Why Gauge

Gauge theories clearly constitute a considerable advance in out understanding of the world. But it is not exactly clear what role the gauge freedom of such theories plays. Consider the following passage form a well known textbook on gauge theories:

[Gauge theories] are theories in which the physical system being dealt with is described by more variables that there are physically independent degrees of freedom. The physically meaningful degrees of freedom then ory. I agree that they can function as such, but note that the reduced phase space is not a necessary part of the holonomy interpretation. I prefer a view that retains all of the gauge freedom in the formalism, but remains neutral about its onto-logical status.

126 A notable exception is Belot's masterly survey of the relation between symmetry and gauge freedom: [Belot, 2003b].

127 Even this connection has only recently come to be appreciated by philosophers; in most presentations, the gauge theoretic aspects of the hole argument were either not noticed at all, or merely hinted at. Not until Belot & Earman [1999] did this situation change; largely, it has to be said, as a result of the connections noticed between the hole argument and the problem of time in quantum gravity.

reemerge as being those invariant under a transformation connecting the variables (gauge transformation). Thus, one introduces extra variables to make the description more transparent, and brings in at the same time a symmetry to extract the physically relevant content. ([Henneaux and Teit-elboim, 1992], p. 1)

We have seen this idea in action in the context of Maxwell's theory, quantum statistical mechanics, and Newtonian mechanics. The claim here is that the introduction of gauge freedom is vindicated by an increase in transparency. They don't further explain the nature of this transparency, but I expect that they are referring to the fact that gauge theories with gauge freedom can make manifest and explicit various properties of physical systems to do with symmetries: invariance, covariance, conservation laws, etc. One can think of other more philosophical reasons too, connected with the previous ones. The idea is that the excess degrees of freedom allow one to understand certain modal claims about the behaviour of physical systems. These will be like the counterfactuals of the Leibniz-Clarke debate, such as 'if we move this system of matter a distance of 5 feet to the West its qualitative properties will be the same'. In formalisms without the excess this doesn't seem to be possible. Hence, the transparency is both technical and philosophical, and the two types are intimately related.

Clearly though, as interpreters of physics, we are faced with the question of how we are to understand the gauge freedom. However useful the apparatus of gauge theory, it does not resolve the problem of where the surplus structure comes from, and how we are to interpret it. At best, it gives us a more precise medium for exposing and investigating surplus structure and symmetries. Here is how Martin describes the same problem:

The received way of characterizing the domain of gauge symmetries is that gauge symmetry concerns the covariance of the fundamental equations of motion for specific interactions, and that the covariance is tied to a certain descriptive freedom related to the presence of non-physical and, therefore, redundant or 'surplus' quantities in the theory. The basic idea is that in describing the physics we introduce too much, and the symmetry under the covariance group effectively rids the theory of the non-physical excess. ([Martin, 2003], p. 49)

Now, as I have argued already, the symmetries do not imply reduction. We can reduce, and get rid of the "non-physical excess", or we can deny that it really is excess and either accept the indeterminism or retain the symmetry as effectively imposing a version of PSR such that non-observable differences are not counted as physically relevant (that is, as far as 'the physics' goes). The idea is that, as far as the non-gauge degrees of freedom are concerned, it doesn't matter which gauge potential plays the role, providing that the gauge potentials are gauge-equivalent.128 Hence, there is the possibility of retaining all of the gauge freedom, giving it a direct interpretation, and nonetheless having an account of the physical structure that is not interfered with by the gauge freedom. I suggested above that one might

128 In Chapters 8 and 9 I shall return to the issue, where I defend the claim that such symmetries fit a structural ontology.

even consider this structure as representing possibilities (or potential states) for the physical structure.

Before I leave this topic, I should point out that what I had to say in the previous chapters, as regards inflation and deflation, passes over into the present context too. The gauge symmetry group generates an inflated possibility set, and direct approaches face the problems concerning commitment to indistinguishable worlds: haecceitism, indeterminism, etc. I argued that supposedly direct approaches like substantivalism and individualistic packages are not so committed. The point was that such interpretations are not necessarily direct: one can choose a selective interpretation; a (non-reductive) indirect interpretation, and thereby deflate the inflated possibility set; or, a direct interpretation modified so that the observables do not constitute an inflated possibility set. It is true, though, that Leibniz's version of relationalism (a reductive interpretation) is deflationary: PII implies that the whole orbit of gauge-related points itself represents a single possibility. But accepting PII in this sense is not a demand enforced by relationalism— a point stressed by Saunders [2003b].

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