Suppose that one thinks that certain symmetries of a theory reveal “surplus structure”. What would a formalism without that surplus structure look like? The conventional answer is that it would be a reduced theory: a theory which traffics only in structures invariant under the relevant symmetry. In this paper, I argue that there is a neglected alternative: one can work with a sophisticated version of the theory, in which the symmetries act as isomorphisms.

The world appears to be well described by gauge theories; why? I suggest that gauge is more than mathematical redundancy. Gauge-dependent quantities can not be predicted, but there is a sense in which they can be measured. They describe “handles” though which systems couple: they represent real relational structures to which the experimentalist has access in measurement by supplying one of the relata in the measurement procedure itself. This observation leads to a physical interpretation for the ubiquity of gauge: it (...) is a consequence of a relational structure of physical quantities. (shrink)

Substantivalists believe that spacetime and its parts are fundamental constituents of reality. Relationalists deny this, claiming that spacetime enjoys only a derivative existence. I begin by describing how the Galilean symmetries of Newtonian physics tell against both Newton's brand of substantivalism and the most obvious relationalist alternative. I then review the obvious substantivalist response to the problem, which is to ditch substantival space for substantival spacetime. The resulting position has many affinities with what are arguably the most natural interpretations of (...) special and general relativity. I move on to consider and reject two recent antisubstantivalist lines of thought. The interim conclusion is that the best argument for relationalism is an appeal to Ockham's razor. However, for this to be successful there must be genuine relationalist theories that share the theoretical virtues of their substantivalist rivals but without the additional ontological commitment. The bulk of the paper is therefore an investigation of various concrete relationalist proposals. I distinguish three options for the relationalist in the face of the success of Galilean invariant physics and trace how these generalise to relativistic physics. One of the options is particularly promising but, since its basic objects end up being spacetime points, this does not help the prospects of relationalism as traditionally conceived. I end with some reflections on the fate of substantivalism in the aftermath of the Hole Argument, concluding that we have as yet to be given good reasons to abandon the natural, substantivalist interpretation of current physics. (shrink)

While empirical symmetries relate situations, theoretical symmetries relate models of a theory we use to represent them. An empirical symmetry is perfect if and only if any two situations it relates share all intrinsic properties. Sometimes one can use a theory to explain an empirical symmetry by showing how it follows from a corresponding theoretical symmetry. The theory then reveals a perfect symmetry. I say what this involves and why it matters, beginning with a puzzle that is resolved by the (...) subsequent analysis. I conclude by pointing to applications and implications of the ideas developed earlier in the paper. (shrink)

This paper expounds the modern theory of symplectic reduction in finite-dimensional Hamiltonian mechanics. This theory generalizes the well-known connection between continuous symmetries and conserved quantities, i.e. Noether's theorem. It also illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. The exposition emphasises how the theory provides insights about the rotation group and the rigid body. The theory's device of quotienting a state space also casts light on philosophical (...) issues about whether two apparently distinct but utterly indiscernible possibilities should be ruled to be one and the same. These issues are illustrated using ``relationist'' mechanics. (shrink)

In a recent paper in this journal, Kosso ([2000]) discussed the observational status of continuous symmetries of physics. While we are in broad agreement with his approach, we disagree with his analysis. In the discussion of the status of gauge symmetry, a set of examples offered by 't Hooft ([1980]) has influenced several philosophers, including Kosso; in all cases the interpretation of the examples is mistaken. In this paper, we present our preferred approach to the empirical significance of symmetries, re-analysing (...) the cases of gauge symmetry and general covariance. Direct and indirect empirical significance Global and local continuous symmetries Gauge symmetry 3.1 Local gauge symmetry 3.1.1 Discussion of the first claim 3.1.2 Discussion of the second claim 3.2 Global gauge symmetry Space-time symmetries Direct and indirect empirical significance again Conclusion. (shrink)

This paper sketches a taxonomy of forms of substantivalism and relationism concerning space and time, and of the traditional arguments for these positions. Several natural sorts of relationism are able to account for Newton's bucket experiment. Conversely, appropriately constructed substantivalism can survive Leibniz's critique, a fact which has been obscured by the conflation of two of Leibniz's arguments. The form of relationism appropriate to the Special Theory of Relativity is also able to evade the problems raised by Field. I survey (...) the effect of the General Theory of Relativity and of plenism on these considerations. (shrink)

