Mathematical Foundations of Quantum Mechanics#No hidden variables proof

The book was originally published in German in 1932 by Springer.{{cite journal|doi=10.1090/S0002-9904-1933-05665-3|title=Book Review: Mathematische Grundlagen der Quantenmechanik|year=1933|last1=Margenau|first1=Henry|author-link=Henry Margenau|journal=Bulletin of the American Mathematical Society|volume=39|issue=7|pages=493–495|mr=1562667|doi-access=free}} It was translated into French by Alexandru Proca in 1946,{{Cite web |title=NEUMANN (von) : Les fondements mathématiques de la Mécanique quantique, 1946 |url=https://www.gabay-editeur.com/NEUMANN-von-Les-fondements-mathematiques-de-la-Mecanique-quantique-1946 |access-date=2024-12-04 |website=www.gabay-editeur.com |language=fr}} and into Spanish in 1949.{{Cite journal |last=Halmos |first=P. R. |date=1973 |title=The Legend of John Von Neumann |url=https://www.jstor.org/stable/2319080 |journal=The American Mathematical Monthly |volume=80 |issue=4 |pages=382–394 |doi=10.2307/2319080 |jstor=2319080 |issn=0002-9890|url-access=subscription }} An English translation by Robert T. Beyer was published in 1955 by Princeton University Press. A Russian translation, edited by Nikolay Bogolyubov, was published by Nauka in 1964. A new English edition, edited by Nicholas A. Wheeler, was published in 2018 by Princeton University Press.{{cite book |title=Mathematical Foundations of Quantum Mechanics. New Edition |author=John von Neumann |translator=Robert T. Beyer |editor=Nicholas A. Wheeler |publisher=Princeton University Press |date=2018 |isbn=9781400889921 |url=https://press.princeton.edu/titles/11352.html|author1-link=John von Neumann}}

According to the 2018 version, the main chapters are:

1. Introductory considerations
2. Abstract Hilbert space
3. The quantum statistics
4. Deductive development of the theory
5. General considerations
6. The measuring process

In chapter 6, von Neumann develops the theory of quantum measurement. Von Neumann addresses measurement by outlining two kind of processes:{{Efn|Some authors consider process III to be the collapse (state reduction) of the wave function.}}{{Cite book |last1=Rédei |first1=Miklós |url=https://books.google.com/books?id=jv-PBAAAQBAJ&q=measurement+problem+Von+Neumann%27s+psycho+physical+parallelism&pg=PT2 |title=John von Neumann and the Foundations of Quantum Physics |last2=Stöltzner |first2=Michael |date=2013-03-09 |publisher=Springer Science & Business Media |isbn=978-94-017-2012-0 |language=en}}

• Process I: during measurement a quantum state of a system evolves of into a mixed state of eigenstates of the measured observable. This process is non-causal (the outcome of a single measurement does not depend only on the initial state) and irreversible.
• Process II: when the system is unobserved, the state evolves according to Schrödinger equation. This process is causal and reversible.

Von Neumann was concerned that having two incompatible processes violated what he called the principle of psycho-physical parallelism, indicating the need that every mental process can be described as a physical process. Von Neumann argues that this issue does not appear in quantum mechanics as it set the border between observed and observer arbitrarily along a sequence of subsystems.

The sequence begins with a quantum system whose observable is to be measured. When the system interacts with a measuring device, they become entangled. As a result, the system does not end up in a definite eigenstate of the observable, and the measuring device does not display a specific value. When the observer is added to the picture, the description implies that their body (including the brain) are also entangled with the measuring apparatus and the system. This sequence is known as the von Neumann chain. The problem then becomes understanding how collapse to one of the eigenstates emerges from this chain.{{Cite book |last=Esfeld |first=Michael |url=https://philpapers.org/rec/ESFERW |title=Essay Review Wigner's View of Physical Reality}} Von Neumann demonstrated that, when it comes to the final outcomes, the chain can be interrupted at any and a wave function collapse can be introduced at any point to explain the results.{{Cite journal |last=Thaheld |first=Fred H. |date=2005-08-01 |title=Does consciousness really collapse the wave function?: A possible objective biophysical resolution of the measurement problem |url=https://linkinghub.elsevier.com/retrieve/pii/S0303264705000237 |journal=Biosystems |volume=81 |issue=2 |pages=113–124 |doi=10.1016/j.biosystems.2005.03.001 |pmid=16009281 |arxiv=quant-ph/0509042 |bibcode=2005BiSys..81..113T |issn=0303-2647|url-access=subscription }}

Von Neumann measurement scheme is part of the orthodox Copenhagen interpretation which postulates a collapse, however alternative interpretations of quantum mechanics have come out of this idea.{{Cite journal |last=Zeh |first=H. D. |date=2000 |title=[No title found] |url=http://link.springer.com/10.1023/A:1007895803485 |journal=Foundations of Physics Letters |volume=13 |issue=3 |pages=221–233 |doi=10.1023/A:1007895803485|url-access=subscription }} Eugene Wigner considered that the von Neumann chain implied that consciousness causes collapse of the wave function. However Wigner rejected this idea after the formalism of quantum decoherence was developed. Hugh Everett III developed the many-worlds interpretation based on von Neumann's processes, by keeping only process II.{{Cite web |title=Hugh Everett III |url=https://www.informationphilosopher.com/solutions/scientists/everett/ |access-date=2025-04-11 |website=www.informationphilosopher.com}}

{{Anchor|Von Neumann's no hidden variables proof|No hidden variables proof}}

One significant passage is its mathematical argument against the idea of hidden variables. Von Neumann's claim rested on the assumption that any linear combination of Hermitian operators represents an observable and the expectation value of such combined operator follows the combination of the expectation values of the operators themselves.

