Mathematical Physics - Atkin, R H

. .

Résumé :
To write this book the author has delved deeply into the histories of both Mathematics and Physics, and the result has been an original analysis of one of mankind's most successful intellectual achievements. The initial impulse for the development of Mathematical Physics began with classical Greek/Euclidean geometry and thence through the ideas of Algebra, founded by Arabian mathematicians, until it has flowered, initially in Western Europe, but now worldwide. In the twentieth century a new level of mathematical thought has arisen by the development of the ideas of Topology, both Point-set Topology and more importantly in that of Algebraic Topology. Although one aspect of Mathematics, viz., Projective Geometry, was hardly noticed in the subject yet it has become clear that a great simplification of the derivation of natural laws can be found in that discipline. By natural laws it is generally understood to be the appearance of geometrical (and thereby algebraic) invariants which are found among the various physical measurements. And this means that we need to examine both the requirements of such measurements as well as of their expression in a suitable algebraic language. The concepts of the Differential/Integral Calculus, introduced by Newton and Leibnitz, allowed the idea of a field (magnetic, electric, electromagnetic, gravitational) to flourish. Otherwise the subject has been founded on the basic idea of a particle - whether it be in classical dynamics (particle-Physics and/or rigid-body-Physics), or the particles of a fluid, or the particles which are postulated as atomic, or the elementary particles which are dragged out of some cosmic field. Since a variety of algebras, and of geometries, are involved in the story the author presents the bare bones of these structures in the appendices. By noticing how the whole range of the Laws of Physics can be encompassed in the modern terms of algebraic topology (based on homological and cohomological properties) the author produces and illustrates the fundamental Cocycle Law which underlies the subject. Finally in Part-4 the author sketches out the Q-Analysis of finite binary arrays which takes us out of the laboratory of Physics and into a more familiar everyday world.