Gauge Symmetry and the Theta-Vacuum

ly, a symmetry of a structure is an automorphism— a transformation that maps the elements of an object back onto themselves so as to preserve the structure of that object. A physical theory speci…es a set of models— mathematical structures— that may be used to represent various di¤erent situations, actual as well as merely possible, and to make claims about them. Any application of a physical theory is to a situation involving some system, actual or merely possible. Only rarely is that system the entire universe: typically, one applies a theory to some subsystem, regarded as a relatively isolated part of its world. The application proceeds by using the theory to model the situation of that subsystem in a way that abstracts from and idealizes the subsystem’s own features, and also neglects or idealizes its interactions with the rest of the world. We can therefore enquire about the symmetries of the class of models of a theory; or we can enquire about the symmetries of a class of situations, whether or not we have in mind a theory intended to model them. The …rst enquiry may reveal some theoretical symmetry : the second may reveal some empirical symmetry. An empirical symmetry can be recognized even without a physical theory to account for it. But it does not cease to be empirical if and when such a theory becomes available. A theory may entail an empirical symmetry. Galileo[2](1967, pp.186-7) illustrated his relativity principle by describing a famous empirical symmetry of this kind. Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some ‡ies, butter‡ies and other small ‡ying animals... When you have observed all these things carefully..., have the ship proceed with any speed you like, so long as the motion is uniform and not ‡uctuating this way and that. You will discover not the least change in all the e¤ects named, nor could you tell from any of them whether the ship was moving or standing still. His implicit claim is that a situation inside the cabin when the ship is in motion is indistinguishable from another situation inside the cabin when the ship is at rest by observations con…ned to those situations. The claim follows from a principle of the relativity of all uniform horizontal motion. While we know today that an unquali…ed form of Galileo’s claim is false, in a modi…ed form it continues to play an important role in physics. Galileo’s implicit claim is that situations related by a uniform collective horizontal motion are empirically symmetrical. Speci…cally