On Arthur Fine's solution to the measurement problem

Arthur Fine has recently proposed a novel solution to the measurement problem. Fine's innovation is to exploit the idea that a measurement is a selective interaction in which the apparatus responds to a probability distribution. I develop Fine's solution in some detail, responding to various potential objections along the way. In the end, however, I suggest that Fine needs to tell us more before we can declare the measurement problem solved.     

Frequency Analysis of the EPR-Bell Argumentation

We perform a frequency analysis of the EPR-Bell argumentation. One of the main consequences of our investigation is that the existence of probability distributions of the Kolmogorov-type which was supposed by some authors is a mathematical assumption which may not be supported by actual physical quantum processes. In fact, frequencies for hidden variables for quantum particles and measurement devices may fluctuate from run to run of an experiment. These fluctuations of frequencies for micro-parameters need not contradict to the stabilization of frequencies for physical observables. If, nevertheless, micro-parameters are also statistically stable, then violations of Bell's inequality and its generalizations may be a consequence of dependence of collectives corresponding to two different measurement devices. Such a dependence implies the violation of the factorization rule for the simultaneous probability distribution. Formally this rule coincides with the well known BCHS locality condition (or outcome independence condition). However, the frequency approach implies totally different interpretation of dependence. It is not dependence of events, but it is dependence of collectives. Such a dependence may be induced by the same preparation procedure.     

Ensemble interpretation of the EPR-Bell correlations

The EPR-Bell correlations between the spins of a pair of particles originally in a singlet state are discussed both on the basis of the mathematical formalism of quantum mechanics and the ensemble interpretation. It is shown that the correlations predicted by the mathematical formalism are in agreement with those expected on the basis of the ensemble interpretation, if the electrons are treated as distinguishable particles after they separate and undergo observation. In this case, the correlations are only in partial agreement with agedanken experiment of Mermin on the subject. It is pointed out, however, that agreement with Mermin’s conclusions is possible if one treats the electrons as indistinguishable even when they are subjected to observation after separation, though there is no obvious theoretical justification for doing so.     

Two proofs of Fine's theorem

Fine's theorem concerns the question of determining the conditions under which a certain set of probabilities for pairs of four bivalent quantities may be taken to be the marginals of an underlying probability distribution. The eight CHSH inequalities are well-known to be necessary conditions, but Fine's theorem is the striking result that they are also sufficient conditions. Here two transparent and self-contained proofs of Fine's theorem are presented. The first is a physically motivated proof using an explicit local hidden variables model. The second is an algebraic proof which uses a representation of the probabilities in terms of correlation functions.     

Attempt to Resolve the EPR-Bell Paradox via Reichenbach's Concept of Common Cause

Reichenbach's common cause principle claims that if there is correlation betweentwo events and none of them is directly causally influenced by the other, thenthere must exist a third event that can, as a common cause, account for thecorrelation. The EPR-Bell paradox consists in the problem that we observecorrelations between spatially separated events in the EPR experiments whichdo not admit common-cause-type explanation, and it must therefore be concludedthat, contrary to relativity theory, in the realm of quantum physics there existsaction at a distance, or at least superluminal causal propagation is possible; thatis, either relativity theory or Reichenbach's common cause principle fails.By means of closer analyses of the concept of common cause and a more precisereformulation of the EPR experimental scenario, I sharpen the conclusion wecan draw from the violation of Bell's inequalities.     

The Problem of Subrayleigh Resolution in Interference Microscopy

A method allowing subwavelength resolution within the framework of optical interferometric microscopy is proposed. It is shown that overcoming the diffraction limit is not a necessary requirement. Generally speaking, subrayleigh resolution and overcoming the diffraction limit are basically different concepts. The method developed uses the all-parameter modulation of the light source and separation of the phase shifts generated due to different reasons. In this method, a topological phase technique is used to separate the phases and therefore determine the corresponding characteristics of an object. The approach developed provides for a way to create a new type of optical devices. Within the framework of a uniform measuring procedure these devices make possible determining both geometrical and material parameters of the object under study.     

The Quantum Phase Problem: Steps Toward a Resolution

Defining the observable canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. In a previous work we have shown how to define the absolute phase observable || by suitably restricting the Hilbert space of x and p like variables. Here we show that also from the classical point of view, there is no rigorous definition for the phase even though it's absolute value is well defined.     

