The gauge argument

For simplicity, consider the example of a free non-relativistic particle with wave-function f(x). Invariance under U(1) means that if we act (by multiplication) on this wave-function by an element of el0 e U(1) (i.e., a phase factor), the resulting wave-function is physically equivalent98 to the original, i.e.

In this case a global U(1) transformation was applied to the wave-function. What this means is that the same operation is applied at every point in space at some time (i.e. 0 does not depend upon x). Quantum mechanics is thus invariant under global U(1) transformations. However, it is by demanding that wave-functions be invariant under local (i.e. position dependent) U(1) transformations that Maxwell's theory results.99 These are transformations of the form f '(x) = ei0 (x)f (x) (3.8)

Such transformations do not result in physically equivalent wave-functions for they differ in terms of their momenta. The charge density is invariant, but the current density is not. In order to achieve local invariance, modifications to the Hamiltonian have to made (or the Lagrangian if one is working in a spacetime description). These modifications require that (1) interactions be turned on between the system and charge;100 and (2) a new field be introduced (the 'gauge potential': A). The system is coupled to this new field, and the modified Hamiltonian (respectively, Lagrangian) is invariant under local U(1) transformations whenever the field is likewise transformed. The form of the transformations then matches those of Maxwell's theory, where A is identified as the vector potential for the magnetic field.101

The gauge argument highlights philosophically interesting issues regarding the application of gauge theories, but it is not the aspect I wish to focus on. I am concerned with the nature of the gauge symmetries, and in particular with the notion of gauge freedom. I think that such a foundational study is required in order to properly understand how the gauge argument works.

One of the characteristic properties of a gauge theory is the conceptual problems faced when setting up a direct, one-to-one correspondence between the mathematical formalism of the theory and the physical system that the theory represents. As Henneaux and Teitelboim put it in their canonical text on gauge systems, "the physical system being dealt with is described by more variables than there are physically independent degrees of freedom" ([1992], p. 1). One extracts

98 Physical equivalence is cashed out in this case in terms of the expectation values of the system's observables—e.g. charge density, p = e&* &, and current density, j = 1 i exp(&*V& — &V&*).

99 The generic name for the argument that demanding local gauge invariance produces an interacting field theory is known as "the gauge argument" or "gauge principle". It is discussed by Martin, and its "profundity" questioned, in [2003].

100 The charge is the conserved current that results from global U(1) invariance. See §3.2.3 below.

101 I should point out that the argument properly requires fields, rather than a single particle. The imposition of local gauge invariance is then underwritten by the fact that field values at spacelike separated points can be specified independently of one another. However, the present example makes the basic idea plain enough without recourse to the complications field theory introduces.

the physical degrees of freedom through the use of a gauge symmetry connecting certain of the variables: either one can factor out the action of the symmetry group giving a formalism without gauge freedom, or one can 'ignore' the gauge freedom by focusing on gauge invariant observables.102 Gauge theories thus supply the machinery for both generating and eradicating (or 'dealing with'—in a way that preserves the 'physically relevant' structure) the 'surplus structure' created by the gauge freedom of the theory.

I think it is appropriate to say something about the case of the permutation symmetry of quantum mechanics at this point. The permutation operation can be construed as a kind of gauge transformation such that states connected by such a transformation are to understood as representing the same physical situation. The quantum statistics are then explained by something akin to gauge freedom in the theory; namely, the freedom to permute indistinguishable particles' labels without thereby altering the structure of the physically measurable quantities (i.e. the expectation values of the observables) of the theory: permuted particles will simply differ by a constant phase factor, and this does not allow for the distinction of wave-functions, as we saw earlier. Hence, the theory is permutation invariant. According to the received view this was supposed to hold some metaphysical significance concerning the individuality of the particles. But I argued that no such view was forced upon us by permutation invariance, and that permutation invariance will be a feature of any reasonable physical theory possessing indistinguishable elements (related to the kinematical and/or dynamical structure of the theory) and a way of generating permutations on these: permutation invariance can be accommodated by 'individualists' and 'non-individualists' alike. Likewise, the symmetries of Newtonian space used in the shift argument can be loosely construed as a kind of gauge transformation too. The generation of haecceitisti-cally differing shifted worlds corresponds to something like gauge freedom: we can globally shift the material contents of a world without altering the physically observable properties. Hence, any physical system defined on this space will be invariant under such shifts. This was supposed to hold some metaphysical significance concerning the ontological status of space (and, mutatis mutandis, time). I argued that this wasn't forced upon us by the invariance: the invariance can be accommodated by substantivalists and relationalists alike.

I argued, and will argue, that with respect to these two cases one should view the underdetermination of metaphysical (interpretive) options by the physics as pointing to a structuralist understanding of quantum states and spacetime: all parties will agree about the structure that is exemplified in these cases. Accepting this does not imply that one should be realist only about this structure, or even about this structure itself, for I prefer to detach the realism/anti-realism issue from the interpretation of physics. What I do wish to defend is the claim that any interpretive options that go beyond this basic structure are straying from physics, and once this happens, the underdetermination is ineluctable. Since I am concerned with the interpretation of physics, I hold that the invariant structures exhaust what phys

102 We shall see in §3.2.3 (and, in more detail, in §4.3) that this gauge freedom manifests itself in the form of constraints on the variables when the theory is cast into Hamiltonian form. In general relativity these constraints lead to problems with the status of space, time, and change in both the classical and quantum theories.

ical theories can sensibly talk about with definiteness as regards ontology. There might be individuals, there might be objects with strange identities, there might be entities with bizarre modal behaviour, but a physical theory cannot tell us much about these aspects. At best they can function as a constraint on these metaphysical positions; but there will be cases where mutually incompatible metaphysical positions are both compatible with some theory. Because of this, what I propose is that we should view physical theories as imposing a version of PSR with respect to such meta-physical' issues, such that only those aspects of (mathematical) structure that can be distinguished by the observables count as ontologically relevant in the physical description of the world-structure to be represented. In cases where a mathematical formalism admits more degrees of freedom than are strictly necessary in the representation of a physical system, the theory (qua states, observables, and laws) should be understood as indifferent to these. Again, I do not rule out that there is more to the world than this invariant structure, only that physical theories cannot 'see' beneath such structure.103 With this claim out in the open, I can now introduce the details of gauge theories and show how it squares up.

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