Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone provide me a reference to such a resource? I don't care if it's a math or physics book as long as it makes use of the language of representation theory mathematicians are familiar with.
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6$\begingroup$ Peter Woit wrote some good lecture notes in this spirit based on some advanced undergraduate classes that he taught last year: math.columbia.edu/~woit/QM/fall2012.html and math.columbia.edu/~woit/QM/spring2013.html $\endgroup$– Paul SiegelCommented Jan 22, 2014 at 21:24
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1$\begingroup$ faculty.virginia.edu/quantum/MathSeminar2003_04_23.pdf $\endgroup$– Carlo BeenakkerCommented Jan 22, 2014 at 21:57
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3 Answers
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Here are some references.
- Hermann Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950
- George W. Mackey, Induced representations of groups and quantum mechanics, W. A. Benjamin, 1968
- M. Gourdin , Unitary Symmetries, John Wiley & Sons, New York, 1968.
- A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton, 1968
- L. C. Biedenharn, James D. Louck, Angular Momentum in Quantum Physics: Theory and Application, Cambridge University Press, 2009
- Gianni Cassinelli, The Theory of Symmetry Actions in Quantum Mechanics: With an Application to the Galilei Group, Springer, 2004
I believe 1 and 2 are classics, 3 and 4 probably nearly so. I'm not familiar with 5 nor 6 in detail, but they look interesting - I'm very curious to see comments by other users on any of the above.
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Basic foundations of quantum mechanics in a representation theoretic language can be found in van den Ban's lectures "Applications of representation theory in classical quantum mechanics" at this page http://www.staff.science.uu.nl/~ban00101/lecnot.html
The following articles can be also relevant: http://link.springer.com/article/10.1023%2FB%3AACAP.0000024199.14167.44 (N. P. Landsman, Quantum Mechanics and Representation Theory: The New Synthesis). http://arxiv.org/abs/0711.2699 (I. Frenkel, M. Libine, Quaternionic Analysis, Representation Theory and Physics)
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There is an upcoming book of Peter Woit, available for download if you click on the link. From the Preface:
These are the course notes prepared for a class taught at Columbia during the 2012-13 academic year. The intent was to cover the basics of quantum mechanics, from a point of view emphasizing the role of unitary representations of Lie groups in the foundations of the subject. [..]
I just noticed that these lecture notes were also mentioned in the comments.