Maxwellian Electromagnetism

As traditionally conceived, Maxwell's equations for electromagnetism describe the behaviour of a pair of vector fields—E (the electric field) and B (the magnetic field)—that are (1) defined at each point of space (taken to possess the structure represented by R3); (2) functions of time t e R; and (3) dependent upon the electric charge density p and current density/. Setting the speed of light c to 1, Maxwell's equations are:

89 In other words, underdetermination is not sufficient for indeterminism; indeterminism requires that no amount of specification of initial data can secure unique future values for some physical quantity or object. In the case of quantum statistical mechanics and Newtonian mechanics, there are enough evolution equations to propagate all of the physical magnitudes once a labeling of particles and localization has been settled on. The indeterminism that lies at the core of gauge theories is an underdetermination of solutions of the equations of motion given an initial data set (cf. [Gotay et al., 1998], p. 1, and §3.2.3 below).

A straightforward interpretation of Maxwellian electrodynamics is as a theory describing the behaviour of these fields, where the electromagnetic field can be represented by a pair of potentials90: the vector potential A and the scalar potential 0, where (setting c = 1 again):

The fields, rather than the potentials, comprise the fundamental ontology of the theory: these are the things we measure. An instantaneous state of the electromagnetic field is given by specifying the values and first derivatives of the electric and magnetic fields at each point of space (at an instant).91 Maxwell's equations then describe how this state evolves deterministically through time. So far there seems to be no problem in representation regarding the ontology of fields, and the match between formalism and system appears to be one-to-one. However, there are two problems: (1) the Aharonov-Bohm effect, and (2) the underdetermination of the potentials by the fields. We deal with both problems in detail in the context of gauge theory in the subsequent sections; in this section we try to make some (intuitive) headway on (2), and simply mention what (1) looks like.

The underdetermination concerns the vector potential A. Since A = A + gradf (for smooth functions f), curl A = curl(A + gradf) = B, and so many formally distinct vector potentials will represent a physically indistinguishable magnetic field (since the curl of a gradient is zero). As Wigner ([1967], p. 19) points out: "the potentials are not uniquely determined by the field; several potentials (those differing by a gradient) give the same field." However, in this case the underdetermination is a mathematical artifact of the formalism employed, it is gauge. The reason is that since we are dealing with B, and not A, as our basic ontology, this will take the same value on all values of A that differ by a gradient. This gives rise to a gauge invariant interpretation of the theory (see §3.3). Classically, this interpretation is fairly un-problematic, though it requires an account of the nature of the gauge invariance; the vector potentials are still not uniquely determined by the magnetic field, nor by the equations of motion, but we are understanding this part of the formalism to be unphysical. However, when one considers the behaviour of a charged quantum particle in a classical electromagnetic field (in a non-simply connected space), this gauge invariant interpretation of Maxwell's theory faces a serious problem: maintaining that the magnetic and electric fields exhaust the ontology, and that

90 An alternative representation is provided by the electromagnetic field tensor F. The equations for this object are: dF = 0 and *d * F = J, where J is the current, and d and * encode the curls, divergences, and time derivatives in the alternative equations given above. See Part I of Baez & Munian [1994] for an excellent presentation of the various representations of Maxwell's equations. Note that a similar problem to the one I am about to detail applies to this representation too.

91 The topology of the space turns out to be a crucial factor in assessing the cogency of this interpretation once quantum mechanical effects are taken in to consideration, as is the case in the Aharonov-Bohm effect (see §3.3.2). If the space is not simply connected, the magnetic field must be seen as acting non-locally.

the vector potential is surplus, leads to the conclusion that the magnetic field acts non-locally (i.e. where it isn't). This is the content of the Aharonov-Bohm effect: the charged quantum particles undergo a phase shift when the magnetic field is present even though the field value is zero throughout the regions of space containing the particles' trajectories. In order to retain locality one must attribute physical reality to the vector potential; this has positive values in the places where the value of the magnetic field is zero, and different vector potentials can give rise to the observable effects of the Aharonov-Bohm effect. This move quite clearly leads right back into the problem of underdetermination of the vector potentials, and the indeter-minism which now manifests itself as a physical indeterminism since we are now conceiving of the previously formal degrees of freedom of the vector potential as physical degrees of freedom.

It will be useful to introduce some relevant terminology at this point.92 A transformation on the potentials that results in the same (i.e. physically indistinguishable) fields is known as a gauge transformation and such potentials are said to be gauge equivalent or gauge related. Gauge equivalent potentials are said to lie within the same gauge orbit. The freedom to choose from a gauge orbit of vector potentials that represent one and the same field is known as gauge freedom. The electromagnetic field vectors (or the electromagnetic field tensor F) are said to be "invariant under gauge transformations", or just gauge invariant—the same also applies to Maxwell's equations themselves, with "covariant" replacing "invariant". However, the vector potential is not gauge invariant, its value is altered by a gauge transformation.

Suppose that we wish to maintain a one-to-one representation relation between the vector potentials and physically possible worlds (or, simply, possibilities)— because of a desire for a literal, realistic interpretation of the formalism or the desire the accommodate the Aharonov-Bohm effect, for example—then seemingly we must countenance both the haecceitistic differences mentioned in the previous chapters (in this case indiscernible field differences) and a form of indeterminism, since the equations of motion cannot determine the behaviour of the potentials uniquely, but only up to a gradient. As Maudlin explains:

if the original dynamics implies that a state of the electromagnetic field E0 — B0 will evolve, after a period of time, into Ei — B1, then we should expect the new dynamics only to demand that a pair of potentials which yields [by Maxwell's equations] E0 — B0 ought to evolve into some pair of potentials which yields E1 — B1. But since many different pairs of potentials yields E1 — B1, we have no reason to expect the dynamics to pick out one of these pairs over any gauge-equivalent pair. ([Maudlin, 2002], pp. 3-4)

Hence, there is a mismatch between the formalism (in terms of the potentials) and physical reality (in terms of observable field values): many distinct potentials give the same fields.93 If we wish to predict how the potentials will evolve into

92 I give more precise, general definitions of this terminology in the next section. For now I give simple qualitative explanations restricted to the theory of electromagnetism.

