On the Sharpness and Bias of Quantum Effects

The question of quantifying the sharpness (or unsharpness) of a quantum mechanical effect is investigated. Apart from sharpness, another property, bias, is found to be relevant for the joint measurability or coexistence of two effects. Measures of bias will be defined and examples given.     

Notes on Joint Measurability of Quantum Observables

For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hold. Also some possible candidates which would accompany joint measurability and generalize these apparently useful properties are discussed.     

Compatibility and Marginality

We present a way of introducing joint distibution function and its marginal distribution functions for non-compatible observables. Each such marginal distribution function has the property of commutativity. Models based on this approach can be used to better explain some classical phenomena in stochastic processes.     

Quantum World with Quantum Time

Since the beginning of quantum mechanics there have been a lot of attempts to quantize time. In this paper we refer to the little known concept of quantum time proposed by E.Kapuscik [Hadronic J. 8 (1985) 75]. We analyze some properties of systems with quantum time. Moreover we comment and discuss the idea of quantum time.     

Geometry of the electron local observables

We show that the complete set of the second power identities for the electron local observables consists of 36 equations. The identities connect the products of the electron bilinear forms and, being considered as geometrically meaningful equations in 3D Euclidean space, are separated over the groups of equations for the scalar, vector and tensor quantities. Considering the complete set of identities as a set of the second power equations, we solve the equations and find the irreducible representation for the electron local observables. The representation defines the 16 electron local observables as functions of 7 basic parameters and can be formulated in 6 various forms. The basic parameters include scalar and pseudoscalar, the time components of a 4-vector and a 4-pseudovector, and three Euler angles which define the angular position of a local 3D frame with respect to the 3D laboratory frame. These 7 parameters completely define the space components of the 4-vector and the 4-pseudovector, as well as the polar and axial vectors. The developed representation shows that the analysis of the any electron wave packet can be considerably simplified by the reduction of the number of analyzed real functions from 16 to 7. As an example, we present the structure of the local observables defined by the irreducible representation in a case of a traveling electron wave.     

Commutative POVMs and Fuzzy Observables

In this paper we review some properties of fuzzy observables, mainly as realized by commutative positive operator valued measures. In this context we discuss two representation theorems for commutative positive operator valued measures in terms of projection valued measures and describe, in some detail, the general notion of fuzzification. We also make some related observations on joint measurements. This article is dedicated to Pekka Lahti on the occasion of his 60th birthday.     

Recoil polarimetry in meson photoproduction at MAMI

The nucleon excitation spectrum remains poorly known, with the masses, widths, EM couplings and even existence of many states not well established. A program of experiments using meson photoproduction off the nucleon is being carried out to improve this situation. A new large acceptance recoil polarimeter has been developed by the Edinburgh group for use in such reactions with the Crystal Ball at MAMI. This work summarizes the procedure used to measure recoil polarization with the new device and presents some preliminary results for the double polarization observable Cx in the reaction γp→pπ, compared to the current partial wave analysis.     

经典守恒定律的量子力学推导

从量子力学出发,推出了经典动量守恒定律和能量守恒定律,并讨论了在微观领域经典守恒定律适用的条件.     

Recoil polarimetry in meson photoproduction at MAMI

The nucleon excitation spectrum remains poorly known, with the masses, widths, EM couplings and even existence of many states not well established. A program of experiments using meson photoproduction off the nucleon is being carried out to improve this situation. A new large acceptance recoil polarimeter has been developed by the Edinburgh group for use in such reactions with the Crystal Ball at MAMI. This work summarizes the procedure used to measure recoil polarization with the new device and presents some preliminary results for the double polarization observable C_x in the reaction γp→pπ~0, compared to the current partial wave analysis.     

On the Structure of the Observable Algebra for QED on the Lattice

We prove that the matter field subalgebra of the observable algebra for QED on a finite lattice is isomorphic to the enveloping algebra of the Lie algebra sl(2N, C), factorized by a certain ideal. Using this result, we give a new proof of the decomposition of the physical Hilbert space into charge superselection sectors.     

