Theory and Practice of Mathematics and Computer Science Vol. 7,
12 February 2021,
Strike a bell, and then we hear the sound of its gong and not before striking. This is statement of Causality is Universal phenomena; and is a ‘matter of fact’- taken very casually. The information about theory of Causality is too scattered, and is not concise, and is lost is ‘theory of complex analysis’-making the understanding of this ‘matter-of-fact’ phenomena very complicated. In the available literatures authors use the mathematical formulas without explaining them thoroughly, and their practical utility seems missing; and gets lost in the complications. The formulas that are used do not carry out much detailed and elaborate steps that give readers jitters. The purpose of this Chapter and its deliberation with detailed derivations is to make stringent presentation of the Principles of Causality and develop the mathematics in a simplified way, and still make the purpose of applications in mind. A simple principle of nature that is: ‘the effect can only happen after the cause’, i.e. called causality has great mathematical treatment and development we term that as Kramer-Kronigs relation, analyticity, Titchmarsh principle etc. Like the statement- ‘a causal response function is analytic in upper-half of the complex plane’-sounds very complicated and abstract. Here in this Chapter we try to give elaborate treatment, on all the mathematical seemingly complicated and abstract statements and expressions.
Though in terms of ‘complex-analysis’ the Causality Principles looks very complicated and too abstract, here in this Chapter we simplify the derivation of analyticity Kramer-Kronigs relations and obtain these expressions in time and frequency domains. We start from the basics of Impulse Response Function or Green’s Function and then we define the generalized susceptibility function. Thereafter we elaborate by use of Fourier transformation techniques the Kramer-Kronig relations in frequency domain and later also in time domain. This method gets applied to various fields, i.e. in impedance studies, in dielectric relaxation/retardation studies, in refractive index studies, in electric polarization studies, in magnetic systems studies, in stress-strain relaxation studies etc. Even we if we make an artificial material with negative permittivity and negative permeability (thus showing negative refraction), it should and must follow the mathematical tests of causality that is Kramer-Kronigs relation. In this Chapter the examples that we consider are for simple Debye systems, however, the theory and principles, which we deliberate can be extended to non-Debye systems-as well. We are not sure about causality theory that is developed and discussed here if it can be applied to non-differentiable systems i.e. response function defined on fractal support? –perhaps a new formal mathematics needs to be developed in this regard. Our discussion is only for continuous and differentiable systems.
We make declaration- the contents of this Chapter are not new rather existed since 1930’s yet, the theory and its mathematics were difficult to grasp and also to teach. That is because information about the details are too scattered. This Chapter will help as teaching matter to the physics, engineering and mathematics students- and readers will find the explanations and detailed derivations useful in their research work.