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.

He came to Berkeley one semester in late sixties (anyway around 1970) and I took his course on Group Representations. The large hall was always packed. A lot went by too fast for me. Left a vivid impression though. He had so much verve.

Would like to hear others’ recollections.

George Mackey’s daughter, Ann, was my roommate in college and remains one of my closest friends. When I was a sophomore in college, interested in both mathematics and physics, I sent him a short note asking him what his field was, and requesting advice. He responded with two long handwritten letters that I am reading with interested after rediscovering them in my files this morning. I can’t imagine that I understood much of it at the time. Here’s a sample, from near the end of the first letter:

What is astounding is that nearly twenty-five years later these forgotten letters hidden in a drawer describe exactly the mathematics that is most important to me. It never would have occurred to me to think that George Mackey sent me on that path, since I got there (like Alice through the looking glass) by heading in what seemed to me the opposite direction.

Also astounding is the generosity of a man who would handwrite twenty or more pages in response to a student he had never met.

Another memory: George supposedly spent eight hours each day on his research, in addition to his teaching and other obligations. He told me once that his method was to work for forty-five minutes of each hour and then spend fifteen minutes of each hour reading a nonmathematical book or taking some other kind of break. Knowing this has always helped me to pace myself.

Thank you, George.

Alice Mackey, George’s wife and Ann’s mother, supported him and put up with his foibles. The French, at least, had the option of sending him home! Alice gave him the space and time to do his work and letter-writing. In the last several years, Alice supported and loved George through a difficult period of declining health. Thank you, Alice.

There will be memorial service for George at Harvard on April 29, 2006, in the afternoon.

Stephanie,

Thanks a lot for your memories of Mackey.

I’m guessing we both arrived at an interest similar to Mackey’s in QM and representation theory via symplectic geometry and its relation to quantization. I recommend Stephanie’s books “Symmetry in Mechanics” and “Linearity, Symmetry and Prediction in the Hydrogen Atom”, to anyone who wants an introduction to these subjects.

I highly recommend Stephanie Singer’s book “Symmetry in Mechanics” to all students who wish to read about symplectic geometry and related topics for the very first time. People should write more such books.

I heartily agree with both Peter and MathPhys: Both of Stephanie Singer’s books are very good, very clear introductions. I wish they were available when I was learning these subjects for the first time, but at least I can recommend them now to any interested beginners who cross my path.

Warm congratulations to Ms. Singer, and I do hope you keep writing!

to stefanie :

as someone who is precisely trying to find what mackey describes, i was wondering if maybe you could type or scan these letters onto your website. (i realise they are pretty long and a lot of it might just be defining lie groups, hilbert spaces, etc. so, maybe atleast some excepts which talk about motivations). it would prove to be very useful, specially for younglings like me who dont know too much.

Stephanie,

There is an incredible amount of material on integrability that remains to be explained to students. Why not write sequels to “Symmetry in mechanics”?

Well, wow, folks, thanks for the feedback. I’m feeling the love! Don’t hesitate to repeat your opinions at Amazon.com.

I will post George Mackey’s letters as soon as I get ten quiet minutes. I’m incredibly busy in 2006, starting two businesses (real estate development and election data consulting) and working to get Chuck Pennacchio elected to the US Senate. Also, I’m working with activists around the state of Pennsylvania to fight the installation of unauditable, unaccountable voting machines. I’m starting to wonder how many Americans really believe that democracy is worth fighting for. I’m talking about fighting laziness, established power brokers and bureaucracy here on the home front.

So I haven’t even found time to post the errata that Tudor Ratiu so kindly sent me for

Linearity, Symmetry, and Prediction in the Hydrogen Atom, much less to write a new book. But do send me ideas of what you’d like to read. The next book I’ve thought of is the story of therealKepler problem, that is, the problem Kepler devoted most of his energy to (according to Koestler’sThe Watershed): why are the planets where they are? I heard a talk by Jerry Marsden a few years ago announcing that there was a geometrical explanation: once Jupiter’s orbit is given, then the positions of the other planets can be predicted geometrically. That’s the book I’m daydreaming about these days. But it will have to wait.A link to Mackey’s letters at Stephanie’s web-site was added.