New Relativistic Wave Equations for Two-Particle Systems

We seek to introduce a mathematical method to derive the relativistic wave equations for two-particle system. According to this method, if we define stationary wave functions as special solutions like $\Psi(\mathbf{r}_1,\mathbf{r}_2,t)=\psi(\mathbf{r}_1,\mathbf{r}_2)e^{-iEt/\hbar},\, \psi(\mathbf{r}_1,\mathbf{r}_2)\in\mathscr{S} (\mathbb{R}^3\times\mathbb{R}^3)$, and properly define the relativistic reduced mass $\mu_0$, then some new relativistic two-body wave equations can be derived. On this basis, we obtain the two-body Sommerfeld fine-structure formula for relativistic atomic two-body systems such as the pionium and pionic hydrogen atoms bound states, using which, we discuss the pair production and annihilation of $\pi+$ and $\pi-$.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1008.422

Stationary Solutions of the Klein-Gordon Equation in a Potential Field

We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like $\Psi(\mathbf{r},t)=\psi(\mathbf{r})e^{-iEt/\hbar}$, and define $m=E/c^2$, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schr\"{o}dinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method

New Properties of Fourier Series and Riemann Zeta Function

We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relations between power series and trigonometric series, and by using such mapping relations we obtain a general method to find the sum function of a trigonometric series. According to this method, if each coefficient of a power series is respectively equal to that of a trigonometric series, then if we know the sum function of the power series, we can obtain that of the trigonometric series, and the non-analytical points of which are also determined at the same time, thus we obtain a general method to find the sum of the Dirichlet series of integer variables, and derive several new properties of $\zeta(2n+1)$.Comment: 28 page

A New Operator Theory of Linear Partial Differential Equations

We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily apply this Laplace transform to n+1 dimensional partial differential equations. Further, all the analytic solutions to an initial value problem of an arbitrary order linear partial differential equation are expressed in these abstract operators. By writing abstract operators in this class into integral forms, the solutions in operator form are represented into integral forms. We thus solved the important problem of representing the solutions of linear higher-order partial differential equations into the integrations of some given functions. By introduction of abstract operators on Hilbert space, we further discuss the solvability of initial-boundary value problem for the linear higher-order partial differential equations.Comment: 28 page

Constructing non-equilibrium statistical ensemble formalism based on Subdynamics

In this work, we present a general non-equilibrium ensemble formalism based on the subdynamic equation (SKE). The constructing procedure is to use a similarity transformation between Gibbsian ensemble formalism and the non-equilibrium ensemble formalism. The obtained density distribution is a projected one that can represent essence part of (irreversible) evolution of the density distribution, by which a generalized reduced density distribution for the quantum canonical ensembles is studied and applications in Cayley tree and spin network are discussed.Comment: 17pages, submitted in Physica

Condensation and evolution of space-time network

In this work, we try to propose, in a novel way using the Bose and Fermi quantum network approach, a framework studying condensation and evolution of space time network described by the Loop quantum gravity. Considering quantum network connectivity features in the Loop quantum gravity, we introduce a link operator, and through extending the dynamical equation for the evolution of quantum network posed by Ginestra Bianconi to an operator equation, we get the solution of the link operator. This solution is relevant to the Hamiltonian of the network, and then is related to the energy distribution of network nodes. Showing that tremendous energy distribution induce huge curved space-time network, may have space time condensation in high-energy nodes. For example, in the black hole circumstances, quantum energy distribution is related to the area, thus the eigenvalues of the link operator of the nodes can be related to quantum number of area, and the eigenvectors are just the spin network states. This reveals that the degree distribution of nodes for space-time network is quantized, which can form the space-time network condensation. The black hole is a sort of result of space-time network condensation, however there may be more extensive space-time network condensation, for example, the universe singularity (big bang).Comment: 8 pages, for Beibjing complex network 200

Some Rapidly Converging Series for ζ(2n+1)\zeta(2n+1) from Abstract Operators

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).$$ The method is based on the mapping relationships between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relationships between power series and trigonometric series, if each coefficient of a power series is respectively equal to that of a trigonometric series. Thus we obtain a general method to find the sum of the Dirichlet series of integer variables. By defining the Zeta function in an abstract operators form, we have further generalized these results on the whole complex plane.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1008.504

On Evaluation of Zeta and Related Functions by Abstract Operators

Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, and by defining the Zeta and related functions including the Hurwitz Zeta function and the Dirichlet L-function in the form of abstract operators, we have obtained many new series expansions associated with these functions on the whole complex plane, and investigate the number theoretical properties of them, including some new rapidly converging series for $\eta(2n+1)$ and $\zeta(2n+1)$. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).$$Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1806.0788

Cellularity of a Larger Class of Diagram Algebras

In this paper, we realize the algebra of $\mathbb{Z}_2$-relations, signed partition algebras and partition algebras as tabular algebras and prove the cellularity of these algebras using the method of \cite{GM1}. Using the results of Graham and Lehrer in \cite{GL}, we give the modular representations of the algebra of $\mathbb{Z}_2$-relations, signed partition algebras and partition algebras.Comment: 3 figure