Interpretive Problems Of Gauge Theories

The key problem in trying to interpret gauge theories, is knowing what to do with the gauge freedom. There are multiple options, and hence, multiple ways of interpreting gauge theories. These differences are interpretively non-trivial in the classical theory, and can also lead to technically non-trivial differences when it comes to quantization of a gauge theory.

Let us call an interpretation that takes each phase point as representing a distinct physically possible state of a system a direct interpretation. Hence, each point x1 in a gauge orbit [x] represents a distinct possibility. However, such a direct interpretation leads to a form of indeterminism for the reasons outlined in §3.2.2. But, since each of the phase points represents a distinct physical possibility, there is (strictly speaking) no surplus structure according to such an interpretation. Recall also that the indeterminism is of a very peculiar kind: the multiple futures that are

107 In other words, there's more mathematical structure than is strictly necessary to represent the physical system. Thus, we have a direct connection to Redhead's notion of surplus structure even within the Lagrangian framework.

108 As I show below in §3.3, there are a number of ways of understanding how this works, some involving a 'reductive' move (analogous to Leibniz's in the space/time context) and some involving 'non-reductive' moves.

109 Preserving these under evolution may require the imposition of higher-order constraints. Once one has a situation where all the constraints are preserved by the motion, one will have defined a submanifold where all of the constraints are satisfied—this is the "constraint surface". See Earman ([2003b], pp. 144-5) for a clear explanation of these constraints and their relation to the singularity of the Hessian. I offer a slightly more elementary presentation in [Rickles, 2006].

compatible with an initial state are physically indistinguishable, for they are represented by points lying within the same gauge orbit. Hence, the indeterminism concerns haecceitistic differences.110 However, for realists the indeterminism will still constitute a problem, though it is not insurmountable. As Belot notes:

if we supplement this account of the ontology of the theory with an account of measurement which implies that its observable quantities are gauge-invariant, then the indeterminism will not interfere with our ability to derive deterministic predictions from the theory. ([1998], p. 538)

I defend an account along similar lines in the following chapters based loosely upon the idea that a form of PSR should be seen as operating on observables, so that the observables are indifferent' as to the roles played by particular individuals. I think that Saunders "non-reductive relationalism" ([2003b; 2003a]) can be seen as implementing much the same idea—a claim I defend in §6.2. Using this method one can help oneself to gauge invariance at the level of observable ontology and remain neutral about the rest (spacetime points, quantum particles, shifted worlds, vector potentials, etc.). Saunders differs on this point; he does not wish to remain neutral, he wants to retain a notion of an individual, regardless of how thin this notion happens to be. This is a consequence of the fact that his proposal flows from his own version of PII.

Let us call an interpretation that takes many phase points (from within the same gauge orbit) as representing a single physically possible state of a system an indirect interpretation. There are two ways of achieving such an interpretation. According to the first method one simply takes the representation relation between phase points from within the same gauge orbit and physically possible states to be many-to-one. Since the points of a gauge orbit represent physically indistinguishable possibilities, there is no indeterminism on this approach. Redhead suggests that "the ' physical' degrees of freedom [i.e. the fields] at [a future] time t are being multiply represented by points on the gauge orbit terms of the 'unphysical' degrees of freedom" ([2003], p. 130).111 The gauge freedom is simply an artifact of the formalism. There are superficial similarities between this approach and the modified direct approach given above. However, the stance taken on this approach is that not all of the phase points represent distinct possibilities. Even on the modified direct approach this is false. The latter approach simply says that the question of whether or not all of the phase points represent distinct possibilities is irrelevant to the observable content of the theory, the observables are indifferent as to what state underlies them provided the states are physically indistinguishable.

The second method involves treating the gauge orbits rather than phase points as the fundamental objects of one's theory. By taking the set of gauge orbits as the points of a new space, and endowing this set with a symplectic structure, one can construct a phase space for a Hamiltonian system—this new space is known as

110 I review this problem of indeterminism below in §3.3.1.

111 Redhead's analysis seems to suggest that this is the only way to interpret the direct formulation (in terms of vector potentials)—though he mentions that a gauge invariant or gauge-fixing account can resolve the indeterminism. But clearly, it is open to us to give a direct interpretation and accept the qualitatively indistinguishable worlds that are represented by the isomorphic futures (points within the gauge orbit).

the reduced phase space,112 and the original is then called the extended phase space.113 Hence, the procedure amounts to giving a direct interpretation of the reduced phase space—i.e. one that takes each gauge orbit as representing a distinct physically possible state—but an indirect interpretation of the extended phase space. The resulting system is deterministic since real-valued functions on the reduced space correspond to gauge invariant functions on the extended space. In effect, the structure of the reduced space encodes all of the gauge invariant information of the extended space even though no gauge symmetry remains (i.e. there is no gauge freedom). Note, however, that complications can arise in reduced space methods: the reduced space might not have the structure of a manifold, and so will not be able to play the role of a phase space; or some phenomenon might arise that requires the gauge freedom to be retained, such as the Aharonov-Bohm effect (cf. Earman [2003b], pp. 158-9 and Redhead [2003], p. 132). If these complications do arise, one can nonetheless stick to the claim that complete gauge orbits represent single possibilities, as per the above method.

