Most of the important equations in physics -- those that concern the flow of a fluid, the action of gravity, the motion of the planets, the transfer of heat, the movement of waves, the action of magnetism, and the propagation of light and sound -- are differential equations (p. 162).
Lie groups also, however, have the property of mapping onto defining features of three of the forces of nature -- the electromagnetic, weak, and strong forces -- and their force particles.
Dan Hooper's book Nature's Blueprint details the mapping of Lie groups onto physical forces and their particles, noting the contributions of Murray Gell-Mann to advancing physicists' understanding of this topic. Before we get to Gell-Mann's contributions, however, we'll start out with some of the more basic ideas. Notes Hooper:
Mathematically speaking, the U(1) group describes the most simple of all possible local gauge symmetries. In terms of particle physics, it corresponds to a single type of boson (the photon) that acts on a single kind of quantum number (electric charge) (p. 70).
In layperson's terms, we can simply think of an ordinary circle. We can rotate this circle by any amount, whether an infinitesimally tiny nudge or a large number of degrees, and the circle will retain its appearance -- which, after all, is the signature definition of symmetry.
To get a little bit mathematical, as described in this document by Clark University's David Joyce, "The unit circle is the circle of radius 1 centered at 0." The unit circle follows the equation x-squared + y-squared = 1. Among the points on the circle would be (1, 0), (-1, 0), (0, 1), (0, -1), (.707, .707), (.707, -.707), etc. Spicing things up a bit, we must also incorporate complex numbers, in particular the imaginary square root of -1, denoted i in this terminology.
As stated on the Wikipedia page on unitary groups (that's where the "U" in "U(1)" comes from), "...the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication."
U(1), as we noted above, pertains to the electromagnetic force. The physical analogue to electricity, as explained by Rachel Thomas in Plus Magazine, is as follows:
"...the electromagnetic field around a conducting wire is symmetrical around the wire -- you can rotate the wire (around an axis running along the wire) and it won't affect the electromagnetic field. This kind of symmetry is described by the group of rotations on a circle, called U(1), and this defines how particles behave in an electromagnetic field."
One other source that readers might find helpful in understanding the U(1) symmetry group is Peter Woit's book Not Even Wrong (pp. 35-41).
The strong and weak forces correspond to much more elaborate Lie groups than U(1). We'll get to that in the next entry...