Continuing with our series on symmetry, in Part IV we discuss a frequently encountered term, gauge symmetry. Terry Tao, who among his many distinctions earned full professor status at UCLA at age 24, has addressed the concept of a gauge, on his blog:
“Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations).
Writers have offered several everyday examples of gauge systems. The main idea I extract from them is that, if you add a constant to every value in a system, the behavior of the system is unchanged.
Dan Hooper, whose previous book Dark Cosmos I reviewed here, uses the example of mountain-climbing and altitude to explain the gauge concept in his new book, Nature's Blueprint. The amount of upward climbing a person does from the base to the top of Mount Everest is the same -- 11,528 feet -- whether the frame of reference is sea level (where the base is said to be at 17,500 feet and the top at 29,028) or the reference is the lowest point on earth beneath the oceans (53,338 to 64,866).
As Hooper notes, "With global gauge symmetries, the arbitrary scale can be changed without affecting anything, but it must be changed at every location at the same time" (p. 64). If, therefore, in our discussions of altitude we added a constant 35,838 feet all throughout the earth in order to change the baseline from the depths of the oceans to sea level, we would have a global gauge symmetry.
Another kind of symmetry, where one type of change is made at one location but a different type of change is made at another, is called local gauge symmetry. Hooper credits Hermann Weyl with realizing the implications of local gauge symmetries for physics.
Ian Stewart, in his book on symmetry, Why Beauty is Truth, uses monetary systems to illustrate local gauge symmetry:
Suppose that you are traveling in a different country -- let us call it Duplicatia -- and you need money. The Duplicatian currency is the pfunnig, and the exchange rate is two pfunnigs to the dollar. ...everything costs twice as many pfunnigs as you would expect to pay in dollars.
This is a kind of symmetry. The "laws" of commercial transactions are unchanged if you double all the numbers. To compensate for the numerical difference, though, you have to pay in pfunnigs, not in dollars.
...Just across the border, in neighboring Triplicatia, the local currency is the boodle, and these are valued at three to the dollar. When you take a day trip to Triplicatia, the corresponding symmetry requires all sums to be multiplied by three. But again the laws of commerce remain invariant.
Now we have a "symmetry" that differs from one place to another. In Duplicatia, it is multiplication by two; in Triplicatia, by three. You would not be surprised to find that on visiting Quintuplicatia the corresponding multiple is five. All of these symmetry operations can be applied simultaneously, but each is valid only in the corresponding country...
This local rescaling of currency transactions is a gauge symmetry of the laws of commerce. In principle, the exchange rate could be different at every point of space and time, but the laws would still be invariant provided you interpreted all transactions in terms of the local value of the "currency field" (p. 231).
If you go to the "Review of the Universe" summary of particle physics, particularly the section entitled, "Gauge Theory and the Standard Model," you will find a graphical representation of the distinction between global and local transformations in Figure 15-06a. The original and globally transformed objects (spheres) look identical, whereas the locally transformed one looks very different.
The next figure, 15-06b, shows (schematically) the process of local transformation. Between steps 1 and 2, the small dial on the right appears from visual inspection to have been rotated 180 degrees, whereas the other two dials have been rotated by amounts other than 180. Yet as step 3's caption says, the gauge field compensates for the change, and as step 4's caption says (pointing upward on the right), "The original field configuration is restored." (Note that the term "compensation" was also used by Stewart in his above quotes about the hypothetical monetary unit, the pfunnig.)
Such local symmetries are behind many of the major concepts of physics. Writes Stewart:
The possibility of changing the phase arbitrarily at each point of space-time, with no global constraint to make the same change everywhere, is a gauge symmetry of Maxwell's equations, and it carries over into the quantum version of those equations, quantum electrodynamics (p. 233).