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We demonstrate that quantum Hamiltonian operator for a free transverse field within the framework of the second quantization reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based upon 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 then naturally takes place of the one of plane or spherical waves in the process of Fourrier or spherical variable separation., Comment: 22 pages

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.

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.

Determinant and higher-loop terms, usually treated by the Pauli–Villars and higher covariant derivatives methods, in the background field method can hardly be regularized simultaneously. At the same time, we observe that introducing a scalar multiplier in front of the quadratic form, which is equivalent to changing the measure in the functional integral, influences only the determinant part of the effective action. This allows one to choose the integration measure and the function in the regularized propagator in such a way as to make all terms in the expansion finite.

We consider the set of closable homogeneous extensions of the quadratic form of the Laplace operator generated by interaction with two point-like sources. We show that this set consists of the trivial (maximal) extension, one point, and a subset equivalent to the Riemann sphere 𝕊2.

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.

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 ℝ3 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.

We study algebraic properties of matrix operators which act on the torsion in the Einstein–Cartan action. Multiplication rules are written and the corresponding equations of motion are explored in different variables. Bibliography: 6 titles.

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.

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.

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.

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.

We review the quadratic form of the Laplace operator in 3 dimensions in spehrical coordinates which acts on the transverse components of vector functions. Operators, acting on the parametrizing functions of one of the transverse components with angular momentum 1 and 2, appear to be fourth order symmetric differential operators with deficiency indices (1,1). We develop self-adjoint extensions of these operators and propose correspondent extensions for the initial quadratic form. The relevant scalar product for the angular momentum 2 differs from the original product in the space of vector functions, but nevertheless it is still local in radial variable., Comment: 33 pages