It was sad to see an announcement today on the Harvard math department web-site of the death earlier this week of emeritus Harvard professor George Mackey.
Mackey’s mathematical work is dear to my heart, since its central concern is the relationship between quantum mechanics and representation theory. He began his career in functional analysis, getting his Ph.D. in 1942 under Marshall Stone. Back in 1930 Stone and von Neumann had proved a crucial theorem about quantum mechanics, a theorem which essentially says that once you choose Planck’s constant, up to unitary equivalence there is only one possible representation of the Heisenberg commutation relations. This uniqueness theorem is what allows one to just define quantum theory in terms of the operator commutation relations, and not worry about which explicit construction of the representation of these operators on a Hilbert space one uses. The theorem is only true for a finite number of degrees of freedom, and thus doesn’t apply to quantum field theory, one reason why quantum field theory is a much more subtle business than quantum mechanics. Stone and von Neumann put their work in the context of representation theory of the Heisenberg group (actually due to Weyl) and this was of great interest to mathematicians since it was one of the first results about the representation theory of non-compact Lie groups. For an excellent history and introduction to this subject, see the paper A Selective History of the Stone von-Neumann Theorem by Jonathan Rosenberg.
Mackey seems to have been the person who gave this theorem its name, in his important paper of 1949 “A Theorem of Stone and von Neumann” which generalized it. Over the next few years Mackey extended this much further in a series of papers on induced representations (representations of a group G “induced” from representations of a subgroup H). The foundation of this work is now known as the Mackey Imprimitivity Theorem, and it provides a powerful tool for studying representations of a large class of non-compact groups, including especially semi-direct products.
Mackey was a wonderful expositor, and over the years I’ve learned a great deal from some of his expository books and papers. His 1963 monograph Mathematical Foundations of Quantum Mechanics is very readable. In 1966-67 he gave a course at Oxford on representation theory and its applications, the notes of which were published in 1978 as Unitary Group Representations in Physics, Probability and Number Theory. This is a fantastic book, covering a wide range of topics relating quantum mechanics, representation theory and even number theory. A later collection of expository material, from 1992, was published by the AMS as The Scope and History of Commutative and Non-Commutative Harmonic Analysis. It contains what is perhaps the best of his expository work, an historical survey first published in the AMS Bulletin in 1980 entitled “Harmonic Analysis as the Exploitation of Symmetry”.
While I never took a course from Mackey, I did get to talk to him on several occasions. I especially remember a conversation in which he described his technique for speaking French during the time he spent in France. He decided to speak his own rationalized version of the language, eliminating extraneous and confusing structure like genders of nouns. Not clear what the French thought of this. He was an original, and I’m sad to hear he’s no longer with us.
Update: Stephanie Singer has put up copies of letters from Mackey on her web-site. A memorial service for Mackey will be held in Cambridge on April 29.
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