# Philosophy of Symmetry and Spacetime

### Gauge Invariance through Gauge Fixing (2024)

*Studies in History and Philosophy of Science*, forthcoming.

Phenomena in gauge theory are often described in the physics literature via a specific choice of gauge. In foundational and philosophical discussions this is often criticized as introducing gauge dependence, and contrasted against (often aspirational) "gauge-invariant" descriptions of the physics. I argue, largely in the context of scalar electrodynamics, that this is misguided, and that descriptions of a physical process within a specific gauge are in fact gauge-invariant descriptions. However, most of them are non-local descriptions of that physics, and I suggest that this ought to be the real objection to such descriptions. I explore the unitary gauge as the exception to this nonlocality and consider its strengths and limitations, as well as (more briefly) its extension beyond scalar electrodynamics.

### First-class constraints generate gauge transformations in electromagnetism (reply to Pitts) (2022)

(Oliver Pooley and DW) Currently online-only.

Brian Pitts has recently claimed to show via straightforward calculation that, at least in the case of Hamiltonian electromagnetism, an arbitrary first-class constraint ``generates not a gauge transformation, but a bad physical change'' (Annals of Physics 351 (2014) pp.382-406; arXiv:1310.2756). We show, via a straightforward calculation, that a transformation generated by an arbitrary first-class constraint relates gauge-equivalent phase space points, vindicating orthodoxy. Pitts, however, is primarily concerned with transformations of entire histories, rather than of instantaneous states. We show that, even in this context, a transformation generated by an arbitrary first-class constraint is also a gauge transformation, once the empirically observed electric field is correctly identified via its dynamical interactions with charge, and not simply given stipulatively as a certain combination of the potential and its derivatives.

### Observability, redundancy and modality for dynamical symmetry transformations (2019)

In James Read, Bryan Roberts and Nic Teh (eds.), *The Philosophy and Physics of Noether's Theorems: A Centenary Volume* (CUP, 2022), 322-353.

I provide a fairly systematic analysis of when quantities that are variant under a dynamical symmetry transformation should be regarded as unobservable, or redundant, or unreal; of when models related by a dynamical symmetry transformation represent the same state of affairs; and of when mathematical structure that is variant under a dynamical symmetry transformation should be regarded as surplus. In most of these cases the answer is `it depends': depends, that is, on the details of the symmetry in question. A central feature of the analysis is that in order to draw any of these conclusions for a dynamical symmetry it needs to be understood in terms of its possible extensions to other physical systems, in particular to measurement devices.

### Isolated Systems and their Symmetries, Part I: General Framework and Particle-Mechanics Examples (2019)

*Studies in the History and Philosophy of Science* 92 (2022), 239-248.

Physical theories, for the most part, should be understood as modelling isolated subsystems of a larger Universe; doing so, among other benefits, greatly clarifies the interpretation of the dynamical symmetries of those theories. I provide a detailed framework for analysing the subsystem structure of physical theories and applying it to the interpretation of their symmetries: the core concept is subsystem-recursivity, whereby interpretative conclusions about a sector of a theory can be deduced from considering subsystems of other models of the same theory. I illustrate the framework by extensive examples from nonrelativistic particle mechanics, and in particular from Newtonian theories of gravity. A sequel to the paper will apply the framework to the local and global symmetries of classical field theory.

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### Isolated Systems and their Symmetries, Part II: Local and Global Symmetries of Field Theories (2019)

*Studies in the History and Philosophy of Science* 92 (2022), 249-259.

Physical theories, for the most part, should be understood as modelling isolated subsystems of a larger Universe; doing so, among other benefits, greatly clarifies the interpretation of the dynamical symmetries of those theories. Building on a general framework for the symmetries of isolated systems developed in the prequel to this paper, I apply that framework to field theory. The resultant analysis provides a general basis for interpreting the physical significance of symmetries according to their topological and asymptotic features: global symmetries in general, and local symmetries insofar as they preserve boundary conditions and are asymptotically nonvanishing and/or topologically nontrivial, can be understood as physical transformations of an isolated system against the assumed background of other systems. Other symmetries in general must be understood as mere redescription, though in certain circumstances even non-boundary-preserving local symmetries can be afforded physical significance. The analysis largely reproduces - and so can be seen as a theoretical justification for - general practice in contemporary physics.

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### Fundamental and Emergent Geometry in Newtonian Physics (2016)

*British Journal for the Philosophy of Science, * 71 (2020) pp.1-32.

Using as a starting point recent and apparently incompatible conclusions by Simon Saunders (Philosophy of Science 80 (2013) pp.22-48) and Eleanor Knox (British Journal for the Philosophy of Science 65 (2014) pp.863-880), I revisit the question of the correct spacetime setting for Newtonian physics. I argue that understood correctly, these two theories make the same claims both about the background geometry required to define the theory, and about the inertial structure of the theory. In doing so I illustrate and explore in detail the view --- espoused by Knox, and also by Harvey Brown (Physical Relativity, OUP 2005) --- that inertial structure is defined by the dynamics governing subsystems of a larger system. This clarifies some interesting features of Newtonian physics, notably (i) the distinction between using the theory to model subsystems of a larger whole and using it to model complete Universes, and (ii) the scale-relativity of spacetime structure.

