Lattice operations between observables in axiomatic quantum mechanics

Observables are treated as-homomorphisms of the Borel sets of the real line into an orthomodular-latticeL. By means of corresponding spectral-resolutions operations meet and join are defined between observables which endow the set of all observables with a lattice structure in caseL is-continuous and which give rise to lattices of observables in caseL is chosen arbitrarily and the observables commute.     

Observables on hypergraphs

Observables on hypergraphs are described by event-valued measures. We first distinguish between finitely additive observables and countably additive ones. We then study the spectrum, compatibility, and functions of observables. Next a relationship between observables and certain functionals on the set of measures M(H) of a hypergraph H is established. We characterize hypergraphs for which every linear functional on M(H) is determined by an observable. We define the concept of an effect and show that observables are related to effect-valued measures. Finally, we define operational transformations from M(H) to itself and show that they can be described as a certain combination of effects.On leave from University of Berne, Institute of Mathematical Statistics, Sidlerstrasse 5, CH-3012 Berne, Switzerland.     

Sensitive observables of quantum mechanics

We prove that there are quantum mechanical observables which are sensitive to the type of state-vector (first type or second type) describing two correlated physical systems, in the sense that the expectation value of these sensitive observables is measurably different in the two cases. The proof centers around Bell's inequality since we show that in quantum mechanics forall state-vectors of the second type (and only for them) sensitive observables exist in the absence of super-selection rules. Experimental verification of the existence of sensitive observables rules out local hidden variables.     

Link invariants and combinatorial quantization of hamiltonian Chern Simons theory

We define and study the properties of observables associated to any link in ×R (where is a compact surface) using the combinatorial quantization of hamiltonian Chern-Simons theory. These observables are traces of holonomies in a non-commutative Yang-Mills theory where the gauge symmetry is ensured by a quantum group. We show that these observables are link invariants taking values in a non-commutative algebra, the so-called Moduli Algebra. When =S ² these link invariants are pure numbers and are equal to Reshetikhin-Turaev link invariants.Laboratoire Propre du CNRS UPR 14.     

The joint measurement problem

According to orthodox quantum theory, the joint measurement of noncommuting observables is impossible. It has been claimed recently that such joint measurements are admitted in a generalized formalism for quantum theory developed by Ludwig and Davies, by means of so-called unsharp observables. It is argued in this paper that this claim has not been substantiated.     

Convergence of observables on quantum logics

We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are close if bounded functions of them are close. We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense.     

A local realistic explanation of EPR correlations

The reality of physical properties is divided into two types: relatively and absolutely real. Concerning the reality of spatial observables, it is proposed to drop the concept of an absolute reality of spatial observables. The resulting relative reality then isnot the observer-dependent reality of the standard interpretation of quantum mechanics, but rather the reference frame-dependent reality implied by the principle of relativity. Within the frame of this relative reality, it is then shown that a local explanation for the existence of EPR correlations can be found when two-valued discrete spatially directed observables (like the spin-1/2 components of electrons, or the polarization states of photons) are assumed. The new explanation is formally analogous to the standard explanation of classical correlations at a distance. Sufficient assumptions to obtain this result are the isotropy of space and the existence of two-valued discrete spatially directed observables. Thus, the possibility to find a both local and realistic explanation of correlations at a distance is extended to quantum variables violating Bell's inequality.     

Remarks on the notion of state in quantum mechanics

In the present paper two kinds of quantum-theoretical states are considered: the physical state determined by a complete observation and the intrinsic state which comprises the values of the observed as well as the unobserved observables. It will be shown that the future values of all these observables are determined. Causality is therefore valid, though not verifiable.     

Coexistence of observables

Recent results in the theory of integration of complex-valued functions with respect to a positive operator-valued measure are used to generalize the usual notion of coexistent observables. This leads to a connection between effects as observables and the quantization scheme of stochastic quantum mechanics. It also leads to a new viewpoint for the concept of a classical apparatus for quantum measurement which does not require a classical mechanical treatment of the apparatus from the outset.     

Tests for planar events in e+e− annihilation

We present a new class of observables which distinguish events containing two or three hadron jets from those containing a larger number. These observables, which essentially measure the coplanary of events, are calculable in QCD perturbation theory. Their use should allow the mechanism of

decay to be determined.     

The charged sectors of Quantum Electrodynamics in a framework of local observables

The construction of charged sectors in Quantum Electrodynamics (QED) is analyzed within a framework of algebras of local observables. It is argued that charged sectors arise by composing a vacuum state with charged * morphisms of an algebra of (neutral) quasi-local observables. Charged * morphisms, in turn, are obtained as weak limits of charge transfer cocycles. These are non-local elements of the algebra of all quasi-local observables obeying topological commutation relations with the local charge operators. It is shown that in this framework, charged sectors are invariant under the time evolution and satisfy the relativistic spectrum condition. The total charge operator is well defined and time-independent (conserved) on all charged sectors. Under an additional hypothesis the spectrum of the total charge operator is shown to be a discrete subgroup of the real line. A generalized Haag-Ruelle scattering theory for charged infra-particles is suggested, and some comments on non-abelian gauge theories are described.     

