On the existence of quantum representations for two dichotomic measurements

Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories.Comment: 16+7 pages, presentation improved and minor errors correcte

Zur allgemeinen Theorie der halbgeordneten Räume

Foreword by K. Kopotun11Correspondence to: K. Kopotun, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada ^^IR3T 2N2. Email: [email protected] paper “On the general theory of semi-ordered spaces” (“Zur allgemeinen Theorie der halbgeordneten Räume”) was written by L.V. Kantorovich and G.R. Lorentz22Until 1946, G.G. (Georg Gunter) Lorentz was using the name Geogrij Rudolfovich (G.R.) Lorentz. sometime in 1937–1939, and this is the first time it appears in print.The following is a short history of this manuscript.In his letter to I.P. Natanson written on October 11, 1937, G.G. Lorentz mentioned a talk on joint work with L.V. Kantorovich that he gave at a Session on Functional Analysis in Moscow earlier that year. The records of the Academy of Sciences of USSR indicate that a Session on Functional Analysis took place in Moscow during September 27–29, 1937, and that G.R. Lorentz gave a talk “Topological theory of semi-ordered spaces” there, and that L.V. Kantorovich was speaking on “Theory of linear operations in semi-ordered spaces”.The manuscript “On the general theory of semi-ordered spaces” was found in the archives of L.V. Kantorovich. According to Vsevolod Leonidovich Kantorovich, L.V. Kantorovich’s son, it was submitted to Trudy Tomskogo Gosudarstvennogo Universiteta imeni V. V. Kuibysheva (Proceedings of Tomsk State University). The typed version33See www.math.ohio-state.edu/~nevai/LORENTZ/KANTOROVICH_LORENTZ_typed.pdf/. of the manuscript has a handwritten note by N. Romanov44N.P. Romanov (1907–1972) was a Professor at Tomsk University from 1935 until 1944. After 1944 he worked in Uzbekistan. His main area of research was Number Theory and Theory of Functions of Complex Variables. For more information see “Nikolaĭ Pavlovich Romanov (on the eightieth anniversary of his birth)”, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1987, no. 3, 92–93, MR0914654 (89b:01069). dated by August 31, 1939 stating that the manuscript is accepted for publication. The manuscript was never published (probably because of the World War II) and around 1945 was returned to L.V. Kantorovich.It has been decided to publish this manuscript in its original language (German), and translate the extended abstract accompanying this manuscript from Russian to English. The manuscript appears here in its original form with only minor editorial corrections.Publication of this historical document would not have been possible without the assistance and effort of many people. In particular, the significant help of C. de Boor, Ya.I. Fet, V.L. Kantorovich, V.N. Konovalov, and S.S. Kutateladze is acknowledged and greatly appreciated.Extended abstract55Translated from Russian by K. Kopotun.The current manuscript is devoted to the investigation of general semi-ordered spaces that are not necessarily linear. Hence, it may be considered a generalization of the work of L.V. Kantorovich [Linear semi-ordered spaces, Mat. Sbornik, 2 (1) 1937, 121–168].We say that a set Y={y} is a semi-ordered space if its elements are partially ordered using a relation “<” so that I.If y1<y2, y2<y3, then y1<y3.II.For any pair y1, y2, there exist elements y3,y4 such that y3⩽y1, y3⩽y2, y1⩽y4, and y2⩽y4.III.Every set E⊂Y bounded above has a least upper bound (supE).IV.For every set E⊂Y, there exists a countable subset E′ that has the same least upper and greatest lower bound as E. The above assumptions allow us to introduce notions of a limit superior, limit inferior, and of a convergent sequence in Y. For example, define lim¯yn=infn(sup(yn,yn+1,…)). It is possible to introduce, e.g., the limit superior differently, for example, by defining lim¯∗yn to be the least element y having the property that, for any subsequence {ynk}, there exists a subsequence {ynki} such that y⩾lim¯i→∞ynki. This type of convergence, ∗-convergence, turns out to be identical with the topological convergence that we arrive at if we turn Y into a topological space using the convergence defined initially. Relationships among various limits which we can define using the above approaches as well as some properties of these limits are studied in § 1 and § 2. In § 3, we study semi-ordered spaces equipped with a nonnegative metric function ρ(y1,y2) defined for all pairs y1, y2 such that y1⩽y2, and satisfying 1∘.ρ(y1,y2)=0 is equivalent to y1=y2.2∘.ρ(y1,y3)⩽ρ(y1,y2)+ρ(y2,y3) (y1⩽y2⩽y3).3∘.ρ(sup(y,y1),sup(y,y2))⩽ρ(y1,y2) (an analogous inequality holds with inf).4∘.If yn→y monotonically, then ρ(yn,y)→0 (or ρ(y,yn)→0).5∘.If yn monotonically tends to infinity, then the condition limn,m→∞ρ(yn,ym)=0 should not hold.Let ρ(y1,y2,…,yn)=ρ(inf(y1,…,yn),sup(y1,…,yn)). Then yn→y turns out to be equivalent to ρ(y,yn,…,yn+p)→0 when n→∞, and yn→y(∗) is equivalent to ρ(y,yn)→0. In addition, Cauchy’s convergence principle holds. Moreover, if Y is distributive, i.e., inf(y,sup(y1,y2))=sup(inf(y,y1),inf(y,y2)), then it is also strongly distributive: inf(y,supnyn)=supn(inf(y,yn)). In § 4, we study similar spaces under weaker assumptions. Particular examples of such spaces are the Hausdorff space of closed sets (see Hausdorff “Set theory”, p. 165) and the space of semicontinuous functions. § 5 is devoted to applications of the general theorems to the theory of semicontinuous functions y=f(x) that map a metric space {x}=X into a semi-ordered space {y}=Y. Under some additional assumptions (Y is regular, distributive, and between any two elements y1 and y2 such that y1<y2 there is a third element y3, y1<y3<y2) it is possible to develop a complete theory of semicontinuous functions including a theorem that every semicontinuous function is a limit of a monotone sequence of continuous functions as well a theorem on separation by a continuous function

