Gruppentheorie und Quantenmechanik explained

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,^[1] basing the text on lectures he gave at ETH Zurich during the 1927–28 academic year.^{[2] [3]} The first edition was published in 1928; a second edition followed in 1931, which was translated into English by Howard P. Robertson.^[4] Dover Publications issued a reprint of this translation in 1950.

Contents

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."^[5] Edward Condon called the text "authoritative".^[6] 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.^{[7] [8] [9] [10]} (Robertson, who would later translate 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.^{[10] [11]}) Moreover, it contains an early description of density matrices and quantum entanglement,^[12] 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.^{[13] [14] [15]}

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.^{[16] [17]}

External links

• 1950 edition at the Internet Archive (registration required)
• Hermann Weyl and the Application of Group Theory to Quantum Mechanics by George Mackey

Notes and References

1. H. . Weyl . Quantenmechanik und Gruppentheorie . Zeitschrift für Physik . 1927 . 46 . 1–2 . 1–46 . 10.1007/bf02055756. 1927ZPhy...46....1W .
2. Book: Speiser, David . Gruppentheorie und Quantenmechanik: The Book and its Position in Weyl's Work . 2011 . Crossroads: History of Science, History of Art . 79–99 . Williams . Kim . Basel . Springer . en . 10.1007/978-3-0348-0139-3_7 . 978-3-0348-0138-6.
3. Scholz . Erhard . Erhard Scholz . Introducing groups into quantum theory (1926–1930) . Historia mathematica . 33 . 4 . 2006 . 440–490 . 10.1016/j.hm.2005.11.007 . math/0409571.
4. M. H. . Stone . Four books on group theory and quantum mechanics . Bulletin of the American Mathematical Society . 42 . 3 . March 1936 . 165–170 . 10.1090/S0002-9904-1936-06266-X .
5. Hermann Weyl and the Unity of Knowledge . John Archibald . Wheeler . John Archibald Wheeler . American Scientist . 74 . July–August 1986 . 4 . 366–375 . 27854250 . 1986AmSci..74..366W .
6. Condon . Edward . Edward Condon . none . Science . 75 . 1953 . 3 June 1932 . 586–588 . 10.1126/science.75.1953.586 . 1657310.
7. Busch. Paul. Paul Busch (physicist) . Lahti. Pekka. Werner. Reinhard F. . Reinhard F. Werner . 17 October 2013. Proof of Heisenberg's Error-Disturbance Relation. Physical Review Letters. en. 111. 16. 160405. 10.1103/PhysRevLett.111.160405. 24182239. 1306.1565 . 2013PhRvL.111p0405B.
8. Appleby. David Marcus. 6 May 2016. Quantum Errors and Disturbances: Response to Busch, Lahti and Werner. Entropy. en. 18. 5. 174. 10.3390/e18050174. 1602.09002. 2016Entrp..18..174A. free.
9. Werner . Reinhard F. . Reinhard F. Werner . Terry . Farrelly . Uncertainty from Heisenberg to today . Foundations of Physics . 49 . 2019 . 6 . 460–491 . 10.1007/s10701-019-00265-z . 1904.06139. 2019FoPh...49..460W .
10. Berthold-Georg . Englert . Berthold-Georg Englert . Uncertainty relations revisited . Physics Letters A . 494 . 2024 . 129278 . 2310.05039 . 10.1016/j.physleta.2023.129278. 2024PhLA..49429278E .
11. Robertson . H. P. . Howard P. Robertson . The Uncertainty Principle . Physical Review . 1929 . 34 . 1 . 163–164 . 10.1103/PhysRev.34.163 . 1929PhRv...34..163R .
12. Adrian . Heathcote . Multiplicity and indiscernability . 10.1007/s11229-020-02600-8 . Synthese . 198 . 8779–8808 . 2021 . 9 . 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..
13. Book: Schwinger, Julian . Julian Schwinger . Hermann Weyl and Quantum Mechanics . Exact Sciences and their Philosophical Foundations . Wolfgang . Deppert . Peter Lang . 1988 . 107–29.
14. Book: Ingemar . Bengtsson . Karol . Życzkowski . Karol Życzkowski . Geometry of Quantum States: An Introduction to Quantum Entanglement . 314 . 2017 . Cambridge University Press . 2nd . 978-1-107-02625-4.
15. Ingemar . Bengtsson . SICs: Some explanations . Foundations of Physics . 2020 . 50 . 12 . 1794–1808 . 10.1007/s10701-020-00341-9 . 2004.08241. 2020FoPh...50.1794B .
16. Helen R. . Quinn . Helen Quinn . The asymmetry between matter and antimatter . Physics Today . 56 . 2 . 2003 . 30–35 . 10.1063/1.1564346. 2003PhT....56b..30Q .
17. weyl . Hermann Weyl . 8 June 2024 . John L. . Bell . John Lane Bell . Herbert . Korté.