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

Any 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 , 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(), 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 Perspective on State-Vector Reduction

A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all M-sets, where M is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a ‘neo-realist’ interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of state-vector reduction.     

A Topos Perspective on State-Vector Reduction

A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all M-sets, where M is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a ‘neo-realist’ interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of state-vector reduction.     

Parity Proofs of the Kochen-Specker Theorem Based on the 24 Rays of Peres

A diagrammatic representation is given of the 24 rays of Peres that makes it easy to pick out all the 512 parity proofs of the Kochen-Specker theorem contained in them. The origin of this representation in the four-dimensional geometry of the rays is pointed out.     

Topos Perspective on the Kochen=nSpeckerTheorem: III. Von Neumann Algebras as theBase Category

We extend the topos-theoretic treatment given in previous papers of assigningvalues to quantities in quantum theory, and of related issues such as theKochen–Specker theorem. This extension has two main parts: the use of vonNeumann algebras as a base category and the relation of our generalized valuationsto (i) the assignment to quantities of intervals of real numbers and (ii) the ideaof a subobject of the coarse-graining presheaf.     

A Continuous Transition Between Quantum and Classical Mechanics. II

Examples are worked out using a new equation proposed in the previous paper to show that it has new physical predictions for mesoscopic systems.     

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.     

Topology and phase transitions II. Theorem on a necessary relation

In this second paper, we prove a necessity theorem about the topological origin of phase transitions. We consider physical systems described by smooth microscopic interaction potentials VN(q)$V_N(q)$, among N   degrees of freedom, and the associated family of configuration space submanifolds _{Mv}v∈R${\{M_v\}}_{v\in \mathbb{R}}$, with Mv={q∈RN|VN(q)?v}$M_v=\{q\in {\mathbb{R}}^N|V_N(q)?v\}$. On the basis of an analytic relationship between a suitably weighed sum of the Morse indexes of the manifolds _{Mv}v∈R${\{M_v\}}_{v\in \mathbb{R}}$ and thermodynamic entropy, the theorem states that any possible unbound growth with N   of one of the following derivatives of the configurational entropy S(−)(v)=(1/N)logMv_∫dNq$S^{(-)}(v)=(1/N)\log {\int }_{M_v}{d}^{N}q$, that is of |k^∂S(−)(v)/∂vk|$|{\partial }^k{S}^{(-)}(v)/\partial v^k|$, for k=3,4$k=3,4$, can be entailed only by the weighed sum of Morse indexes. Since the unbound growth with N of one of these derivatives corresponds to the occurrence of a first- or of a second-order phase transition, and since the variation of the Morse indexes of a manifold is in one-to-one correspondence with a change of its topology, the Main Theorem of the present paper states that a phase transition necessarily stems from a topological transition in configuration space. The proof of the theorem given in the present paper cannot be done without Main Theorem of paper I.     

Comment on “Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics,” by A. Chubykalo et al.

Fundamental errors in a paper by Chubykalo et al. [2] are highlighted. Contrary to their claim that “… the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously,” it is shown that this instantaneous component is physically irrelevant because it is always canceled by a term contained in the solenoidal component. This result follows directly from the solution of the wave equation that satisfies the solenoidal component. Therefore the subsequent inference of these authors that there are two mechanisms of transmission of energy and momentum in classical electrodynamics, one retarded and the other one instantaneous, has no basis. The example given by these authors in which the full electric field of an oscillating charge equals its instantaneous irrotational component on the axis of oscillations is proven to be false.     

A General Framework for Localization of Classical Waves: II. Random Media

We study localization of classical waves in random media in the general framework introduced in Part I of this work. This framework allows for two random coefficients, encompasses acoustic waves with random position dependent compressibility and mass density, elastic waves with random position dependent Lamé moduli and mass density, electromagnetic waves with random position dependent magnetic permeability and dielectric constant, and allows for anisotropy. We show exponential localization (Anderson localization) and strong Hilbert–Schmidt dynamical localization for random perturbations of periodic media with a spectral gap. This revised version was published online in July 2006 with corrections to the Cover Date.     

Helmholtz Theorem and the V-Gauge in the Problem of Superluminal and Instantaneous Signals in Classical Electrodynamics

In this work we substantiate the applying of the Helmholtz vector decomposition theorem (H-theorem) to vector fields in classical electrodynamics. Using the H-theorem, within the framework of the two-parameter Lorentz-like gauge (so called v-gauge), we show that two kinds of magnetic vector potentials exist: one of them (solenoidal) can act exclusively with the velocity of light c and the other one (irrotational) with an arbitrary finite velocity v (including a velocity more than c). We show also that the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously.     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

Interacting Quantum and Classical Continuous Systems II. Asymptotic Behavior of the Quantum Subsystem

In the framework of event-enhanced quantum theory the dynamical equation for the reduced density matrix of a quantum system interacting with a continuous classical system is derived. The asymptotic behavior of the corresponding dynamical semigroup is discussed. The example of a quantum–classical coupling on Lobatchevski space is presented.     

A Brief Remark on Energy Conditions and the Geroch-Jang Theorem

The status of the geodesic principle in General Relativity has been a topic of some interest in the recent literature on the foundations of spacetime theories. Part of this discussion has focused on the role that a certain energy condition plays in the proof of a theorem due to Bob Geroch and Pong-Soo Jang [“Motion of a Body in General Relativity.” Journal of Mathematical Physics 16(1) (1975)] that can be taken to make precise the claim that the geodesic principle is a theorem, rather than a postulate, of General Relativity. In this brief note, I show, by explicit counterexample, that not only is a weaker energy condition than the one Geroch and Jang state insufficient to prove the theorem, but in fact a condition still stronger than the one that they assume is necessary.     

A Note on the Adiabatic Theorem Without Gap Condition

We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of Avron and Elgart by providing an elementary solution of the commutator equation. In addition, a minor modification of their argument allows for more direct treatment of eigenvalue crossings. We also obtain simple, explicit conditions that yield information on the rate of convergence in the adiabatic limit.     

Sine-Gordon Theory for the Equation of State of Classical Hard-Core Coulomb Systems. II. High-Temperature Expansion

We perform a high-temperature expansion of the grand potential of the restrictive primitive model of electrolytes in the frame of the extended sine-Gordon theory exposed in the companion paper. We recover a result already obtained by Stell an Lebowitz (J. Chem. Phys. 49, 3706 (1968)) by means of diagrammatic expansions.     

A Polar Representation and the Morse Index Theorem

In this Letter we study the behavior of the eigenvalues of an operator defined by the action associated to a generic quadratic time-dependent Hamiltonian. This is done using a polar representation of the solutions of the corresponding linear Hamiltonian system. A proof of the Morse index theorem is given.     

A Continuous Transition Between Quantum and Classical Mechanics. I

In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative formulation of mechanics which provides a continuous transition between quantum and classical mechanics via environment-induced decoherence.     

Aspects of the correlation between raw material and ferrite properties,II

Starting materials, especially iron oxides, for ferrites contain anions as accompanying substances. Information on the type and content of anions is essential for judging the properties of these raw materials and their influence on ferrite properties and on the possibility of controlling these influences. In general the type and quantity of anions influences the reactivity of the raw material.