Local gauge theories are in a complicated relationship with boundaries. Whereas fixing the gauge can often shave off unwanted redundancies, the coupling of different bounded regions requires the use of gauge-variant elements. Therefore, coupling is inimical to gauge-fixing, as usually understood. This resistance to gauge-fixing has led some to declare the coupling of subsystems to be the \textit{raison d'\^etre} of gauge \cite{RovelliGauge2013}. Indeed, while gauge-fixing is entirely unproblematic for a single region without boundary, it introduces arbitrary boundary conditions on the (...) gauge degrees of freedom themselves---these conditions lack a physical interpretation when they are not functionals of the original fields. Such arbitrary boundary choices creep into the calculation of charges through Noether's second theorem, muddling the assignment of physical charges to local gauge symmetries. The confusion brewn by gauge at boundaries is well-known, and must be contended with both conceptually and technically. It may seem natural to replace the arbitrary boundary choice with new degrees of freedom, for using such a device we resolve some of these confusions while leaving no gauge-dependence on the computation of Noether charges \cite{DonnellyFreidel}. But, concretely, such boundary degrees of freedom are rather arbitrary---they have no relation to the original field-content of the field theory. How should we conceive of them? Here I will explicate the problems mentioned above and illustrate a possible resolution: in a recent series of papers \cite{GomesRiello2016, GomesRiello2018, GomesHopfRiello} the notion of a connection-form was put forward and implemented in the field-space of gauge theories. Using this tool, a modified version of symplectic geometry---here called `horizontal'---is possible. Independently of boundary conditions, this formalism bestows to each region a physically salient, relational notion of charge: the horizontal Noether charge. Meanwhile, as required, the connection-form mediates an irenic composition of regions, one compatible with the attribution of horizontal Noether charges to each region. Thes aims of this paper are to highlight the boundary issues in the treatment of gauge, and to discuss how gauge theory may be conceptually clarified in light of a resolution to these issues. (shrink)

Symmetries in physics are most commonly recognized and discussed in terms of their function in the mathematical formalism of the theories. Discussion of the observation of symmetries in nature is less common. This paper analyses the observation of particular symmetries such as Lorentz and gauge symmetries, distinguishing between direct observation of the symmetry itself and indirect evidence, the latter being the observation of some consequence of the symmetry are, in an important sense, directly observed, while local symmetries such as gauge (...) symmetry are not. (shrink)

On standard accounts of scientific theorizing, the role of idealizations is to facilitate the analysis of some real world system by employing a simplified representation of the target system, raising the obvious worry about how reliable knowledge can be obtained from inaccurate descriptions. The idealizations involved in the Aharonov–Bohm effect do not, it is claimed, fit this paradigm; rather the target system is a fictional system characterized by features that, though physically possible, are not realized in the actual world. The (...) point of studying such a fictional system is to understand the foundations of quantum mechanics and how its predictions depart from those of classical mechanics. The original worry about the use of idealizations is replaced by a new one; namely, how can actual world experiments serve to confirm the AB effect if it concerns the behavior of a fictional system? Struggle with this issue helps to account for the fact that almost three decades elapsed before a consensus emerged that the predicted AB effect had received solid experimental support. Standard accounts of idealizations tout the role they play in making tractable the analysis of the target system; by contrast, the idealizations involved in the AB effect make its analysis both conceptually and mathematically challenging. The idealizations required for the AB effect are also responsible for the existence of unitarily inequivalent representations of the canonical commutation relations and of the current algebra, representations which an observer confined to the electron’s configuration space could invoke to ‘explain’ AB-type effect without the need to posit a hidden magnetic field. The goal of this paper is to bring to the attention of the philosophers of science these and other aspects of the AB effect which are neglected or inadequately treated in literature. (shrink)

The symmetries of a physical theory are often associated with two things: conservation laws and representational redundancies. But how can a physical theory's symmetries give rise to interesting conservation laws, if symmetries are transformations that correspond to no genuine physical difference? In this article, I argue for a disambiguation in the notion of symmetry. The central distinction is between what I call "analytic" and "synthetic" symmetries, so called because of an analogy with analytic and synthetic propositions. "Analytic" symmetries are the (...) turning of idle wheels in a theory's formalism, and correspond to no physical change; "synthetic" symmetries cover all the rest. I argue that analytic symmetries are distinguished because they act as fixed points or constraints in any interpretation of a theory, and as such are akin to Poincaré's conventions or Reichenbach's 'axioms of co-ordination', or 'relativized constitutive a priori principles'. (shrink)

Classical and quantum field theory provide not only realistic examples of extant notions of empirical equivalence, but also new notions of empirical equivalence, both modal and occurrent. A simple but modern gravitational case goes back to the 1890s, but there has been apparently total neglect of the simplest relativistic analog, with the result that an erroneous claim has taken root that Special Relativity could not have accommodated gravity even if there were no bending of light. The fairly recent acceptance of (...) nonzero neutrino masses shows that widely neglected possibilities for nonzero particle masses have sometimes been vindicated. In the electromagnetic case, there is permanent underdetermination at the classical and quantum levels between Maxwell's theory and the one-parameter family of Proca's electromagnetisms with massive photons, which approximate Maxwell's theory in the limit of zero photon mass. While Yang–Mills theories display similar approximate equivalence classically, quantization typically breaks this equivalence. A possible exception, including unified electroweak theory, might permit a mass term for the photons but not the Yang–Mills vector bosons. Underdetermination between massive and massless (Einstein) gravity even at the classical level is subject to contemporary controversy. (shrink)

