#### 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 __\( m \)__ is given by
its position and momentum. The motion or dynamics of a set of
particles with associated forces is determined by Newton’s second law
of motion, a system of ordinary differential equations. This yields a
picture of the macroscopic world which matches our intuition. The
ultramicroscopic world requires a quite different description. The
state of a particle of mass __\( m \)__ in __\( \mathbb{R}^3 \)__ is a complex valued function __\( \psi \)__ on
__\( \mathbb{R}^3 \)__. Its momentum is similarly described by the function __\( \phi \)__, the
*Fourier transform* of __\( \psi \)__, normalized by the presence in the exponent of
the ratio __\( \frac{h}{m} \)__, where __\( h \)__ is Planck’s constant. (Using standard
properties of the Fourier transform, one can deduce Heisenberg’s
uncertainty principle.) For __\( n \)__ particles, __\( \psi \)__ is defined on __\( \mathbb{R}^{3n} \)__. This is
a brilliant extrapolation of the initial ideas of DeBroglie. The
Schrödinger equation determines the dynamics. If both __\( \psi \)__ and __\( \phi \)__ are
largely concentrated around __\( n \)__ points in position *and* momentum space
respectively, then the quantum state resembles a blurry picture of the
classical state. The more massive the particles, the less the
blurriness (protons versus baseballs).

The fundamental interpretation
of the “wave function” __\( \psi \)__ is the work of
Max Born.1
His paper analyzing collisions of particles ends with the conclusion
that __\( |\psi |^2 \)__ should be interpreted as the probability distribution
for the positions of the particles. Therefore the wave function must
be a unit vector in __\( L^2 \)__. Thus did Hilbert space enter quantum
mechanics.

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 __\( H \)__. Periodic orbits correspond to the eigenvalues of __\( H \)__,
i.e., the discrete energy levels. The calculation of the eigenvalues
is very difficult, save for a few simple systems. The energy levels
for the hydrogen atom were ingeniously calculated by Wolfgang Pauli;
his results agreed with
Bohr’s
calculations done at the very beginning
of the “old” quantum theory.

Born was quite familiar with Hilbert’s theory of integral equations in
__\( L^2 \)__. Accordingly, he was able to interpret Heisenberg’s matrices as
Hermitian symmetric kernels with respect to some orthonormal basis,
which might just as well be regarded as the corresponding integral
operators on __\( L^2 \)__. Formally, every Hermitian matrix could be regarded
as an integral operator, usually with a very singular kernel. (The
most familiar example is the identity, with kernel the Dirac delta
function.) In this way, Born initiated the standard picture of
observables as Hermitian operators A on __\( L^2 \)__. But at that time, the
physicists did not grasp the important distinction between unbounded
Hermitian operators and unbounded self-adjoint operators. That was
greatly clarified by
John von Neumann,
major developer of the theory
of unbounded self-adjoint operators.

Having interpreted __\( |\psi |^2 \)__ as the probability distribution for the
positions of particles, Born went on to devise what immediately became
the standard interpretation of measurements in quantum mechanics: the
probability that a measurement of a quantum system will yield a
particular result.

Born’s line of thought was this. A state of a quantum system
corresponds to a unit vector __\( \psi \in L^2 \)__. What are the possible
values of a measurement of the observable represented by the operator
A, and what is the probability that a specific value is observed? Born
dealt only with operators with a discrete spectrum, namely, the set of
all its eigenvalues. For simplicity, assume that there are no multiple
eigenvalues. Let __\( \phi_n \)__ be the unit eigenvector with eigenvalue
__\( \lambda_n. \)__ These form an orthonormal basis of __\( L^2 \)__. Expand __\( \psi \)__ as a
series __\( \sum_k c_k \phi_k \)__. Since __\( \|\psi \|^2 = 1 \)__, we get __\( \sum_k |c_k |^2 = 1 \)__.
Born’s insight was that any measurement must yield one of the
eigenvalues __\( \lambda_n \)__ of __\( A \)__, and __\( |c_n |^2 \)__ is the probability that the result of
the measurement is __\( \lambda_n \)__. This is known as *Born’s rule*. It
follows that the expected value of a measurement of __\( A \)__ is __\( \sum_k |c_k |^2 \lambda_k \)__. Note that
this sum equals the inner product __\( (A\psi , \psi) \)__. This is the same as
__\( \operatorname{trace}(P A) \)__, where __\( P \)__ is the projection onto the one-dimensional
subspace spanned by __\( \psi \)__. (To jump ahead,
George Mackey
wondered if
Born’s rule might involve some arbitrary choices. Gleason ruled this
out.)

