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Gauge Theory

GAUGE THEORY

Gauge theory is concerned with the problem of comparing physical states at different space-time locations. To get a feel for the problem, it is best to begin with a simple example. Quantum chromodynamics is the theory of the force that binds quarks together. An initial presentation of the theory might begin by stating that the color force (a fanciful name having nothing to do with visual colors) comes in three color charges red, blue, and green, and their anticharges anti-red, anti-blue, and anti-green. Every quark has one of these charges, and a stable collection of quarks must have no net color. Thus, a stable three-quark object, such as a proton, can be formed from a red, a blue, and a green quark (red + green + blue = white, which is colorless), or a stable two-quark object, such as a pion, can be formed from a red quark and an anti-red quark. This would explain why quarks are never seen in isolation. Just as electric charge comes in two forms, positive and negative, color charges come in six species. The interaction between any pair of quarks will depend on their charges.

Note an immediate consequence of the little story just told. It suggests that given any two arbitrarily specified quarks, no matter where they happen to be, there is a fact about what color charge they have, and, a fortiori, a fact about whether their color charges are the same or different. Most metaphysical accounts of properties have the same consequence. If there is a universal corresponding to the color charge red, for example, then there is a fundamental fact about any pair of quarks whether they both instantiate this universal or not. Or if one prefers a theory of tropes, then there is still a fundamental metaphysical fact about whether the trope that is part of one quark is qualitatively identical to the trope that is part of the other. These metaphysical facts would obtain no matter where the two quarks were located even if they were located in different space-times.

Gauge theory rejects this metaphysics. It is correct, according to gauge theory, that there is a variety of color charges possible for each quark and that the structure of these physical possibilities is that of a genus with six species (speaking roughly). But there is no natural identification between the particular color states available to one quark and the particular color states available to a distantly located quark. Here an analogy will help.

Consider the surface of a sphere. At any point on the sphere, there is a set of directions one can move in. And the generic structure of that set at any one point is exactly the same as the structure of the set at any other point: Wherever you are, you have a full 360 degrees of different directions available. But there is no fact about whether a particular direction at one point on the sphere is the same or different from a particular direction at another point on the sphere. One could, for example, lay down a circle with degree markings from zero to 360 at the North Pole and lay down an identical circle somewhere on the equator. But there is no further sensible constraint that the two circles be oriented the same way or that the zero-degree direction at the North Pole point the same way as the zero-degree direction at the equator. One cannot sensibly ask of two arrows, one at the North Pole and the other at the equator, whether they point the same way or not.

Having placed the circle of degrees at the North Pole in one orientation, one is still free to place the circle at the equator at any orientation one likes. Such an arbitrary choice of orientation for the degree numbers is called picking a gauge, that is, fixing on a convention for assigning numbers to different directions at different points. Once one has picked a gauge, one can talk about an arrow at the North Pole and one at the equator pointing the same way (e.g., both pointing at thirty-seven degrees), but since the gauge itself was an arbitrary choice, the sameness carries no ontological weight.

When considering distant points on the sphere, it is obvious that there is no sameness or difference of direction: The set of directions one can go in at one point are, as it were, specifically different from the set of directions one can go at another. And one might then be tempted to simply index any direction by the point it is attached to: There are a set of North Pole directions and a set of Eiffel Tower directions, and so on, with none of these being intrinsically comparable to any other. But the situation is not so simple. Suppose a person is standing at the North Pole holding a rod out in a certain direction and is told to walk forward keeping the rod pointed the same way, that is, the person is to walk forward without letting the rod twist. This is a sensible demand, and a physically meaningful one: If the rod is allowed to twist as the person walks, the force will be felt in the hands. But twisting is just changing direction. So if there is a fact about whether the rod is twisting, there must be a fact about whether it is changing direction even though at every moment the person is located at a different point on the surface of the earth.

