We give an explicit construction of a positive-definite invariant inner-product for the Klein-Gordon fields, thus solving the old problem of the probability interpretation of Klein-Gordon fields without having to restrict to the subspaces of the positivefrequency solutions. Our method has a much wider domain of application and may be used to obtain the most general invariant inner-product on th...

In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a onedimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on so...

The method of optimal prediction is applied to calculate the future means of solutions to the Klein-Gordon equation. It is shown that, in an appropriate probability space, the difference between the average of all solutions that satisfy certain constraints at time t = 0 and the average computed by an approximate method is small with high probability.

We study reparametrization-invariant systems, mainly the relativistic particle and its D-dimensional extended object generalization d-brane. The corresponding matter Lagrangians naturally contain background interactions, like electromagnetism and gravity. For a dbrane that doesn’t alter the background fields, we define non-relativistic equations assuming integral sub-manifold embedding of the d...

In this paper, we investigate the fractional-order Klein–Fock–Gordon equations on quantum dynamics using a new iterative method and residual power series based Caputo operator. The equation is generalization of traditional that allows for non-integer orders differentiation. This has been used in study to model behavior particles with fractional spin. Laplace transform employed into simpler form...