## Causality Principle- Simplified Deliberation and Explanation of Its Complex Mathematics

Strike a bell, and then the sound of the gong is heard, and not until it strikes. This is the Universal Phenomenon assertion of Causality; and it’s a ‘matter of fact’- taken very lightly. The knowledge on causality theory is too fragmented and not concise, and the ‘complex analysis theory’ is lost, rendering the interpretation of this ‘matter-of-fact’ phenomenon very difficult. Authors use the mathematical formulas in the available literature without adequately describing them, and their functional usefulness seems to be missing; and in the complexities it gets lost. The formulas used do not perform very detailed and elaborate measures that provide readers with jitters. The aim of this chapter and its deliberation with thorough derivations is to present the Concepts of Causality in a strict manner and to establish mathematics in a simpler way, while still taking into account the purpose of applications. A basic definition of nature that is:’ the result can only arise after the cause’, i.e. called causality has excellent mathematical treatment and development that we name as the relationship between Kramer-Kronigs, analyticity, the theory of Titchmarsh, etc. Like the sentence in the upper half of the complex plane,’ a causal answer mechanism is analytical,’ sounds very complicated and abstract. We try to provide elaborate care here in this chapter on all the seemingly complicated and abstract mathematical statements and expressions. Although the Causality Concepts look very complicated and too abstract in terms of ‘complex-analysis, here in this chapter we simplify the derivation of Kramer-Kronigs relationships of analyticity and obtain these expressions in domains of time and frequency. We start from the basics of the Impulse Response Function or the Function of Green and then describe the function of generalized susceptibility. The Kramer-Kronig relationships in the frequency domain and later in the time domain are then established using Fourier transformation techniques. This approach is used in various fields, such as impedance studies, dielectric relaxation/retardation studies, refractive index studies, electric polarization studies, studies of magnetic systems, studies of stress-strain relaxation, etc. Even if we render an artificial material with negative permittivity and negative permeability (thus showing negative refraction), the statistical causality checks, the Kramer-Kronigs relationship, can and must be observed. The examples we take into account in this chapter are for simple Debye systems, but the theory and concepts we intentionally apply can also be applied to non-Debye systems. We are not aware of the theory of causality formed and discussed here whether it can be extended to non-differentiable systems, i.e. the fractal support response function? In this respect, maybe a new formal mathematics must be developed. Our debate is just about continuous and distinguishable structures. We make a point that the contents of this chapter have not been fresh since the 1930’s, but that the theory and its mathematics have been difficult to understand and also to teach. That’s because the data on the information is too dispersed. This chapter will assist students in physics, engineering and mathematics as a teaching subject, and readers will find the descriptions and detailed derivations helpful in their academic work.

**Author (s) Details**

**Shantanu Das
**BARC, Mumbai, India (Reteired).

View Book :- https://stm.bookpi.org/TPMCS-V7/issue/view/18

analyticity causal systems conjugate Fourier transform convolution Fourier transform Hilbert-transform impulse response function Kramer-Kronigs relation non-causal systems relaxation retardation susceptibility