Two-Dimensional Fluids Via Matrix Hydrodynamics

A direct method for the reduction of inner products of irreducible representations (irreps) of unitary groups has been proposed using the duality between the permutation and unitary groups. A canonical tensor basis set has been used to obtain a closed expression for the Clebsch–Gordan coefficients of U(n). This expression involves the subduction coefficients arising in the outer-product reduction of SN1⊗SN2→SN1+N2 of the permutation groups, the symmetrization coefficients of U(n), and matrix elements of the standard representation of SN. The expression holds good for an inner-product reduction of irreps of U(n), and is independent of n. The method has been illustrated with examples.

THE WAVE-PARTICLE DUALITY: TOWARDS AN INTEGRATIVE FRAMEWORK FOR INTERNATIONAL BUSINESS

Introduction. The purpose of the present paper is to determine the decomposition of the Kronecker product of two irreducible representations of the real 2X2 unimodular group into a continuous direct sum of irreducible representations. The irreducible unitary representations of this group have been determined first by V. A. Bargmann [l](1), and those of the 2X2 complex unimodular group by I. M. Gel'fand and M. A. Nalmark [3]. In both cases the list of these representations contains two continuous series; first, the principal continuous series, the members of which can be described by a pair (m, p) of two variables, m with a discrete, p with a continuous range; and secondly, the representations of the exceptional interval, characterized by a single parameter, varying over a finite interval. In the real case in addition to these there exists a discrete series of representations characterized by integers. Concerning the representations of the exceptional interval it has been proved that they do not occur in the decomposition of the left regular representations of these groups into a continuous direct sum of irreducible representations. The problem of finding the irreducible parts for the Kronecker product of two of these representations by the Reduction Theory of von Neumann [9] was taken up first by G. W. Mackey, in the complex case, for two factors taken from the principal series [4; 5]. W. F. Stinespring applied the same method to the discussion of the analogous case for the real group(2). Recently, M. A. Nalmark attacked the same problem in the complex case, and gives a complete discussion of all possibilities [10](3). In Parts I, II, and III of the present work we give the decomposition of the product of any two irreducible unitary representations of the real 2X2 unimodular group. To sketch our method, we restrict ourselves, for the sake

Unifying quantum and classical physics has proved difficult as their postulates are conflicting. Using the notion of counts of the fundamental measures—length, mass, and time—a unifying description is resolved. A theoretical framework is presented in a set of postulates by which a conversion between expressions from quantum and classical physics can be made. Conversions of well-known expressions from different areas of physics (quantum physics, gravitation, optics and cosmology) exemplify the approach and mathematical procedures. The postulated integer counts of fundamental measures change our understanding of length, suggesting that our current understanding of reality is distorted.

International audience

Back in 1967, Clifford Gardner, John Greene, Martin Kruskal and Robert Miura published a seminal paper in Physical Review Letters which was to become a cornerstone in the theory of integrable systems. In 2006, the authors of this paper received the AMS Steele Prize. In this award the AMS pointed out that

We describe a new approach to the general theory of unitary representations of Lie groups which makes use of the Gelfand-Segal construction directly on the universal enveloping algebra of any Lie algebra. The crucial observation is that Nelson's theory of analytic vectors allows the characterisation of certain states on the universal enveloping algebra such that the corresponding representations of the universal enveloping algebra are the infinitesimal part of unitary representations of the associated simply connected Lie group. In the first section of the paper we show that with the aid of Choquet's theory of representing measures one can derive a simple new approach to integral decomposition theory along these lines.In the second section of the paper we use these methods to study the irreducible unitary representations of general semi-simple Lie groups. We give a simple proof that theK-finite vectors studied by Harish-Chandra [5] are all analytic vectors. We also give new proofs of some of Godement's results [2] characterising spherical functions of height one, at least for unitary representations. Compared with [2] our method has the possible advantage of obtaining the characterisations by infinitesimal methods instead of using an indirect argument involving functions on the group. We point out that while being purely algebraic in nature, this approach makes almost no use of the deep and difficult theorems of Harish-Chandra concerning the universal enveloping algebra [5].Our work is done in very much the same spirit as that of Power's recent paper [8]. The main difference is that by concentrating on a more special class of positive states we are able to carry the analysis very much further without difficulty.

Quantized Vershik–Kerov theory and quantized central measures on branching graphs

The standard interpretation of quantum physics (QP) and some recent generalizations of this theory rest on the adoption of a rerificationist theory of truth and meaning, while most proposals for modifying and interpreting QP in a “realistic” way attribute an ontological status to theoretical physical entities (ontological realism). Both terms of this dichotomy are criticizable, and many quantum paradoxes can be attributed to it. We discuss a new viewpoint in this paper (semantic realism, or briefly SR), which applies both to classical physics (CP) and to QP. and is characterized by the attempt of giving up verificationism without adopting ontological realism. As a first step, we construct a formalized observative language L endowed with a correspondence truth theory. Then, we state a set of axioms by means of L which hold both in CP and in QP. and construct a further language Lv which can express bothtestable andtheoretical properties of a given physical system. The concepts ofmeaning andtestability do not collapse in L and Le hence we can distinguish between semantic and pragmatic compatibility of physical properties and define the concepts of testability and conjoint testability of statements of L and Le. In this context a new metatheoretical principle (MGP) is stated, which limits the validity of empirical physical laws. By applying SR (in particular. MGP) to QP, one can interpret quantum logic as a theory of testability in QP, show that QP is semantically incomplete, and invalidate the widespread claim that contextuality is unavoidable in QP. Furthermore. SR introduces some changes in the conventional interpretation of ideal measurements and Heisenberg’s uncertainty principle.

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple group is a classical group, we prove that the new theory is essentially equivalent to Howe's theory of N-rank (see [Ho4], [L2], [Sc]). Therefore our results provide a systematic generalization of the notion of a small representation (in the sense of Howe) to exceptional groups. However, unlike previous works that used ad hoc methods to study different types of classical groups (and some exceptional ones; see [We], [LS]), our definition is simultaneously applicable to both classical and exceptional groups. The most important result of this article is a general “purity” result for unitary representations which demonstrates how similar partial results in these authors' works should be formulated and proved for an arbitrary semisimple group in the language of Kirillov's theory. The purity result is a crucial step toward studying small representations of exceptional groups. New results concerning small unitary representations of exceptional groups will be published in a forthcoming paper [S]

Probability in Classical and Quantum Physics

We use the method of group contractions to relate wavelet analysis and Gabor analysis. Wavelet analysis is associated with unitary irreducible representations of the affine group while the Gabor analysis is associated with unitary irreducible representations of the Heisenberg group. We obtain unitary irreducible representations of the Heisenberg group as contractions of representations of the extended affine group. Furthermore, we use these contractions to relate the two analyses, namely, we contract coherent states, resolutions of the identity, and tight frames. In order to obtain the standard Gabor frame, we construct a family of time localized wavelet frames that contract to that Gabor frame. Starting from a standard wavelet frame, we construct a family of frequency localized wavelet frames that contract to a nonstandard Gabor frame. In particular, we deform Gabor frames to wavelet frames.

A general theory of a unified construction of the oscillator-like unitary irreducible representations (UIR) of non-compact groups and supergroups is presented. Particle state as well as coherent state bases for these UIRs are given and the case of SU(m,p/n+q) is treated in detail. Applications of this theory to the construction of unitary representations of non-compact groups and supergroups of extended supergravity theories, with particular emphasis on E7(7) and OSp(8/4,IR) are also discussed.

Whence chemistry?

Non-unitary representations of nilpotent groups, I: Cohomologies, extensions and neutral cocycles