by Paul R. Chernoff
About Andy
The course was both a challenge and a pleasure. I can only echo what others have said about Andy’s luminous clarity and massive abstract power. But I must admit that the lectures, always exciting, weren’t absolutely perfect; in the course of a year Andy made one genuine blunder. As to his famous speed, John Schwarz, the well-known string theorist, once said after class that Andy had “the metabolism of a hummingbird”.
I was extremely lucky that Andy was affiliated with Lowell House, my undergraduate residence. Every week Andy came for lunch, where we sat around a large circular table. That’s how Andy and I became friends. Of course we discussed a lot of mathematics around that table, but lots of other things, including Andy’s “war stories”. I am not surprised that someone kept a great treasure: all of Andy’s napkin manuscripts.
Almost any mathematical problem could intrigue Andy. At one of the annual math department picnics, he had fun figuring out how to do cube roots on an abacus. But most important was his unpretentiousness, openness, and great interest in students. I suppose that all teachers are impatient at times; no doubt Andy was sorely tried on occasion. But he rarely, if ever, showed it. The students in one of his classes gave him a framed copy of Picasso’s early painting Mother and Child. Perhaps they chose this gift to symbolize Andy’s nurturing of them. It’s regrettable that there are some teachers for whom Guernica would be more appropriate.
Quantum mechanics

In this section we set the stage for a discussion of Andy’s unique contribution to physics: his remarkable paper “Measures on the closed subspaces of a Hilbert space” [1]. It’s interesting in several ways: its history; its influence in mathematics; and especially its unexpected importance to the analysis of “hidden variable” theories of quantum mechanics by the physicist John Bell.
In classical mechanics, the state of a particle of mass
The fundamental interpretation
of the “wave function”
Prior to Schrödinger’s wave mechanics,
Heisenberg
had begun to develop
a theory in which observable quantities are represented by
Hermitian-symmetric infinite square arrays. He devised a “peculiar”
law for multiplying two arrays by an ingenious use of the physical
meaning of their entries. Born had learned matrix theory when he was a
student and realized (after a week of “agony”) that Heisenberg’s
recipe was just matrix multiplication. Hence the Heisenberg theory is
called matrix mechanics.
(Schrödinger
showed that matrix mechanics
and wave mechanics are mathematically equivalent.) As in classical
mechanics, the dynamics of a quantum system is determined from its
energy
Born was quite familiar with Hilbert’s theory of integral equations in
Having interpreted
Born’s line of thought was this. A state of a quantum system
corresponds to a unit vector
John von Neumann was the creator of the abstract theory of quantum
mechanics. In his theory, a pure state is a unit vector in a Hilbert
space
The eigenvalues of a projection operator are 1 and 0; those are the
only values a measurement of the corresponding observable can yield.
That is why Mackey calls a projection a question; the answer is always
either 1 or 0: “yes” or “no”. The fundamental example is the
following. Given a self-adjoint operator
Quite generally, consider a positive operator
Gleason’s theorem
Mackey didn’t try very hard to solve his problem for the excellent
reason that he had no idea how to attack it. But he discussed it with
a number of experts, including
Irving Segal,
who mentioned Mackey’s
problem in a graduate class at Chicago around 1949 or 1950. Among the
students was
Dick Kadison,
who realized that there are counterexamples
when
There matters stood for some years. Then Gleason entered the story. In 1956 he sat in on Mackey’s graduate course on quantum mechanics at Harvard. To Mackey’s surprise, Andy was seized by the problem “with intense ferocity”. Moreover, Kadison was visiting MIT at the time, and his interest in Mackey’s problem was rekindled. He quickly perceived that there were many “forced inter-relations” entailed by the intertwining of the great circles on the sphere and in principle a lot could be deduced from an analysis of these relations, though the problem still looked quite tough. He mentioned his observation to Andy, who found it a useful hint. (But Kadison informed me that his observation did not involve anything like Andy’s key “frame function” idea.)
The proof has three parts. First, using countable additivity and induction, it is easy to reduce the case of any separable real Hilbert space of dimension greater than 2 to the 3-dimensional case. (The complex case follows from the real case.)
Next, consider a vector
The proof comes down to showing that the frame function
Quite a few people have worked on simplifying the proof. The paper by Cooke, Keane, and Moran [e13] is interesting, well written, and leads the reader up a gentle slope to Gleason’s theorem. The authors use an important idea of Piron [e9]. (The CKM argument is “elementary” because it does not use harmonic analysis.)
Generalizations of Gleason’s theorem
In his paper Andy asked if there were analogues of his theorem for countably additive probability measures on the projections of von Neumann algebras other than the algebra of bounded operators on separable Hilbert spaces.
A von Neumann algebra, or
A state of a von Neumann algebra
Let
The converse for countably additive measures is due to A. Paszkiewicz [e14]. See E. Christensen [e10] and F. J. Yeadon [e11], [e12] for finitely additive measures. Maeda has a careful, thorough presentation of the latter in [e16].
It is not surprising that the arguments use the finite-dimensional
case of Gleason’s theorem. A truly easy consequence of Gleason’s
theorem is that
A great deal of work has been done on Gleason measures which are unbounded or complex valued. A good reference is [e19]. Bunce and Wright [e18] have studied Gleason measures defined on the lattice of projections of a von Neumann algebra with values in a Banach space. They prove the analogue of the results above. A simple example is Paszkiewicz’s theorem for complex-valued measures, which had been established only for positive real-valued measures.
Nonseparable Hilbert spaces
Gleason’s theorem is true only for separable Hilbert spaces. Robert Solovay has completely analyzed the nonseparable case. (Unpublished. However, [e22] is an extended abstract.) I consider Solovay’s work to be the most original extension of Gleason’s theorem.
A countable set is not gigantic. Indeed, gigantic sets are very, very large. Also, in standard set theoretic terminology, a gigantic cardinal is called a measurable cardinal.
Gleason’s theorem states that every Gleason measure on a separable
Hilbert space is standard. But suppose
On the other hand, it can be shown that if
If
Observe that Gleason’s theorem is analogous;
Solovay also proves a beautiful formula giving a
canonical representation of a Gleason measure
Hidden variables and the work of John Bell
The major scientific impact of Gleason’s theorem is not in mathematics but in physics, where it has played an important role in the analysis of the basis of quantum mechanics. A major question is whether probabilistic quantum mechanics can be understood as a phenomenological theory obtained by averaging over variables from a deeper nonprobabilistic theory. The theory of heat exemplifies what is wanted. Heat is now understood as due to the collisions of atoms and molecules. In this way one can understand thermodynamics as a phenomenological theory derived by averages over “hidden variables” associated with the deeper particle theory; hence the term “statistical mechanics”. Einstein sought an analogous relation between quantum mechanics and — what? He is supposed to have said that he had given one hundred times more thought to quantum theory than to relativity.
The fourth chapter of John von Neumann’s great book [e7] is devoted to his famous analysis of the hidden variable question. His conclusion was that no such theory could exist. He writes, “The present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one may be possible.” That seemed to settle the question. Most physicists weren’t much interested in the first place when exciting new discoveries were almost showering down.
But in 1952 there was a surprise. Contrary to von Neumann,
David Bohm
exhibited a hidden variable theory by constructing a system of
equations with both waves and particles which exactly reproduced
quantum mechanics. But Einstein rejected this theory as “too easy”,
because it lacked the insight Einstein was seeking. Worse yet, it had
the feature Einstein most disliked. Einstein had no problem
understanding that there can easily be correlations between the
behavior of two distant systems,
In fact, soon after its publication, von Neumann’s argument was
demolished by
Grete Hermann
[e8],
a young student of
Emmy Noether.
Her point was that in quantum mechanics the expectation of the sum of
two observables
The outstanding Irish physicist John Bell was extremely interested in the hidden variable problem. Early on he discovered a simple example of a hidden variable theory for a two-dimensional quantum system; it’s in chapter 1 of [e15], which is a reprint of [e4]. This is another counterexample for von Neumann’s “impossibility” theorem. (Bell did a great deal of important “respectable” physics. He said that he studied the philosophy of physics only on Saturdays. An interesting essay on Bell is in Bernstein’s book [e17].)
When Bell learned of Gleason’s theorem he perceived that in Hilbert spaces of dimension greater than 2, it “apparently” establishes von Neumann’s “no hidden variables” result without the objectionable assumptions about noncommuting operators. Bell is reported to have said that he must either find an “intelligible” proof of Gleason’s theorem or else quit the field. Fortunately Bell did devise a straightforward proof of a very special case: nonexistence of frame functions taking only the values 0 and 1. Such frame functions correspond to projections. This case sufficed for Bell’s purposes [e4]. See the first chapter of [e15].2

