XXVI INTERNATIONAL WORKSHOP ON GEOMETRICAL METHODS IN PHYSICS

Quantization of dissipative systems—some irresponsible speculations
View Description Hide DescriptionThe Newton‐Lagrange equations of motion represent the fundamental law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses classical and quantal dynamics of such systems and presents some irresponsible speculations by introducing a certain canonical two‐form Ω. By its construction Ω embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle is introduced, where variation is performed over a set of “umbilical surfaces” instead of system histories. It provides correct Newton‐Lagrange equations of motion and something more. The quantization is inspired by the Feynman functional integral approach. The quintessence is to rearrange path integral into an “umbilical world‐sheet” integral in accordance with the proposed variational principle. In the case of potential‐generated forces, the new approach reduces to the standard quantum mechanics.

Topological spectrum of classical configurations
View Description Hide DescriptionFor any classical field configuration or mechanical system with a finite number of degrees of freedom we introduce the concept of topological spectrum. It is based upon the assumption that for any classical configuration there exists a principle fiber bundle that contains all the physical and geometric information of the configuration. The topological spectrum follows from the investigation of the corresponding topological invariants. Examples are given which illustrate the procedure and the significance of the topological spectrum as a discretization relationship among the parameters that determine the physical meaning of classical configurations.

BRST quantization in the canonical setting
View Description Hide DescriptionAfter an introduction to the canonical BRST quantization procedure for a first class constrained system on a symplectic manifold, the modification required for a system with reducible symmetry is described. A topological model which exhibits this method is discussed.

Quantization of Algebraic Reduction
View Description Hide DescriptionFor a Poisson algebra obtained by algebraic reduction of symmetries of a quantizable system we develop an analogue of geometric quantization based on the quantization structure of the original system.

A family of star products and its application
View Description Hide DescriptionStar products with a real deformation parameter ℏ are considered. By means of complex symmetric matrices {K}, a family of star products is given on the space of complex polynomials. By taking completion of the set of polynomials, star products are extended to star products on the spaces of certain entire functions. Then one can construct star exponential functions in these spaces, which produce various interesting identities. As an example, Clifford algebras are constructed explicitly in terms of the extended star product algebra.

Operatorial Methods in Noncommutative Field Theory
View Description Hide DescriptionWe review the operatorial quantization of noncommutative field theory, with emphasis on the fundamentally bilocal nature of the degrees of freedom. Interactions and IR/UV mixing are discussed from this point of view.

Lagrangian and non‐Lagrangian approaches to electrodynamics including supersymmetry
View Description Hide DescriptionWe take a general approach to nonlinear electrodynamics that includes non‐Lagrangian as well as Lagrangian theories. We introduce the constitutive tensors which, together with Maxwell's equations, describe nonlinear electrodynamics in an extremely general way. We show how this approach specializes to particular cases that were previously considered, and indicate how it generalizes to supersymmetric electrodynamics.

Noncommutative Noether theorem
View Description Hide DescriptionNoncommutative (NC) analogue of classical Noether theorem for field theory within the Moyal algebra framework is considered. Relevant spacetime and gauge infinitesimal transformations are classified and used to formulate a NC version of Noether procedure by the mean of Ward identity operators. As an illustration, the method is performed on the recent Grosse‐Wulkenhaar scalar model.

Conformal symmetry and quantum localization in space‐time
View Description Hide DescriptionThe classical procedures which define the relativistic notion of space‐time can be implemented in the framework of Quantum Field Theory. Only relying on the conformal symmetries of field propagation, time‐frequency transfer and localization lead to the definition of time‐frequency references and positions in space‐time as quantum observables. Quantum positions have a non vanishing commutator identifying with spin, both observables characterizing quantum localization in space‐time. Frame transformations to accelerated frames differ from their classical counterparts. Conformal symmetry nevertheless allows to extend the covariance rules underlying the formalism of general relativity under an algebraic form suiting the quantum framework.

Solutions of Euler‐Lagrange equations in fractional mechanics
View Description Hide DescriptionIn the paper Euler‐Lagrange equations of fractional mechanics are studied. The characteristic feature of such equations is mixing of left‐ and right‐sided fractional derivatives due to the rules of integration by parts in fractional calculus. We propose to solve a class of equations of this type using transformation to equivalent fractional integral equations and then applying Banach theorem on fixed point of a contractive mapping. The method is explained in detail for nonlinear fractional oscillator equation in two versions, then the results for a wide class of fractional differential equations are reported.

