Unification through Generalised Proper Time: The Elementary Construction of the Theory

Unification based upon the generalisation of proper time is proposed as a comprehensive framework to account for the fundamental structure of matter, in a manner contrasting with the more familiar approach based on extra dimensions of space. The elementary properties of matter to be incorporated include the Standard Model of particle physics together with a source for the dark sector and a coherent formalism for quantum gravity. We elaborate upon the manner in which all such material phenomena and empirical properties as distributed in an extended 4-dimensional spacetime can be encompassed within, and derived from, the continuous flow of time alone via a generalised arithmetic form for infinitesimal intervals of proper time. This approach will also be compared and contrasted with the basic structure of causal set theory as a means of demonstrating how it is possible to construct a full physical theory essentially from elements of time alone, as explicitly developed from the most elementary level. The conception of time as utilised and elucidated in this theory, with emphasis upon the causal continuum properties and as the basis for unification, will be clarified.

On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues

Let S denote the unit circle on the complex plane and ★:S2→S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S,★) is isomorphic to the group (S,·); thus, it is a group, where · is the usual multiplication of complex numbers. However, an elementary construction of such isomorphism has not been published so far. We present an elementary construction of all such continuous isomorphisms F from (S,·) into (S,★) and obtain, in this way, the following description of operation ★: x★y=F(F−1(x)·F−1(y)) for x,y∈S. We also provide some applications of that result and underline some symmetry issues, which arise between the consequences of it and of the analogous outcome for the real interval and which concern functional equations. In particular, we show how to use the result in the descriptions of the continuous flows and minimal homeomorphisms on S.

A Bond Charge Model Ansatz for Intrinsic Bond Energies: Application to C-C Bonds

A simple Bond Charge Model is proposed to predict <i>intrinsic</i> bond energies. Model parameters can be derived from the topology of the Electron Localization Function and optimized geometries through classic considerations. Results for carbon-carbon covalent bonds are shown to be very accurate in different chemical environments. Insight can be extracted from the application of the model due to its elementary construction and simple mathematical formulation. The remarkable robustness of the fitted model highlights how different Density Functional Approximations relate geometries, densities and energies.

A Bond Charge Model Ansatz for Intrinsic Bond Energies: Application to C-C Bonds

A simple Bond Charge Model is proposed to predict <i>intrinsic</i> bond energies. Model parameters can be derived from the topology of the Electron Localization Function and optimized geometries through classic considerations. Results for carbon-carbon covalent bonds are shown to be very accurate in different chemical environments. Insight can be extracted from the application of the model due to its elementary construction and simple mathematical formulation. The remarkable robustness of the fitted model highlights how different Density Functional Approximations relate geometries, densities and energies.

Uniform decomposition of probability measures: quantization, clustering and rate of convergence

The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimensiond=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimensiond≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.

Twistor theory at fifty: from contour integrals to twistor strings

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space–time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold—the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics—anti-self-duality equations on Yang–Mills or conformal curvature—can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang–Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose’s proposal for a role of gravity in quantum collapse of a wave function.