Christopher Isham

Canonical quantum gravity and the problem of time

The aim of this paper is to provide a general introduction to the problem of time in quantum gravity. This problem originates in the fundamental conflict between the way the concept of ‘time’ is used in quantum theory, and the role it plays in a diffeomorphism-invariant theory like general relativity. Schemes for resolving this problem can be sub-divided into three main categories: (I) approaches in which time is identified before quantising; (II) approaches in which time is identified after quantising; and (III) approaches in which time plays no fundamental role at all. Ten different specific schemes are discussed in this paper which also contain an introduction to the relevant parts of the canonical decomposition of general relativity.

Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations

AbstractAny attempt to construct a realistinterpretation of quantum theory founders on theKochen–Specker theorem, which asserts theimpossibility of assigning values to quantum quantitiesin a way that preserves functional relations between them. We constructa new type of valuation which is defined on alloperators, and which respects an appropriate version ofthe functional composition principle. The truth-values assigned to propositions are (i) contextual and(ii) multivalued, where the space of contexts and themultivalued logic for each context come naturally fromthe topos theory of presheaves. The first step in our theory is to demonstrate that theKochen–Specker theorem is equivalent to thestatement that a certain presheaf defined on thecategory of self-adjoint operators has no globalelements. We then show how the use of ideas drawn from the theory ofpresheaves leads to the definition of a generalizedvaluation in quantum theory whose values are sieves ofoperators. In particular, we show how each quantum state leads to such a generalized valuation. Akey ingredient throughout is the idea that, in asituation where no normal truth-value can be given to aproposition asserting that the value of a physical quantity A lies in a subset

Lectures on Quantum Theory: Mathematical and Structural Foundations

\Delta \subseteq \mathbb{R}

A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues

, it is nevertheless possible toascribe a partial truth-value which is determined by theset of all coarse-grained propositions that assert thatsome function f(A) lies in f(Δ), and that are true in a normalsense. The set of all such coarse-grainings forms asieve on the category of self-adjoint operators, and ishence fundamentally related to the theory ofpresheaves.

A topos foundation for theories of physics: I. Formal languages for physics

An extensive analysis is made of the Gell‐Mann and Hartle axioms for a generalized ‘histories’ approach to quantum theory. Emphasis is placed on finding analogs of the lattice structure employed in standard quantum logic. Particular attention is given to ‘quasitemporal’ theories in which the notion of time‐evolution in conventional Hamiltonian physics is replaced by something that is much broader; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space–time. The quasitemporal structure is coded in a partial semigroup of ‘temporal supports’ that underpins the lattice of history propositions. Nontrivial examples include quantum field theory on a non‐globally‐hyperbolic space–time, and a possible cobordism approach to a theory of quantum topology. A key result is the demonstration that the set of history propositions in standard quantum theory can be realized in such a way that each such proposition is represented by a genuine projection operator. This gives valuable insight into the possible lattice structure in general history theories.

Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory

Vector spaces linear operators properties in classical physics the general formalism of quantum theory technical developments unitary operators in quantum theory some conceptual issues in quantum theory properties in quantum physics problems and answers.

Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity

Abstract The study of conformal group symmetries within the framework of nonlinear realizations is extended and reexpressed in terms of metric tensors and connections on spacetime. The standard Vierbein formalism of general relativity is then reinterpreted in terms of nonlinear realizations of the group GL(4, R). Throughout we emphasise the connection between massless Goldstone bosons and the preferred fields of nonlinear realizations.

“What is a Thing?”: Topos Theory in the Foundations of Physics

In a previous paper, we proposed assigning asthe value of a physical quantity in quantum theory acertain kind of set (a sieve) of quantities that arefunctions of the given quantity. The motivation was in part physical — such a valuationilluminates the Kochen–Specker theorem — andin part mathematical — the valuation arisesnaturally in the topos theory of presheaves. This paperdiscusses the conceptual aspects of this proposal. We also undertake two othertasks. First, we explain how the proposed valuationscould arise much more generally than just in quantumphysics; in particular, they arise as naturally in classical physics. Second, we give anothermotivation for such valuations (that applies equally toclassical and quantum physics). This arises fromapplying to propositions about the values of physical quantities some general axioms governingpartial truth for any kind of proposition.