Indeterminism

The main interpretive problem facing direct interpretations of gauge theories is the underdetermination that results from the gauge freedom. How one chooses to deal with this problem leads quickly into many distinct interpretive problems: indeterminism, non-locality and frozen dynamics (to name the most important for our purposes). I save the latter to a detailed examination in Chapter 7; the non-locality problem that results from a particular attitude to this problem is reviewed in the next subsection, §3.3.2; for now I restrict my attention to the indeterminism problem itself.

I have already briefly mentioned how the gauge freedom in Maxwell's equations gives rise to a form of indeterminism or, more accurately, underdetermina-tion in the evolution of certain kinds of data off an initial surface. Let me expand on this idea some more, for it has a direct bearing on the following chapters. In particular, I argue that the hole argument and the problem of time in canonical quantum gravity are simply special cases of this problem.

The indeterminism flows from a direct (one-to-one) interpretation of the formalism I outlined in the previous section. Here, each point of the phase space is taken to represent a distinct physically possible state of the theory, even for those points occupying the same gauge orbit and which, therefore, describe qualitatively identical physical states. In theories with gauge freedom, there will be cases where a completely specified initial state (even an entire history up to some instant) is not sufficient to uniquely determine the evolution of data. Multiple evolutions are equally compatible with the initial state and the laws of the theory. The best that one can do, is to predict which gauge orbit the data will lie in, the points of which represent isomorphic structures. Hence, for two futures that are compatible with an initial state x(0), there will be two corresponding states x(t') and x(t") respectively. The physical structures (fields, in the case of the electromagnetism) that x(t') and x(t") represent will be physically indistinguishable since the states are isomorphic: there is a structure preserving map $ : x(t') ^ x(t") (a gauge transformation) connecting the evolutes; in other words x(t'), x(t") e [x], where [x] is a gauge orbit 'to the future' of x(0).

it as surplus or not. However, the notion of what counts as observable in such theories is crucial here, for denying that observables are fully gauge invariant (i.e. with respect to all constraints) can escape the problem, but only at the price of reintroducing indeterminism.

There are a variety of 'tried and tested' methods for dealing with this apparent violation of determinism that results from the gauge freedom of Maxwell's equations; they apply more generally to any gauge theory, and we will meet them again in the chapters that follow. They are as follows:

• Gauge-invariance: 'Ostrich Style'. If one sticks to the view that the electric and magnetic fields comprise the basic ontology of the theory, then the inde-terminism that results is simply understood to be a by-product of the freedom to choose from a gauge equivalence class of potentials: it is not a physical form of indeterminism. As Earman nicely sums it up:

The fix for determinism is to blame the apparent failure [of determinism] on redundancy of the descriptive apparatus, which is simply another way of saying that the variational symmetries containing arbitrary functions of the independent variables connect equivalent descriptions of the same physical situation, i.e. are gauge transformations. ([Earman, 2006], p. 450)

Thus, in terms of the electric and magnetic fields (and, mutatis mutandis, for fields associated with other gauge theories), the theory is completely deterministic. This state of affairs might lead to one to simply ignore the indeterminism, to bury one's head in the sand. As Maudlin puts it, "if one regards the fields as the real ontology, then one knows that the dynamics of that ontology is deterministic" ([2002], p. 5). Pragmatically, perhaps, this move is admissible; and, in fact, I have some sympathy with this view. Clearly, however, more needs to be said about the origin and nature of the mismatch between the formal representation (with its 'mathematical indeterminism') and the deterministic physical system.

• Gauge fixing: Restrict the constraint surface. In this case the indeterminism is removed by imposing a certain condition on the potentials so as to select a single member from each gauge equivalence class. This is tantamount to selecting a submanifold of the constraint surface that contains just one point from each gauge orbit. Fixing a gauge in this way implies that the potentials can be understood as being in a one-to-one correspondence with the fields without any surplus structure: since the field dynamics is deterministic, so is the dynamics for potentials. Hence, in addition to fixing the indeterminism problem, another interpretive problem is resolved; namely, the representation relation between the formalism (potentials) and reality (fields) can be read as direct once again (though such a conception required an original selective step). Aside from the technical difficulties I mentioned earlier, this method also loses all of the information contained in the gauge freedom: information about invariances, co-variance, conservation laws, etc. In many cases this loss will be too high a price to pay for eradicating indeterminism.