The gist of von Neumann’s proof is an argument that dispersion-free
states do not exist. Here a state
Bell writes: “That so much follows from such apparently innocent
assumptions leads one to question their innocence.” He points out
that if
Finally, a few words about the famous “Bell’s Inequality”.
The second chapter of Bell’s book is a reprint of [e3] (actually written after [e4]). In this very important paper, Bell derives a specific inequality satisfied by certain “local” hidden variable theory for nonrelativistic quantum mechanics. (“Locality” excludes “weird” correlations of measurements of widely separated systems.) There are many similar but more general inequalities. Moreover, the study of the “entanglement” of separated quantum systems has opened a new field of mathematical research.

Starting in 1969, difficult experimental work began, using variants of Bell’s inequality, to test if very delicate predictions of quantum mechanics are correct. Of course, quantum mechanics has given superb explanations of all sorts of phenomena, but these experiments waterboard quantum mechanics. Many experiments have been done; so far there is no convincing evidence that quantum mechanics is incorrect. In addition, experiments have been done which suggest that influence from one system to the other propagates enormously faster than light. These experiments point toward instantaneous transfer of information.
Bell’s papers on quantum philosophy have been collected in his book Speakable and Unspeakable in Quantum Mechanics [e15]. The first paper [e4] discusses Gleason’s theorem and the second “Bell’s inequality”. The entire book is a pleasure to read.3
Anagrams
Among his many talents, Andy was a master of anagrams. His fragmentary 1947 diary records a family visit during Harvard’s spring break:
March 30. …We played anagrams after supper and I won largely through the charity of the opposition.
April 1. …Played a game of anagrams with Mother and won.
April 2. …Mother beat me tonight at anagrams.
So we know a little about where he honed that talent.
Many years ago Andy and I had a little anagram “contest” by mail. (Dick Kadison said then, “You’re having an anagram competition with Andy Gleason? That’s like arm wrestling with Gargantua.”) Anyhow, I figured out ROAST MULES, and I was proud to come up with I AM A WONDER AT TANGLES, which is an anagram of ANDREW MATTAI GLEASON. Unfortunately, it should be MATTEI. But I didn’t have the chutzpah to ask Andy to change the spelling of his middle name. I am grateful for very interesting correspondence and conversations with the late Andy Gleason and George Mackey, together with Dick Kadison, Si Kochen, and Bob Solovay.