93 Hence, we have an example of what Redhead calls surplus structure since "there are more degrees of freedom in the mathematical description than in the physical system itself" ([2003], p. 124).

the future then this mismatch manifests itself as an indeterminism: the dynamics can only determine the evolution of potentials up to a gauge transformation. This problem inspires the move to the traditional gauge invariant account involving only an ontology of fields, for we can predict which gauge orbit potentials will lie in, and the elements of a gauge orbit give the same field configuration. But anti-realism about the vector potential renders the traditional account unable to deal adequately with the Aharonov-Bohm effect: it leads to a violation of locality. Hence, we have an interpretational dilemma consisting of a trade-off between locality and determinism.

However, it is possible to give a deterministic account using vector potentials, provided we incorporate gauge invariance into the description of the observables. This can be achieved either by using the holonomies of vector potentials (qua connections) or the Wilson loops of the holonomies.94 Both holonomies and Wilson loops contain all of the gauge invariant information contained in the vector potentials. According to the approach in terms of vector potentials, the canonical variables are standardly given by pairs (Aa(x), Ea(x)), representing vector potentials and electric fields respectively on a spacelike hypersurface. But, as we saw above, in these variables the phase space contains a lot of surplus structure corresponding to potentials differing by a gradient: good for locality but bad for determinism. However, as I have just noted, all of the gauge invariant content contained in the potentials (where the potential is understood to be a connection on the (fiber bundle) space) is encoded in the holonomies (Hy(A) = expi§adlaAa, where y is a loop). (In the case of the electric fields the physical content is encoded in functionals of the form f d3xEa(x)fa(x).) Pairs of holonomies and functionals form a (complete) set of configuration and momentum variables for the theory's phase space. Hence, we have in fact shifted from a description in terms of vector potentials, to one that carries a vector potential around a loop in space (or, rather, a particle is carried around a loop); the holonomy exemplifies the invariant structure of the vector potentials in the sense that many vector potentials underlie one and the same holonomy. More precisely, the same holonomy is represented by gauge related vector potentials: Hy (A) = Hy (A + gradf); the gauge-equivalence class of vector-potentials is, on-tologically speaking, what the holonomy is like. Thus, the holonomies are gauge invariant, and vector potentials give the same holonomies iff they are gauge related; moreover, they give a local account. Thus, holonomies give us, interpretively speaking, the best of both worlds: like the magnetic field, they are indifferent to

94 These objects occur in the fiber bundle formalism, the arena for Yang-Mills theories (in which Maxwell's theory belongs) and theories of connections (e.g. vector potentials) in general. Connections in such a formalism permit the (frame independent, invariant) comparison of points in neighbouring fibers. This notion of parallel transport then allows for the definition of the directional derivative of some mathematical object, a vector, for example. The parallel transport of an object around a closed curve y is, in general, dependent upon the choice of curve. The parallel transport itself may be seen as a map (a homomorphism) from closed curves (living in the base space) to the Lie group of the bundle: so, to each curve there is associated a group element H e G. The action of H induces parallel transportation. We know that the result of parallel transportation is path dependent, so we write H(y). This object is known as the holonomy. We can connect this to general relativity by noting that a notion of curvature is obtained by considering the failure of some element of the fiber to return to its pre-parallel transported (around some closed curve) value. This curvature is equivalent to the holonomy when the latter is evaluated on an infinitesimal closed curve. As Gambini and Pullin point out ([1996], p. 2), "[k]nowledge of the holonomy for any closed curve ...allows one ...to reconstruct the connection at any point of the base manifold up to a gauge transformation".

certain smooth changes in the vector potential (namely, the gauge transformations); and, since the vector potential resides outside of the solenoid as well, they provide an explanation of the Aharonov-Bohm effect that does not result in nonlocal action.

Setting Maxwell's theory up as a theory of connections (and 1-forms) allows for two known representations: (1) the connection representation using holonomies (as outlined above), in which states are represented by holomorphic functionals95 of the complex connections Aa; and (2) a loop representation, according to which the states are represented by functionals of closed loops on a 3-manifold.96 The gauge invariant content contained in the connections is encoded in the holonomies, and this is in turn reflected in the traces of holonomies of the connections (i.e. Wilson loops WY (A): maps from loops to complex numbers). The configuration variables are then given by functionals of the form q[a](A) := Tr P exp G§adlaAa = Tr Ua (where P means 'path ordered', Ua is a group element given by the holonomy around the loop a, and the index a ranges over closed loops). The momentum variables are given by functionals P[s](A, E) := fs dSab nabc Tr UvEc (where S is a closed 'strip' (with topology S1 x R), Uv is a group element representing the holonomy of the connection Aia, and v labels a specific loop).

Although determinism is restored in these two representations—since the canonical variables are gauge invariant—non-locality re-enters as a result. However, the non-locality is of a rather curious kind, for it isn't of the action-at-a-distance' variety that plagued the traditional account. The non-locality concerns the spread outness' of the variables: they are not-localized at points. In the case of holonomies and Wilson loops it is better to understand them as living in loop space (cf. Redhead [2003], p. 138). I return to this issue in §3.3.2.

0 0

Post a comment