• Reducing phase space. This method eliminates the indeterminism by forming a new phase space out of equivalence classes of gauge equivalent phase points (i.e. the gauge orbits). (Recall that the elements of gauge orbits are isomorphic, representing indistinguishable physical states.) A new space, the quotient space, is formed so that the gauge orbits of the extended space are the phase points for this smaller space. The gauge freedom is thus "quotiented out", the new 'reduced' phase space being the quotient of the original space by the gauge orbits. With no gauge freedom remaining, the problem of indeterminism is eradicated. Unlike the previous method, this method retains information about the symmetries of the extended phase space in its geometrical structure.119 I mentioned earlier that the reduced phase space method can face severe technical difficulties. For example, the set of gauge orbits does not necessarily have the structure of a manifold. In such cases it may be impossible to construct the reduced space. There are obvious similarities between this strategy and the gauge-fixing strategy: in both cases one gets rid of the gauge freedom, thereby getting rid of the symmetry. The difference: gauge-fixation methods select a single state from a gauge orbit while reduced space methods identify all states from a gauge orbit.

• Gauge invariance: 'Giraffe style'. The final method involves utilizing the presymplectic structure of the phase space by restricting the observables to those that are gauge invariant. I mentioned that the presymplectic structure ensures that any two trajectories passing through a single phase point on the constraint surface will intersect the same gauge orbits to the future of that point. On this approach, each gauge orbit can nominally be taken to represent a single physically possible world of the theory, but there is no reduction and no 'ostriching': the focus is simply shifted from the states to the observables. By choosing only gauge invariant functions we eradicate the in-determinism, since the initial-value problem is well posed for these functions. (There are two common classes of gauge invariant functions for the theories we deal with, these are holonomies and Wilson loops. Such objects are essentially 'blind' to the differences within gauge orbits, which are retained but treated as whole objects (i.e. one does not have to worry about their individual elements). There is, then, a large amount of surplus structure that is left largely unexplained. But, as I argue in Chapter 8, this is not necessarily a bad thing.)

To summarize. Any direct interpretation of a gauge theory will lead to indeter-minism with regard to the states: fixing the initial state (given the equations of motion) will not fix a possible future uniquely, it will do so only up to gauge. The clear advantage of these approaches is that they are, interpretatively speaking, non-arbitrary. Each element of the phase space is assigned the same ontological weight; each represents a physically distinct possibility. One can restore determinism, moreover, by modifying the notion of observables so that one's measurement theory only concerns gauge invariant magnitudes. However, in addition to the indeterminism, there is the problem that the physically distinct states remain, and these physically distinct possibilities are qualitatively indistinguishable: those

119 As Maudlin [2002] points out, this method has an further advantage over the previous method, for the gauge condition may be too stringent on two counts: (1) if more than one point in the extended ("inflated", in Maudlin's terminology) phase space satisfies the condition then determinism may be lost, since the dynamics will not determine which of the points the trajectory will pass through. (2) if some physical state does not meet the gauge condition then certain physical possibilities cannot be represented in the formalism. In the case of quotienting, however, "one is automatically guaranteed that each physical state will correspond to exactly one gauge orbit, since the orbits by definition contain all the points in the phase space that represent the state" (p. 5).

with respect to which the theory was initially indeterministic. They differ haec-ceitistically, in terms of which individual elements of the ontology play which roles (which vector potential it is that determines the fields, for example). Regardless of one's theory of measurement, this problem will remain on a direct account of the representation relation. This might be enough to call the interpretation into question, but I think it can be dealt with (as I try to show in Chapter 8).

Indirect gauge invariant approaches come in two types: reductive and non-reductive. The former remove the gauge freedom by quotienting it out, avoiding both the indeterminism and the qualitatively indistinguishable possibilities— such reductive interpretations are the analogue of Leibnizian relationalism. Each gauge orbit represents a single physical possibility, and they are conceived of as phase points in a new space. Non-reductive approaches likewise treat gauge orbits as representing (from the perspective of the observables) a single physical possibility—thus avoiding indeterminism and qualitatively indistinguishable possibilities—but nonetheless treat the elements of gauge orbits as distinct: they are seen simply as different modes of presentation of the same state of affairs.