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 __\( \mathcal{H} \)__ . Observables are self-adjoint operators,
unbounded in general, whose spectrum may be any Borel subset of
__\( \mathbb{R} \)__. Von Neumann also developed the important concept of a
*mixed state*. A mixed state __\( \mathbf{D} \)__ describes a situation in which
there is not enough information to determine the pure state __\( \psi \)__ of the
system. Usually physicists write __\( \mathbf{D} \)__ as a convex combination
of orthogonal pure states, __\( \sum_k w_k \psi_k \)__ . This notation is
confusing; __\( \mathbf{D} \)__ is *not* a vector in __\( \mathcal{H} \)__! It may be
interpreted as a list of probabilities __\( w_k \)__ that the corresponding
pure state is __\( \psi_k \)__. Associated with the state __\( \mathbf{D} \)__ there is
a positive operator __\( D \)__ with trace 1, given by the formula
__\[
D =
\sum_k w_k P_k
\]__
where __\( P_n \)__ is the projection on the eigenspace of __\( \mathbf{D} \)__
corresponding to the eigenvalue __\( w_n \)__. The expected
value of an observable __\( A \)__ is quite clearly
__\[
E (A) = \sum_k w_k (A\psi_k , \psi_k ) =\operatorname{trace}(D A).
\]__
This is von Neumann’s general Born rule.

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 __\( A \)__, we will apply the spectral
theorem. Let __\( S \)__ be any Borel subset of __\( \mathbb{R} \)__ and let __\( P_S \)__ be the
corresponding “spectral projection” of __\( A \)__. (If the set __\( S \)__ contains only
some eigenvalues of __\( A \)__, then __\( P_S \)__ is simply projection onto the subspace
spanned by the corresponding eigenvectors.) Now suppose the state of
the system is the mixed state __\( \mathbf{D} \)__. From the general Born rule, the
probability that a measurement of __\( A \)__ lies in __\( S \)__ is the expected value of
__\( P_S \)__, namely, __\( \operatorname{trace}(DP_S ) \)__. That is the obvious generalization of
Born’s formula for the probability that a measurement of __\( A \)__ is a
particular eigenvalue of __\( A \)__ or a set of isolated eigenvalues.

Quite generally, consider a positive operator __\( D \)__
with __\( \operatorname{trace}(D ) = 1 \)__. The nonnegative real-valued
function __\( \mu (P ) = \operatorname{trace}(D P ) \)__ is a countably additive
probability measure on the lattice of projections on
__\( \mathcal{H} \)__. This means that if __\( \{P_n \} \)__ is a countable family
of mutually orthogonal projections,
__\[
\mu\biggl(\sum_n
P_n\biggr) =
\sum_n
\mu(P_n).
\]__
Also __\( \mu (I ) = 1 \)__. Mackey asked whether every such
measure on the projections is of this form, i.e.,
corresponds to a state __\( D \)__ . We already mentioned
Mackey’s interest in Born’s rule. A positive answer
to Mackey’s question would show that the Born
rule follows from his rather simple axioms for
quantum mechanics
[e2],
[e1], and thus, given
these weak postulates, Born’s rule is not ad hoc
but inevitable.

#### 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 __\( \mathcal{H} \)__ is two-dimensional. The higher-dimensional case
remained open.

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.)

