Abstract
For study of quantum self-frictional (SF) relativistic nucleoseed spinor-type tensor (NSST) field theory of nature (SF-NSST atomic-molecular-nuclear and cosmic-universe systems) we use the complete orthogonal basis sets of -component column-matrices type SF -relativistic NSST orbitals (-RNSSTO) and SF -relativistic Slater NSST orbitals (-RSNSSTO) through the -nonrelativistic scalar orbitals (-NSO) and -nonrelativistic Slater type orbitals (-NSTO), respectively. Here or and , are the integer ) or noninteger ) SF quantum numbers, where . We notice that the nonrelativistic -NSO and -NSTO orbitals themselves are obtained from the relativistic -RNSSTO and -RSNSSTO functions for , respectively. The column-matrices-type SF -RNSST harmonics (-RNSSTH) and -modified NSSTH (-MNSSTH) functions for arbitrary spin introduced by the author in the previous papers are also used. The one- and two-center one-range addition theorems for -NSO and noninteger -NSTO orbitals are presented. The quantum SF relativistic nonperturbative theory for -RNSST potentials (-RNSSTP) and their derivatives is also suggested. To study the transportations of mass and momentum in nature the quantum SF relativistic NSST gravitational photon (gph) with is introduced.
1. Introduction
Construction of combined quantum approach for AMN and CU systems of nature is the most important because the classical aspect of field theories leads to contradictions (see review papers [1, 2] and references therein). These contradictions, as shown in Figures 1 and 2 for nonrelativistic and SF relativistic NSST potentials and forces, arise for some values of distance (, , , , and ) between fields of nature. The quantities , ( and ), , NW, and NUA in Figures 1 and 2 describe the gravitational, nuclear, electromagnetic, nuclear weak, and Newtonian (Newtonian universal attraction law) fields, respectively (see [3, 4] on observation of gravitational () and nuclear ( and ) fields). The difficulties arising for these values of are not explained by classical field theories. Taking into account all values of distance from nucleus (for ), such a problem can be solved only using quantum SF relativistic NSST field theory of nature presented in this work. We note that the , , , NW, and NUA fields are obtained from the single quantum SF relativistic NSST field when the SF properties are disappearing from sight.
According to the theory introduced by Lorentz in classical electrodynamics [5–7], the electrons move around the atomic point-charge nuclei under total nuclear attraction forces , where is the time derivative of the acceleration of the electron. We note that the inclusion of the third derivative of displacement leads to the radiation and self-force problems in classical electrodynamics. These problems do not arise in the case of quantum SF relativistic NSST field theory. The analytical formulas for the quantum attraction forces suggested in the previous papers [8–10] are the extensions of Lorentz theory to the quantum cases in standard and nonstandard conventions (see [11, 12] and references therein to our works on standard and nonstandard conventions). In the quantum cases, the SF particles move around the nucleus under relativistic NSST attraction forces. These forces depend on quantum numbers and scaling parameters and distance from nucleus of SF-NSST AMN and CU systems. We note that the new fields, which may be discovered in the future for , depended on the productive capacity of science and technology. The presented quantum nonrelativistic and SF relativistic NSST field theory of nature is the generalization introduced by the author of AMN approach to the CU systems.
The purpose of this work is to construct the combined quantum nonrelativistic and relativistic NSST field theory of nature in position space for arbitrary values of parameters. This theory may open new avenues of approach to the solution of problems related to the properties of AMN-CU systems.
2. Gravitational Photon
To study the quantum SF relativistic NSST field theory of nature, we use the Einstein classical relativistic relation between mass and energy in the following form:where (for ) is the mass of fermion; and are the energies of fermion and gph boson with , respectively. The gph boson moves with the velocity of light () and carries the mass and momentum . We note that, in the case of classical electrodynamics, the SF properties of fermions and bosons disappear and the radiation problems arise.
3. Quantum Nonrelativistic Field Theory in Standard Convention
It is easy to show that the SF relativistic NSST functions are expressed through the corresponding nonrelativistic basis sets. Therefore, the SF relativistic NSST particles can be described by the use of nonrelativistic functions. Now we investigate at first the nonrelativistic cases.
In order to study the scalar quantum field theory of nature in standard convention, we use the following formulas (see [8–10]):
(1) The complete orthogonal sets are as follows:whereand , , , and are the NSST polynomials (-NSSTP), are the nonrelativistic Slater type orbitals, and the is the complex (for ) or real spherical harmonic (see [13]). We note that our definition of phases for the complex spherical harmonics () differs from the Condon-Shortley phases [14] by the sign factor .
The confluent hypergeometric function [15] occurring in ((2a) and (2b)) can be determined bywhereAs we see, all of the functions , , , and are expressed through the Pochhammer symbols.
The orthogonality relations are defined as
(2) The eigenvalues corresponding to the scalar functions are the same and determined bywhere () is the screening constant. We note that the parameters can be chosen properly according to the nature of corresponding field under consideration.
(3) The scalar potentials are as follows:
(4) The scalar forces are as follows:
(5) The one- and two-center one-range addition theorems for nonrelativistic -NSO and noninteger -NSTO functions in standard convention are determined by the following relations.
-NSO Functions. For whereFor See [16] for the calculation of overlap integrals.
Noninteger -NSTO Functions. For whereSee (19) for definition of quantities occurring in (29).
For Herewhere
We note that the (27) and (30) are obtained by the use of complete orthogonal functions -NSO.
4. Quantum Nonrelativistic Field Theory in Nonstandard Convention
Now we investigate the properties of nonrelativistic functions for and . In this case, these functions are determined by the following.
4.1. The Scalar Energies
4.2. The Scalar Potentials
4.3. The Scalar Forces
4.4. The Scalar Eigenfunctions
where are the associated Laguerre polynomials (-ALP) defined by Here , and are the Schrödinger’s eigenvalue, potential, force, and eigenfunction for the hydrogen-like atoms in nonstandard convention. As we see from (37), the -ALP polynomials are the special cases of -NSSTP for and . The similar calculations can be also performed in the case of nonrelativistic standard convention. It should be noted that the eigenfunctions and Laguerre polynomials obtained in nonrelativistic standard conventions are not complete basis sets. Therefore these functions cannot be used especially in the series expansion studies (see [17–20]).
5. Quantum SF Relativistic NSST Field Theory
In order to construct the complete orthogonal basis sets of SF relativistic -RNSSTO and -RSNSSTO orbitals, we use the properties of eigenfunctions of operators , and which are determined by the following column-matrices [21, 22]: Here are the Clebsch-Gordan coefficients [23], where It is easy to show that the -RNSSTH and -MRNSSTH harmonics for are transformed into the scalar spherical harmonics:
It should be noted that (see [21, 22, 24, 25]) the fermions (for ) and bosons (for ) are the special cases of SF particles when the SF properties disappear from the sight.
5.1. Study of Properties of Quantum SF Relativistic NSST Functions
The -RNSSTH and -MRNSSTH harmonics for fixed satisfy the following orthonormality relations [21, 22]:
Using (38)–(41), we obtain for Clebsch-Gordan coefficients the following properties:where
The SF relativistic NSST functions for arbitrary spin are defined by the following column-matrices:where See (53) and [21, 22, 24–32] for the exact definition of functions occurring in ((43a) and (43b))–((45a) and (45b)).
The relativistic orbitals of arbitrary spin ) and nonrelativistic functions are determined by the following relations:
Using ((46a) and (46b))–((48a) and (48b)) it is easy to show that the SF relativistic NSST functions for spin and simple are reduced to the corresponding nonrelativistic scalar basis sets:The orthogonality relations for SF relativistic NSST functions are determined by