Gruppentheorie und Quantenmechanik

{{Short description|1928 textbook by Hermann Weyl}}

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| name = Gruppentheorie und Quantenmechanik

| image = Gruppentheorie und Quantenmechanik 1928.jpg

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| alt = Title page of Gruppentheorie und Quantenmechanik 1928

| caption = Title page of Gruppentheorie und Quantenmechanik, 1928

| author = Hermann Weyl

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| translator = H.P. Robertson (1950; English)

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| language = {{hlist|German|English}}

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| subjects = {{hlist|Group theory|quantum mechanics}}

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| publisher = {{ill|S. Hirzel Verlag|de|lt=S. Hirzel}}, Leipzig (1st edition)

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| pub_date = 1928 (1st edition in German)

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| pages = 286

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| notes = [https://archive.org/details/theoryofgroupsqu1950weyl Internet Archive]

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Gruppentheorie und Quantenmechanik, or The Theory of Groups and Quantum Mechanics, is a textbook written by Hermann Weyl about the mathematical study of symmetry, group theory, and how to apply it to quantum physics. Weyl expanded on ideas he published in a 1927 paper,{{cite journal|first=H. |last=Weyl |title=Quantenmechanik und Gruppentheorie |journal=Zeitschrift für Physik |year=1927 |volume=46 |issue=1–2 |pages=1–46 |doi=10.1007/bf02055756|bibcode=1927ZPhy...46....1W }} basing the text on lectures he gave at ETH Zurich during the 1927–28 academic year.{{cite book |first=David |last=Speiser |chapter=Gruppentheorie und Quantenmechanik: The Book and its Position in Weyl's Work |date=2011 |title=Crossroads: History of Science, History of Art |pages=79–99 |editor-last=Williams |editor-first=Kim |place=Basel |publisher=Springer |language=en |doi=10.1007/978-3-0348-0139-3_7 |isbn=978-3-0348-0138-6}}{{cite journal|last=Scholz |first=Erhard |author-link=Erhard Scholz |title=Introducing groups into quantum theory (1926–1930) |journal=Historia mathematica |volume=33 |number=4 |year=2006 |pages=440–490 |doi=10.1016/j.hm.2005.11.007 |arxiv=math/0409571}} The first edition was published by {{ill|S. Hirzel Verlag|de|lt=S. Hirzel}} in Leipzig in 1928; a second edition followed in 1931, which was translated into English by Howard P. Robertson.{{cite journal|first=M. H. |last=Stone |title=Four books on group theory and quantum mechanics |journal=Bulletin of the American Mathematical Society |volume=42 |number=3 |date=March 1936 |pages=165–170 |doi=10.1090/S0002-9904-1936-06266-X |url=https://www.ams.org/journals/bull/1936-42-03/S0002-9904-1936-06266-X/S0002-9904-1936-06266-X.pdf}} Dover Publications issued a reprint of this translation in 1950.{{MacTutor|class=Extras |id=Weyl_introduction |title=H Weyl: Theory of groups and quantum mechanics Introduction |mode=cs1}}

