BasicEquation...further considerations
more about the Basic Equation
The basic equation is a scalar equation and does not contain any information neither about time nor about space. It is thus independent of space and time and holds anywhere and anytime. Accordingly, it is also invariant under translations and rotations in space and also invariant under translations in time. Following the Noether theorem such invariances directly imply the conservation of quantities like momentum, spin and energy in any equation being derived from this Basic Equation.
The Basic Equation also accounts for the requirements of unitarity and locality being fundamental principles of quantum mechanics. The sequence of summing up things does not change the result. This corresponds to the commutative law in mathematics and is important as it allows for sorting of things and correlations between neighbored things.
The Basic Equation has already unveiled its power in multi-phase-fields models* describing the evolution of complex structures **. In these concepts it holds everywhere and any time i.e.:
\[\sum_{i=0}^{N_\Phi}\Phi_i=\sum_{i=0}^{N_\Phi}\Phi_i(x,t)=1 \: for\:all\:x \:and\:all\: t \]
* I. Steinbach, F. Pezzolla, B. Nestler, M. Seeßelberg, R. Prieler, G.J. Schmitz, J.L.L. Rezende: „A phase field concept for multiphase systems” Physica D 94(1996), p.135-147