Properties of Boolean orthoposets

A Boolean orthoposet is the orthoposetP fulfilling the following condition: Ifa, b P anda b = 0, thena b. This condition seems to be a sound generalization of distributivity in orthoposets. Also, the class of (orthomodular) Boolean orthoposets may play an interesting role in quantum logic theory. This class is wide enough and, on the other hand, enjoys some properties of Boolean algebras. In this paper we summarize results on Boolean orthoposets involving distributivity, set representation, properties of the state space, existence of Jauch-Piron states, and results concerning orthocompleteness and completion.     

A simple model of resonance-dominated quasi-elastic scattering

We examine a highly simplified model of projectile-nucleus scattering in which the elementary interactions are resonance dominated. The essential element introduced into the nuclear dynamics is a shift of spectral strength to an energy comparable with or greater than the resonance half-width. The interplay of the energy dependences yields strong sensitivity to the underlying dynamics. Phenomenological treatments based on simplified nuclear dynamics, such as the -hole model, are seen to reproduce inclusive observables. They do less well on more exclusive observables and can be misleading in their implications for the underlying physics.Communicated by F. Lenz     

Simple Realism and Canonically Conjugate Observables in Non-Relativistic Quantum Mechanics

In this paper we list some minimal requirements for a physically natural, straightforwardly realist interpretation of non-relativistic quantum mechanics. The goal is to characterize what one might call a simple realism of quantum systems, and of the observables associated with them.Simple realism as developed here is a generalized interpretation-scheme, one that abstracts important shared features of Einsteinian naive realism, the so-called modal interpretations, and the orthodox interpretation itself. Some such schemes run afoul of the classic no-go theorems, while others do not. The role of non-commuting observables plays a major role in this success or failure. In particular, we show that if a simple-realist interpretation attributes simultaneously definite values to canonically conjugate observables, then it necessarily falls prey to Kochen-Specker contradictions.This exercise provides some insight into why modal interpretations work, while more generally placing limits on the scope of simple realism itself. In particular, we find that within the framework of simple realism, the only consistent interpretation of the uncertainty relations is the orthodox one. What's more, we point out that similar conclusions are bound to hold for many other non-commuting observables as well.     

Ergodic Properties of Microcanonical Observables

The problem of the existence of a strong stochasticity threshold in the FPU- model is reconsidered, using suitable microcanonical observables of thermodynamic nature, like the temperature and the specific heat. Explicit expressions for these observables are obtained by exploiting rigorous methods of differential geometry. Measurements of the corresponding temporal autocorrelation functions locate the threshold at a finite value of the energy density, which is independent of the number of degrees of freedom.     

Representations of the CCR in the (φ4)3 model: Independence of space cutoff

The algebra of observables for the renormalized ⁴ interaction in 3-dimensional space-time is constructed. It is shown that the von Neumann algebras associated with observables in a bounded regionB are independent of the space cutoff which is used in the construction of a Hamiltonian for this interaction. This result is shown to be useful in the construction of a translation invariant ⁴ theory in three dimensions. It also gives a physical criterion for the equivalence of non-Fock representations of the canonical commutation relations.     

Maximal structures of determinate propositions in quantum mechanics

I formulate and answer some questions concerning maximal structures of determinate quantum propositions, i.e., maximal structures of propositions that can be taken as having definite (but perhaps unknown) truth values for a given quantum state. The basic constraint on such structures is the Kochen and Specker no-go hidden-variables theorem, which demonstrates that no value assignment to certain finite sets of observables can preserve the functional relations between commuting observables. The problem I want to consider is how large we can take the set of determinate observables without violating the functional relationship constraint. I show how to construct maximal determinate sublattices of quantum propositions that are unique, subject to certain constraints, and I comment on the relevance of this go theorem for the interpretation of quantum mechanics.     

On clustering states

Clustering properties of states over the algebra of local observables are discussed under the weak form of asymptotic abelianness.     

Symmetries of the Einstein Hamiltonian

The vacuum Einstein equations for metrics that have two commuting spacelike Killing vector fields are studied from a Hamiltonian point of view using the Ashtekar variables. It is shown that the evolution equations are equivalent to those of a modified SL(2) principal chiral model with a time dependent coupling constant. This fact is used to extract an infinite set of symmetries of the Einstein Hamiltonian via a generalized zero-curvature formulation. These symmetries give evolving observables explicitly on the phase space, and may be viewed as providing an infinite set of solutions of the Hamiltonian Einstein equations. The possibility of quantization using these observables is briefly discussed.     

General formulae for polarization observables in two-body break-up of deuteron photodisintegration

The formal expressions of all possible polarization observables ind(,N)N with polarized photons and oriented deuterons are derived in terms of thet-matrix elements. Furthermore, using the multipole expansion of thet-matrix, all observables are expanded in terms of Legendre polynomials or associated functions, the coefficients of which are given as bilinear forms of the multipole moments and allow a model independent analysis of experimental data.Supported by the Deutsche Forschungsgemeinschaft (SFB 201)