The selective interpretations will eradicate the indeterminism (and qualitatively indistinguishable worlds), but at the high price of losing the utility that gauge freedom affords (including not being able to deal with the Aharonov-Bohm effect). The technical problems are also severe when compared to the problems of the other approaches. We see in §5.2.2 that Butterfield's One [Butterfield, 1989] as a response to the indeterminism of the hole argument corresponds to a selective interpretation, and must therefore shoulder the burden of these problems. He seeks to avoid the loss of gauge freedom by distinguishing between the 'models' of a theory and the 'worlds' the models represent. His selectivism then applies to worlds but not models. In the case of electromagnetism, this would amount to retaining a formalism with surplus structure given by the vector potentials, but insisting that only one of these represents a physical possibility (i.e. only one corresponds to the magnetic induction). This move would also avoid the Gribov obstructions, since that problem applies to the models. I will discuss and argue against this idea when we reach the responses to the hole argument in Chapter 5. The next subsection looks at how the various responses to the indeterminism (corresponding to the various interpretations of a gauge theory) fare with regard to the Aharonov-Bohm effect.

3.3.2 Non-locality

I mentioned in various places above that indirect interpretations of the relation between gauge potentials and a physical system lead to non-locality. In the case of Maxwell's theory, this afflicts the approach that takes the electric and magnetic fields as the ontology of the theory, and views the underdetermination of the vector potential by the magnetic field as highlighting the fact that the field is being multiply represented by the potentials. The non-locality enters the picture when we consider the behaviour of electrons in the neighbourhood of a classical elec tromagnetic field confined within a solenoid. This is the arena for the Aharonov-Bohm effect.120

The setup consists of a two-slit apparatus with a solenoid (long—ideally, infinitely so—and thin) sitting beyond and in between the slits (1 and 2, standing distance d apart). This produces a magnetic field confined within the solenoid when it is turned on. Outside of the solenoid the value of the magnetic field is zero. Electrons are fired through the slits at the screen (separated by length L), and when the solenoid is turned on, they undergo a phase shift—this is manifested by the shift in the interference pattern on the detection screen. This phase shifting is known as the "Aharonov-Bohm effect" (see [Aharonov and Bohm, 1959] for the original presentation). Formally, the (relative) phase shift 8 = 0\ — 02 (for amplitudes C1e101 and C2e102 from each slit) is computed as follows.121 Firstly we note that B = 0 outside of the solenoid, but A = 0 in such regions. The presence of a magnetic field in the solenoid alters the phase of an electron by the integral of the vector potential along the path taken by the electron, multiplied by the charge e of the electron divided by Planck's constant h.

Let 01 be the phase of the wave that travels through slit one (let [1] denote the whole trajectory of the wave from source to screen), and let 0\(B = 0) be the phase when the solenoid is turned off. When the solenoid is turned on, and a magnetic field is produced, confined within the solenoid, the phase is given by:

Replacing instances of '1' with '2' gives us the expression for the phase of the wave that passes through slit 2 (again, let [2] denote the trajectory of the wave passing through slit 2). The interference of waves is then given by the phase difference:

8 = 01 (B = 0) — 02 (B = 0) + e-j A • dl — e-j A • dl (3.10)

We can now write this as a loop integral going along [1] to the screen, and back along [2] (where 8(B = 0) is the phase difference when the solenoid is off, and y labels the loop formed by stringing [1] and [2] together):

Hence, the electrons undergo a phase shift, yet the magnetic field does not extend out beyond the boundary of the solenoid and the electrons cannot penetrate the boundary of the solenoid. As Drieschner et al. put it,

What makes the well known AB effect so astonishing is that there seems to exist "nothing" in the region of space where the electron's possible paths

120 See Nounou [2003] for a clear exposition of the effect followed by a nice philosophical analysis utilizing the fiber-bundle approach to gauge theories. Leeds [1999] offers an in depth and general discussion of a variety of interpretive options that have been suggested in response to the effect.