Von Neumann's makes the following assumptions:{{cite journal |last=Ballentine |first=L. E. |date=1970-10-01 |title=The Statistical Interpretation of Quantum Mechanics |url=https://link.aps.org/doi/10.1103/RevModPhys.42.358 |journal=Reviews of Modern Physics |language=en |volume=42 |issue=4 |pages=358–381 |doi=10.1103/RevModPhys.42.358 |bibcode=1970RvMP...42..358B |issn=0034-6861|url-access=subscription }}

1. For an observable $R$, a function $f$ of that observable is represented by $f(R)$.
2. For the sum of observables $R$ and $S$ is represented by the operation $R+S$, independently of the mutual commutation relations.
3. The correspondence between observables and Hermitian operators is one to one.
4. If the observable $R$ is a non-negative operator, then its expected value $\backslash langle\; R\backslash rangle\backslash geq0$.
5. Additivity postulate: For arbitrary observables $R$ and $S$, and real numbers $a$ and $b$, we have $\backslash langle\; aR+bS\backslash rangle=a\backslash langle\; R\backslash rangle+b\backslash langle\; S\backslash rangle$ for all possible ensembles.

Von Neumann then shows that one can write

: $\backslash langle\; R\backslash rangle=\backslash sum\_\{m,n\}\backslash rho\_\{nm\}\; R\_\{mn\}=\backslash mathrm\{Tr\}(\backslash rho\; R)$

for some $\backslash rho$, where $R\_\{mn\}$ and $\backslash rho\_\{nm\}$are the matrix elements in some basis. The proof concludes by noting that $\backslash rho$ must be Hermitian and non-negative definite ($\backslash langle\; \backslash rho\backslash rangle\backslash geq0$) by construction. For von Neumann, this meant that the statistical operator representation of states could be deduced from the postulates. Consequently, there are no "dispersion-free" states:{{Efn|A dispersion-free state $|\backslash psi\backslash rangle$ has the property $\backslash sigma\_R=\backslash langle\backslash psi|\; R^2|\backslash psi\backslash rangle-\backslash langle\backslash psi|\; R|\backslash psi\backslash rangle^2=0$ for all $R$ (eigenstate or not).}} it is impossible to prepare a system in such a way that all measurements have predictable results. But if hidden variables existed, then knowing the values of the hidden variables would make the results of all measurements predictable, and hence there can be no hidden variables. Von Neumann's argues that if dispersion-free states were found, assumptions 1 to 3 should be modified.

Von Neumann's concludes:{{cite journal |last=Albertson |first=James |date=1961-08-01 |title=Von Neumann's Hidden-Parameter Proof |url=https://pubs.aip.org/aapt/ajp/article-abstract/29/8/478/1037469/Von-Neumann-s-Hidden-Parameter-Proof?redirectedFrom=fulltext |journal=American Journal of Physics |volume=29 |issue=8 |pages=478–484 |doi=10.1119/1.1937816 |bibcode=1961AmJPh..29..478A |issn=0002-9505|url-access=subscription }}

{{Quote|text=if there existed other, as yet undiscovered, physical quantities, in addition to those represented by the operators in quantum mechanics, because the relations assumed by quantum mechanics would have to fail already for the by now known quantities, those that we discussed above. It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics, the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.|title=pp. 324-325}}

This proof was rejected as early as 1935 by Grete Hermann who found a flaw in the proof.{{cite journal |last1=Mermin |first1=N. David |last2=Schack |first2=Rüdiger |date=2018 |title=Homer Nodded: Von Neumann's Surprising Oversight |journal=Foundations of Physics |language=en |volume=48 |issue=9 |pages=1007–1020 |doi=10.1007/s10701-018-0197-5 |issn=0015-9018|doi-access=free |arxiv=1805.10311 |bibcode=2018FoPh...48.1007M }} The additive postulate above holds for quantum states, but it does not need to apply for measurements of dispersion-free states, specifically when considering non-commuting observables. Dispersion-free states only require to recover additivity when averaging over the hidden parameters. For example, for a spin-1/2 system, measurements of $(\backslash sigma\_x\; +\backslash sigma\_y)$ can take values $\backslash pm\; \backslash sqrt\{2\}$ for a dispersion-free state, but independent measurements of $\backslash sigma\_x$ and $\backslash sigma\_y$ can only take values of $\backslash pm\; 1$ (their sum can be $\backslash pm\; 2$ or {{tmath|1= 0 }}).{{cite journal |last=Bub |first=Jeffrey |date=2010 |title=Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal |url=http://link.springer.com/10.1007/s10701-010-9480-9 |journal=Foundations of Physics |language=en |volume=40 |issue=9–10 |pages=1333–1340 |doi=10.1007/s10701-010-9480-9 |issn=0015-9018|arxiv=1006.0499 |bibcode=2010FoPh...40.1333B }} Thus there still the possibility that a hidden variable theory could reproduce quantum mechanics statistically.