Covariant fuzzy observables and coarse-graining

A fuzzy observable is regarded as a smearing of a sharp observable, and the structure of covariant fuzzy observables is studied. It is shown that the covariant coarse-grainings of sharp observables are exactly the covariant fuzzy observables. A necessary and sufficient condition for a covariant fuzzy observable to be informationally equivalent to the corresponding sharp observable is given.     

Axial momentum for the relativistic Majorana particle

The Hilbert space of states of the relativistic Majorana particle consists of normalizable bispinors with real components, hence the usual momentum operator $?i\mathrm{?}$ can not be defined in this space. For this reason, we introduce the axial momentum operator, $?i{\gamma }_5\mathrm{?}$ as a new observable for this particle. In the Heisenberg picture, the axial momentum contains a component which oscillates with the amplitude proportional to $m/E$, where E is the energy and m the mass of the particle. The presence of the oscillations discriminates between the massive and massless Majorana particle. Furthermore, we show how the eigenvectors of the axial momentum, called the axial plane waves, can be used as a basis for obtaining the general solution of the evolution equation, also in the case of free Majorana field. Here a novel feature is a coupling of modes with the opposite momenta, again present only in the case of massive particle or field.     

不确定广义组合大系统的分散输出反馈鲁棒镇定及脉冲分析

考虑了一类不确定广义组合大系统,利用Lyapunov稳定性理论和矩阵范数性质研究了该类系统的分散镇定问题.给出了一种分散输出反馈鲁棒镇定控制器的设计,得到了系统可输出反馈鲁棒镇定的不确定量的范数界.同时分析了广义组合大系统及各个孤立子系统的脉冲控制问题,给出了其闭环系统无脉冲的不确定量的范数界.最后获得了广义组合大系统及各个孤立子系统的闭环系统同时渐近稳定和无脉冲的范数界. 关键词： 广义系统 鲁棒镇定 分散控制 脉冲能观     

A note on the Pauli problem in light of approximate joint measurements

We show that there exist informationally incomplete phase space observables such that the Cartesian margins are informationally equivalent with position and momentum. This shows that it is possible to reconstruct the position and momentum distributions of a quantum system from the statistics of a single observable, and thus a single measurement, even though the state of the system is not uniquely determined by the statistics.     

对量子力学中角动量的一点认识

指出了角动量算符Ｊ不存在本征值和本征矢量完全集,角动量算符Ｊ不描写一个可观察量以及角动量Ｊ不宜作为整体来讨论。     

Absorbing boundary condition as limiting case of imaginary potentials

Imaginary potentials such as V(x)=-iσ1_Ω(x)(with σ> 0 a constant,Ω a subset of 3-space,and 1_Ω its characteristic function) have been used in quantum mechanics as models of a detector.They represent the effect of a ’soft’ detector that takes a while to notice a particle in the detector volume Ω.In order to model a ’hard’ detector(i.e.one that registers a particle as soon as it enters Ω),one may think of taking the limit σ→∞ of increasing detector strength σ.However,as poin...     

Statistics of Local Value in Quantum Mechanics

Given a quantum mechanical observable and a state, one can construct a classical observable, that is, a real function on the configuration space, such that it is the optimal estimate of the quantum observable, in the sense of minimum variance. This optimal estimate turns out to be the quantum mechanical local value, which arises from several contexts such as de Broglie–Bohm's casual approach to quantum mechanics, instantaneous frequency in time–frequency analysis, Nelson's quantum fluctuations formalism, and phase-space approach to quantum mechanics. Accordingly, any observable can be decomposed into a local value part and a quantum fluctuation part, which are independent, both geometrically and statistically. Furthermore, the current density in quantum mechanics, the osmotic velocity in stochastic mechanics, and the Fisher information in classical statistical inference, arise naturally in connection with local value. In particular, Heisenberg uncertainty principle can be quantified more precisely by virtue of local value.