121 The following presentation is derived from Feynman ([1962], Ch. 15).

extend, yet the electron is discernibly influenced by "something". ([2002], p. 305)

Of course, the vector potential does extend out beyond the solenoid, there is lots of A circulating around it; if it were a physically real field then we would have "something" living in the region of space through which the electrons travel, and this would explain the phase shifts. Classically of course, A has no discernible effect, since the forces, given by the Lorentz force law F = q(E + v x B), determine the motion of particles inside and outside of the solenoid. Quantum mechanically, however, the force law is not sufficient to determine the shift; the electrons are subject to some force in the regions where the electromagnetic field is not. But note that we still have the problems of gauge freedom associated with the vector potential. The expression for the phase shift is itself invariant under changes of the form A' = A + grad/: both A and A' give the same phase difference, and therefore the same interference pattern. As Feynman puts it "it is only the curl of A that matters; any choice of the function of A which has the correct curl gives the correct physics" ([1962], §15-10).

Hence, thus far we seem to have two options regarding the interpretation of the AB-effect: Either the magnetic field acts non-locally or the vector potential acts locally. The latter option would grant physical status to the vector potential, thus introducing indeterminism at the level of ontology; the former option would introduce action-at-a-distance into a field theory. We thus have an appearance of the trade off between locality and determinism again. Let me develop these rough views some more.

Now, the magnetic field is confined within the solenoid, so if we view it as being responsible for the phase shift (and, hence, accept an indirect, gauge invariant view), then we must accept that it acts non-locally. The vector potential is not so confined, when the current flows, it extends outside of the solenoid. If we take the potential as responsible then we can see how it affects electron phases locally. This clearly requires that we view the vector potential as a physically real field. It seems that we must do this if we don't wish to have the magnetic field acting at a distance. It is, however, going to be an indeterministic account unless we apply one of the above methods for removing gauge freedom. The fact that the vector potential (a gauge potential) is not gauge invariant is enough, on gauge invariance accounts of observables (such as Dirac's [1958]) to show that it is not physically real. The problem is that gauge potentials are not measurable; there is no way to determine which gauge potential one has (c/. Nounou [2003], pp. 176-7). No realist is likely to be happy with this. In the present case, the fields and the AB-effect un-derdetermine the value of the vector potential. However, in spite of this problem, the view of many physicists, including Feynman, Bohm, and Aharonov, is that the vector potential is a physically real field more fundamental that the magnetic field, and the reason seems to boil down to the fact that it provides an explanation of the AB-effect that does not violate the local action principle. Weighed against the indeterminism of the vector potential ontology, the non-locality of the magnetic field loses out: indeterminism is traded in for locality.

Where the traditional accounts given above fail, on pain of violating non-locality and determinism, is where alternative gauge invariant accounts come into their own. The idea is, rather than taking the vector potential field as being responsible for the effect (resulting in a local but non-gauge invariant and, therefore, indeterministic account), one takes certain gauge invariant quantities constructed from the potential as responsible. Hence, one can work in terms of the holonomy of the vector potential around a loop encircling the solenoid, or the corresponding Wilson loops (traces of holonomies). These quantities are gauge invariant, and hence determinism is secured, and they extend beyond the solenoid (i.e. they are defined on paths outside of the solenoid), encircling it, and so can be called upon to explain the phase shifts—indeed, the description of the holonomy is more or less identical to Eq. (3.11).

This is, it is true, non-local too, but in a very different sense from the gauge invariant account involving the magnetic induction, which involved action-ata-distance. In the holonomy case, the non-locality is simply a manifestation of the fact that holonomies are given as functions on a space of loops rather than a point-manifold (cf. Belot [2003b], p. 204). They are therefore not localized to manifold points as the magnetic field and vector potential are. This does not make the holonomy interpretation non-separable (cf. [Healey, 2004], pp. 646-7—and see below), for that requires that a physical process in a certain region of spacetime is not supervenient on assignment of qualitative intrinsic properties at the spacetime points in that region; if spacetime points make a showing at all, it is as elements of a loop that wraps around the solenoid. The physical processes are not grounded in spacetime regions in the first place. Hence, strictly speaking, pace Healey ([1997]— and see below), there is neither non-locality (qua action-at-a-distance) nor non-separability in the holonomy interpretation. Redhead's otherwise superb account of the interpretation of gauge symmetry is marred by his conflation of these two forms of non-locality (see Redhead [2003], pp. 132 and 138).122