On the Problem of Electromagnetic-Field Quantization

We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg’s equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.     

On the Crepant Resolution Conjecture in the Local Case

In this paper we analyze four examples of birational transformations between local Calabi–Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov–Witten invariants, proving the Coates–Iritani–Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan’s original Crepant Resolution Conjecture should be modified, by including appropriate “quantum corrections”, and that there is no straightforward generalization of either Ruan’s original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds.     

On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation

In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as ²0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.     

On the Problem of Universal Density Functional

A performed analysis shows that the density functional theory for the inhomogeneous electron gas, which is based on the hypothesis of the existence of the density functional for the external field potential is incorrect. This means that the density functional theory is not an ab initio theory.     

On the Bosonization of the Many Electron Problem

The Garbaczewski bosonization procedure is applied to the many-electron problem. It leads to a non-local c-number path integral representation for the trace of the evolution operator which also depends on a mapping of the lattice points in the natural numbers. This mapping is needed for the implementation of the bosonization. We argue that the non-local interactions and the dependence on the mapping can be disregarded in the limit of small electron c-number field. We can then discuss the physics of the saddle point contribution. In the discussion the effects of the Pauli exclusion principle are introduced phenomenologically.     

On a Homogenization Problem

We study the average Green’s function of stochastic, uniformly elliptic operators of divergence form on \(Zd\mathbb {Z}^d\). When the randomness is independent and has small variance, we prove regularity of the Fourier transform of the self-energy. The proof relies on the Schur complement formula and the analysis of singular integral operators combined with a Steinhaus system.     

On the Problem of Initial Conditions for Inflation

I review the present status of the problem of initial conditions for inflation and describe several ways to solve this problem for many popular inflationary models, including the recent generation of the models with plateau potentials favored by cosmological observations.     

On the Problem of Random Walk without Self-intersections

The particle random walk without self-intersections in the n-dimensional Euclidean space is investigated. The distribution function for the end-to-end distance of a walking particle trajectory is obtained up to the second order in the excluded volume V₀.     

巧用圆讨论力的合成和分解问题

力的合成与分解遵循平行四边形定则或三角形定则,有时巧用圆可以有效地讨论有关力的合成和分解的解数、极值、大小和方向等问题.1求解数【例1】如图1,已知合力F和一个分力F1的方向以及另一个分力F2的大小(F2的大小可根据解题需要自取),问F可以分解为几组分力?     

On the Gribov Problem for Generalized Connections

 The bundle structure of the space of Ashtekar's generalized connections is investigated in the compact case. It is proven that every stratum is a locally trivial fibre bundle. The only stratum being a principal fibre bundle is the generic stratum. Its structure group equals the space of all generalized gauge transforms modulo the constant center-valued gauge transforms. For abelian gauge theories the generic stratum is globally trivial and equals the total space . However, for a certain class of non-abelian gauge theories – e.g., all SU(N) theories – the generic stratum is nontrivial. This means, there are no global gauge fixings – the so-called Gribov problem. Nevertheless, for many physical measures there is a covering of the generic stratum by trivializations each having total measure 1. Finally, possible physical consequences and the relation between fundamental modular domains and Gribov horizons are discussed. Received: 4 March 2002 / Accepted: 20 August 2002 Published online: 30 January 2003 Communicated by H. Nicolai     

On the Problem of the Amenability of the Gauge Group

Let G be one of the local gauge groups C(X, U(n)),C (X, U(n)), C(X, SU(n)) or C (X, SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which it is amenable, viz. the relative weak topology of C(X, M(n)). This topology seems more useful than other known amenable topologies for G. We construct a simple fermionic model containing an action of G, continuous w.r.t. this amenable topology.     

On the Second Law of Thermodynamics and the Piston Problem

The piston problem is investigated in the case where the length of the cylinder is infinite (on both sides) and the ratio m/M is a very small parameter, where m is the mass of one particle of the gaz and M is the mass of the piston. Introducing initial conditions such that the stochastic motion of the piston remains in the average at the origin (no drift), it is shown that the time evolution of the fluids, analytically derived from Liouville equation in a previous work, agrees with the Second Law of thermodynamics. We thus have a non equilibrium microscopical model whose evolution can be explicitly shown to obey the two laws of thermodynamics.