Feb 24 2015
Most books about fundamental quantum mechanics use results in the theory of Hilbert space operators without proving them, while most books about Hilbert space operators do not treat quantum mechanics. In addition, they often use fairly advanced results from other branches of mathematics assuming the reader to be already familiar with them, but this is seldom true.
The cover of the book pays homage to David Hilbert and John von Neumann, the founding fathers of the mathematical foundations of quantum mechanics. (Credit: World Scientific, 2015)
In Franco Gallone's latest book published with World Scientific, "Hilbert Space and Quantum Mechanics" aims to give a really self-contained treatment of all the elements of the theory of Hilbert space operators that are necessary for a sound and mathematically accurate exposition of the principles of quantum mechanics.
This exposition is the object of the final part of the book, where the principles of quantum mechanics are discussed with complete mathematical accuracy and with an unstinting effort to trace them back to the experimental reality that lies at their root. No previous knowledge of quantum mechanics is required. The statistical aspects of quantum mechanics are compared and contrasted with the same aspects of classical physics.
An important characteristic of the book is that the mathematical theory is developed only assuming familiarity with elementary analysis. To make cross-reference as easy as possible, almost every bit of the book is marked with a three numbers code. In all the mathematical proofs, logical deduction is thoroughly assisted with this system of cross-reference.
All the proofs are written in minute detail; in them, however, previous results are always quoted by just their three numbers codes, without spelling them out; this enables experts to pursue the logic of a proof without too many diversions, and beginners to receive all the support they might need.
The book is of interest to researchers and graduate students in functional analysis, who can see how closely an important part of their chosen field is linked with quantum mechanics, and also to physicists, who can see how the abstract language of functional analysis brings unity to the apparent distinct approaches employed in quantum theory.