On standard accounts of scientific theorizing, the role of idealizations is to facilitate the analysis of some real world system by employing a simplified representation of the target system, raising the obvious worry about how reliable knowledge can be obtained from inaccurate descriptions. The idealizations involved in the Aharonov–Bohm effect do not, it is claimed, fit this paradigm; rather the target system is a fictional system characterized by features that, though physically possible, are not realized in the actual world. The (...) point of studying such a fictional system is to understand the foundations of quantum mechanics and how its predictions depart from those of classical mechanics. The original worry about the use of idealizations is replaced by a new one; namely, how can actual world experiments serve to confirm the AB effect if it concerns the behavior of a fictional system? Struggle with this issue helps to account for the fact that almost three decades elapsed before a consensus emerged that the predicted AB effect had received solid experimental support. Standard accounts of idealizations tout the role they play in making tractable the analysis of the target system; by contrast, the idealizations involved in the AB effect make its analysis both conceptually and mathematically challenging. The idealizations required for the AB effect are also responsible for the existence of unitarily inequivalent representations of the canonical commutation relations and of the current algebra, representations which an observer confined to the electron’s configuration space could invoke to ‘explain’ AB-type effect without the need to posit a hidden magnetic field. The goal of this paper is to bring to the attention of the philosophers of science these and other aspects of the AB effect which are neglected or inadequately treated in literature. (shrink)

Highlighting main issues and controversies, this book brings together current philosophical discussions of symmetry in physics to provide an introduction to the subject for physicists and philosophers. The contributors cover all the fundamental symmetries of modern physics, such as CPT and permutation symmetry, as well as discussing symmetry-breaking and general interpretational issues. Classic texts are followed by new review articles and shorter commentaries for each topic. Suitable for courses on the foundations of physics, philosophy of physics and philosophy of science, (...) the volume is a valuable reference for students and researchers. (shrink)

To systematically answer the generalized Kretschmann objection, I propose a mean to make operational a criterion widely recognized as allowing one to decide whether the gauge symmetry of a theory is artificial or substantial. My proposition is based on the dressing field method of gauge symmetry reduction, a new simple tool from mathematical physics. This general scheme allows one in particular to straightforwardly argue that the notion of spontaneous symmetry breaking is superfluous to the empirical success of the electroweak theory. (...) Important questions regarding the context of justification of the theory then arise. (shrink)

It is often said that the Aharonov-Bohm effect shows that the vector potential enjoys more ontological significance than we previously realized. But how can a quantum-mechanical effect teach us something about the interpretation of Maxwell's theory—let alone about the ontological structure of the world—when both theories are false? I present a rational reconstruction of the interpretative repercussions of the Aharonov-Bohm effect, and suggest some morals for our conception of the interpretative enterprise.

In the philosophical literature on Yang-Mills theories, field formulations are taken to have more structure and to be local, while curve-based formulations are taken to have less structure and to be nonlocal. I formalize the notion of locality at issue and show that theories with less structure are nonlocal. However, the amount of structure had by some formulation is independent of whether it uses fields or curves. The relevant difference in structure is not a difference in set-theoretic structure. Rather, it (...) is a difference in the structure of the category of models of the theory. (shrink)

This article examines the implications of the holonomy interpretation of classical electromagnetism. As has been argued by Richard Healey and Gordon Belot, classical electromagnetism on this interpretation evinces a form of nonseparability, something that otherwise might have been thought of as confined to nonclassical physics. Consideration of the differences between this classical nonseparability and quantum nonseparability shows that the nonseparability exhibited by the classical electromagnetism on the holonomy interpretation is closer to separability than might at first appear.

Richard Healey argues that the Aharonov- Bohm effect demands the recognition of either nonlocal or nonseparable physics in much the way that violations of Bell's inequality do. A careful examination of the effect and the arguments, though, shows that Healey's interpretation of the Aharonov- Bohm effect depends critically on his interpretation of gauge theories, and that the analogy with violations of Bell's inequalities fails.

It is widely recognized that ‘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, conventional wisdom 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 conventional wisdom. We develop a framework for analysing the relationship between Galileo-ship empirical phenomena on the one hand, and (...) physical theories that model such phenomena on the other, 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 an analogue of Galileo’s ship for the local gauge invariance of electromagnetism. 1 Introduction2 Analogues of Galileo’s Ship? Faraday’s Cage and t’Hooft’s Beam-Splitter2.1 Faraday’s cage2.2 t’Hooft’s beam-splitter3 A Framework for Symmetries I: Systems and Subsystems4 An Example: Coulombic Electrostatics5 A Framework for Symmetries II: The Relationship between Theoretical and Empirical Symmetries6 Newtonian Gravity7 Local Symmetries that Are Not Boundary-Preserving: Classical Electromagnetism and Faraday’s Cage8 Local Boundary-Preserving Symmetries: Klein-Gordon-Maxwell Gauge Theory and t’Hooft’s Beam-Splitter9 Summary10 Conclusions. (shrink)