__(Gleason’s theorem)__Let

__\( \mathcal{H} \)__be a separable Hilbert space of dimension greater than 2. Let

__\( \mu \)__be a countably additive probability measure on the projections of

__\( \mathcal{H} \)__. Then there is a unique nonnegative self-adjoint operator

__\( D \)__, with

__\( \operatorname{trace}(D ) = 1 \)__, such that, for every projection

__\( P \)__,

__\[ \mu (P ) = \operatorname{trace}(D P ). \]__

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 __\( x \)__ on the unit sphere. Let __\( P_x \)__ be the
one-dimensional subspace containing __\( x \)__, and define __\( f (x) = \mu (P_x
) \)__. This function is called a *frame function*. The additivity of the
measure __\( \mu \)__ implies that for any three mutually orthogonal unit
vectors,
__\[ f (x) + f (y ) + f (z ) = 1. \]__

The proof comes down to showing that the frame function __\( f \)__ is
quadratic and therefore is of the form __\( f (x) = \operatorname{trace}(D
P_x ) \)__, where __\( D \)__ is as in the statement of the theorem. Gleason
begins his analysis by showing that a *continuous* frame
function is quadratic via a nice piece of harmonic analysis on the
sphere. The centerpiece of the paper is the proof that __\( f \)__ is
continuous. Andy told me that this took him most of the summer. It
demonstrates his powerful geometric insight. However, despite Andy’s
talent for exposition, much effort is needed to really understand his
argument.

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 __\( W^{\ast} \)__ algebra, is an algebra
__\( \mathcal{A} \)__ of bounded operators on a Hilbert space __\( \mathcal{H} \)__,
closed with respect to the adjoint operation. Most importantly,
__\( \mathcal{A} \)__ is closed in the *weak operator topology*. The
latter is defined as follows: a net of bounded operators __\( \{a_i \} \)__
converges weakly to __\( b \)__ provided that, for all vectors __\( x, y \in
\mathcal{H} \)__,
__\[ \lim_{n\to\infty} (a_nx, y) = (b_x, y). \]__

A state of a von Neumann algebra __\( \mathcal{A} \)__ is a positive linear
functional __\( \phi : A \to \mathbb{C} \)__ with __\( \phi(I ) = 1 \)__. This means
that __\( \phi(x) \geq 0 \)__ if __\( x \geq 0 \)__ and also __\( \|\phi\| = 1 \)__. The state
__\( \phi \)__ is normal provided that if __\( a_i \)__ is an increasing net of
operators that converges weakly to __\( a \)__ , then __\( \phi(a_i ) \)__ converges
to __\( \phi(a) \)__. The normal states on __\( B (\mathcal{H}) \)__ are precisely
those of the form __\( \operatorname{trace}(D x) \)__, where __\( D \)__ is a positive
operator with trace 1.

Let __\( P (\mathcal{A}) \)__ be the lattice of orthogonal projections in
__\( \mathcal{A} \)__. Then the formula
__\[ \mu (P ) = \phi(P ) \]__
defines a
finitely additive probability measure on __\( P (\mathcal{A}) \)__. If __\( \phi \)__
is normal, the measure __\( \mu \)__ is countably additive.

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 __\( \mu \)__ is a uniformly continuous function on the
lattice of projections __\( P \)__, equipped with the operator norm.

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.

__\( \mathcal{H} \)__is a countably additive probability measure on the lattice of projections of

__\( \mathcal{H} \)__. We say that a Gleason measure

__\( \mu \)__is standard provided there is a positive trace-class operator

__\( D \)__with trace 1 such that

__\( \mu (P ) =\operatorname{trace}(D P ) \)__. Otherwise,

__\( \mu \)__is exotic.

__\( X \)__is gigantic if there is a continuous probability measure

__\( \rho \)__defined on

*all*the subsets of

__\( X \)__. Continuity means that every point has measure 0.

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 __\( \mathcal{H} \)__ is a nonseparable Hilbert
space with a gigantic orthonormal basis __\( \{e_i : i \in I \} \)__. Let __\( \rho \)__ be the
associated measure on __\( I \)__. Then the formula
__\[
\mu(P) = \int_I (Pe_i , e_i)\,d\rho(i)
\]__
defines an exotic Gleason measure, because __\( \mu (Q) = 0 \)__ for
every projection __\( Q \)__ with finite-dimensional range.

