It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

Using octooctonions (i.e., octonions with octonion coefficients O×O), this paper expresses select findings from nonassociative quantum theory in harmonized notation: Nonrelativistic and relativistic spin operators, Pauli and Dirac matrices, Dirac equation with electromagnetic and gravitational field, and dimensional reduction from quaternionic spin. A generalization of the dimensional reduction program is proposed to argue that octooctonion algebra is wide enough to model a speculated quantum theory that contains all symmetries of the Standard Model, together with four-dimensional Euclidean quantum gravity. The most narrow candidate for such a formulation consists of four generalized Dirac matrices and a four-dimensional operator space with associated fields and charges. Algebraic properties of this relation will be discussed, together with a landscape choice between all possible octooctonionic relations of similar kind.

We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time evolution of the expectation values of the energy-related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.

By using the Hamiltonian form of the Maxwell equations, a biquaternionic model of electro-gravimagnetic (EGM) field is proposed. The EGM-field equations, generating different charges and currents, are constructed. The field analogs of three Newton's laws are formulated for free and interacting charges and currents, as well as the total field of interaction. The Lorentz invariance of the EGM-field equations is investigated (in particular, the charge-current conservation law). It is shown that at the presence of field interaction, this law differs from the well-known one. A new modification of the Maxwell equations is proposed with the scalar resistance field in the biquaternion EGM-field tension. Relativistic transformations of mass and charge-current densities, forces, and their powers are constructed. The solution of the Cauchy problem is given for equation of charge-current transformations.

The use of the data assimilation technique to identify optimal topography is discussed in frames of time-dependent motion governed by nonlinear barotropic ocean model. Assimilation of artificially generated data allows to measure the influence of various error sources and to classify the impact of noise that is present in observational data and model parameters. The choice of length of the assimilation window in 4DVar is discussed. It is shown that using longer window lengths would provide more accurate ocean topography. The topography defined using this technique can be further used in other model runs that start from other initial conditions and are situated in other parts of the model's attractor.

In this paper we prove that Bose gas with weak pair interaction is nonergodic system. In order to prove this fact we consider the divergences in some nonequilibrium diagram technique. These divergences are analogous to the divergences in the kinetic equations discovered by Cohen and Dorfman. We develop the general theory of renormalization of such divergences and illustrate it with some simple examples. The fact that the system is nonergodic leads to the following consequence: to prove that the system tends to the thermal equilibrium we should take into account its behavior on its boundary. In this paper we illustrate this thesis with the Bogoliubov derivation of the kinetic equations.