Physical Mathematics

The fibre bundle construct defined in our previous work continues to be the context for this paper; quantum fields composed of fibre algebras become liftings of; or sections through; a fibre bundle with base space a subset of curved space-time. We consider a compact Lie group such as SU(n) acting as a local gauge group of automorphisms of each fibre algebra A(x). Compact Lie groups, represented as gauge groups acting locally on quantum fields, are key elements in electroweak and strong force unification. In our recent joint work we have focused on the translational subgroup of the Poincare group as the generator of local diffeomorphism invariant quantum states. Here we extend those algebraic non-perturbative approaches to address the other half of unification by considering the existence of quantum states of the fibre algebra A(x) invariant to the action of compact non-abelian Lie groups. Wigner sets are complementary to little groups and we prove they have the finite intersection property. Exploiting this then allows us to show that invariant states are common in the sense that the weakly closed convex hull of every normal (density matrix) state contains such an invariant state. From these results and our related research emerges the existence of a locally invariant density matrix quantum state of the field.

This work discusses the Mean First Passage Time (MFPT) and Mean Residence Time (MRT) of continuous-time nearest-neighbor random walks in a finite one-dimensional system with a trap at the origin and a reflecting barrier at the other end. The asymptotic results of the MFPT for random walks that start, for example, at the reflecting point, have a variety of dependencies with respect to its size N. For example, for the case of birth and death processes the MFPT~N; for the case of symmetric random walks the MFPT~N² and for the case of biased random walks the MFPT~α^N where α is a constant that depends on the system’s rates. In this work a transition matrix is derived in such a way that the MRT of the system is equal to (m+1)^d where m is the site number, and d is any arbitrary number satisfying the condition d>0. Since the MFPT is the sum of MRTs then the corresponding MFPT for such a transition matrix is MFPT~N^(1+d). Thus, one can determine the asymptotic result of the MFPT to be N^1+d for any arbitrary d>0, and based on it, obtain the corresponding transition matrix. Several examples of fractional and high order asymptotic results of the MFPT such as N^3.5, N⁵, N⁶, are presented.

There has been a growing interest in investigating the approximate solutions of the Klein-Gordon equation and relativistic wave equations for some physical potential models. This is due to the fact that the analytical solutions contain all the necessary information for the quantum system under consideration. In this paper, we obtained the solutions of the Klein-Gordon equation with more general exponential screened coulomb (MGESC), Yukawa potential (YP) and the sum of the mixed potential (MGESCY) using the Parametric Nikiforov-Uvarov Method (PNUM). The bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions are obtained.

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In this article we apply three mathematical methods for solving the Maccari system, namely, the exp(-ϕ(ξ))- expansion method, the sine-cosine approach and the Riccati-Bernoulli sub-ODE method. These methods are used to construct new and general exact periodic and soliton solutions of the Maccari system. This nonlinear system can be turned into another nonlinear ordinary differential equation by suitable transformation. It is shown that the exp(-ϕ(ξ))-expansion method, the sine-cosine method and the Riccati-Bernoulli sub-ODE method provide a powerful mathematical tool for solving a great many systems of nonlinear partial differential equations in mathematical physics.

In this paper, we apply the homotopy analysis method for solving the fourth-order initial value problems by reformulating them as an equivalent system of first-order differential equations. The analytical results of the differential equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy analysis method.