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Mathematical Foundations of Quantum Mechanics
Mathematical Foundations of Quantum Mechanics was a revolutionary book that caused a sea change in theoretical physics. Here, John von Neumann, one of the leading mathematicians of the twentieth century, shows that great insights in quantum physics can be obtained by exploring the mathematical structure of quantum mechanics. He begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which von Neumann regards as the definitive form of quantum mechanics. Using this theory, he attacks with mathematical rigor some of the general problems of quantum theory, such as quantum statistical mechanics as well as measurement processes. Regarded as a tour de force at the time of publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.
464 pages, Hardcover
First published January 1, 1955
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About the author
John von Neumann
77 books358 followersJohn von Neumann (Hungarian: margittai Neumann János Lajos) was a Hungarian American[1] mathematician who made major contributions to a vast range of fields,[2] including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is generally regarded as one of the foremost mathematicians of the 20th century. The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians." Even in Budapest, in the time that produced Szilárd (1898), Wigner (1902), and Teller (1908) his brilliance stood out. Most notably, von Neumann was a pioneer of the application of operator theory to quantum mechanics, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory and the concepts of cellular automata and the universal constructor. Along with Edward Teller and Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.
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May 13, 2022
We who enjoy the advantage of having grown up with quantum mechanics have to bear in mind how strange the revolutionary theory appeared to its discoverers, who had been professionally trained to think along the well worn lines of classical mechanics and electrodynamics. If one consults, for instance, Max Planck’s addresses to scientific audiences during the period of the old quantum theory (up to 1925), one can gain an impression of how persistent classical habits of thought in fact were (see our review here) – one held fast to them dearly, and let them be prized away only in the face of incontrovertible experimental evidence. The advent of matrix mechanics and wave mechanics in 1925 threw everyone into tumult, in that it was clear that the outlines of a coherent framework had arrived but not immediately so clear what all to make of it. The consolidation of the theoretical physics community around the Copenhagen interpretation was due, in no little part, to the efforts of the young and genial John von Neumann to shore up the conceptual foundations of the nascent theory, in his Mathematische Grundlagen der Quantenmechanik, originally published in 1932. Von Neumann got his start as an assistant to David Hilbert in Göttingen, who organized a seminar on mathematical methods in quantum mechanics during the academic year of 1926/27.
In the preface, von Neumann explains his attitude towards Dirac:
Die erwähnte, infolge ihrer Durchsichtigkeit und Eleganz heute in einen großen Teil der quantenmechanischen Literatur übergegangene Methodik von Dirac wird den Anforderungen der mathematischen Strenge in keiner Weise gerecht – auch dann nicht, wenn diese natürlicher- und billigerweise auf das sonst in der theoretischen Physik übliche Maß reduziert werden. So wird z.B. konsequent an der Fiktion festgehalten, daß jeder selbstadjungierte Operator auf die Diagonalform gebracht werden kann, was bei denjenigen Operatoren, für die dies tatsächlich nicht der Fall ist, das Einführen ‘uneigenlicher’ Funktionen mit selbstwidersprechenden Eigenschaften notwendig macht. Ein solches Einschalten mathematischer ‘Fiktionen’ ist u.U. selbst dann unvermeidlich, wenn es sich nur darum handelt, das Resultat eines anschaulich definierten Versuches numerisch zu berechnen. Dies wäre kein Einwand, wenn diese in den heutigen Rahmen der Analysis nicht passenden Begriffsbildungen für die neue physikalische Theorie wirklich wesentlich wären. So wir die Newtonsche Mechanik zunächst das Entstehen des in seiner damalign Form zweifellos selbstwidersprechenden Infinitesimalkalküls mit veranlaßte, würde die Quantenmechanik einen Neuaufbau unserer ‘Analysis der unendlich vielen Variablen’ nahelegen – d.h. der mathematische Apparat müßte geändert werden, und nicht die physikalische Theorie. Das ist aber keineswegs der Fall, es soll vielmehr gezeigt werden, daß die Transformationstheorie auf eine ebenso klare und einheitliche Weise auch mathematisch einwandfrei begründet werden kann. Dabei ist zu betonen, daß der korrekte Aufbau nicht etwa aus einer mathematischen Präzisierung und Explizierung der Diracschen Methode besteht, sondern daß er ein von vornherein abweichendes Vorgehen nötig macht, nämlich das Anlehnen an die Hilbertsche Spektraltheorie der Operatoren. [p. 2]
Today, after three generations of physicists have gone by, little has changed. Most physicists still prefer Dirac’s approach and so do not worry about the technicalities involved – with the difference that we now know that they can be satisfactorily resolved. It comes perhaps as a surprise, though, that despite his brilliance von Neumann failed to discover for himself Laurent Schwartz’s ingenious theory of distributions! The likely explanation for this omission is that the (for most everyone) frighteningly complicated theory of unbounded operators was so easy for him that he felt no need to try to refine Dirac’s sloppy methods, just as later Julian Schwinger would have no truck with Feynman diagrams.
