In this paper I shall extend the formulation of the hole argument to permutable theories. As covariant theories provides a general mathematical framework for classical physics, permutable theories provide the language for quantum physics. This analogy is deeply founded: as covariant theories are defined functorially on the category of manifolds and local diffeomorphisms with values into the category of fibered manifolds and fiber-preserving morphisms, and some rule of selecting sections, permutable theories are functorially defined on the category of sets and permutations with values into the category of fibered sets and fiber-preserving automorphisms, and rules of selecting sections. One of the main features of covariant theories, in particular general relativity, is that the field equation possesses gauge freedom associated with global diffeomorphisms of the underlying manifold. I shall explain here how the hole argument is a reflection of this gauge freedom. Finally I shall point out some implications of the hole argument to current development in physics.
In this paper I shall extend the formulation of the hole argument to permutable theories. As covariant theories provides a general mathematical framework for classical physics, permutable theories provide the language for quantum physics. This analogy is deeply founded: as covariant theories are defined functorially on the category of manifolds and local diffeomorphisms with values into the category of fibered manifolds and fiber-preserving morphisms, and some rule of selecting sections, permutable theories are functorially defined on the category of sets and permutations with values into the category of fibered sets and fiber-preserving automorphisms, and rules of selecting sections. One of the main features of covariant theories, in particular general relativity, is that the field equation possesses gauge freedom associated with global diffeomorphisms of the underlying manifold. I shall explain here how the hole argument is a reflection of this gauge freedom. Finally I shall point out some implications of the hole argument to current development in physics.
The relation between the manifold topology, observables, and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated with positions and displacements on the manifold. The guiding, physically motivated, principles are (i) locality, i.e., the generating role of the algebras localized in small, topologically trivial, regions, (ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, and (iii) the exclusion of additional local degrees of freedom with respect to the Schrödinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, “topological,” subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering M̃ of M and on the decomposition of L2(M̃) according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the permutation group in particle statistics are discussed.
We treat the mechanics of point particles as a one-dimensional general-relativistic gauge field theory, which may be referred to as mechanical field theory (MFT), exploiting the bundle geometry of mechanical field space (MFS). The diffeomorphism covariance of MFT encodes its relational character, arising—as in all general-relativistic physics—via the conjunction of a hole and a point-coincidence argument. Any putative “boundary problem,” meaning the claim that “spacetime” boundaries break diffeomorphism and gauge symmetries, thereby dissolves. It is highlighted that the standard path integral (PI) on the MFS, the exact analog of the PI used in gauge field theory, is conceptually and technically from the standard PI of quantum mechanics. We then use the dressing field method to give a manifestly invariant and relational reformulation of MFT, which reproduces the standard textbook formulation when a clock field is chosen as a (natural) dressing field. The dressed, or basic, PI on the MFS, defining i.e., the quantization of invariant relational degree of freedom—is shown to reproduce the standard PI of quantum mechanics. This establishes the soundness of relational quantization as a general guiding principle: We outline it for general-relativistic gauge field theories.
General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)-dimensional manifolds representing ‘space ’ and whose morphisms are n-dimensional cobordisms representing ‘spacetime’. Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe ‘states’, and whose morphisms are bounded linear operators used to describe ‘processes’. Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are ∗-categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime. 1
This article develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles. In GR, the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature—a theory's ‘not caring which point, or particle, is which’—supported a structuralist ontology. Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution. We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of particle symmetry, in which states (vectors, rays, or density operators) are not fixed by all permutations (called ‘paraparticle states’). Thus Stachel's analogy is revived. 1 Introduction2 Structuralism Applied to General Relativity and Quantum Mechanics 2.1 Structuralism and individuality 2.2 The semantics of the structuralist 2.3 General relativity and Leibniz equivalence3 Anti-haecceitism and Quantum Mechanics 3.1 Pure and mixed states; permutations 3.2 Symmetrization and indistinguishability 3.3 Quantum statistics and the bad argument for anti-haecceitism4 The Generalized Hole Argument for Sets 4.1 Models and possible worlds 4.2 Permutations and permutes 4.3 State descriptions and structure descriptions 4.4 Theories; permutability, fixity, and general permutability 4.5 Stachel's argument for structuralism5 Fixed Theories and Metaphysical Under-determination 5.1 Pooley's objection; amending Stachel's premises 5.2 Is SP anti-haecceitistic?6 Superselection and Paraparticles 6.1 Quantum permutability 6.1.1 Symmetry types and symmetric operators 6.1.2 Statistics and symmetrization; superselection 6.1.3 The two arguments for quantum permutability 6.2 Quantum general permutability 6.3 Examples of quantum permutability 6.3.1 Two quantum coins, again 6.3.2 A generalized ray
This chapter examines the nature of physical reality in the context of unavoidable unobservables in physics. One of the features of science, especially physical science, is a powerful urge to construct theories of phenomena, instead of just cataloging and classifying phenomena and their spatio-temporal patterns. The concept of fields' of various kinds dominates physical theory and modern physics, and fields are essential to both classical and quantum theories. The concept of physical reality in the context of the Einstein, Podolsky and Rosen (EPR) assertion of the incompleteness of quantum mechanics is perhaps appropriate. Quantum theory of physical systems consisting of large number of particles led to the concept of a quantum field associated with the particles, a generalisation of the wave function itself. Space and time are supreme and primary unobservables in physics, without any reality independent of matter, notwithstanding their modern status as the dynamical arena of gravitational effects as described in Einstein's general theory of relativity.
Classically, a gauge potential was merely a convenient device for generating a corresponding gauge field. Quantum-mechanically, a gauge potential lays claim to independent status as a further feature of the physical situation. But whether this is a local or a global feature is not made any clearer by the variety of mathematical structures used to represent it. I argue that in the theory of electromagnetism (or a non-Abelian generalization) that describes quantum particles subject to a classical interaction, the gauge potential is best understood as a feature of the physical situation whose global character is most naturally represented by the holonomies of closed curves in space-time.