The mathematical foundations of quantum mechanics;: A lecture-note volume (The Mathematical physics monograph series) by George Whitelaw Mackey (PDF)

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Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I guess it was my fault. This book is very wordy. Not enough math. I though that mathematical foundations referred to the math but it is more like a lot of history and then a little bit of math. Buy it if you like reading about math.

⭐I was very excited to order this book until I perused its pages. It does not deliver what it promises, in my opinion. Perhaps, people are impressed because of the word “Harvard”. This is NOT a thorough treatment of QM mathematics. It is at best a thumb nail sketch of the subject. Schaum’s outline of QM is much better and there isn’t much difference in the price.

⭐Note that the review of this treatise (Bulletin of The Mathematical Society) is one of glowing recommendation.Note that Loomis and Sternberg in their text Advanced Calculus (page 571), reference Mackey’s book “which treats classical mechanics in the spirit of this chapter.” (Their chapter 13, Classical Mechanics). Loomis and Sternberg provide collateral reading to these lecture notes. Allow me to return to the preface of Mackey’s book: “In accordance with our wish to demand no significant physical prerequisites, we have made an attempt to define all physical concepts used in terms of those of pure mathematics and the basic ones of space and time.” That explains my inability to comprehend what Mackey has accomplished here. Whereas his aim is to present a course “designed for students with a reasonably high degree of mathematical facility in dealing with mathematical concepts and little or no knowledge of physics,”I wonder if the converse could be attempted—that is, could one present an advanced course of physics presuming little or no knowledge of mathematics ? In all likelihood, I am at a loss to imagine that such a course could be beneficial ! What remains ? I let Mackey speak:(1) ” Indeed, it seems, in principle at least, to reduce the whole of chemistry to a set of problems in pure mathematics.” (page 132).(2) ” The statistical theory of heat and temperature here discussed extends rather completely to quantum mechanics.” (page 112).(3) “It is, of course, easy to understand Schrodinger’s discovery now,” and “By vague and mystical but inspired heuristic reasoning he (Heisenberg) was led to consider analogs of the differential equations of mechanics in which the varying elements were infinite matrices.” (page 98).(4) ” The reader may find it interesting to write Schrodinger’s equation as a system of two real equations, especially in the case of one particle. He will get equations that are very closely analogous to the equations of hydrodynamics”(page 96).(5) “The constant h-bar may be looked upon as a conversion factor from the natural mass to the one adopted before the advent of quantum mechanics.” (page 91).(6) “In spite of the acausal character of quantum mechanics, as reflected in the impossibility of making more than statistical statements about the results to be expected from the measurement of observables, the relationship between states at different times is strictly causal.” (page 81).(7) “It is somewhat harder to see the significance of choosing our Hilbert Space to be complex rather than real. However, we shall see that the existence of multiplication by i makes it possible to carry over to quantum mechanics one of the most striking formal features of classical mechanics–namely, the natural correspondence between observables and one-parameter groups of symmetries.” (page 73).(8) “We should like to emphasize that the quantum rules are a deduction and not a fundamental feature of the formulation of the theory.” and ” It (quantum mechanics) is basically a revision of statistical mechanics” (page 61).(9) ” The essential feature of the Hamiltonian (or, Lagrangian) formalism of course is the fact that it provides an algorithm for passing from a single function on phase space to the infinitesimal generator of the dynamical group.” (Page 39).(10) “In quantum mechanics the states can never be described by a finite number of coordinates–even when the corresponding classical states could be. Thus, in quantum mechanics we always have a partial differential equation (or, a system of such).” (page 3).There you have a few words from our esteemed author. What I have not presented is the full panoply of mathematical concepts in which he frames those physical concepts. For that, one must turn to the book. Of particular mention is the synopsis of the relationship between information theory and statistical mechanics (Sections 1-5, pages 47-55).A tour de force, if providing for enough background ! And, thus, much food for thought offered by Mackey.(As a precursor: Read Mackey’s Quantum Mechanics In Hilbert Space, American Mathematical Monthly, Vol. 64, Issue 8, pages 45-57). In any event, I am on my third go-around of the text. Perhaps one day it will all sink in.Thus, it is a text which demands very careful study. Mackey has offered an interesting viewpoint–which is why I strive to understand it ! In Conclusion, I do recommend its perusal, but, I am uncertain as to who to recommend it to !

⭐This book tries to cover, (but not with success) the needed mathematics for learning or teaching Quantum Mechanics. The contains of a course in Mathematical Foundations of Quantum Mechanics must include mainly two branches of Mathematics: Group theory and Operator theory. The corresponded chapters in the book are short. This book, written in a high level, is difficult to be understood by undergraduate students. It is proper for postgraduates

⭐I wish this out-of-print book would be in the collection of Dover. Now it’s hard to find it, but still many physics & math people are looking for it and discussing about it.Geroge Mackey, Hermann Weyl, and John von Neumann are the mathematical foundations of Q.M.

⭐very good and clear book in mathematical foundations of QM, although a little bit dated.

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