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arXiv1992

Lie-algebraic approach to the theory of polynomial solutions. I. Ordinary differential equations and finite-difference equations in one variable

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial element of the universal enveloping algebra of $sl_2({\bf R})$ (for differential equations) or $sl_2({\bf R})_q$ (for finite-difference equations) in the "projectivized" representation possessing an invariant subspace. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

hep-th↗
arXiv1992

Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations

A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue problem for a polynomial element of the universal enveloping algebra of the algebra $osp(2,2)$ in the "projectivized" representation (in differential operators of the first order) possessing an invariant subspace. A classification of 2 x 2 matrix differential equations in one real variable possessing polynomial solutions is described. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

hep-th↗
arXiv1992

Lie-algebraic approach to the theory of polynomial solutions. III. Differential equations in two real variables and general outlook

Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the eigenvalue problem for a polynomial elements of the universal enveloping algebras of the algebras $sl_3({\bf R})$, $sl_2({\bf R})\oplus sl_2({\bf R})$ and $gl_2 ({\bf R})\ \triangleright\!\!\!< {\bf R}^{r+1}\ , r>0$ taken in the "projectivized" representations (in differential operators of the first order in two real variables) possessing an invariant subspace. General insight to the problem of a description of linear differential operators possessing an invariant sub-space with a basis in polynomials is presented. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

hep-th↗
arXiv1994

Diffeomorphisms of the Klein Bottle and Projective Plane

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem.

hep-th↗
arXiv1994

Infinite Dimensional Geometry and Quantum Field Theory of Strings. I. Infinite Dimensional Geometry of Second Quantized Free String

There are investigated several objects of an INFINITE DIMENSIONAL GEOMETRY appearing from the second quantization of a free string. The paper contains 2 chapters: 1st is devoted to the infinite dimensional geometry of flag, fundamental and $Π$-spaces for Virasoro-Bott group and its nonassociative deformation defined by Gelfand-Fuchs 3-cocycle (Gelfand-Fuchs loop) as well as of infinite-dimensional non-Euclidean symplectic grassmannian, to the constructions of Verma modules, their models and skladens over Virasoro algebra; an infinite dimensional geometry of the configuration space for the second quantized free string in flat and curved backgrounds as well as author version of Bowick- Rajeev formalism of the separation of internal and external degrees of freedom of a closed string are described in 2nd chapter. In the 1st chapter the main objects are infinite dimensional Lie algebras, groups and loops, homogeneous, Kähler, Finsler, contact and symmetric spaces, complex, real and CR-manifolds, determinant sheaves, manifolds with subsymmetries, polarizations and Fock spaces, bibundles and objects of integral geometry, nonholonomic spaces, deformations of geometric structures and moduli spaces. In the 2nd chapter they are gauge fields, Faddeev-Popov ghosts, Gauss-Manin connections, Kostant-Blattner-Sternberg pairings, BRST-operators.

hep-th↗
arXiv1995

Triangular dissections, aperiodic tilings and Jones algebras

The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph of type $A_n$ determines planar fractal sets obtained by infinite dissections of a given triangle. All triangles appearing in the dissection process have angles that are multiples of $π/ (n+1).$ There are usually several possible infinite dissections compatible with a given $n$ but a given one makes use of $n/2$ triangle types if $n$ is even. Jones algebra with index $[ 4 \ \cos^2{π\over n+1}]^{-1}$ (values of the discrete range) act naturally on vector spaces associated with those fractal sets. Triangles of a given type are always congruent at each step of the dissection process. In the particular case $n=4$, there are isometric and the whole structure lead, after proper inflation, to aperiodic Penrose tilings. The ``tilings'' associated with other values of the index are discussed and shown to be encoded by equivalence classes of infinite sequences (with appropriate constraints) using $n/2$ digits (if $n$ is even) and generalizing the Fibonacci numbers.

hep-th↗
arXiv1994

Invariant Indentities in the Heisenberg Algebra

Polynomial relations between the generators of $q$--deformed Heisenberg algebra invariant under the quantization and $q$-deformation are discovered. One of the examples of such relations is the following: if two elements $a$ and $b$, obeying the relation \[ ab - q ba = p, \] where $p, q$ are any complex numbers, then for any $p,q$ and natural $n$ \[ (aba)^n = a^n b^n a^n \]

hep-th↗
arXiv1995

Solutions of the Hamilton--Jacobi equation for one component two dimensional Field Theories

The Hamilton--Jacobi formalism generalized to 2--dimensional field theories according to Lepage's canonical framework is applied to several covariant real scalar fields, e.g. massless and massive Klein--Gordon, Sine--Gordon, Liouville and $ϕ^4$ theories. For simplicity we use the Hamilton--Jacobi equation of DeDonder and Weyl. Unlike mechanics we have to impose certain integrability conditions on the velocity fields to guarantee the transversality relations between Hamilton--Jacobi wave fronts and the corresponding families of extremals embedded therein. Bäcklund Transformations play a crucial role in solving the resulting system of coupled nonlinear PDEs.