Redhead ([2003], p. 138) suggests that in adopting a gauge invariant account using holonomies, one thereby waves goodbye to the principle of gauge invariance, thus tarring such an account with the same brush as I tarred the selective interpretation with. His reasoning is that in shifting attention to holonomies one eschews non-gauge invariant quantities, such as the gauge potentials. But gauge transformations are only defined on non-gauge invariant quantities, so gauge invariance cannot be accommodated. Some might think that this is a good thing since such quantities are indeterministic. But even if one doesn't, Redhead's claim does not hold water. Firstly, there are reductive accounts that can incorporate the principle of gauge invariance in the very structure of the phase space. Secondly, if we don't wish to reduce, then we can still adopt a direct interpretation, but modified so that only gauge invariant quantities are measured.

There is a sizeable philosophical literature on the interpretation of the Aha-ronov-Bohm effect. It is fair to say that no consensus has been reached. This work is largely tangential to my brief though, which has been to examine the general subject of gauge freedom and symmetries and their relation to metaphysical issues. However, I briefly mention some of the most important proposals. Richard

122 The same conflation is made in the accounts of Nounou ([2003], p. 178), Lyre [2001] (corrected in his [2004]), and Healey [1997].

Healey [1999] argues for a view whereby the effect shows, by analogy with quantum non-locality, that there is non-separability in the world; he uses this to defend holism.123 As I noted above, the non-locality of the holonomies concerns the fact that they are not localized at spacetime points, rather than any violation of local action or separability. Healey defines these two notions as follows:

Local Action

If A and B are spatially distant things, then an external influence on A has no immediate effect on B. ([Healey, 1997], p. 23)

Separability

Any physical process occurring in spacetime region R is supervenient upon an assignment of qualitative intrinsic physical properties at spacetime points in R. ([Healey, 1997], p. 24)

Healey claims that "no interpretation gives a completely local account of the effect" ([1997], p. 32), some violate the local action principle, some violate the principle of separability, and others violate both. Now, I agree with Healey that a completely local account cannot be given, if by that is meant an account in terms of interactions at spacetime points, or in infinitesimally close regions of spacetime points. But this is not what Healey means; his idea of locality is exhausted in the two principles above. However, if we view the holonomies as defined with respect to a space of loops, not spacetime points, then we find that the account of the AB-effect given in terms of them will trivially satisfy both of Healey's conditions for locality, for the physical processes do not happen in spacetime regions and they are local in loop space. Indeed, I think this is the right way to view the holonomies: spacetime points do not enter into their description; this is precisely why they are so useful in general relativity, where we have the freedom to arbitrarily permute points of the manifold. But the account is non-local in the additional sense I mentioned above. Holger Lyre formulates the principle of separability slightly differently to Healey, as follows:

Lyre-Separability

Given any physical system S and its exhaustive, disjoint decomposition into spatiotemporally divided subsystems, it is possible to retrieve the properties of S from the properties of these subsystems. ([Lyre, 2004], p. 608)

This has the quite definite advantage of avoiding the issue of supervenience (though it isn't clear that the notion of 'property retrieval' is any less of a problem), and of assuming an initial region of spacetime in which the physical occurs. Again, the notion of spatiotemporally decomposing a holonomy defined in loop space does not make much sense, so that Lyre's definition is not applicable. However, what is violated is something very close to both Healey's and Lyre's notions of separability, though more general; namely, Lewis's thesis of Humean Supervenience (cf., e.g., [1986b], pp. IX-X). This is the thesis that all there is supervenes on the arrangement of intrinsic properties localized to spacetime points. Thus, the entailment

123 The type of holism that results is close in many respects to Teller's "relational holism", argued for in the context of the Bell inequalities (see Teller [1989]).

would be that the magnetic interaction, and the Aharonov-Bohm effect, supervenes on some local matters of fact, on the distribution of intrinsic properties over a set of spacetime points. It seems that even an action-at-a-distance account that restores separability will violate Humean supervenience, since the causal connection doesn't admit a reduction of the kind that is required by that thesis; we seem to have something over and above a 'mosaic' of properties attached to points of spacetime. Healey's notion of separability is most commonly associated to many systems bearing relations to one another, and the issue is whether or not the whole (the composite system) is greater than the sum of its parts. This is put to the test by quantum entanglement since that implies that the joint state of the composite system is what determines the states of the components, rather than the other way around. This naturally suggests a holism, and that is where Teller's notion of relational holism finds a home, since it denies the thesis he calls "particularism", the idea that individuals are completely characterized by the intrinsic properties, such that any relations that there are supervene on the intrinsic properties of the relata (see Teller [1989], p. 213). But this is not anything like what we have with holonomies; they are simply singular beasts, not composites. For this reason, Lyre's further notion of holism does not apply either. It is as follows:

