Quantum Self-Frictional Relativistic Nucleoseed Spinor-Type Tensor Field Theory of Nature

For study of quantum self-frictional (SF) relativistic nucleoseed spinor-type tensor (NSST) field theory of nature (SF-NSST atomicmolecular-nuclear and cosmic-universe systems) we use the complete orthogonal basis sets of 2(2s + 1)-component columnmatrices type SF Ψ(δ∗)s nljm푗 -relativistic NSST orbitals (Ψ(δ∗)s-RNSSTO) and SF Xs nljm푗 -relativistic Slater NSST orbitals (Xs-RSNSSTO) through the ψ(δ∗) nlm푙 -nonrelativistic scalar orbitals (ψ(δ∗)-NSO) and χnlm푙 -nonrelativistic Slater type orbitals (χ-NSTO), respectively. Here δ∗ = p∗ l or δ∗ = α∗ and p∗ l = 2l + 2 − α∗, α∗ are the integer (α∗ = α, −∞ < α ≤ 2) or noninteger (α∗ ̸ = α, −∞ < α∗ < 3) SF quantum numbers, where s = 0, 1/2, 1, 3/2, 2, . . .. We notice that the nonrelativistic ψ(δ∗)-NSO and χ-NSTO orbitals themselves are obtained from the relativisticΨ(δ∗)s-RNSSTO andXs-RSNSSTO functions for s = 0, respectively.The column-matrices-type SF 1Yls jm푗 -RNSST harmonics (1Yls-RNSSTH) and 2Yls jm푗 -modified NSSTH (2Yls-MNSSTH) functions for arbitrary spin s introduced by


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  (10 −16 ≤  ≤ 10 −15 , 10 −14 ≤  ≤ 10 −13 , 10 −12 ≤  ≤ 10 −11 , 10 −9 ≤  ≤ 10 −8 , and 10 −6 ≤  ≤ 10 −5 ) between fields of nature.The quantities , ( 1 and  2 ), , 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 ( 1 and  2 ) 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 0 <  < ∞), 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][6][7], the electrons move around the atomic point-charge nuclei under total nuclear attraction forces ⃗   = ⃗  + (2 2 /3 3 ) ... ⃗ , 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 selfforce 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][9][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 0 <  < 10 −18 , 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.

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 0 <  < ∞) is the mass of fermion;  2 and () are the energies of fermion and gph boson with  = 1, respectively.The gph boson moves with the velocity of light (V = ) and carries the mass  = ()/ 2 and momentum .We note that, in the case of classical electrodynamics, the SF properties of fermions and bosons disappear and the radiation problems arise.

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.
The confluent hypergeometric function 1  1 [15] occurring in ((2a) and (2b)) can be determined by where As we see, all of the functions  ( * )  푙 (, ⃗ ),  The orthogonality relations are defined as (2) The eigenvalues corresponding to the  ( * )  푙 (, ⃗ ) scalar functions are the same and determined by where  (0 <  < ∞) is the screening constant.We note that the parameters  can be chosen properly according to the nature of corresponding field under consideration.
(15b) (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. where For ⃗   = 0 See [16] for the calculation of overlap integrals.
For   = 0 Here where We note that the ( 27) and ( 30) are obtained by the use of complete orthogonal functions  ( * ) -NSO.

Quantum Nonrelativistic Field Theory in Nonstandard Convention
Now we investigate the properties of nonrelativistic functions for  * = 1 and  = /.In this case, these functions are determined by the following.

The Scalar Eigenfunctions
where   푙  ((2/)) are the associated Laguerre polynomials (  푙 -ALP) defined by Here , (), (), and   푙 (/, ⃗ ) are the Schrödinger's eigenvalue, potential, force, and eigenfunction for the hydrogen-like atoms in nonstandard convention.As we see from (37), the   푙 -ALP polynomials are the special cases of L ( * ) -NSSTP for  * = 1 and  = /.The similar calculations can be also performed in the case of nonrelativistic standard convention.It should be noted that the eigenfunctions   ((2/)) obtained in nonrelativistic standard conventions are not complete basis sets.Therefore these functions cannot be used especially in the series expansion studies (see [17][18][19][20]).
The orthogonality relations for SF relativistic NSST functions are determined by where  ≥

Conclusion
The construction of quantum self-frictional relativistic nucleoseed spinor-type tensor field theory of nature is based on the generalization of AMN approach to the CU systems introduced by the author in the previous papers.It has been shown that the gravitational, nuclear ( 1 and  2 ), electromagnetic, nuclear weak, and Newtonian fields are the special cases of quantum SF relativistic NSST field presented in this work.We note that the fermions and bosons are obtained from the SF particles of arbitrary spin ( = 0, 1/2, 1, 3/2, 2, . ..) when their SF properties disappear.
The one-and two-center one-range addition theorems for nonrelativistic noninteger  -NSTO orbitals have been suggested.The SF relativistic NSST field through the nonrelativistic field, and vice versa, has been presented.The quantum self-frictional relativistic nonperturbative theory has been also suggested.
The anomaly in [3,4] could lead to a fundamental revision of the Quantum Electrodynamics theory known as the besttested and best-understood theory in all of science until now.We believe that the presented quantum SF relativistic NSST field approach will be of interest in the quantum mechanics of cosmic sciences and combined open shell Hartree-Fock theory suggested by the author (see [38] and references therein to our papers).
The application of presented theory is in progress in our group for the study of SF-NSST atomic-molecular-nuclear and cosmic-universe systems.