In surveying the then-prevailing approaches to quantum mechanics, von Neumann contrasts Heisenberg-Born-Jordan’s matrix mechanics with Schrödinger’s wave mechanics and declares his preference for Dirac-Jordan’s transformation theory [pp. 4-5]. To any student of physics today, much of the ensuing exposition must appear very familiar – one must bear in mind that the ideas are being discussed here for the first time in print. Dirac, for instance, is usually credited with having shown the identity of Heisenberg’s and Schrödinger’s superficially very different-seeming formalisms, yet his argumentation can scarcely be held rigorous. Here, von Neumann applies the Riesz-Fischer theorem to establish an isomorphism between square summable sequences and square integrable functions in abstract Hilbert space: equivalence of matrix and wave mechanics follows as a logical consequence – what is not necessarily new, it goes back to work of Fréchet twenty years earlier. Another point that flows from von Neumann’s more stringent demands concerning rigor is that the eigenvalue problem becomes non-trivial as unlike what happens with Sturm-Liouville theory the eigenfunctions do not form a complete orthonormal set, failing to include free particles or scattered waves. Max Jammer’s critical remarks pertaining to this problem and von Neumann’s resolution of it [pp. 321-322 in The conceptual development of quantum mechanics]: i) it is unclear that separable Hilbert spaces are sufficient for treatment of the continuous spectrum but there is no canonical set of basis functions in larger function spaces; ii) there is no rigorous treatment of the S-matrix operator within von Neumann’s formalism; iii) furthermore, it is inapplicable to quantum field theory. Perhaps von Neumann himself was conscious of criticisms such as these and motivated thereby to engage in his path-breaking work on normed rings of operators, which in the decades after his death would prove so useful to the mathematical formulation of quantum physics.
As an aside, it will be of interest to quote von Neumann on causality in quantum mechanics, compare with Cassirer’s opposing judgment in his Determinismus und Indeterminismus in der modernen Physik (q.v., our review here):
Aber bei Beachtung aller dieser Kautelen dürfen wir doch sagen: es gibt gegenwärtig keinen Anlaß und keine Entschuldigung dafür, von der Kausalität in der Natur zu reden – denn keine Erfahrung stützt ihr vorhandensein, da die makroscopischen dazu prinzipiell ungeeignet sind, und die einzige bekannte Theorie, die mit unseren Erfahrungen über die Elementarprozesse verträglich ist, die Quantenmechanik, widerspricht ihr. [pp. 172-173]
Probably the typical reader who is indifferent to the technicalities will appreciate most in what von Neumann does with the foundations of quantum mechanics his treatments of the density matrix and of measurement theory, to which the remainder of this review will be devoted. Indeed, among early practitioners von Neumann was the first to have a clear view of the need for the density matrix to describe impure states and in the present work gave its definition and basic properties, along with a justification of his expression for the entropy of an ensemble in terms of the corresponding density matrix. On the paradox that in his formulation entropy is constant and increases only upon a measurement, versus what happens in classical mechanics, where for instance if a partition be removed, the entropy must increase to match the increase in volume, von Neumann’s resolution: für einen klassischen Beobachter, der alle Koordinaten und Impulse kennt, ist also die Entropie konstant, and zwar 0….die zeitlichen Variationen der Entropie rühren also daher, daß der Beobachter micht alles weiß, bzw. daß er nicht alles ermitteln (messen) kann, was prinzipiell meßbar ist. [p. 213]
Two features of von Neumann’s measurement theory are worth highlighting, as they are seldom commented upon: first, pp. 216-217 on why all macroscopic observables practically speaking can be treated as if they commute with one another. Second, on the necessity of a ‘cut’ between system and observer:
….Aber einerlei, wie weit wir rechnen: bis ans Quecksilbergefäß, bis an die Skala des Thermometers, bis an die Retina, oder bis ins Gehirn, einmal müssen wir sagen: und dies wird vom Beobachter wahrgenommen. D.h. wir müßen die Welt immer in zwei Teile teilen, der eine ist das beobachtete System, der andere der Beobachter….Die Grenze zwischen beiden ist weitgehend willkürlich, [p. 223]
The purpose of the following section on zusammengesetzte Systeme [pp. 225-232] is to justify the above statement: that it doesn’t matter whether one puts the cut between I + II, III or I, II + III. Introduces the partial trace and seeks consistency conditions.
