Quantum Mechanics and Quantum Field Theory: A Mathematical Primer

Webster defines “primer” as (1) “a small book for teaching children to read,” (2) “a small introductory book on a subject,” and (3) “a short informative piece of writing.” Jonathan Dimock’s book is accordingly well named: the book is small, a little over 220 pages; it is indeed an introduction to what I’ll just refer to as QM and QFT, with Dimock stating in his Preface that “the book does help prepare the reader for a journey in any of [a number of further] directions”; finally, “informative” is unquestionably the mot juste for the book. Written in a welcome terse style, Dimock covers both non-relativistic and relativistic QM, statistical QM, and, of course, as the title guarantees, QFT.

Regarding the latter, in the last part of the book, after covering classical and quantum fields on a manifold, Dimock presents three chapters starting with the life’s blood of QFT, the Feynman method of path integrals, ending with a discussion of non-linear field theory. The very last section of the book deals with a reformulation of Dimock’s foregoing discussion of the 2-dimensional model of a field theory governed by a nonlinear field equation, equipped with Hamiltonian whose potential energy part looks like ∫φ₄, where φ is the corresponding scalar field on 2-dimensional spacetime. This final reformulation concerns “a variation which involves constructing the Hilbert space [of the Wightman reconstruction theorem] directly at imaginary spacetime,” and Schwinger functions are featured prominently.

It’s already abundantly clear from the preceding snippet that in the orbit of a little over 200 pages Dimock deals with a lot of what Feynman used to call “the good stuff,” and covers a good deal of serious functional analysis and geometry in the process. Of course, for us mathematicians, this is really the most attractive feature of QM and QFT, isn’t it? These aspects of physical reality provide occasions for the application of spectacularly pretty mathematics, including functional analysis, group representation theory, and differential geometry, and the development of machinery and methodology of great mathematical interest.

Thus, this Mathematical Primer presents a decent chunk of functional analysis already in its opening chapter: the focus falls on operators on a Hilbert space of all four flavors, viz. bounded, unbounded, self-adjoint, and compact operators. Later, in the middle part of the book, relativity is presented, with Einstein’s formulation heavily favored (well, there is really no competition, of course) and Minkowski spacetime featured, but soon it’s on to charged scalar fields and Dirac fields, and all the attendant mathematics. After a discussion of the electromagnetic field and the according reformulation of the Maxwell equations it’s on to interaction of (classical) fields and the gauge principle, setting the stage for some real fireworks including probabilistic methods (on steroids) and QFT proper.

Finally, dotting mathematical i’s and crossing mathematical t’s, the book sports three appendices on, respectively, normed spaces, tensor products, and distributions.

I think this is an excellent book that more than lives up to what its title promises. It is well-written and is a very good introduction to QM and QFT for mathematicians keen on getting quickly off the ground in this direction. There are ca. 100 problems scattered throughout the book, and it’s really non-negotiable that in order to get all it offers out of Dimock’s presentation, the reader should go at them with guns blazing. Dimock is a mathematician writing for mathematicians about themes that the physicists have crafted using their own patois. To reclaim it as mathematics is quite a task in itself: Dimock deserves a lot of praise for taking it on and doing such a wonderful job.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.