Lyre-Holism

Given any physical system S and its exhaustive, disjoint decomposition into subsystems, it is impossible to retrieve the properties of S from the properties of these subsystems. ([Lyre, 2004], p. 609)

This definition evades both supervenience (though with the same proviso as for Lyre-Separability) and any explicit dependence on spacetime points or regions. But the holonomies do not class as Lyre-holistic either, for the simple reason that it does not make sense to speak of decomposing a single holonomy into subsystems; it has no subsystems. However, Lyre is right in saying that "[t]raditional ontology, which thinks of objects as being spatiotemporally fixed [localized], is clearly undermined" ([Lyre, 2004], p. 620); we have something wholly more abstract than magnetic fields or vector potentials with holonomies, yet I think that the interpretive benefits outweigh their prima facie ontologically puzzling nature.

Maudlin [1998] argues that a local (in both mine and Healey's senses) and separable interpretation can be given if we take the vector potential to be gauge fixed, described by "one true gauge".124 It is doubtful that Maudlin wishes to take this view seriously, he mentions it as a counterexample to Healey's claim that non-locality afflicts any approach to the AB-effect. Leeds [1999] adopts a view similar to Aharonov and Bohm, such that the effect is understood as an interaction between matter and a gauge potential. Such a view has, I think, been superseded by holonomy interpretations.125 If one is willing to give up on the idea that the ontology has to involve localization to spacetime points, then such an account gives

124 As evidence for my claim that Butterfield's counterpart theoretic substantivalist should be seen as adopting a selective interpretation, note that an objection of Martin's against Maudlin's account exactly parallels one given against Butterfield by Norton. Martin argues that "nothing in the physics can reveal this one true gauge" ([2003], p. 49); Norton argues that there is no way to distinguish the one true model that represents a general relativistic spacetime from the "imposters" ([1988], p. 63).

125 See Belot ([1998]; also §7 of Belot [2003b]) for the best account and defense of such a view. Belot claims that the holonomies of A also "provide a good set of coordinates" ([2003b], p. 204) for the reduced phase space of Maxwell's the-

a local (in Healey's sense) and deterministic account of the effect. This view also sits best with the claim that observables should be indifferent to isomorphic states: holonomies, you will recall, contain all of the gauge invariant information of A.

It is somewhat curious that the interpretive issues surrounding the substan-tivalism/relationalism debate are not mentioned in connection with gauge theories.126 True, the translations of the Leibniz-shift argument are not gauge in the strict sense, but the philosophical issues are pretty much of a kind nonetheless. The PII wielding relationalist response to the translation invariance by modding out by the action of the group of translations would seem to offer a similar move to the realist about gauge potentials. The obvious move is to regard those states with vector potentials related by a gradient as the same physical state. The physical ontology is then given once one mods out by the action of the gauge group. The (Newtonian) substantivalist position would then correspond to treating the states as distinct though indistinguishable. (I have already mentioned that the idea of 'one true gauge' corresponds to Butterfield's idea that only one model out of a gauge equivalence class represents a physical possibility for the system.)

Gauge or no gauge, the central issue is a representational one: how should we understand the relation between a piece of mathematical formalism and physical reality? Gauge theory makes this issue more urgent, but it is really a feature of any theory with non-trivial symmetries. In gauge theory, too, the interpretive options are more closely allied to particular formal choices (reducing, gauge fixing, etc.). This is how the connection to general relativity and the hole argument becomes apparent, for general relativity is a spacetime theory with gauge freedom. We then find that the substantivalism/relationalism is explicitly connected to the interpretive moves taken with respect to gauge freedom in exactly the way I suggested above.127 For this reason, I reserve discussion of the relationship between these interpretive moves and issues in the metaphysics of identity and modality until Chapters 4 and 5.

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