On Casimir's Ghost

We define on the universal enveloping superalgebra of a nonstandard adjoint action, endowing it with a module structure. This allows, in particular, to construct a bosonic operator which anticommutes with all the fermionic generators and which appears to be the square root of a certain Casimir operator. Received: 23 May 1996 / Accepted: 6 January 1997     

Fairy QCD

We discuss QCD at nonstandard values of parameters (quark masses, vacuum angle θ, the number of colors and flavors) and/or at nonstandard physical conditions (when the system is essentially heated and/or squezed. Studying these “fairy” version of the theory provides a broader perspective and allows one to understand better the dynamics of actual QCD.     

On affine extension of splint root systems

Splint of root system of simple Lie algebra appears naturally in the study of (regular) embeddings of reductive subalgebras. It can be used to derive branching rules. Application of splint properties drastically simplifies calculations of branching coefficients. We study affine extension of splint root system of simple Lie algebra and obtain relations on theta and branching functions.     

Mathematics of dispersive water waves

A commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed. The modified water wave hierarchy is a bi-Hamiltonian system; its modification bifurcates. The water wave hierarchy, and the hierarchies of the Korteweg-de Vries and the modified Korteweg-de Vries equations, as well as the classical Miura map, are given new representations through various specializations of nonstandard systems.     

Equilibrium statistical distributions for subchains in an entangled polymer melt

In de Gennes-Doi-Edwards theory for entangled polymeric melts, a length scale r₀ is introduced, giving the equilibrium mesh size of the physical network of chains. Each polymer molecule is then represented as a random walk, with a step size r₀ (a “subchain”, made up of n₀ Kuhn segments) dictated by the existence of entanglements. Progressing from this simple picture, an issue that has been constantly overlooked so far, despite its potential relevance, is that of finite-size effects at the de Gennes-Doi-Edwards characteristic length scale. Actually, since a subchain in a melt is a “small”, nonmacroscopic system, fluctuations of both its length and its number of Kuhn segments are certainly nonnegligible. An ad hoc theoretical treatment from nonstandard (nano) statistical mechanics and thermodynamics seems then required, to find the anticipated equilibrium statistical distributions of the subchain population. In this contribution, we carefully discuss this topic. Some predictions from the nonstandard fluctuation-inclusive approach on the statistics of subchains are here obtained, and compared with existing simulations, even down to the atomistic level.     

Classification of representations of the algebra U′q(so 3) through examples II

This paper completes series of articles devoted to classification of the representations of the nonstandard deformation U′_q(so ₃) providing examples of such representations in low dimensions. The classification differs substantially when the deformation parameter q is/is not root of unity (q ⁿ=1). When it is a root of unity, the situation differs for odd and even n. The examples presented here cover the first nontrivial case when n is even (namely, n=4), from which the general case follows easily.     

Simultaneity,conventialism, general covariance,and the special theory of relativity

In a recent article, De Ritis and Guccioni claimed to give an easy way of falsifying the conventionalist interpretation of the definition of simultaneity of distant events within an inertial system. For a particular closed light path, they derived a necessary condition and claimed that it could be violated by a nonstandard choice of the definition of simultaneity. It is pointed out here that (as shown by many authors) the possibility of a nonstandard definition is implicitly contained in a generally covariant formulation of the special theory of relativity, and it is shown that conditions such as the one they derived are satisfied automatically. The most general position-independent definition of simultaneity is given, and an example of a position- as well as direction-dependent definition is exhibited in an Appendix. A number of objections against the possibility of a nonstandard definition of simultaneity raised by various authors are discussed and are found to lack any physical or mathematical justification.     

Nonstandard Fock spaces

A theory of nonstandard inner product spaces is developed using methods of nonstandard analysis. Various results concerning nonstandard operators and their spectra are proved. The theory is applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Moreover, a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

Classification of representations of the algebraU q ′ (so3) through examples

Classification of the representations of the nonstandard deformationU _q ^′ (so₃) is provided through examples of low dimensions. Classification differs substantially when deformation parameterq is/is not root of unity (q ⁿ = 1). When it is a root of unity, situation differs for odd and evenn. Examples are presented for the first nontrivial cases (namelyn=3), from which general case is easy to follow. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. The work was partially supported by Commitee for collaboration of the Czech Republic with CERN, partially by research grant IG CTU 300010004/2000.     