Geometry of mixing and degeneracy of resonances in a two‐channel model
View Description Hide DescriptionSome geometric aspects of the mixing and degeneracy of two unbound energy eigenstates of a quantum system depending on two control parameters will be described. It will be shown that the surface that represents the complex resonance energy eigenvalues in parameter space has a singularity of square root type at the degeneracy point, and branch cuts in its real and imaginary parts that start at the exceptional point but extend in opposite directions. The rich phenomenology of crossings and anticrossings of energies and widths of the resonances in an isolated doublet of unbound states as well as the changes of identity of the poles of the S–matrix, observed when one control parameter is varied while the other is kept constant is fully explained in terms of sections of the eigenenergy surfaces.

Quantum field theory in curved spacetime and the dark matter problem
View Description Hide DescriptionQuantum field theory in nonstationary curved Friedmann spacetime leads to the phenomenon of creation of massive particles. The hypothesis that in the end of inflation gravitation creates from vacuum superheavy particles decaying on quarks and leptons leading to the observed baryon charge is investigated. Taking the complex scalar field for these particles in analogy with ‐meson theory one obtains two components—the long living and short living ones, so that the long living component after breaking the Grand Unification symmetry has a long life time and is observed today as dark matter. The hypothesis that ultra high energy cosmic rays occur as manifestation of superheavy dark matter is considered and some experimental possibilities of the proposed scheme are analyzed.

A Global Operator Approach to Wess‐Zumino‐Novikov‐Witten models
View Description Hide DescriptionWe present a global operator approach to Wess‐Zumino‐Novikov models for compact Riemann surfaces of arbitrary genus g with N marked points. The approach is based on the multipoint Krichever‐Novikov algebras of global meromorphic functions and vector fields, and the global algebras of affine type and their representations. Using the global Sugawara construction and the identification of a certain subspace of the vector field algebra with the tangent space to the moduli space of the geometric data, the Knizhnik‐Zamalodchikov connection is defined. For fermionic representations it defines a projectively flat connection on the vector bundle of conformal blocks. The presented work is joint work with Oleg Sheinman.

Why Lagrangians?
View Description Hide DescriptionWe argue that Feynman's Integral imposes the condition of being mutually unbiased on pairs of bases that are causally proximal. This sheds light on the nature of Lagrangian theories from the emergent space‐time perspective.

Lorentz Covariant Distributions with Spectral Conditions
View Description Hide DescriptionThe properties of the vacuum expectation values of products of the quantum fields are formulated in the book [1]. The vacuum expectation values of quantum fields products would be the Fourier transforms of the Lorentz covariant tempered distributions with supports in the product of the closed upper light cones. Lorentz invariant distributions are studied in the papers [2]—[4]. The authors of these papers wanted to describe Lorentz invariant distributions in terms of distributions given on the Lorentz group orbit space. This orbit space has a complicated structure. It is noted [5] that a tempered distribution with support in the closed upper light cone may be represented as the action of the wave operator in some power on a differentiable function with support in the closed upper light cone. For the description of the Lorentz covariant differentiable functions the boundary of the closed upper light cone is not important. The measure of this boundary is zero.

On the Equations of Conformally‐Projective Harmonic Mappings
View Description Hide DescriptionIn this paper we study compositions of conformal and geodesic diffeomorphisms, which are at the same time harmonic mappings (conformally‐projective harmonic mappings). The equations of conformally‐projective harmonic mappings are shown. We obtained the fundamental equations of these mappings in form of a system of differential equations of Cauchy type. Solutions of this system depend on at most independent parameters.

Operators on superspaces and generalizations of the Gelfand–Kolmogorov theorem
View Description Hide DescriptionGelfand and Kolmogorov in 1939 proved that a compact Hausdorff topological space X can be canonically embedded into the infinite‐dimensional vector space the dual space of the algebra of continuous functions as an “algebraic variety”, specified by an infinite system of quadratic equations.
Buchstaber and Rees have recently extended this to all symmetric powers using their notion of the Frobenius n‐homomorphisms.
We give a simplification and a further extension of this theory, which is based, rather unexpectedly, on results from super linear algebra.

N‐commutators for simple Lie algebras
View Description Hide DescriptionThe N‐commutator of N vector fields (differential operators of order 1) is defined as the skew‐symmetric sum of the N! compositions for all permutations In general it is a differential operator of order more than 1, but for some special cases of N it might happen that will be a well‐defined operation on the space of vector fields. We construct such N‐commutators for simple Lie algebras for the four classical series and for the exceptional algebra We establish that has well‐defined 2‐ and 10‐commutators.

ON THE ALGEBRAIC ASPECTS OF SAXON‐HUTNER THEOREM
View Description Hide DescriptionHere we present some necessary and sufficient conditions for the validity of the Saxon‐Hutner conjecture concerning the preservation of the energy gaps into an infinite one‐dimensional lattice.