Nonlocality and Gleason's lemma. Part 2. Stochastic theories

I derive a Gleason-type contradiction from assumptions weaker than those needed to reach a Bell inequality. By establishing the inconsistency between local realism and QM's perfect EPR-type anticorrelations, the proof fills in a gap left open by Bell arguments.     

Nonlocality and conservation laws in hidden variable theories

It is shown that any hidden variable model that reproduces quantum mechanics for a single particle must either be nonlocal or violate conservation of momentum. This is established by deriving an inequality which must hold in any local, momentum-conserving hidden variable model for a modified form of the double-slit experiment. It is then shown that any hidden variable model that reproduces quantum mechanics must violate the inequality. The inconsistency between the classical and quantum views of the world is therefore demonstrated in a new way.     

Gleason's theorem and completeness criteria

We give some applications of Gleason's theorem to completeness criteria of inner product spaces using different families of subspaces, measures on them, and frame functions. Some open criteria problems are formulated.     

Nonlocality and Entanglement

By constructing a series of mixture, which is inseparable but satisfies the Mermin-Klyshko inequality, for the N-qubit system, we prove that ＂violation of the Mermin-Klyshko inequality＂ is not equivalent to ＂the state is inseparable＂.     

Interacting Quantum and Classical Continuous Systems I. The Piecewise Deterministic Dynamics

A mathematical construction of a Markov–Feller process associated with a completely positive coupling between classical and quantum systems is proposed. The example of the free classical particle on the Lobatchevski space Q interacting with the quantum system characterized by coherent states on Q is considered.     

Realism and Single-Quanta Nonlocality

We show that local realism applied to states characterized by a single quantum equally and coherently shared between a number of qubits (so-called W states) produces predictions incompatible with quantum theory. The origin of this incompatibility is shown to originate from the destructive interference of amplitude probabilities associated with nonlocal states, a phenomenon that has no classical analog.     

Propagators and renormalization transformations for lattice gauge theories. I

Lattice gauge theories may be looked at as perturbations of the theory of a vector field with a Gaussian action. We study this theory here and in following papers obtaining crucial results for understanding the renormalization group method in more complicated non-Abelian gauge field theories.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Gleason's theorem and Cauchy's functional equation

We study measures on the effect algebras of the closed interval [0, 1] and we describe regular or bounded measures. Applying Gleason's theorem for measures on the system of all closed subspaces of a Hilbert space, we show that any bounded measure m has the formm(t)=tm 1, t [0, 1], and as a by-product it gives a solution of Cauchy's basic functional equationf (x+ y)=f(x) + f(y) forx, y,x + y [0, 1].     

Three-Slit Experiments and Quantum Nonlocality

An interesting link between two very different physical aspects of quantum mechanics is revealed; these are the absence of third-order interference and Tsirelson’s bound for the nonlocal correlations. Considering multiple-slit experiments—not only the traditional configuration with two slits, but also configurations with three and more slits—Sorkin detected that third-order (and higher-order) interference is not possible in quantum mechanics. The EPR experiments show that quantum mechanics involves nonlocal correlations which are demonstrated in a violation of the Bell or CHSH inequality, but are still limited by a bound discovered by Tsirelson. It now turns out that Tsirelson’s bound holds in a broad class of probabilistic theories provided that they rule out third-order interference. A major characteristic of this class is the existence of a reasonable calculus of conditional probability or, phrased more physically, of a reasonable model for the quantum measurement process.     

The conceptual analysis (CA) method in theories of microchannels: Application to quantum theory. Part I. Fundamental concepts

A method is proposed that should facilitate the construction of theories of submicroscopic particles (denoted as theories of microchannels) in a way similar to the use of group-theoretical methods. The conceptual analysis (CA) method is based on the analysis of the basic concepts of a theory; it permits a determination of necessary conditions imposed on the mathematical apparatus (of the theory) which then appear as a mathematical representation of the structures obtained in a formal scheme of a theory. A pertinent conceptual analysis leads to a new definition (relativization) of the concept empirical implication. The approach may be characterized as realistic and operational. The application of the CA method is illustrated on the example of quantum theory. In Part I the algebraic structure of a partially ordered, up-ward directed, bounded set is deduced from the rudimentary concepts. In Parts II and III, we shall deduce the Hilbert-space structure (well established in quantum mechanics) from postulates on some essential idealizations accepted in the theory. Whereas Part II is concerned with the idealizations of existing quantum theories based on the Hilbert-space formalism, Part I may be considered as a general basis for a wider class of theories.Dedicated to Prof. G. Ludwig on the occasion of his 60th birthday.     

