Gérard G. Emch

Scaling and Renormalization

We begin with the static scaling laws which have the objective of reducing the number of independent critical exponents to a couple, and the proposal is to achieve this by focussing on ever smaller neighborhoods of the critical point. The basic ideas of scaling are to be traced back to [Essam and Fisher, 1963, Widom, 1965, Kadanoff, 1966, Fisher, 1967a]; their justification was found later in the renormalization program we discuss in the next section. In a nutshell, the idea was that in the neighborhood of the critical point, the thermodynamic functions can be assumed to be generalized homogenous functions; see e.g. [Stanley, 1971, Fisher, 1983, Cardy, 1996].

A rejoinder on quaternionic projective representations

In a series of papers published in this journal, a discussion was started on the significance of a new definition of projective representations in quaternionic Hilbert spaces. In the present paper we give what we believe is a resolution of the semantic differences that had apparently tended to obscure the issues.

On Wigner’s different usages of models

An extensive reading of Wigner’s scholarly contributions brings to light how he was led to pursue two radically different purposes when he appealed to models. The present paper sketches the logical background for this distinction, and illustrates it with some specific models Wigner considered.

Geometric quantization: Regular representations and modular algebras

A link is established between the geometric quantization programme and the decomposition theory of the regular representation of the Weyl group for homogeneous manifolds of constant curvature K ≤ 0.

Quantum Anosov flows: A new family of examples

A quantum version is presented for the Anosov system defined by the time evolution implemented by the geodesic coflow on the cotangent bundle of any compact quotient manifold obtained from the Poincare half-plane. While the canonical Weyl algebra does not close under time evolution, the symplectic structure of these classical systems can be exploited to produce objects akin to the CCR algebras encountered in quantum field theory. This construction allows one to lift both the geodesic and the horocyclic flows to a Weyl algebra describing the quantum dynamics corresponding to the systems under consideration. The Anosov relations as proposed in Ref. Reference 1 are found to be valid for these models. A quantum version of the classical ergodicity of these systems is discussed in the last section.

Comments on a recent paper by S. Adler on projective group representations in quaternionic Hilbert spaces

A version of Schur{close_quote}s lemma is proven that elucidates some issues raised by that paper. {copyright} {ital 1996 American Institute of Physics.}

On the need to adapt de Finetti's probability interpretation to QM

von Neumann’s reliance on the von Mises frequentist interpretation is discussed and compared with the Dutchbook approach proposed by de Finetti.

Beyond irreducibility and back

Abstract One of the tacit assumptions in the geometric quantization literature seems to be that irreducible representations are the representations of relevance to physics. We show why and how this assumption deserves further scrutiny.

Modular Structures in Geometric Quantization

The purpose of this lecture is to show how certain modular structures, borrowed from the theory of von Neumann algebras, can be exploited to extract primary representations (prequantization) and irreducible representations (quantization) from the regular representation of the symmetry group of the physical systems to be considered. The emphasis is on presenting specific examples for which the solution is exhibited explicitly.

Contributions of Indian Mathematicians to Quantum Statistics

While modern Algebra and Number Theory have well documented and established roots deep into India’s ancient scholarly history, the understanding of the springing up of statistics, specifically quantum statistics, demands a closer inquiry. My project is two-fold. Firstly, I explore and delineate the cultural and educational circumstances that presided over the inception of the very concept that quantum theory required its own dedicated statistical analysis. My quest therefore is anchored in a brief review of the pioneering contributions of personalities as diverse as those of Bose and Chandrasekhar, or Raman and Krishnan, and Mahalanobis. Secondly, I examine how the intellectual climate and some of the local mathematical traditions have fostered the ongoing development of quantum probability and stochastic processes theories in India.