Deterministic Walks in Quenched Random Environments of Chaotic Maps

This paper concerns the propagation of particles through a quenched random medium. In the one- and two-dimensional models considered, the local dynamics is given by expanding circle maps and hyperbolic toral automorphisms, respectively. The particle motion in both models is chaotic and found to fluctuate about a linear drift. In the proper scaling limit, the cumulative distribution function of the fluctuations converges to a Gaussian one with system dependent variance while the density function shows no convergence to any function. We have verified our analytical results using extreme precision numerical computations.Comment: 18 pages, 9 figure

Persistent random walk on a one-dimensional lattice with random asymmetric transmittances

We study the persistent random walk of photons on a one-dimensional lattice of random asymmetric transmittances. Each site is characterized by its intensity transmittance t (t') for photons moving to the right (left) direction. Transmittances at different sites are assumed independent, distributed according to a given probability density Distribution. We use the effective medium approximation and identify two classes of probability density distribution of transmittances which lead to the normal diffusion of photons. Monte Carlo simulations confirm our predictions.Comment: 7 pages, submitted to Phys. Rev.

A Paradox of State-Dependent Diffusion and How to Resolve It

Consider a particle diffusing in a confined volume which is divided into two equal regions. In one region the diffusion coefficient is twice the value of the diffusion coefficient in the other region. Will the particle spend equal proportions of time in the two regions in the long term? Statistical mechanics would suggest yes, since the number of accessible states in each region is presumably the same. However, another line of reasoning suggests that the particle should spend less time in the region with faster diffusion, since it will exit that region more quickly. We demonstrate with a simple microscopic model system that both predictions are consistent with the information given. Thus, specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system, even assuming the absence of external forces. We propose an alternative framework for modelling diffusive dynamics in which both the diffusion rate and equilibrium probability density for the position of the particle are specified by the modeller. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and is suitable for discontinuous diffusion coefficients.Comment: 21 pages, 6 figures. Second round of revisions. This is the version that will appear in Proc Roy So

Performance and selection of winter durum wheat genotypes in different European conventional and organic fields

Sustainability is a key factor for the future of agriculture. Productivity in agriculture has more than tripled in developed countries since the 1950s. Beyond the success of plant breeding, the increased use of inorganic fertilizers, application of pesticides, and spread of irrigation also contributed to this success. However, impressive yield increases started to decline in the 1980s because of the lack of sustainability. One of the most beneficial ways to increase sustainability is organic agriculture. In such agro-ecosystem-based holistic production systems the prerequisite of successful farming is the availability of crop genotypes that perform well. However, selection of winter durum wheat for sub-optimal growing conditions is still mainly neglected, and the organic seed market also lacks of information on credibly tested winter durum varieties suitable for organic agriculture

