The phenomenon of broken spacetime symmetry in the quantum theory of infinite systems forces us to adopt an unorthodox ontology. We must abandon the standard conception of the physical meaning of these symmetries, or else deny the attractive “liberal” notion of which physical quantities are significant. A third option, more attractive but less well understood, is to abandon the existing (Halvorson-Clifton) notion of intertranslatability for quantum theories.

We pose and resolve a seeming paradox about spontaneous symmetry breaking in the quantum theory of infinite systems. For a symmetry to be spontaneously broken, it must not be implementable by a unitary operator. But Wigner's theorem guarantees that every symmetry is implemented by a unitary operator that preserves transition probabilities between pure states. We show how it is possible for a unitary operator of this sort to connect the folia of unitarily inequivalent representations. This result undermines interpretations of quantum (...) theory that hold unitary equivalence to be necessary for physical equivalence. (shrink)

Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don't we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. We (...) argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher-Haag-Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality-inspired principle we call Charge Recombination. (shrink)

In this paper I reconstruct and critically examine the reasoning leading to the famous prediction of the ‘omega minus’ particle by M. Gell-Mann and Y. Ne’eman (in 1962) on the basis of a symmetry classification scheme. While the peculiarity of this prediction has occasionally been noticed in the literature, a detailed treatment of the methodological problems it poses has not been offered yet. By spelling out the characteristics of this type of prediction, I aim to underscore the challenges raised by (...) this episode to standard scientific methodology, especially to the traditional deductive-nomological account of prediction. (shrink)

Symmetry, intended as invariance with respect to a transformation (more precisely, with respect to a transformation group), has acquired more and more importance in modern physics. This Chapter explores in 8 Sections the meaning, application and interpretation of symmetry in classical physics. This is done both in general, and with attention to specific topics. The general topics include illustration of the distinctions between symmetries of objects and of laws, and between symmetry principles and symmetry arguments (such as Curie's principle), and (...) reviewing the meaning and various types of symmetry that may be found in classical physics, along with different interpretative strategies that may be adopted. Specific topics discussed include the historical path by which group theory entered classical physics, transformation theory in classical mechanics, the relativity principle in Einstein's Special Theory of Relativity, general covariance in his General Theory of Relativity, and Noether's theorems. In bringing these diverse materials together in a single Chapter, we display the pervasive and powerful influence of symmetry in classical physics, and offer a possible framework for the further philosophical investigation of this topic. (shrink)

In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassen’s elegant presentation of how symmetry principles can be used as problem-solving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problem-solving tool. Finally, I shall explore one consequence of (...) this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]). (shrink)

Bertrand Russell famously argued that causation is not part of the fundamental physical description of the world, describing the notion of cause as "a relic of a bygone age." This paper assesses one of Russell’s arguments for this conclusion: the ‘Directionality Argument’, which holds that the time symmetry of fundamental physics is inconsistent with the time asymmetry of causation. We claim that the coherence and success of the Directionality Argument crucially depends on the proper interpretation of the ‘time symmetry’ of (...) fundamental physics as it appears in the argument, and offer two alternative interpretations. We argue that: (1) if ‘time symmetry’ is understood as the time-reversal invariance of physical theories, then the crucial premise of the Directionality Argument should be rejected; and (2) if ‘time symmetry’ is understood as the temporally bidirectional nomic dependence relations of physical laws, then the crucial premise of the Directionality Argument is far more plausible. We defend the second reading as continuous with Russell’s writings, and consider the consequences of the bidirectionality of nomic dependence relations in physics for the metaphysics of causation. (shrink)

The long established but infrequently discussed dependence of Lorentz boost generators on the presence and nature of interactions is reviewed in this tutorial note. The last third of the note presents a discussion of the covariant transformation and evolution equations for the non-conserved partial generators of the inhomogeneous Lorentz group for interacting subsystems.

"Symmetry" was one of the most important methodological themes in 20th-century physics and is probably going to play no lesser role in physics of the 21st century. As used today, there are a variety of interpretations of this term, which differ in meaning as well as their mathematical consequences. Symmetries of crystals, for example, generally express a different kind of invariance than gauge symmetries, though in specific situations the distinctions may become quite subtle. I will review some of the various (...) notions of "symmetry" and highlight some of their uses in specific examples taken from Pauli's scientific oevre. This paper is based on a talk given at the conference "Wolfgang Pauli's Philosophical Ideas and Contemporary Science", May 20.-25. 2007, at Monte Verita, Ascona, Switzerland. (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)

Physical theories continue to be interpreted in terms of particles. The idea of a particle required modification with the advent of quantum theory, but remains central to scientific explanation. Particle ontologies also have the virtue of explaining basic epistemic features of the world, and so remain appealing for the scientific realist. However, particle ontologies are untenable when coupled with the empirically necessary postulate of permutation invariance—the claim that permuting the roles of particles in a representation of a physical state results (...) in a representation of the same physical state. I demonstrate that any theory which is permutation invariant in this sense is incompatible with a particle ontology. (shrink)

This paper, part II of a two-part project, continues to explore the meaning of spontaneous symmetry breaking (SSB) by applying and expanding the general notion we obtained in part I to some more complex and, from the physics point of view, more important models (in condensed matter physics and in quantum field theories).

