# Hilbert Space and Quantum Mechanics

Quantum mechanics has several fathers: Planck, Bohr, Heisenberg, Einstein, Schrödinger, Dirac, Hilbert, von Neumann, and many others. As a consequence, there are different approaches to the theory. Dirac's approach is rather popular among physicists because it requires relatively little mathematics. In this book, the author follows the ideas of von Neumann and Hilbert who thought that quantum mechanics needed an abstract framework which we now call a Hilbert space. It turned out that spectral theory of self adjoint operators in Hilbert space were substantial in the development of quantum mechanics.

So this book brings a self-contained axiomatic mathematical foundation in which quantum mechanics can be studied. All the mathematics needed to develop a spectral theory of self adjoint operators in Hilbert space are built up from scratch. Eventually, in the last two chapters, observables can be introduced as self-adjoint operators and physical compatibility nicely matches commutativity of the operators etc. However, these two chapters, are only an introduction of the basic elements and structures with which quantum mechanics can take off. It certainly is not a summary of the whole theory. Thus most of the book is functional analysis and everything needed for the last chapters. And that is a whole lot of mathematics indeed.

The approach is axiomatic and very thorough. Starting with the basics of sets and relations and the notion of a group, the proper work starts with metric spaces, linear operators, measure theory and integration, Hilbert spaces (in particular L²), self-adjoint operators and orthogonal projections. Then the topics become somewhat less standard discussing integration with respect to these projection valued measures (giving resolution of the identity), spectral theorems, commuting operators, trace class operators, and finally statistical operators. All this to prepare for the last two chapters in which quantum mechanics in Hilbert spaces are introduced and the concepts of position and momentum in non-relativistic quantum mechanics.

So, in the end, there is only 126 pages of quantum mechanics but 610 pages of mathematical introduction, not that there is much difference. The quantum mechanics are treated in the same strict axiomatic approach and all the previous notation and structures are used. To name just one example, the bra-ket notation dominating the standard books on quantum mechanics is not used. The quantum mechanics is still using the same structures and operators but it has its own terminology. Hence, there are fewer theorems and many more definitions in the penultimate chapter to introduce the terminology of the new subject.

The stunning achievement of this book is however that, even though the book starts with rules of logic to construct a proof and an explanation what the usual mathematical symbols mean, all the theorems from the first till the last chapter are fully proved. This is fairly unusual for mathematics books where proofs often rely on results from different fields. Thus it is no wonder that the bibliography of this book is relatively short (barely 2 pages), given all the topics covered. Since one does not read such a book cover to cover in one day, it needs often backtracking to look up a definition, a theorem, or a proof. Then of course a detailed subject index is essential. The index here is not as extensive at it could have been, but fortunately it is not needed very often because whenever in the text a theorem, definition, or relation is used that was introduced a few pages earlier or in a previous chapter, then a precise pointer is given. The discussion however is abstract. That is *very* abstract. Trained mathematicians are perhaps used to this, but if this book is intended to enlighten and convince the 'otherwise-not-so-mathematically-oriented' physicists about the advantages of this approach, then that might be a bit of a problem. There are some examples to illustrate the abstract concepts in the text, but they are very rare. In my opinion, there could, and perhaps should have been more. That would have made the book even thicker (and heavier) than it is now, which might require two volumes. Another fact, illustrating the level of abstraction is that there are no figures either, except for one page with figures in the quantum mechanics chapter.

But all in all, it is an excellent tool for self study. A mathematics student can just enjoy the mathematics, even if not interested in quantum mechanics. The last two chapters can easily be dropped. A trained mathematician interested in quantum mechanics will be familiar with most of the mathematics in the book, but will find the trailing chapters most informative. He or she will need however another *matching* book to embark on the full quantum mechanics. Not all books on quantum mechanics will hook up with the approach used here. The book too thick to be used as a handbook for one course, but some chapters could be selected for such a course. In fact some suggestions are made in the introduction for a course in real analysis, or in operator theory. There are no exercises though.

**Submitted by Adhemar Bultheel |

**18 / May / 2015