Overall comment: a curious work which, true to its title, doesn’t solve any example problems, compute any spectra or cross-sections or discuss the physics in the usual sense at all. Von Neumann focuses all but exclusively on two things: i) the mathematical definition of Hilbert space and unbounded operators, ii) the statistical interpretation and measurement theory. Re. i), the text more or less presumes Hilbert’s spectral theory already known, there is no systematic presentation of it and one couldn’t pick up very much about it here. Re. ii), pretty good, should be read in connection with Max Born’s original papers from 1926 (which are far more motivated by consideration of actual problems of physics). Justifies what a density operator is, collapse postulate, excursion into thermodynamics, mixed systems, explanation of the meaning of the uncertainty principle, the cut, supposed disproof of hidden variables (criticized by J.S. Bell as involving a non-physical hypothesis to the effect that the expectation value may be taken of a linear combination of observables, whether or not these commute).
As to style, it alternates unevenly between great generality and concrete calculations to support a point (which though often difficult were of course straightforward for von Neumann!). Not the customary mathematician’s definition-theorem-proof format but more akin to the logician’s calling out propositions by alphanumerical symbol and minute analysis thereof in relation to others. Let the venturesome reader beware: one couldn’t possibly learn quantum mechanics from this work but it should prove very instructive to one who wants to think more carefully about what the theory means as opposed to embracing the ‘shut up and calculate’ attitude!
In the preface, von Neumann explains his attitude towards Dirac:
Die erwähnte, infolge ihrer Durchsichtigkeit und Eleganz heute in einen großen Teil der quantenmechanischen Literatur übergegangene Methodik von Dirac wird den Anforderungen der mathematischen Strenge in keiner Weise gerecht – auch dann nicht, wenn diese natürlicher- und billigerweise auf das sonst in der theoretischen Physik übliche Maß reduziert werden. So wird z.B. konsequent an der Fiktion festgehalten, daß jeder selbstadjungierte Operator auf die Diagonalform gebracht werden kann, was bei denjenigen Operatoren, für die dies tatsächlich nicht der Fall ist, das Einführen ‘uneigenlicher’ Funktionen mit selbstwidersprechenden Eigenschaften notwendig macht. Ein solches Einschalten mathematischer ‘Fiktionen’ ist u.U. selbst dann unvermeidlich, wenn es sich nur darum handelt, das Resultat eines anschaulich definierten Versuches numerisch zu berechnen. Dies wäre kein Einwand, wenn diese in den heutigen Rahmen der Analysis nicht passenden Begriffsbildungen für die neue physikalische Theorie wirklich wesentlich wären. So wir die Newtonsche Mechanik zunächst das Entstehen des in seiner damalign Form zweifellos selbstwidersprechenden Infinitesimalkalküls mit veranlaßte, würde die Quantenmechanik einen Neuaufbau unserer ‘Analysis der unendlich vielen Variablen’ nahelegen – d.h. der mathematische Apparat müßte geändert werden, und nicht die physikalische Theorie. Das ist aber keineswegs der Fall, es soll vielmehr gezeigt werden, daß die Transformationstheorie auf eine ebenso klare und einheitliche Weise auch mathematisch einwandfrei begründet werden kann. Dabei ist zu betonen, daß der korrekte Aufbau nicht etwa aus einer mathematischen Präzisierung und Explizierung der Diracschen Methode besteht, sondern daß er ein von vornherein abweichendes Vorgehen nötig macht, nämlich das Anlehnen an die Hilbertsche Spektraltheorie der Operatoren. [p. 2]
Today, after three generations of physicists have gone by, little has changed. Most physicists still prefer Dirac’s approach and so do not worry about the technicalities involved – with the difference that we now know that they can be satisfactorily resolved. It comes perhaps as a surprise, though, that despite his brilliance von Neumann failed to discover for himself Laurent Schwartz’s ingenious theory of distributions! The likely explanation for this omission is that the (for most everyone) frighteningly complicated theory of unbounded operators was so easy for him that he felt no need to try to refine Dirac’s sloppy methods, just as later Julian Schwinger would have no truck with Feynman diagrams.
