Mathematicians really understand what mathematics is. Theoretical physicists really understand what physics is. No matter how fruitful the interplay between the two subjects, the deep intersection of these two understandings seems to me to be quite modest. Of course, many theoretical physicists know a lot of mathematics. And many mathematicians know a fair amount of theoretical physics. This is very different from a deep understanding of the other subject. There is great advantage in the prospect of each camp increasing its appreciation of the other's goals, desires, methodology, and profound insights. I do not know how to really go about this in either case. However, the book in hand is a good first step for the mathematicians.

The method of the text is to explain the meaning of a large number of ideas in theoretical physics via the splendid medium of mathematical communication. This means that there are descriptions of objects in terms of the precise definitions of mathematics. There are clearly defined statements about these objects, expressed as mathematical theorems. Finally, there are logical step-by-step proofs of these statements based on earlier results or precise references. The mathematically sympathetic reader at the graduate level can study this work with pleasure and come away with comprehensible information about many concepts from theoretical physics … quantization, particle, path integral … After closing the book, one has not arrived at the kind of understanding of physics referred to above; but then, maybe, armed with the information provided so elegantly by the authors, the process of infusion, assimilation, and deeper insight based on further rumination and study can begin.