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### Who's Afraid of Coordinate Systems? An essay on the representation of spacetime structure (2016)

*Studies in the History and Philosophy of Modern Physics* 67 (2019) pp.125-136.

Coordinate-based approaches to physical theories remain standard in mainstream physics but are largely eschewed in foundational discussion in favour of coordinate-free differential-geometric approaches. I defend the conceptual and mathematical legitimacy of the coordinate-based approach for foundational work. In doing so, I provide an account of the Kleinian conception of geometry as a theory of invariance under symmetry groups; I argue that this conception continues to play a very substantial role in contemporary mathematical physics and indeed that supposedly "coordinate-free" differential geometry relies centrally on this conception of geometry. I discuss some foundational and pedagogical advantages of the coordinate-based formulation and briefly connect it to some remarks of Norton on the historical development of geometry in physics during the establishment of the general theory of relativity.

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### Fields as Bodies: a unified presentation of spacetime and internal gauge symmetry (2015)

Online only (cite as arxiv:1502.06539)

Using the parametrised representation of field theory (in which the location in spacetime of a part of a field is itself represented by a map from the base manifold to Minkowski spacetime) I demonstrate that in both local and global cases, internal (Yang-Mills-type) and spacetime (Poincare) symmetries can be treated precisely on a par, so that gravitational theories may be regarded as gauge theories in a completely standard sense.

### Deflating the Aharonov-Bohm Effect (2014)

Online only (cite as arxiv:1407.5073).

I argue that the metaphysical import of the Aharonov-Bohm effect has been overstated: correctly understood, it does not require either rejection of gauge invariance or any novel form of nonlocality. The conclusion that it does require one or the other follows from a failure to keep track, in the analysis, of the complex scalar field to which the magnetic vector potential is coupled. Once this is recognised, the way is clear to a local account of the ontology of electrodynamics (or at least, to an account no more nonlocal than quantum theory in general requires); I sketch a possible such account.

### Empirical Consequences of Symmetries (2011)

(Hilary Greaves and DW) *British Journal for the Philosophy of Science*65 (2014), 59-89

`Global' symmetries, such as the boost invariance of classical mechanics and special relativity, can give rise to direct empirical counterparts such as the Galileo-ship phenomenon. However, a widely accepted line of thought holds that `local' symmetries, such as the diffeomorphism invariance of general relativity and the gauge invariance of classical electromagnetism, have no such direct empirical counterparts. We argue against this line of thought. We develop a framework for analysing the relationship between Galileo-ship empirical phenomena and physical theories that model such phenomena that renders the relationship between theoretical and empirical symmetries transparent, and from which it follows that both global and local symmetries can give rise to Galileo-ship phenomena. In particular, we use this framework to exhibit analogs of Galileo's ship for both the diffeomorphism invariance of general relativity and the gauge invariance of electromagnetism.

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### The Relativity and Equivalence Principles for Self-Gravitating Systems (2009)

In D Lehmkuhl, G. Schiemann and E. Scholz (eds.), *Towards a Theory of Spacetime Theories* (Springer, 2017).

I criticise the view that the relativity and equivalence principles are consequences of the small-scale structure of the metric in general relativity, by arguing that these principles also apply to systems with non-trivial self-gravitation and hence non-trivial spacetime curvature (such as black holes). I provide an alternative account, incorporating aspects of the criticised view, which allows both principles to apply to systems with self-gravity.

### QFT, Antimatter, and Symmetry (2009)

*Studies in the History and Philosophy of Modern Physics* 40 (2009) 209-222.

A systematic analysis is made of the relations between the symmetries of a classical field and the symmetries of the one-particle quantum system that results from quantizing that field in regimes where interactions are weak. The results are applied to gain a greater insight into the phenomenon of antimatter.

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### Time-dependent symmetries: the link between gauge symmetries and indeterminism (2003)

In K. Brading and E. Castellani (eds.), *Symmetries in physics: philosophical reflections* (CUP, 2003)

Mathematically, gauge theories are extraordinarily rich --- so rich, in fact, that it can become all too easy to lose track of the connections between results, and become lost in a mass of beautiful theorems and properties: indeterminism, constraints, Noether identities, local and global symmetries, and so on.

One purpose of this short article is to provide some sort of a guide through the mathematics, to the conceptual core of what is actually going on. Its focus is on the Lagrangian, variational-problem description of classical mechanics, from which the link between gauge symmetry and the apparent violation of determinism is easy to understand; only towards the end will the Hamiltonian description be considered.

The other purpose is to warn against adopting too unified a perspective on gauge theories. It will be argued that the meaning of the gauge freedom in a theory like general relativity is (at least from the Lagrangian viewpoint) significantly different from its meaning in theories like electromagnetism. The Hamiltonian framework blurs this distinction, and orthodox methods of quantization obliterate it; this may, in fact, be genuine progress, but it is dangerous to be guided by mathematics into conflating two conceptually distinct notions without appreciating the physical consequences.