In 2000 the first two thirds of the author set for the book under review, which is to say, Johnson and Lapidus, published *The Feynman Integral and Feynman’s Operational Calculu*s with Oxford “at the Clarendon Press” (as the familiar qualifier goes — it always makes me think of Hardy and Wright’s *The Theory of Numbers*, one of the timeless classics of mathematics). The book under review, putatively even going beyond Feynman’s operational calculus (let alone his path integral), is thus something of a sequel to the former now fifteen-year-old book.

This said, however, Johnson, Lapidus, and Nielsen make it a point in their Introduction to the present book to stress that the earlier (800 page) work is actually not a prerequisite — or not really. The earlier book does provide “a number of different approaches to the Feynman path integral (or ‘sums over histories’), in both ‘real’ and ‘imaginary’ time,” and they note that Chapters 14–18 of this book — let’s just call it JL 2000 — deal with Feynman’s operational calculus while its Chapter 19 “begins to build a bridge [to] … a possible more general operational calculus valid for abstract operators (acting on Banach or Hilbert spaces) not necessarily arising *via* Wiener or Feynman functionals and associated path integrals.” So it’s obviously useful to be familiar with JL 2000 prior to hitting what we’ll just call JLN 2015. This latter book is intended to be self-contained, but this is all still sticky very stuff, especially for us mathematicians, as opposed to physicists, given the hugely increased demands we would place on real rigor. Thus, the more one knows beforehand about this important area where modern physics meets mathematics the better: JLN 2015 deals with a lot of difficult, intricate, and subtle operator theory and analysis, and the reader should be properly prepared.

The good news about all this, and also about JL 2000, is the fact that both of these books are written by mathematicians for mathematicians, and can therefore be characterized as rigorous functional analysis, albeit of a rather *avant* *garde* variety. We encounter a good deal of sophisticated measure theory (as must be the case given the notoriously problematical nature of the set of paths over which Feynman integrals are defined), and Banach and Hilbert spaces, as already noted, and of course operators acting on them.

Compared to the functional analysis provided by von Neumann in his definitive “classical” characterization of quantum mechanics in Hilbert spaces (i.e., the theory of densely defined unbounded operators), in Feynman’s theory of operators there are serious twists in the game: now the focus falls on the subtle business of “disentangling” operators. This problem was already identified by Feynman himself in 1951 when he launched this business, and it is part and parcel of a paper 2001 by Jeffries and Johnson (the same one) written in order to develop “Feynman’s operational calculus in a mathematically rigorous way” (cf. p. 31 of JLN 2015).

Naturally, Feynman’s operators are rather different from those on finds in the von Neumann treatment of QM. Specifically, whereas in the latter theory, by now almost three quarters of a century old, one famously attaches operators to observables (or measurables) and gets probabilistic data about the according states living in the Hilbert space for the QM system by direct spectral analysis, Feynman’s operator theory is focused on time evolution of a system *per se *and one follows the following heuristics: “(1) Attach time indices to the operators … to specify the order of operators in products [given that *ab initio* we are lacking commutativity] … (2) With time indices attached, form functions of these operators as though they were commuting [!] … (3) ‘Disentangle’ the resulting expressions, i.e. restore the conventional ordering of the operators.” So there: with Feynman’s famous intuition and pragmatism on display, it’s now all about doing justice to the mathematics.

Thus, of the eleven chapters in JLN 2015 six deal specifically with explicit features of disentangling, and the last two chapters of the book comprise something of a return to perhaps more prosaic quantum mechanical (or perhaps quantum electrodynamical or quantum field theoretic) ground in that, for instance, spectral theory for non-commuting operators is handled: shades of the original Born rules.

Both books under discussion are distinguished by an obvious love for their subject on the part of the authors, making for expansive explanations, excursions into physics and its history (of a modern sort: Feynman, who died in 1988, was nothing if not a modern visionary). But the mathematics presented is very serious and very rigorous, particularly so in light of the all-too-frequent negative reaction on the part of mathematicians in the face of what physicists tend to do. Accordingly, unlike physics books, JLN 2015 is full of theorems and proofs, as well as, to paraphrase Feynman, “a lot of the good stuff” about some modern quantum physics that makes for some very good mathematics. It’s serious mathematics about serious physics.

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