Q: What is the physicist's concept of symmetry? (Submitted by Jessica Williamson, Birmingham, Alabama)A: In mathematics and physics, symmetry has a very precise and specific meaning that is related to, but narrower than, the everyday use of the word. Physicists say that something has symmetry if there are transformations that might have changed it, but in fact do not.
This general formulation, which turns out to be very fruitful and powerful, may at first seem a little abstract. Some simple examples will make the meaning clear. For our transformations, well consider rotations of a plane around a chosen point C. What kind of geometric figures are left unchanged by all these transformations? The most basic ones are the circles centered at C; one can also combine such circles, for example to make disks or annuli. On the other hand, a square centered at C is changed by most rotations, but not by a rotation through 90°. So circles have more symmetry than squares; but squares do have significant symmetry.
Illustration: Symmetry of geometric figures. (a)Circles, (b)disks, and (c) annuli centered at a point C are left unchanged by rotations of the plane around C. (d) Square centered at C is (e) changed by most rotations, but (f) not by a rotation through 90°. Add to 'My Saved Images'
Physical systems often settle into symmetrical configurations. For example, if we allow large numbers of identical molecules to cool and condense they tend to arrange themselves into regular lattices, which have a lot of symmetry. When this occurs, it greatly simplifies the mathematical description of the system.
A different use of the symmetry concept plays a central role in modern physics. Rather than geometric objects, we consider the fundamental laws describing the behavior of a system, or equivalently its governing equations. We say that the laws have symmetry if there are transformations of the system that might have changed the laws, but in fact do not. For example, the idea that the same laws of physics that work in our neighborhood apply everywhere amounts to postulating symmetry under transformations that rigidly translate the universe, bringing our neighborhood anywhere we please. As another example, the basic postulate of (special) relativity is that transformations involving moving the universe at a constant velocity do not change the laws of physics.
In modern theoretical physics, symmetry has proved to be a reliable guiding principle for formulating the most fundamental laws of Nature. By assuming that the basic equations will have a lot of symmetry, one can go a long way toward determining these equations. This strategy has been very successful. The equations of general relativity (our fundamental theory of gravity) and of quantum chromodynamics (our fundamental theory of the strong subnuclear interaction), in particular, were constructed as consistent embodiments of symmetry principles.
Frank Wilczek, Ph.D.