基于USB2.0的非标准视频图像采集系统的设计

为了实现非标准视频信号的实时采集,设计了以DSP和CPLD为核心的基于USB2.0的图像采集系统。DSP通过数据分析能够产生高精度的同步信号,同步信号经过倍频锁相得到用于图像采集的采样脉冲。CPLD器件内部编程由DSP中断和IO引脚控制实现非标准视频信号的图像采集逻辑。系统与主机之间采用EZ USB接口芯片Cy7c68013,利用芯片的SlaveFIFO工作方式,针对视频图像在场消隐期间无有效数据的特点,提出了一种以场为单位分块传输的实时视频图像传输方案,同时实现了读写数据的双向传输。系统能够适应非标准视频图像同步频率的变化,实现预定格式的视频图像实时采集。     

定向耦合法测量非标准接口元件驻波系数

 利用网络分析仪,根据需要在设定的频率范围进行扫频测量,网络分析仪的端口1输出的微波信号经过微波传输转换器进入与待测非标准元件匹配连接的连接段（如波导）;网络分析仪的端口2连接定向耦合器,用于监测传输段内的反射微波信号。首先,在非标准接口端面连接短路面,通过网络分析仪测量反射信号在设定频点的相对幅度值;然后去掉短路面,在非标准接口端面连接待测元件,再次测量反射信号在特定频点的相对幅度值;最后根据本文推导的公式得出驻波系数。该方法的测量误差与定向耦合器的方向性（方向性系数越大越好）和待测元件驻波系数有关。将该方法运用在非标准接口匹配负载驻波系数的测量中,定向耦合器方向性系数取为40 dB,测得其驻波系数小于1.2,误差小于20%。该方法简便可行,可以用于测量常用的非标准接口元件尤其是非标准低驻波系数元件的驻波系数。     

Nonstandard extension of quantum logic and Dirac's bra-ket formalism of quantum mechanics

An extension of the quantum logical approach to the axiomatization of quantum mechanics usingnonstandard analysis methods is proposed. The physical meaning of a quantum logic as a lattice of propositions is conserved by its nonstandard extension. But not only the usual Hubert space formalism of quantum mechanics can be derived from the nonstandard extended quantum logic. Also the Dirac bra-ket quantum mechanics can be derived as a consequence of such an extended quantum logic.     

Toward a rigorous quantum field theory

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

Perturbation theory of neutrino oscillations with and without nonstandard interactions

After making brief comments on the Kamioka-Korea two-detector setting, I delivered a pedagogical seminar on perturbation theory of neutrino oscillations with and without nonstandard interactions. I also described a general property of neutrino flavor transformation of such a system which includes useful relationships between S matrix elements and a phase reduction theorem.     

Twist of Lie algebras by a rank-3 subalgebra

A new nonstandard deformation of all types of classical Lie algebras is constructed by means of twisting based on a six-dimensional subalgebra. This is an extension of extended twists introduced by Kulish et al. It is also shown that the new nonstandard so(3, 2) has a close connection with the symmetry of a discrete analog of the Klein-Gordon equation in (1+2) spacetime.     

Some Key Concepts in Nonstandard Analysis Theory of Turbulence

Some important concepts in the nonstandard analysis theory of turbulence are presented. The point structure, on which differential equations are defined, is analysed. The distinction between the uniform point and the non-uniform point, as well as between the standard point and the nonstandard point, is shown. A new kind of equations, which differ essentially from those in existent theory, is emphasized. These new equations can hold at non-uniform points. The applicability of the Navier-Stokes equations to turbulence is discussed. Some illustrations of the nonstandard analysis theory of turbulence are also presented.     

Numerical and analytical study of wave processes in periodic stratified media

We study the behaviour of the solutions of the Cauchy problem with discontinuous initial data for nonstandard linear partial differential equations modeling wave processes in periodic stratified media. Asymptotic formulas at large t are derived. The found asymptotic formulas are in a good agreement with the results of numerical experiments done by using the analytical computation system REDUCE 3.8.     

Calorimetric method for determination of <Superscript>51</Superscript>Cr neutrino source activity

Experimental study of nonstandard neutrino properties using high-intensity artificial neutrino sources requires the activity of the sources to be determined with high accuracy. In the BEST project, a calorimetric system for measurement of the activity of high-intensity (a few MCi) neutrino sources based on ⁵¹Cr with an accuracy of 0.5–1% is created. In the paper, the main factors affecting the accuracy of determining the neutrino source activity are discussed. The calorimetric system design and the calibration results using a thermal simulator of the source are presented.     

Nonstandard Methods in Quantum Field Theory I: A Hyperfinite Formalism of Scalar Fields

A nonstandard approach to axiomatic quantum field theory is given. Nonstandard axioms for a Hermitian scalar field is proposed, where the field operators act on a hyperfinite-dimensional Hilbert space. The axioms are shown to be equivalent to the Gårding–Wightman axioms. An example of a model of the nonstandard axioms is examined.