However, Hermann's critique remained relatively unknown until 1974 when it was rediscovered by Max Jammer. In 1952, David Bohm constructed the Bohmian interpretation of quantum mechanics in terms of statistical argument, suggesting a limit to the validity of von Neumann's proof. The problem was brought back to wider attention by John Stewart Bell in 1966.{{cite journal |last=Bell |first=John S. |date=1966-07-01 |title=On the Problem of Hidden Variables in Quantum Mechanics |url=https://link.aps.org/doi/10.1103/RevModPhys.38.447 |journal=Reviews of Modern Physics |language=en |volume=38 |issue=3 |pages=447–452 |doi=10.1103/RevModPhys.38.447 |bibcode=1966RvMP...38..447B |osti=1444158 |issn=0034-6861}} Bell showed that the consequences of that assumption are at odds with results of incompatible measurements, which are not explicitly taken into von Neumann's considerations.

It was considered the most complete book written in quantum mechanics at the time of release.{{Cite journal |last=Hove |first=Léon van |date=1958 |title=Von Neumann's contributions to quantum theory |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-64/issue-3.P2/Von-Neumanns-contributions-to-quantum-theory/bams/1183522374.full |journal=Bulletin of the American Mathematical Society |volume=64 |issue=3.P2 |pages=95–99 |doi=10.1090/S0002-9904-1958-10206-2 |issn=0002-9904|doi-access=free }} It was praised for its axiomatic approach. A review by Jacob Tamarkin compared von Neumann's book to what the works on Niels Henrik Abel or Augustin-Louis Cauchy did for mathematical analysis in the 19th century, but for quantum mechanics.{{Cite web |title=John von Neumann books |url=https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_books/#1 |access-date=2024-12-04 |website=Maths History |language=en}}{{Cite journal |last=Tamarkin |first=J. D. |date=1935 |title=Review of Mathematische Grundlagen der Quantenmechanik |url=http://www.jstor.org/stable/2302105 |journal=The American Mathematical Monthly |volume=42 |issue=4 |pages=237–239 |doi=10.2307/2302105 |jstor=2302105 |issn=0002-9890|url-access=subscription }}

Freeman Dyson said that he learned quantum mechanics from the book.{{Cite journal |last=Dyson |first=Freeman |date=2013-02-01 |title=A Walk through Johnny von Neumann's Garden |url=http://www.ams.org/jourcgi/jour-getitem?pii=noti942 |journal=Notices of the American Mathematical Society |language=en |volume=60 |issue=2 |pages=154 |doi=10.1090/noti942 |issn=0002-9920}} Dyson remarks that in the 1940s, von Neumann's work was not very well cited in the English world, as the book was not translated into English until 1955, but also because the worlds of mathematics and physics were significantly distant at the time.

• {{cite journal |last1=von Neumann |first1=J. |author1-link= |date=1927 |title=Mathematische Begründung der Quantenmechanik [Mathematical Foundation of Quantum Mechanics] |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse |pages=1–57}}
• {{cite journal |last1=von Neumann |first1=J. |author1-link= |date=1927 |title=Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik [Probabilistic Theory of Quantum Mechanics] |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse |pages=245–272}}
• {{cite journal |last1=von Neumann |first1=J. |author1-link= |date=1927 |title=Thermodynamik quantenmechanischer Gesamtheiten [Thermodynamics of Quantum Mechanical Quantities] |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse |volume=102 |pages=273–291}}
• {{cite journal |last1=von Neumann |first1=J. |author1-link= |date=1929 |title=Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren [General Eigenvalue Theory of Hermitian Functional Operators] |journal=Mathematische Annalen |pages=49–131|doi=10.1007/BF01782338}}
• {{cite journal |last1=von Neumann |first1=J. |author1-link= |date=1931 |title=Die Eindeutigkeit der Schrödingerschen Operatoren [The uniqueness of Schrödinger operators] |journal=Mathematische Annalen |volume=104 |pages=570–578 |doi=10.1007/bf01457956 |s2cid=120528257}}

• Dirac–von Neumann axioms
• The Principles of Quantum Mechanics by Paul Dirac

• Heisenberg cut

{{notelist}}

{{reflist}}

{{wikiquote}}

• [http://resolver.sub.uni-goettingen.de/purl?PPN379400774 Full online text] of the 1932 German edition (facsimile) at the University of Göttingen.

{{Authority control}}

Category:1932 non-fiction books

Category:Mathematics books

Category:Physics textbooks

Category:Works about quantum mechanics

Category:Hidden variables