Customer Reviews

Classical Mathematical Physics: Dynamical Systems and Field Theories

5 star:		(3)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(1)

 

 

Walther Thirring is a very well known quantum field theorist. He made important contributions to applications of dispersion relations to particle physics, wrote a book on quantum electrodynamics that was so good that Dyson compared it to Pauli's famous quantum mechanics article in the Handbuch der Physik, invented the famous Thirring model, a two-dimensional quantum field with exact (that is, non perturbative) solutions, and produced, with Elliot Lieb, the best demonstration of the stability of matter.At a point in his career he decided to show his fellows what they were losing by ignoring the modern mathematics. Having lectured in mathematical physics, he published his lecture notes, and, later, transformed them into a book of 4 volumes. The present book is a translation, improvement and fusion of the two first volumes, covering Dynamical Systems, that is advanced mechanics and field theories, meaning electrodynamics, gravitation and a little of classical gauge field t! heory. Having done work qualified as high-class mathematics, he is one of the very few scientists of our day who excelled both in physics and mathematics. The book reflects this virtue. I would venture to say that his personal basic reference was the monumental "Traite d'analyse" by Jean Dieudonne'. Not only is this the first of his references, but the way of introducing differential manifolds, and, particularly, tangent spaces, is very close to Dieudonne's. Once you learn what Thirring is offering you, you will adopt the new methods. They are much more natural, which, in mathematics, is tantamount to being deeper. And the methods are also much more efficient for calculations. Take this problem: given a (semi-)riemannian metric, compute the components of the curvature tensor (a problem central to general relativity). You can do it by using classical tensors, as most textbooks do (take Weinberg, for instance), or use Cartan structural equations, which use exterior di! fferential forms (as in Thirring). I benchmarked it: for th! e usual metrics, you gain, in speed, a factor 5 (by following Thirring). Still more important, being much shorter, the calculation is much less prone to errors. In this text I especially liked the sections on the Action Principle and the Noether theorem, in the garb of differential forms, which is, no doubt, their natural language. This is a very compact book. You are supposed to work hard, and it is absolutely essential that you work out the exercises (all of them) and check the solutions. But, of course, this advice applies to every book!

This book represents how a graduate course in mathematical physics ought to look like. It deals with two topics with which the reader should be very familiar, physics-speaking: particle dynamics and classical fields (e.g. Maxwell's equations). Therefore the author is justified in neglecting the physics and concentrating on the mathematics; in fact, introducing the mathematical tools using such old friends makes it easier for the reader to fully understand the mathematics, and the way it relates to the physics.
There are many books on the market that teach differential geometry - Frankel's "geometry and physics", Bishop and Goldberg's "Tensor analysis on manifolds" and countless others - but this book is something more: it deals as much with mathematical physics as it deals with the mathematics. For instance, it formulates hamiltonian mechanics for a particle in an electromagnetic field and proceeds to solve the cases of the constant field, coloumb field, travelling plane disturbances and more, all using the modern language of differential geometry all physicists should know.
Although the book contains an introduction to differential geometry - which is very nice, actually, with plenty of examples and such - I strongly advise the reader to use another book as a main source book for diff. geometry, and to use Thirring's book as a supplement. Thirring's strength is not in teaching diff. geometry, it is in showing how to apply it to physics.
It almost goes without saying, p.s., that the reader should have a good grasp of calculus in R^n, topology and linear algebra before approaching this book.

This book introduces the physicists applications of differential geometry in physics.It is a complete book in classical field theory.Moreover it is very interesting to see the geometry of relativity.I highly recommend this book for the theoretical physicists and the mathematicians interested in physics.In short this is a very useful book.

For theoretical physicists, Thirring has provided an updated third edition of his successful monograph. It offers you an indepth treatment of dynamical systems, presented with good rigour. The emphasis is on the behaviour of Hamiltonian systems and electromagnetic systems, on a manifold.

He discusses both non-relativistic and relativistic particle motion. For the latter, the text does not treat just the special relativity case of zero gravity. Einstein's Field Equations are used to derive motion in a Schwarzschild field around a large mass. A entire chapter is devoted to General Relativity. The treatment here provides a stronger math framework than that given by the standard GR texts of Misner, Thorne and Wheeler or Weinberg.

If you enjoy maths, which you undoubtedly do as a theorist, then Thirring's presentation can be a pleasure to study.

I appreciate mathematical rigor insofar as it serves as a deterrent to sloppy reasoning, but I think this book represents a case of the author focusing on the mathematics to the point of obscuring its physical significance. Maybe I owe it to my relative ignorance of differenial geometric concepts, but I found this book practically worthless to my understanding of dynamical systems. I tend to agree with David Griffiths when he states that mathematics is a tool to the physicist; this text makes it the sole focus.