On some recent interactions between mathematics and physics. (English) Zbl 0569.58019

Bott, Raoul

Can. Math. Bull. 28, 129-164 (1985).

This article is the transcription of a speech the author delivered as the Jeffrey-Williams lecture of the Canadian Mathematical Society. As the author stresses, a more appropriate title would have been: ”A topologist marvels at physics”. In this respect the author explains: ”What there is to marvel at from the perspective of the geometer and topologist is, that the equations which the physicists after many ’supple confusions’ arrive at for their description of the fundamental particles, make such good sense in topology and geometry - and are indeed so inevitable that it is a scandal that the mathematicians had not studied them in their own right years ago. Of course we mathematicians have missed the boat before...”.
In this connection, the author starts ”with a lightning review of classical dynamics from the ultimate ’free enterprise’ point of view embodied in Hamilton’s principle of least action”. His first conclusion is about ”the paradigm of writing down a Lagrangian for the equations of physics”, which leads him to the ’nearly inevitable’ character of Einstein’s gravitational equations, ”once one has had the inspiration to geometrize the whole concept of gravity”. Next he explores the equations of electromagnetism (Maxwell’s equations), fitted ”into this geometric and topological framework”. His ”ultimate goal”, the Yang-Mills equations, then also result ”nearly inevitable”, as the detailed and involved considerations of the author show. The last section is concerned with Yang-Mills fields (applications to geometry and topology). His final conclusions are: ”All in all this then is as beautiful and successful a confluence of different mathematical ideas as one can hope for and an appropriate place to end my lecture.... In short there still is here a great opportunity for cross fertilization between physics and mathematics which I hope will continue to strengthen the bonds between these two great disciplines in the future.”

Reviewer: W.G.Engel

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58-03	History of global analysis
70H25	Hamilton’s principle
01A65	Development of contemporary mathematics
54-03	History of general topology
83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
78A25	Electromagnetic theory (general)
53A45	Differential geometric aspects in vector and tensor analysis
81T08	Constructive quantum field theory

Keywords:

mathematics physics interactions; topology and geometry; classical dynamics; Hamilton’s principle; Einstein’s gravitational equations; concept of gravity; Maxwell’s equations; Yang-Mills equations; Yang-Mills fields