Quaternionic analysis, representation theory and physics

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.     

Gaudin model,Bethe Ansatz and critical level

We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensoproducts of Wakimoto modules. This gives explicit formulas for the eigenvectors via bosonic correlation functions. Analogues of the Bethe Ansatz equations naturally appear as equations on the existence of singular vectors in Wakimoto modules. We use this construction to explain the connection between Gaudin's model and correlation functios of WZNW models.     

Central extensions of current groups in two dimensions

The paper is devoted to generalization of the theory of loop groups to the two-dimensional case. To every complex Riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex Lie group G by the Jacobian of this surface. This extension is topologically nontrivial, as in the loop group case. Orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalG-bundles over the surface. When the surface is the torus (elliptic curve), classification of coadjoint orbits is related to linear difference equations in one variable, and to classification of conjugacy classes in the loop group. We study integral orbits, for which the Kirillov-Kostant form is a curvature form for some principal torus bundle. The number of such orbits for a given level is finite, as in the loop group case; conjecturedly, they correspond to analogues of integrable modules occurring in conformal field theory.     

Four dimensional realization of two dimensional current groups

The quotient realization of the central extensions of the current groups over Riemann surfaces is achieved by means of the Leray residue theory. This approach replaces de Rham cohomology in the classical WZNW construction for affine Lie groups.     

Chaotic scattering and transmission fluctuations

We discuss the phenomenon of chaotic scattering and its application in the study of transmission of electrons in mesoscopic devices as well as the transmission of microwaves through junctions. We show that the fact that the ray optics (classical dynamics) is chaotic, implies fluctuations in the observed transmission coefficients, whose statistics is determined by the theory of random matrices. We also show how the classical distribution functions which reflect the chaotic nature of the classical dynamics, determine the dependence of the correlations observed in the fluctuating transmission coefficients on external parameters. The time domain properties of chaotic scattering systems are also examined, and are shown to depend on the chaotic nature of the classical dynamics, together with a wave mechanical enhancement in time reversal invariant systems. Finally, we study the role of absorption and discuss its effects on the transmission fluctuations and their statistics.     

Character formulae for classical groups

We give formulae relating the value X_λ (g) of an irreducible character of a classical group G to entries of powers of the matrix g ε G. This yields a far-reaching generalization of a result of J.L. Cisneros-Molina concerning the GL ₂ case [1]. Partially supported by OTKA grants T 042769 and T 046365     

Asymptotic states and infrared divergences in non-abelian theories

We give a general definition of a class of asymptotic states in non-abelian gauge theories. We argue, using unitarity, that they give infrared-finite S-matrix elements. We discuss the energy of the soft gluons in these states.     

Complex Cabibbo angle and CP violation in a class of gauge theories

A class of gauge theories which incorporate CP violation is discussed. The CP violating effects arise as a consequence of performing a different rotation in the SU(3) space for the left- and right-handed strangeness conserving charged currents. It is suggested that it is possible to understand the smallness of these effects in the $\text{K}{}_0\text{?}\text{K}{}_0$ system even in the presence of a large relative phase between the left- and right-handed weak currents. The electric dipole moments of hadrons are of fifth order in the semi-weak coupling constant whereas for leptons they are of seventh order. These electric dipole moments are considerably smaller than the experimental upper limits.