Single-Particle Nonlocality and Conditional Measurements

I suggest that quantum mechanical nonlocality may in a certain sense allow a particle to be in two places at the same time, without violating causality. I discuss the measurable consequences of such a feat, and speculate about possible statistical tests which could distinguish this view of quantum mechanics from a corpuscular one. In particular, I describe some experiments being set up at Toronto which will investigate atomic tunneling, looking among other things for a signature of such alkali schizophrenia.     

Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories

The super-Hamiltonian and supermomentum in canonical geometrodynamics or in a parametried field theory on a given Riemannian background have Poisson brackets which obey the Dirac relations. By smearing the supermomentum with vector fields V^→∈ L Diff Σ on the space manifold Σ, the Lie algebra L Diff Σ of the spatial diffeomorphism group Diff Σ can be mapped antihomomorphically into the Poisson bracket algebra on the phase space of the system. The explicit dependence of the Poisson brackets between two super-Hamiltonians on canonical coordinates (spatial metrics in geometrodynamics and embedding variables in parametrized theories) is usually regarded as an indication that the Dirac relations cannot be connected with a representation of the complete Lie algebra L Diff M of spacetime diffeomorphisms.We show how this difficulty may be overcome and construct a homomorphic mapping of spacetime vector fields V ∈ L Diff M into the Poisson bracket algebra on the phase space of the system. In the present paper, I, we explain how the technique works in the case of a parametrized field theory and in the following paper, II, we generalize it to canonical geometrodynamics. In a parametrized theory, the phase space of the system is the ordinary phase space of the field augmented by the embedding variables X: Σ →Mand their conjugate momenta. The dynamical variable H(V) which represents V ∈ L Diff M generates a deformation of the embedding along the flow lines of V accompanied by the correct dynamical evolution of the field data and preserves the constraints in the extended phase space of the system. We also establish the relation between the representations of Diff Σ and DiffM.     

Piron's and Bell's Geometric Lemmas and Gleason's Theorem

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.     

Nonlocality and the Kochen-Specker paradox

A new proof of the impossibility of reconciling realism and locality in quantum mechanics is given. Unlike proofs based on Bell's inequality, the present work makes minimal and transparent use of probability theory and proceeds by demonstrating a Kochen-Specker type of paradox based on the value assignments to the spin components of two spatially separated spin-1 systems in the singlet state of their total spin. An essential part of the argument is to distinguish carefully two commonly confused types of contextuality; we call them ontological and environmental contextuality. These in turn are associated with two quite distinct senses of nonlocality. We indicate the relevance of our treatment to other related discussions in recent literature on the philosophy of quantum mechanics.     

Nonlocality and short-range wetting phenomena

We propose a nonlocal interfacial model for 3D short-range wetting at planar and nonplanar walls. The model is characterized by a binding-potential functional depending only on the bulk Ornstein-Zernike correlation function, which arises from different classes of tubelike fluctuations that connect the interface and the substrate. The theory provides a physical explanation for the origin of the effective position-dependent stiffness and binding potential in approximate local theories and also obeys the necessary classical wedge covariance relationship between wetting and wedge filling. Renormalization group and computer simulation studies reveal the strong nonperturbative influence of nonlocality at critical wetting, throwing light on long-standing theoretical problems regarding the order of the phase transition.     

Boundedness below for fermion model theories. Part II. The linear lower bound

The Generalised Yukawa Model (GY ₂), ± ^N+^2M, M>N>1, is bounded below, uniformly in a momentum cutoff, and linearly in the volume of a spatial cutoff.This article includes the major part of a doctoral dissertation submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Horace H. Rackham School of Graduate Studies at the University of Michigan.Junior Fellow, Michigan Society of Fellows.