hep-th↗
arXiv1995

Modular Covariance, PCT, Spin and Statistics

The notion of modular covariance is reviewed and the reconstruction of the Poincaré group extended to the low-dimensional case. The relations with the PCT symmetry and the Spin and Statistics theorem are described.

hep-th↗
arXiv1996

Analog of Lie Algebra and Lie Group for Quantum Non-Hamiltonian Systems

Quantum mechanics of Hamiltonian (non-dissipative) systems uses Lie algebra and analytic group (Lie group). In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra and analytic quasigroup (loop). The author derives that analog of Lie algebra for quantum non-Hamiltonian systems is commutant Lie algebra and analog of Lie group for these systems is analytic commutant associative loop (Valya loop). A commutant Lie algebra is an algebra such that commutant (a subspace which is generated by all commutators) is a Lie subalgebra. Valya loop is a non-associative loop such that the commutant of this loop is associative subloop (group). We prove that a tangent algebra of Valya loop is a commutant Lie algebra. It is shown that generalized Heisenberg-Weyl algebra, suggested by the author to describe quantum non-Hamiltonian (dissipative) systems, is a commutant Lie algebra. As the other example of commutant Lie algebra, it is considered a generalized Poisson algebra for differential 1-forms. Note that non-Hamiltonian (dissipative) quantum theory has a broad range of application for non-critical strings in "coupling constant" phase space and bosonic string in non-Riemannian (for example, affine-metric) curved space which are non-Hamiltonian (dissipative) systems.

hep-th↗
arXiv1996

Complex structure and the construction of the $:ϕ^4_4:$ quantum field theory in four-dimensional space-time

We announce results about the nonperturbative mathematically rigorous construction of the $:\!ϕ^4_4\!:$ quantum field theory in four-dimensional space-time. The complex structure of solutions of the classical nonlinear (real-valued) wave equation and quantization are closely connected among themselves and allow to construct non-perturbatively the quantum field theory with interaction $:\!ϕ^4_4\!:$ in four-dimensional space-time. We consider vacuum averages, in particular, we construct Wightman functions and matrix elements of the scattering operator as generalized functions for finite energies. The constructed theory is obviously nontrivial.

hep-th↗
arXiv1996

New index formulas as a meromorphic generalization of the Chern-Gauss-Bonnet theorem

Laplace operators perturbed by meromorphic potential on the Riemann and separated type Klein surfaces are constructed and their indices are calculated by two different ways. The topological expressions for the indices are obtained from the study of spectral properties of the operators. Analytical expressions are provided by the Heat Kernel approach in terms of the functional integrals. As a result two formulae connecting characteristics of meromorphic (real meromorphic) functions and topological properties of Riemann (separated type Klein) surfaces are derived.

hep-th↗
arXiv1996

The $:ϕ^4_4:$ quantum field theory, II. Integrability of Wick kernels

We continue the construction of the $:ϕ^4_4:$ quantum field theory. In this paper we consider the Wick kernel of the interacting quantum field. Using the complex structure and the Fock-Bargmann-Berezin-Segal integral representation we prove that this kernel defines a unique operator--valued generalized function on the space $\Sc^α(\R^4)$ for any $α<6/5,$ i.e. the constructed quantum field is the generalized operator-valued function of localizable Jaffe class. The same assertion is valid for the outgoing quantum field. These assertions about the quantum field allow to construct the Wightman functions, the matrix elements of the quantum scattering operator and to consider their properties (positivity, spectrality, Poincare invariance, locality, asymptotic completeness, and unitarity of the quantum scattering).

hep-th↗
arXiv1997

A Direct Method for Obtaining the Differential Conservation Laws

A direct method for obtaining the differential conservation laws in the field theory from the principle of stationary action is proposed. The method is based on a variation of field functions through small local transformation of a special kind. The action of general theory of relativity are considered.

hep-th↗
arXiv2009

Definition of the Dirac Sea in the Presence of External Fields

It is shown that the Dirac sea can be uniquely defined for the Dirac equation with general interaction, if we impose a causality condition on the Dirac sea. We derive an explicit formula for the Dirac sea in terms of a power series in the bosonic potentials. The construction is extended to systems of Dirac seas. If the system contains chiral fermions, the causality condition yields a restriction for the bosonic potentials.

hep-th↗
arXiv1996

Topics in hidden symmetries. IV

This note being devoted to some aspects of the inverse problem of representation theory explicates the links between researches on the Sklyanin algebras and the author's (based on the noncommutative geometry) approach to the setting free of hidden symmetries in terms of "the quantization of constants". Namely, the Racah-Wigner algebra for the Sklyanin algebra is constructed. It may be considered as a result of the quantization of constants in the Racah-Wigner algebra for the Lie algebra $sl(2,C)$. The Racah-Wigner algebra for the Sklyanin algebra is an example of the noncommutative weighted shift operator algebras (NWSO-algebras), which generalize the mho-algebras introduced by the author earlier. If the Sklyanin algebra is interpreted as an algebra of anomalous spins then the Racah-Wigner algebra for it may be regarded as an enlargement of the Sklyanin algebra by operators of the anomalous spin-spin interaction (of tensor type).

q-alg↗