In surveying the then-prevailing approaches to quantum mechanics, von Neumann contrasts Heisenberg-Born-Jordan’s matrix mechanics with Schrödinger’s wave mechanics and declares his preference for Dirac-Jordan’s transformation theory [pp. 4-5]. To any student of physics today, much of the ensuing exposition must appear very familiar – one must bear in mind that the ideas are being discussed here for the first time in print. Dirac, for instance, is usually credited with having shown the identity of Heisenberg’s and Schrödinger’s superficially very different-seeming formalisms, yet his argumentation can scarcely be held rigorous. Here, von Neumann applies the Riesz-Fischer theorem to establish an isomorphism between square summable sequences and square integrable functions in abstract Hilbert space: equivalence of matrix and wave mechanics follows as a logical consequence – what is not necessarily new, it goes back to work of Fréchet twenty years earlier. Another point that flows from von Neumann’s more stringent demands concerning rigor is that the eigenvalue problem becomes non-trivial as unlike what happens with Sturm-Liouville theory the eigenfunctions do not form a complete orthonormal set, failing to include free particles or scattered waves. Max Jammer’s critical remarks pertaining to this problem and von Neumann’s resolution of it [pp. 321-322 in The conceptual development of quantum mechanics]: i) it is unclear that separable Hilbert spaces are sufficient for treatment of the continuous spectrum but there is no canonical set of basis functions in larger function spaces; ii) there is no rigorous treatment of the S-matrix operator within von Neumann’s formalism; iii) furthermore, it is inapplicable to quantum field theory. Perhaps von Neumann himself was conscious of criticisms such as these and motivated thereby to engage in his path-breaking work on normed rings of operators, which in the decades after his death would prove so useful to the mathematical formulation of quantum physics.
As an aside, it will be of interest to quote von Neumann on causality in quantum mechanics, compare with Cassirer’s opposing judgment in his Determinismus und Indeterminismus in der modernen Physik (q.v., our review here):
Aber bei Beachtung aller dieser Kautelen dürfen wir doch sagen: es gibt gegenwärtig keinen Anlaß und keine Entschuldigung dafür, von der Kausalität in der Natur zu reden – denn keine Erfahrung stützt ihr vorhandensein, da die makroscopischen dazu prinzipiell ungeeignet sind, und die einzige bekannte Theorie, die mit unseren Erfahrungen über die Elementarprozesse verträglich ist, die Quantenmechanik, widerspricht ihr. [pp. 172-173]
Probably the typical reader who is indifferent to the technicalities will appreciate most in what von Neumann does with the foundations of quantum mechanics his treatments of the density matrix and of measurement theory, to which the remainder of this review will be devoted. Indeed, among early practitioners von Neumann was the first to have a clear view of the need for the density matrix to describe impure states and in the present work gave its definition and basic properties, along with a justification of his expression for the entropy of an ensemble in terms of the corresponding density matrix. On the paradox that in his formulation entropy is constant and increases only upon a measurement, versus what happens in classical mechanics, where for instance if a partition be removed, the entropy must increase to match the increase in volume, von Neumann’s resolution: für einen klassischen Beobachter, der alle Koordinaten und Impulse kennt, ist also die Entropie konstant, and zwar 0….die zeitlichen Variationen der Entropie rühren also daher, daß der Beobachter micht alles weiß, bzw. daß er nicht alles ermitteln (messen) kann, was prinzipiell meßbar ist. [p. 213]
Two features of von Neumann’s measurement theory are worth highlighting, as they are seldom commented upon: first, pp. 216-217 on why all macroscopic observables practically speaking can be treated as if they commute with one another. Second, on the necessity of a ‘cut’ between system and observer:
….Aber einerlei, wie weit wir rechnen: bis ans Quecksilbergefäß, bis an die Skala des Thermometers, bis an die Retina, oder bis ins Gehirn, einmal müssen wir sagen: und dies wird vom Beobachter wahrgenommen. D.h. wir müßen die Welt immer in zwei Teile teilen, der eine ist das beobachtete System, der andere der Beobachter….Die Grenze zwischen beiden ist weitgehend willkürlich, [p. 223]
The purpose of the following section on zusammengesetzte Systeme [pp. 225-232] is to justify the above statement: that it doesn’t matter whether one puts the cut between I + II, III or I, II + III. Introduces the partial trace and seeks consistency conditions.
