“Bethe-Salpeter” is a classic. When it comes to the definitive texts in quantum mechanics, a few titles with special impact come to mind right away, even to mathematicians. It suffices just to give the author(s): Dirac, Pauli, Landau-Lifshitz, Feynman, and of course the book under review. Maybe this familiarity is because physicists are our cousins, and first cousins at that: I dare say they, or at least the theoretical ones, are genetically closer to us than even computer scientists (at best second cousins, I think). With physicists you can at least chat about Hilbert space and the spectral theorem, differential geometry and Riemann-Christoffel tensors, and nowadays even such a hybrid as Witten’s approach to Morse theory. Yes, there are differences of temperament and even philosophy — just consider what has been happening in mathematics in order to lend much-needed rigor to such things as the Feynman path integral: text after text aiming at putting physicists’ heuristics and stipulations on a mathematically acceptable basis. But there is so much common ground, that, to be sure, we are dealing with, yes, first cousins.
That said, it is not unreasonable for a mathematician to learn some physics from physicists, even if some of it will be painful and unnatural. After all, if we need their stuff, we had better know how to go about getting at it. Again, just think of how much physics, properly so-called, is involved in the mathematical subjects mentioned above, and think of the obvious influence of such physicists as Einstein, Heisenberg, Schrödinger, Dirac, Feynman, and Witten on mathematics properly so-called. And quantum mechanics is an indispensable subject in this connection.
So we come to Bethe-Salpeter: about 350 pages of quantum mechanics, and apparently pretty narrow QM at that: we’re dealing with atoms with one or two electrons. But is it really as narrow as all that? We note (https://en.wikipedia.org/wiki/Two-electron_atom) that merely going with two electrons implies the entry of the Pauli exclusion principle into the game, and this obviously engenders a major complication as regards the analysis of the system we’re dealing with. Just consider the history of QM, given the early work of Bohr and Rutherford, ca. 1913, the planetary model of the atom, and what it set into motion: after over a decade of Sturm und Drang the Pauli exclusion principle didn’t come along until 1925, the same year that gave us the Dreimännerarbeit formulating matrix mechanics, as well as the Schrödinger wave equation. Together with these other breakthrough works the Pauli exclusion principle constitutes one of the pillars of quantum mechanics in the Copenhagen Interpretation, and it is evident that the two electron case is a deeply significant affair, imbued with mathematical complexities that must be properly dealt with if one is to go on to general atomic theory.
Bethe and Salpeter, writing in 1957, present a masterful discussion of these fundamental themes. The over thirty-year lag between the formulation of the fundamental atomic theory mentioned above and the appearance of the book under review is explained in the Preface by the comment that the source for the present book is an article “on the same subject written by one of us (HAB) about 25 years ago for the Geiger-Scheel Handbuch der Physik.” The authors go on to say that their book has two aims, namely, to serve as a reference book on “[the behavior of] hydrogen-like and helium-like atoms and their comparison with experiments,” and “to be of some use to graduate students who wish to learn ‘applied quantum mechanics.’” It is also worth noting that, in Bethe and Salpeter’s words, “[a] large fraction of the present book is devoted to the Dirac theory of the electron and to radiative effects, including short discussions of the relevant experiments. These topics are … treated from the ‘low brow’ or ‘practical’ point of view. In particular no formal derivations of quantum electrodynamics are presented …” This is an exceedingly interesting point, seeing that quantum electrodynamics (QED) goes back to the work done on the Lamb shift the 1940s and 1950s: it was none other than Bethe who early on addressed this matter (cf. https://en.wikipedia.org/wiki/Lamb_shift ) in a now legendary manner — he did the critical Lamb shift calculations on a train from New York to Schenectady and the now-famous Shelter Island Conference (https://en.wikipedia.org/wiki/Shelter_Island_Conference ) and thereby unchained the developments in theoretical physics that led to the formulation of QED as the most successful theory in modern physics in the hands of Tomonaga, Schwinger, Feynman, and Dyson, the latter being a converted transcendental number theorist (a sibling, not just a cousin). In this connection see, e.g. QED and the Men Who Made It (by Sylvain Schweber) and Disturbing the Universe (by, yes, Freeman J. Dyson).
Thus, Bethe-Salpeter is both historically and scientifically pivotal, and is anything but irrelevant even today, when QM and QED have morphed into QFT (quantum field theory: another story, and a very long one, that has by no means reached its conclusion). The book is split into four parts, first hydrogen and helium without external fields, then atoms in external fields, and lastly interaction with radiation. And here, truly is what Feynman would call “the good stuff,” including Dirac theory as chapter 1b, relativistic QM as chapter 2b, the Zeeman and Stark effects comprising chapter 3, and chapter 4 presenting the “photoeffect” and Brehmstrahlung: the stuff that legends are made of, or, equivalently, the themes of classical quantum theory in its golden age.
Accordingly, Bethe-Salpeter is one of the major works in quantum physics, and provides a boots-on-the-ground approach to one of the most important subjects in science, and one with deep resonance for us mathematicians. Enough said.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.