Most scientists and philosophers of science recognize that, when it comes to accepting and rejecting theories in science, considerations that have to do with simplicity, unity, symmetry, elegance, beauty or explanatory power have an important role to play, in addition to empirical considerations. Until recently, however, no one has been able to give a satisfactory account of what simplicity (etc.) is, or how giving preference to simple theories is to be justified. But in the last few years, two different but (...) related accounts have appeared, both of which address the above issues. On the one hand, James McAllister has argued that aesthetic criteria in science reflect scientists' judgements about what kind of theory is most likely to be empirically successful, based on the relative empirical success and failure of different kinds of theories in the past. Scientists employ what McAllister dubs "the aesthetic induction". On the other hand, I have argued that we need to see science as making a hierarchy of metaphysical assumptions about the comprehensibility and knowability of the universe, these assumptions asserting less and less as one ascends the hierarchy. One of the more substantial of these assumptions is that the universe is physically comprehensible. The key non-empirical feature a body of fundamental theories in physics must possess to be acceptable is unity. The better such a body of theory exemplifies the metaphysical thesis that the universe is physically comprehensible, in the sense that it has a unified dynamic structure, so the more acceptable such a body of theory is, from this standpoint. This affects not just theoretical physics, but the whole of natural science. In this paper I compare and contrast, and try to assess impartially the relative merits of, these two views. (shrink)

We are used to talking about the “structure” posited by a given theory of physics. We say that relativity is a theory about spacetime structure. Special relativity posits one spacetime structure; different models of general relativity posit different spacetime structures. We also talk of the “existence” of these structures. Special relativity says the world’s spacetime structure is Minkowskian: it posits that this spacetime structure exists. Understanding structure in this sense seems important for understanding what physics is telling us about the (...) world. But it is not immediately obvious just what this structure is, or what we mean by the existence of one structure, rather than another. The idea of mathematical structure is relatively straightforward. There is geometric structure, topological structure, algebraic structure, and so forth. Mathematical structure tells us how abstract mathematical objects t together to form different types of mathematical spaces. Insofar as we understand mathematical objects, we can understand mathematical structure. Of course, what to say about the nature of mathematical objects isn’t easy. But there seems to be no further problem for understanding mathematical structure. Modern theories of physics are formulated in terms of these mathematical structures. In order to understand “structure” as used in physics, then, it seems we must simply look at the structure of the mathematics that is used to state the physics. But it is not that simple. Physics is supposed to be telling us about the nature of the world. If our physical theories are formulated in mathematical language, using mathematical objects, then this mathematics is somehow telling us about the physical make-up of the world. What is.. (shrink)

John Earman's recent proposal that a substantive version of general covariance consists in the requirement that diffeomorphism invariance be a gauge symmetry is critically assessed. I argue that such a principle does not serve to differentiate general relativity from pre-relativistic theories. A model-theoretic characterization of two formulations of specially-relativistic theories is suggested. Diffeomorphisms are symmetries of only one such style of formulation and, I argue, Earman's proposal does not provide a reason to deny diffeomorphisms the status of gauge transformations relative (...) to this formulation. Carlo Rovelli's distinction between "passive" and "active" diffeomorphism invariance is also clarified. (shrink)

In this paper I examine the connection between symmetry and modality from the perspective of `reduction' methods in geometric mechanics. I begin by setting the problem up as a choice between two opposing views: reduction and non-reduction. I then discern four views on the matter in the literature; they are distinguished by their advocation of distinct geometric spaces as representing `reality'. I come down in favour of non-reductive methods.

This paper addresses a central question of contemporary theoretical physics: Can a unified account be provided for the known forces of nature? The issue is brought into focus by considering the recently revived Kaluza-Klein approach to unification, a program entailing dimensional transformation through cosmogony. First it is demonstrated that, in a certain sense, revitalized Kaluza-Klein theory appears to undermine the intuitive foundations of mathematical physics, but that this implicit consequence has been repressed at a substantial cost. A fundamental reformulation of (...) the Kaluza-Klein strategy is then undertaken, one that casts it within a new intuitive context. This is followed by a provisional application of the suggested approach to the specific problem of cosmic evolution. The paper concludes by exploring the far-reaching epistemological implications of the "neo-intuitive" proposal set forth. (shrink)

The violation of parity in weak interactions demonstrated in 1956 was an event that shook the foundations of physics. Since that time, the status of physical symmetry has been very much in doubt. The problem is presently addressed by first examining the essential relation between symmetry and asymmetry. Then, through the medium of qualitative mathematics, an attempt is made to show how these opposites may be fused in a topological structure expressing a new principle, that of "synsymmetry".

A dialectical, double-aspect model of general interaction is proposed as a means of visualizing existence. It is constructed from observation of the transformational properties of the Moebius surface, with hyperdimensional extrapolation to the Klein bottle. The interaction of systems is viewed as a process of circumversion (i.e. turning inside-out) whereby systems exchange relations, become mutually negated, then exchange identities. Interaction is discussed in the context of the PCT problem, symmetry, creation-annihilation, and uncertainty.

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.

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.

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. (shrink)