Overall comment: a curious work which, true to its title, doesn’t solve any example problems, compute any spectra or cross-sections or discuss the physics in the usual sense at all. Von Neumann focuses all but exclusively on two things: i) the mathematical definition of Hilbert space and unbounded operators, ii) the statistical interpretation and measurement theory. Re. i), the text more or less presumes Hilbert’s spectral theory already known, there is no systematic presentation of it and one couldn’t pick up very much about it here. Re. ii), pretty good, should be read in connection with Max Born’s original papers from 1926 (which are far more motivated by consideration of actual problems of physics). Justifies what a density operator is, collapse postulate, excursion into thermodynamics, mixed systems, explanation of the meaning of the uncertainty principle, the cut, supposed disproof of hidden variables (criticized by J.S. Bell as involving a non-physical hypothesis to the effect that the expectation value may be taken of a linear combination of observables, whether or not these commute).
As to style, it alternates unevenly between great generality and concrete calculations to support a point (which though often difficult were of course straightforward for von Neumann!). Not the customary mathematician’s definition-theorem-proof format but more akin to the logician’s calling out propositions by alphanumerical symbol and minute analysis thereof in relation to others. Let the venturesome reader beware: one couldn’t possibly learn quantum mechanics from this work but it should prove very instructive to one who wants to think more carefully about what the theory means as opposed to embracing the ‘shut up and calculate’ attitude!
March 16, 2025
There are books that explain the universe, and then there are books that rewrite the structure of reality itself. John von Neumann’s Mathematical Foundations of Quantum Mechanics is not a mere textbook—it is a blueprint for the hidden architecture of existence.
Quantum mechanics is already a field that breaks the human mind—it replaces certainty with probability, causality with indeterminacy, and classical logic with something so alien that even Einstein recoiled in horror. But where physicists saw paradox, von Neumann saw structure. This book is the moment where the incomprehensible becomes mathematically inevitable.
Von Neumann does not describe quantum mechanics; he constructs it from first principles. He formalizes wave functions as vectors in Hilbert space, measurement as projection operators, and quantum evolution as unitary transformations—all with a level of precision so absolute that it renders intuition obsolete. His work does not rely on metaphor or approximation; it is a pure mathematical scaffolding upon which all quantum reality is built.
And then, in one move, he detonates the final paradox: the measurement problem. The moment an observer measures a quantum system, its state collapses. Why? Is consciousness involved? Is reality itself undefined until interaction occurs? Von Neumann lays out the equations, and in doing so, he forces us to confront the terrifying possibility that reality is fundamentally observer-dependent.
This book is not written for understanding—it is written for survival in a post-classical reality. To engage with it is to walk the event horizon of human cognition, where every equation strips away illusion and replaces it with the raw machinery of existence.
By the time you finish, you will not simply have read a book. You will have glimpsed the source code of the universe itself.
Quantum mechanics is already a field that breaks the human mind—it replaces certainty with probability, causality with indeterminacy, and classical logic with something so alien that even Einstein recoiled in horror. But where physicists saw paradox, von Neumann saw structure. This book is the moment where the incomprehensible becomes mathematically inevitable.
Von Neumann does not describe quantum mechanics; he constructs it from first principles. He formalizes wave functions as vectors in Hilbert space, measurement as projection operators, and quantum evolution as unitary transformations—all with a level of precision so absolute that it renders intuition obsolete. His work does not rely on metaphor or approximation; it is a pure mathematical scaffolding upon which all quantum reality is built.
And then, in one move, he detonates the final paradox: the measurement problem. The moment an observer measures a quantum system, its state collapses. Why? Is consciousness involved? Is reality itself undefined until interaction occurs? Von Neumann lays out the equations, and in doing so, he forces us to confront the terrifying possibility that reality is fundamentally observer-dependent.
This book is not written for understanding—it is written for survival in a post-classical reality. To engage with it is to walk the event horizon of human cognition, where every equation strips away illusion and replaces it with the raw machinery of existence.
By the time you finish, you will not simply have read a book. You will have glimpsed the source code of the universe itself.
Want to read
October 11, 2014"The first person to suggest that quantum theory implies that reality iscreated by human consciousness was not some crank on the fringes of physics but the eminent mathematician John von Neumann."
"His logic leads to a particular conclusion: that the world is not objectively real but depends on the mind of the observer."
-- Physicist Nick Herbert
"His logic leads to a particular conclusion: that the world is not objectively real but depends on the mind of the observer."
-- Physicist Nick Herbert
Read
October 18, 2023Only understood the basic gist of it.
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February 18, 2024Bleble
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