John Archibald Wheeler wrote of learning quantum mechanics from Weyl's book, "His style is that of a smiling figure on horseback, cutting a clean way through, on a beautiful path, with a swift bright sword."{{cite journal|title=Hermann Weyl and the Unity of Knowledge |first=John Archibald |last=Wheeler |author-link=John Archibald Wheeler |journal=American Scientist |volume=74 |date=July–August 1986 |issue=4 |pages=366–375 |jstor=27854250 |bibcode=1986AmSci..74..366W |url=http://www.weylmann.com/wheeler.pdf}} Edward Condon called the text "authoritative".{{cite journal|last=Condon |first=Edward |author-link=Edward Condon |title=none |journal=Science |volume=75 |number=1953 |date=3 June 1932 |pages=586–588 |doi=10.1126/science.75.1953.586 |jstor=1657310}} Julian Schwinger said of it, "I read and re-read that book, each time progressing a little farther, but I cannot say that I ever – not even to this day – fully mastered it." The book was one of the first works to give a quantitative statement of the uncertainty principle, which Werner Heisenberg had previously introduced in a less precise way. Weyl credited the idea to Wolfgang Pauli.{{Cite journal|last1=Busch|first1=Paul|author-link1=Paul Busch (physicist) |last2=Lahti|first2=Pekka|last3=Werner|first3=Reinhard F. |author-link3=Reinhard F. Werner |date=17 October 2013|title=Proof of Heisenberg's Error-Disturbance Relation|journal=Physical Review Letters|language=en|volume=111|issue=16|pages=160405|doi=10.1103/PhysRevLett.111.160405|pmid=24182239|arxiv=1306.1565 |bibcode=2013PhRvL.111p0405B}}{{Cite journal|last=Appleby|first=David Marcus|date=6 May 2016|title=Quantum Errors and Disturbances: Response to Busch, Lahti and Werner|journal=Entropy|language=en|volume=18|issue=5|pages=174|doi=10.3390/e18050174|arxiv=1602.09002|bibcode=2016Entrp..18..174A|doi-access=free}}{{cite journal|last1=Werner |first1=Reinhard F. |author-link1=Reinhard F. Werner |first2=Terry |last2=Farrelly |title=Uncertainty from Heisenberg to today |journal=Foundations of Physics |volume=49 |year=2019 |issue=6 |pages=460–491 |doi=10.1007/s10701-019-00265-z |arxiv=1904.06139|bibcode=2019FoPh...49..460W }}{{cite journal|first=Berthold-Georg |last=Englert |author-link=Berthold-Georg Englert |title=Uncertainty relations revisited |journal=Physics Letters A |volume=494 |year=2024 |page=129278 |arxiv=2310.05039 |doi=10.1016/j.physleta.2023.129278|bibcode=2024PhLA..49429278E }} (Robertson, who later translated Weyl's book into English, cited the argument Weyl gave as the basis for his own generalization of the uncertainty principle to arbitrary noncommuting observables.{{cite journal|last=Robertson |first=H. P. |author-link=Howard P. Robertson |title=The Uncertainty Principle |journal=Physical Review | year=1929 |volume=34 |number=1 |pages=163–164 |doi = 10.1103/PhysRev.34.163 |bibcode=1929PhRv...34..163R }}) Moreover, it contains an early description of density matrices and quantum entanglement,{{cite journal|first=Adrian |last=Heathcote |title=Multiplicity and indiscernability |doi=10.1007/s11229-020-02600-8 |journal=Synthese |volume=198 |pages=8779–8808 |year=2021 |issue=9 |quote=For Weyl clearly anticipated entanglement by noting that the pure state of a coupled system need not be determined by the states of the composites [...] Weyl deserves far more credit than he has received for laying out the basis for entanglement—more than six years before Schrödinger coined the term.}} and it uses what quantum information theory would later call the Weyl–Heisenberg group to give a finite-dimensional version of the canonical commutation relation.{{cite book|first=Julian |last=Schwinger |author-link=Julian Schwinger |chapter=Hermann Weyl and Quantum Mechanics |title=Exact Sciences and their Philosophical Foundations |editor-first=Wolfgang |editor-last=Deppert |publisher=Peter Lang |year=1988 |pages=107–29}}{{cite book|first1=Ingemar |last1=Bengtsson |first2=Karol |last2=Życzkowski |author-link2=Karol Życzkowski |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |page=314 |year=2017 |publisher=Cambridge University Press |edition=2nd |isbn=978-1-107-02625-4}}{{cite journal|first=Ingemar |last=Bengtsson |title=SICs: Some explanations |journal=Foundations of Physics |year=2020 |volume=50 |issue=12 |pages=1794–1808 |doi=10.1007/s10701-020-00341-9 |arxiv=2004.08241|bibcode=2020FoPh...50.1794B }}

Weyl noted that Paul Dirac's relativistic quantum mechanics implied that the electron should have a positively charged anti-particle. The only known particle with a positive charge was the proton, but Weyl was convinced that the anti-electron had to have the same mass as the electron, and physicists had already established that protons are much more massive than electrons. Weyl wrote, "I fear that the clouds hanging over this part of the subject will roll together to form a new crisis in quantum physics." The discrepancy was resolved in 1932 with the discovery of the positron.{{cite journal|first=Helen R. |last=Quinn |author-link=Helen Quinn |title=The asymmetry between matter and antimatter |journal=Physics Today |url=https://inspirehep.net/files/1781ac28d4e51d4c4c3ac186c0b6a014 |volume=56 |number=2 |year=2003 |pages=30–35 |doi=10.1063/1.1564346|bibcode=2003PhT....56b..30Q }}{{cite SEP|url-id=weyl |title=Hermann Weyl |date=8 June 2024 |author-first1=John L. |author-last1=Bell |author-link1=John Lane Bell |first2=Herbert |last2=Korté}}