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The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing at the origin together with first derivatives is a symmetric operator with deficiency indices (3). Krein's theory allows one to derive an expression for the resolvent kernel of a self-adjoint extension of the operator in question as a sum of the Green's function of the vector Laplace operator and some additional kernel of finite rank. [ABSTRACT FROM AUTHOR]

We show how Pauli–Villars regularization works in the construction of renormalized Hamiltonian for two exemplars of quantum systems with singular perturbations. The systems are the scalar 3-dimensional particle interacting with δ-potential and the infrared extensions of the quadratic forms of the Gaussian functional of the ground state in the quantum field theory. [ABSTRACT FROM AUTHOR]

We extend the quadratic form of the Gaussian functional of the free quantum scalar field theory to the set of functions decreasing in the infinity as x → − 1 . We use the momentum-space representation (after the Fourier transform) and the scalar product is generated by the quadratic form of the Laplace operator (potential term of the quantum Hamiltonian). [ABSTRACT FROM AUTHOR]

It is demonstrated that within the framework of the second quantization, the quantum Hamiltonian operator for a free transverse field reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based on extensions of the quadratic form of the transverse Laplace operator, which are used as a source of spherical basis functions with singularity at the origin. This basis naturally replaces the basis of plane or spherical waves, which is used to separate variables with the help of the Fourier transform or transition to spherical coordinates. [ABSTRACT FROM AUTHOR]

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing together with their first derivatives at selected points xn, n = 1,...,N, is a symmetric operator with deficiency indices (3N, 3N). The calculation of the scalar products of its regular analytic vectors is the key step in the construction of the resolvents of its self-adjoint extensions via Krein's formula. [ABSTRACT FROM AUTHOR]

We present a diagram technique used to calculate the Seeley–DeWitt coefficients for a covariant Laplace operator. We use the combinatorial properties of the coefficients to construct a matrix formalism and derive a formula for an arbitrary coefficient. [ABSTRACT FROM AUTHOR]

We consider the action of the quadratic form of the Laplace operator and its extensions in subspaces of linear combinations of the “transverse” and “longitudinal” functions with the fixed orbital momentum with respect to the coordinate origin. In the statement of the problem, it is required that the extensions obtained, after the transfer back to the space of vector functions, can be represented as simple limit expressions with two coefficients. We study the behavior of these coefficients with respect to the initial choice of the linear subspace. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

The theoretical studies of phase states with a linear dispersion of the spectrum of low-energy electron excitations have been reviewed. Some main properties and methods of experimental study of these states in socalled Dirac materials have been discussed in detail. The results of modern studies of symmetry-protected electronic states with nontrivial topology have been reported. Combination of approaches based on geometry with homotopic topology methods and results of condensed matter physics makes it possible to clarify new features of topological insulators, as well as Dirac and Weyl semimetals. [ABSTRACT FROM AUTHOR]

We develop self-adjoint extensions of the radial part of the Laplace operator for l = 1 in a special scalar product. The product arises under the passage of the standard product from ℝ to the set of functions parametrizing one of two components of the transverse vector field. Similar extensions are treated for the square of the inverse operator of the radial part in question. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

We study the quadratic form of the Laplace operator in 3 dimensions written in spherical coordinates and acting on transverse components of vector-functions. Operators which act on parametrizing functions of one of the transverse components with angular momentum 1 and 2 appear to be fourth-order symmetric operators with deficiency indices (1, 1). We consider self-adjoint extensions of these operators and propose the corresponding extensions for the initial quadratic form. The relevant scalar product for angular momentum 2 differs from the original product in the space of vector-functions, but, nevertheless, it is still local in radial variable. Eigenfunctions of the operator extensions in question can be treated as stable soliton-like solutions of the corresponding dynamical system whose quadratic form is a functional of the potential energy. [ABSTRACT FROM AUTHOR]

The chiral parametrization of gluons in SU(3) QCD is proposed extending an approach developed earlier for SU(2) case. A color chiral field is introduced, gluons are chirally rotated, and vector component of rotated gluons is defined on condition that no new color variables appeared with the chiral field. This condition associates such a vector component with SU(3)/U(2) coset plus a U(2) field. The topological action in SU(3) QCD is derived. It is expressed in terms of axial vector component of rotated gluons. The vector field in CP2 sector is studied in new variables of chiral parametrization. [ABSTRACT FROM AUTHOR]

We consider the matrix σ-model in the background field formalism. In the two-loop approximation, we demonstrate the equality of “running coupling constants” in the momentum cutoff regularization and in the dimensional regularization by direct calculation. We verify that the β-function coincides with the previously obtained data. [ABSTRACT FROM AUTHOR]

We consider inhomogeneous current states in low-dimensional systems characterized by spatial separation of phase states with ordered spin and charge degrees of freedom. We show that near the self-duality point in the Ginzburg-Landau spinor model, the inhomogeneity degree of non-Abelian states is higher than that of states with an Abelian distribution of degrees of freedom. [ABSTRACT FROM AUTHOR]

We discuss the specificity of charge renormalization in Yang—Mills theory. We show that the values of the running coupling constant in dimensional regularization and in momentum truncation coincide. Dimensional transmutation is interpreted as replacing the dimensionless coupling constant with a dimensional invariant of the renormalization group equation. [ABSTRACT FROM AUTHOR]

Introducing a concept of twisted tubes, we provide a speculative interpretation of exponent 3/4 in the Vakulenko—Kapitansky estimate for the Faddeev—Skyrme model. This allows us to make an assumption about the domain of applicability of the fit derived from the VK estimate. We propose an example of a sequence of knotted configurations which may be utilized in construction of static solutions with high energies. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

Several aspects of extension of the Faddeev–Niemi decomposition for the SU(2) Yang–Mills field to the case of the SU(N) gauge group are discussed. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

We generalize the self-dual parameterization of the SU(2) Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group SU(3). We demonstrate that the duality property intrinsic to the SU(2) gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products SU(2)⊗3. [ABSTRACT FROM AUTHOR]

In the present paper, the Faddeev-Popov determinant arising in the partially-dual decomposition of the SU(2) Yang-Mills field is studied. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

We study algebraic properties of matrix operators which act on the torsion in the Einstein-Cart an action. Multiplication rules are written and the corresponding equations of motion are explored in different variables. Bibliography: 6 titles., UCD 517.9 INTRODUCTION The Hilbert-Einstein action for the gravitational field [1] is written in terms of the Riemannian connection [Γ.sub.λ,νμ]: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1) This connection (also [...]