Transport Properties of the Diluted Lorentz Slab

We study the behavior of a point particle incident from the left on a slab of a randomly diluted triangular array of circular scatterers. Various scattering properties, such as the reflection and transmission probabilities and the scattering time are studied as a function of thickness and dilution. We show that a diffusion model satisfactorily describes the mentioned scattering properties. We also show how some of these quantities can be evaluated exactly and their agreement with numerical experiments. Our results exhibit the dependence of these scattering data on the mean free path. This dependence again shows excellent agreement with the predictions of a Brownian motion model.Comment: 14 pages of text in LaTeX, 7 figures in Postscrip

The Averaging Problem in Cosmology and Macroscopic Gravity

The averaging problem in cosmology and the approach of macroscopic gravity to resolve the problem is discussed. The averaged Einstein equations of macroscopic gravity are modified on cosmological scales by the macroscopic gravitational correlation tensor terms as compared with the Einstein equations of general relativity. This correlation tensor satisfies a system of structure and field equations. An exact cosmological solution to the macroscopic gravity equations for a constant macroscopic gravitational connection correlation tensor for a flat spatially homogeneous, isotropic macroscopic space-time is presented. The correlation tensor term in the macroscopic Einstein equations has been found to take the form of either a negative or positive spatial curvature term. Thus, macroscopic gravity provides a cosmological model for a flat spatially homogeneous, isotropic Universe which obeys the dynamical law for either an open or closed Universe.Comment: 8 pages, LaTeX, ws-ijmpa.cls, few style and typo corrections. Based on the plenary talk given at the Second Stueckelberg Workshop, ICRANet Coordinating Center, Pescara, Italy, September 3-7, 2007. To appear in International Journal of Modern Physics A (2008

Minimum Distance Estimation of Search Costs using Price Distribution

Hong and Shum (2006) show equilibrium restrictions in a search model can be used to identify quantiles of the search cost distribution from observed prices alone. These quantiles can be difficult to estimate in practice. This paper uses a minimum distance approach to estimate them that is easy to compute. A version of our estimator is a solution to a nonlinear least squares problem that can be straightforwardly programmed on softwares such as STATA. We show our estimator is consistent and has an asymptotic normal distribution. Its distribution can be consistently estimated by a bootstrap. Our estimator can be used to estimate the cost distribution nonparametrically on a larger support when prices from heterogeneous markets are available. We propose a two-step sieve estimator for that case. The first step estimates quantiles from each market. They are used in the second step as generated variables to perform nonparametric sieve estimation. We derive the uniform rate of convergence of the sieve estimator that can be used to quantify the errors incurred from interpolating data across markets. To illustrate we use online bookmaking odds for English football leagues' matches (as prices) and find evidence that suggests search costs for consumers have fallen following a change in the British law that allows gambling operators to advertise more widely

A real Lorentz-FitzGerald contraction

Many condensed matter systems are such that their collective excitations at low energies can be described by fields satisfying equations of motion formally indistinguishable from those of relativistic field theory. The finite speed of propagation of the disturbances in the effective fields (in the simplest models, the speed of sound) plays here the role of the speed of light in fundamental physics. However, these apparently relativistic fields are immersed in an external Newtonian world (the condensed matter system itself and the laboratory can be considered Newtonian, since all the velocities involved are much smaller than the velocity of light) which provides a privileged coordinate system and therefore seems to destroy the possibility of having a perfectly defined relativistic emergent world. In this essay we ask ourselves the following question: In a homogeneous condensed matter medium, is there a way for internal observers, dealing exclusively with the low-energy collective phenomena, to detect their state of uniform motion with respect to the medium? By proposing a thought experiment based on the construction of a Michelson-Morley interferometer made of quasi-particles, we show that a real Lorentz-FitzGerald contraction takes place, so that internal observers are unable to find out anything about their `absolute ' state of motion. Therefore, we also show that an effective but perfectly defined relativistic world can emerge in a fishbowl world situated inside a Newtonian (laboratory) system. This leads us to reflect on the various levels of description in physics, in particular regarding the quest towards a theory of quantum gravity.Comment: 6 pages, no figures. Minor changes reflect published versio