There is a nice mathematical object that handles this situation. The set of directions one can go in at any given point of the sphere is called its tangent space. The tangent spaces are all generically identical (360 degrees around) but specifically different: Each tangent space is glued to a point on the surface of the sphere. The mathematical object that will now be introduced is called a connection on the tangent spaces, and what it allows, intuitively, is for one to make comparisons between directions in the tangent space at one point with directions in tangents spaces at points infinitesimally nearby. So, as one moves continuously from one point to another on the sphere, the connection will determine whether the direction of the rod is changing or not. There are no absolute comparisons of distant directions, but there are comparisons of nearby ones mediated by the connection. More precisely, the connection provides a notion of parallel transport, that is, of carrying a direction from one tangent space to another without twisting along a specified path. It does not underwrite any absolute comparison of directions in different tangent spaces.

This is the sort of structure used in the gauge theories of physics. There are various charge states available to a quark at one location, and a similarly structured set of charge states available to a quark at another location, but no absolute comparison between the two: There is no fact about whether the states of the quarks are the same or different. How, then, can there be any forces associated with the color charges? In the case of the electric force, it is critical to know whether two particles have the same or different charge: Like charges repel and unlike charges attract. How can one say, as was said above, that a stable collection of quarks must have no net color if there is no fact about exactly which color charge each quark has?

It is not enough to say that in a proton or a pion the quarks are nearby so there is a way of comparing their charges: Nearby is evidently not a mathematically precise term. The story is rather this. In modern particle theory, every force is mediated by a set of particles. The electromagnetic force is mediated by the photon, and the color force is mediated by particles called gluons. Furthermore, unlike the case of the photon, which carries no electric charge, the gluons themselves carry the color charge. And the very same remarks about the impossibility of absolute judgments about which charge a quark carries can be made about which charge a gluon carries. For heuristic purposes, it helps to think of gluons as carrying two charges: a color and an anticolor.

Now, suppose there is a bound state of two quarks, as in a pion. The quarks are only bound to each other through the mediating effect of a gluon. Originally, it was said that the pion as a whole must have no color, so the quarks can be, for example, one red and the other anti-red. But the gauge freedom, the freedom to identify different states at different points as the red state or the blue state means that there must be an equally valid description according to which the one quark is blue and the other anti-red. This seems to violate the demand that the pion have no net color.

Here, the gluon comes to the rescue. When gauge is changed, it must be done in a locally smooth way, and this means that not only do the color charges ascribed to the quarks change, but the color charges ascribed to the mediating gluon will change, too. So, while in one choice of gauge the pion will be described as a red quark bound to an anti-red quark by (say) a mediating blue/anti-blue gluon, in another choice of gauge, the very same pion will be described as a blue quark bound to an anti-red quark by a mediating red/anti-blue gluon. In each case there is no net color charge even though the particular charges ascribed to the constituents change. Evidently, while there is a gauge freedom involved in ascribing charges to particlesfree enough so that any particle can be ascribed any chargethere are global constraints on the choice of gauge, and changing the gauge for one particle will have to have consequences for the charges ascribed to others. Because the gauge can be changed in different ways at different points, this is called a local gauge freedom.

The key point is this: The gauge freedom is wide enough that there is no objective, gauge-independent fact even about whether two particles have either same or different color charge. So no metaphysics that tries to associate the color charges with universals or tropes in the usual way can succeed. This result is of particular significance for David Armstrong's project of justifying belief in universals by appeal to scientific accounts of the world. The fundamental structures employed by the best scientific theories simply do not correspond to the ontology of substance/universal that Armstrong proposes.

Gauge theories provide a novel approach to the fundamental ontological problem of sameness and difference. A metaphysics of universals or tropes entails that there be certain absolute facts about whether two individuals have similar or different qualities or properties, facts that obtain independently of where the individuals are located or even whether they are located in the same space-time at all. According to gauge theories, comparisons of properties are always mediated by a gauge connection. This means both that comparisons between individuals that inhabit disconnected space-time cannot be made at all and that even within a single space-time, ascription of charges to individuals always requires a somewhat arbitrary global choice of a gauge. The gauge connection itself is objective, but the particular charges assigned to individuals are not. This is an ontological structure that does not fit neatly in any traditional metaphysical category.

See also Metaphysics; Relativity Theory.

Bibliography

Armstrong, David. Universals and Scientific Realism. Cambridge, U.K.: Cambridge University Press, 1978.

Moriyasu, K. An Elementary Primer for Gauge Theory. Singapore: World Scientific, 1983.

Tim Maudlin (2005)

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