Hidden Symmetry

By Bruce A. Schumm




The following is excerpted, with permission of the author, from Deep Down Things—The Breathtaking Beauty of Particle Physics.

Most of us who are unfortunate enough to remember what life was like when we were thirteen years old will recall that one of the predominant influences that shaped our lives at that age was the unrelenting coercion of peer pressure.

Picture a schoolyard on a crisp December morning, with a delicate layer of frost adorning the otherwise brown grass. Add to the picture several hundred barefoot teenage boys, each one confronted by a choice between one of four pairs of jogging shoes produced by four different manufacturers. Frankly, even in the best of conditions, it doesn’t really matter which pair one chooses; they are all comfortable and well constructed. And for the situation at hand, a schoolyard full of young men shuffling around barefoot in the cold, it should matter even less. Any of the four pairs will separate the soles of their feet from the numbing cold.

The situation is symmetric with respect to the choice of shoes; the desired effect (saving one’s toes from frostbite) is independent of which company’s shoes each student chooses. But each student does have to make a choice.

Teenage mentality being as it is, none of the boys dares to put on a pair of shoes, for fear of picking the pair that exposes their lack of awareness of fashion. Eventually, some student decides he can’t take the cold any longer, and he grabs one of the pairs of shoes and puts them on as quickly as his numbed fingers permit. His neighbors, accepting his cue, quickly follow suit, and in relatively short order, the entire field of adolescents stands in a state of relax, uniformly clad in the same brand of shoe.

At this point, a department store employee on his way to work happens to glance over to the schoolyard and sees the multitude of young men standing there, all wearing the same brand of shoe, surrounded by discarded pairs from the three other manufacturers. His gut reaction is to rush into work to tell the manager to order a large number of the selected brand’s shoes, but on further reflection, he thinks better of it. Why?

The reason is that the symmetry, the intrinsic underlying equality of all four brands, still pertains. A student from another school who hadn’t been at this particular schoolyard at the time of the arbitrary selection of the chosen brand would be no more likely to purchase that brand than any of the other three. In our schoolyard on that frigid December morning, the symmetry was not broken by the capricious selection of one of the brands over the others, but it was masked, or hidden, by the choice. A careful study, in this case, appropriate interviews by a person well schooled in adolescent psychology, would reveal that the symmetry is still there. Eventually, the school counselor would get the students to admit they couldn’t have cared less which of the pairs of shoes ended up on their feet.

This is the essence of the notion of hidden symmetry. One can be presented with a physical system whose appearance is one of broken symmetry (the selection of the same pair of shoes by all the students seems to suggest that pair is somehow preferable) that belies the true underlying symmetry of the system (the fact that there really is no difference whatsoever in the quality of the four choices of shoes, and the choice of a particular pair was completely arbitrary). Some additional external factor, in our example, the discomfort of standing around barefoot, required that some choice, no matter how arbitrary, be made by the system. Once the choice was made, some internal property of the system—the psychological interdependence of the students—required that the entire system adopt the same arbitrary choice, leading to the masking, or hiding, of the true underlying symmetry. But the symmetry is still there. No one really could give a damn about which pair of shoes are on his feet.

Bruce A. Schumm is a professor of physics at the University of California at Santa Cruz