equation, for it spells out the rules of the game. But there is a much more appealing way to see what is going on and to understand how the particles interact with each other. This approach involves pictures, and it was introduced into physics by Richard Feynman.

FIGURE 14

What happens when two electrons come close to each other? Or two quarks? Or a neutrino gets close to an antimuon? And so on. What happens is that the particles interact with each other, in the precise way specified in the master equation. In the case of two electrons, they will push against each other because they have equal electric charge, whereas an electron and antielectron are attracted to each other because they have opposite electric charge. All of this physics resides in the first two lines of the master equation, and all of it can be summarized in just a handful of rules that we can draw pictorially. It really is a very simple business to get a basic grasp of, although the details take a bit more effort to appreciate. We’ll stick to the basics.

FIGURE 15

Looking again at the second line, the term that involves two ψ symbols and a G is the only portion of the equation that is relevant when quarks interact with each other via the strong force. Two quark fields and a gluon are interacting at the same point in spacetime—that is what the master equation is telling us. More than that, that is the only way they can interact with each other. That single portion of the master equation tells us how quarks and gluons interact, and it is prescribed precisely for us once we decide to make our theory gauge symmetric. We have absolutely no choice in the matter. Feynman appreciated that all of the basic interactions are this simple in essence, and he took to drawing pictures for each of the possible interactions that the theory allows. Figure 14 illustrates how particle physicists usually draw the quark-gluon interaction. The curly line represents a gluon and the straight line represents a quark or antiquark. Figure 15 illustrates the other allowed interactions in the Standard Model that come about from the first two lines of the master equation. Don’t worry about the finer points of the pictures. The message is that we can write them down and