There is another method that involves taking only a single phase point from each gauge orbit as representing a physically possible state of a system. To do this one must introduce gauge fixing conditions that pick out a subset of phase points (a gauge slice) such that each element of this subset is a unique representative from each gauge orbit (cf. Govaerts [2002], p. 63). Gauge fixing thus 'freezes out' the gauge freedom of the extended phase space. In more detail, for a constrained system with constraints fa(q,p) = 0 (i = 1,...,N), one must impose M further conditions Fj(q, p) = 0 (j = 1,...,M, where M = N). This defines another surface in the phase space that should, if successful, intersect each gauge orbit just once; for all practical purposes, this is the same as the reduced phase space, it is composed of just those points that satisfy both the constraints and the gauge fixing conditions.114 This method leads to an interpretation that is neither direct nor indirect, I shall call it a selective interpretation. There is a serious problem—known as a Gribov obstruction (ibid., p. 64)—facing certain gauge fixing procedures. The obstruction implies that the gauge conditions do not result in a unique 'slicing' of phase space, but may result in the selection of two or more points from within the same gauge orbit.115 Thus, suppose we impose the conditions Fa(p) = 0, as best we can; then the Gribov obstruction will manifest itself as there being numerous pi, all of which are related by gauge-transformations, and all of which satisfy

112 In order to distinguish this approach from the previous one, let us call it a reductive interpretation from now on. Note that this matches Leibniz's form of relationalism since it can be seen as enforcing PII on phase points within the same gauge orbit. Thus, to complete the analogy, an extended phase space r would correspond to that containing phase points related by the symmetries associated with Gn (representing indistinguishable shifted, rotated, and boosted worlds) and the reduced phase space r would correspond to the space with symmetries removed: r = r/Gn.

113 Thus the points of the reduced space correspond to gauge orbits of the original extended space. Curves in the reduced space contain information about which gauge orbits the system (as represented by the extended space) passes through— i.e. about which qualitatively distinct states it passes through.

114 Referring back to the Leibniz-Clarke dispute, the present interpretive move would correspond to keeping absolute spacetime, but imposing a condition such that exactly one localization of the matter relative to it was chosen—i.e. where the point particle p1 is at point x, point particle p2 is at point y, and so on. However, in this case, it is difficult to see what could be gained by such a move; there is no symmetry or geometrical structure available to explain the various invariance principles and conservation laws.

115 As Redhead notes ([2003], p. 132), in the case of non-Abelian gauge theories, the application of the gauge fixing method leads to a breakdown of unitarity (in perturbative field theory) that has to be dealt with by the ad hoc introduction of "fictitious" ghost fields—thus replacing one type of surplus structure with another.

the gauge fixings. Even in spite of the Gribov obstruction—which is very serious for any non-Abelian gauge theory (cf. Singer [1978])—there is an obvious problem with using this formal method of dealing with gauge freedom as a way of interpreting gauge theories. The method involves the singling out of a point from each gauge-equivalence class, giving a gauge-fixed submanifold, and carries out the physics on that. But since the elements of the equivalence class are deemed to be physically (qualitatively) indistinguishable, it is difficult to know what could ground this choice of gauge, other than perhaps the ease with which physics can be done with respect to it, at least in the general case.

Finally, there is an approach, named a "coarse-grained gauge-invariant" interpretation by Belot ([1998], p. 538), according to which the gauge orbits themselves stand many-to-one with physically possible states.116 This would class as an indirect interpretation of an indirect (or, possibly, reductive) interpretation according to my definitions above. I don't have reason to consider such interpretations in what follows. Presumably, though, we could construe the many-to-one nature of the representation relation as pointing to gauge freedom again, and so apply one of the above interpretations here as well.

Each of these interpretive options is seen to be applicable in both general relativity and quantum gravity; indeed, they are seen to play a crucial role in both their technical and philosophical foundations—though I am skeptical about the scope of their philosophical import (cf. [Rickles, 2005]). In the next chapter, I turn to a specific argument (the "hole argument") based upon a direct, local interpretation of general relativity. The argument is connected to the nature of spacetime since the gauge freedom is given by (active) diffeomorphisms of spacetime points (or by 'drag-alongs' of fields over spacetime points). What we appear to have in the hole argument, is an expression of the Leibniz-shift argument couched in the language of the models of general relativity, with diffeomorphisms playing the role of the translations. Earman and Norton [1987] see a direct, local interpretation as being implied by spacetime (manifold) substantivalism (i.e. the view that spacetime points, as represented by a differentiable manifold, exist independently of material objects). Clearly, this view is then going to be analogous to the interpretation of Maxwell's theory that takes the vector potential as a physically real field. Such an interpretation is indeterministic: the time-evolution of the potential can only be specified up to a gauge transformation. Earman and Norton extract a similar indeterminism from the direct interpretation in the spacetime case, and use this conclusion to argue against substantivalism.117 The "problem of time" (the subject of Chapter 7) applies the reasoning of the hole argument (as broadly catalogued in the direct, indirect, reductive, and selective interpretations) to the evolution of data off an initial spatial slice. One's interpretation of the gauge freedom then has an impact on the question of whether or not time and change exist!118

116 He calls "simply gauge-invariant" those interpretations that take whole gauge orbits as representing a single physical possibility (ibid.). As is evident from the many-to-one and reduced space methods, there is more than one way to understand such interpretations.

117 In Chapter 5 I show how both the claim that substantivalism implies a direct interpretation and that the direct interpretation leads to indeterminism can be questioned in a variety of ways. These "ways" are more or less on all fours with the general interpretive options outlined above.

118 Or so the received wisdom goes. I argue that the problems of time and change affect any approach that treats general relativity as a gauge theory. All interpretations will face timelessness whether they get rid of the gauge freedom or retain

Let us next consider the two basic problems that face direct and indirect interpretations respectively: indeterminism and non-locality. (I don't consider the problems with selective interpretations here, since they can be lumped with reduced phase space methods for the purposes of the following investigations. I have said enough about the problems in my comments above, to show why I don't think they are good options interpretively speaking. I return to the idea of a selective interpretation in §5, where I argue that Butterfield's application of counterpart theory to spacetime points is the philosophical analogue of such an interpretation of the gauge freedom.)

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