On the other hand, it can be shown that if __\( \mathcal{H} \)__ is any
Hilbert space of nongigantic dimension greater than 2, then every
Gleason measure on __\( \mathcal{H} \)__ is standard. Solovay presents a
proof. (A consequence is that an exotic Gleason measure exists if and
only if a measurable cardinal exists.)

If __\( I \)__ is any set, gigantic or not, and __\( \rho \)__ is any probability
measure, continuous or not, defined on all
the subsets of __\( I \)__, then the formula above defines
a Gleason measure. Solovay’s main theorem says
that every Gleason measure is of this form.

__(Solovay.)__Let

__\( \mathcal{H} \)__be a nonseparable Hilbert space, and let

__\( \mu \)__be a Gleason measure on

__\( \mathcal{H} \)__. Then there is an orthonormal basis

__\( \{e_i : i \in I \} \)__of

__\( \mathcal{H} \)__and a probability measure

__\( \rho \)__on the subsets of

__\( I \)__such that

__\( \mu \)__is given by the formula above.

Observe that Gleason’s theorem is analogous; __\( \rho \)__
is a discrete probability measure on the integers;
the numbers __\( \rho (n ) \)__ are the eigenvalues, repeated
according to multiplicity, of the operator __\( D \)__.

Solovay also proves a beautiful formula giving a
canonical representation of a Gleason measure
__\( \mu \)__ as an integral over the set __\( \mathcal{T} \)__ of positive trace-class
operators __\( A \)__ of trace 1: there is a measure __\( \nu \)__
defined on all subsets of __\( \mathcal{T} \)__ such that, for all __\( P \)__,
__\[
\mu(P) =\int_{\mathcal{T}}\operatorname{trace}(AP) \,d\nu(A).
\]__
Moreover, there is a *unique* “pure, separated” measure __\( \nu \)__ such that the
formula above holds. These two technical terms mean that __\( \nu \)__ is similar
to the sort of measure that occurs in spectral multiplicity theory for
self-adjoint operators. The reader may enjoy proving this formula when
__\( \mathcal{H} \)__ is finite-dimensional; this simple case sheds some light on the
general case.

#### 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, __\( A \)__ and __\( B \)__. If there is a
correlation due to interaction when the systems are close, it can
certainly be maintained when they fly apart. His objection to standard
quantum mechanics was that in some cases a measurement of system __\( A \)__
*instantly* determines the result of a related measurement of
system __\( B \)__. Einstein dubbed this “weird action at a distance.”
Bohm’s model has this objectionable property.

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 __\( A \)__ and __\( B \)__ is the sum of the expectations: __\( E (A + B ) = E
(A) + E (B ) \)__, even if __\( A \)__ and __\( B \)__ don’t commute. This is a “miracle”
because the eigenvalues of __\( A + B \)__ have no relation to those of __\( A \)__ and __\( B \)__
unless __\( A \)__ and __\( B \)__ commute. It is true only because of the special formula
for expectations in quantum mechanics. It is not a “law of thought”.
Yet von Neumann postulated that additivity of expected values must
hold for all underlying hidden variable theories. That is the fatal
mistake in von Neumann’s argument. However, although Heisenberg
immediately understood Hermann’s argument when she spoke with him, her
work was published in an obscure journal and was forgotten for
decades.

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 __\( D \)__ is *dispersion-free*
provided __\( E (A^2 ) = E (A)^2 \)__ for any observable __\( A \)__. In other words, every
observation of __\( A \)__ has the value __\( E (A) \)__, its mean value. Quantum
mechanics is supposedly obtained by averaging over such states. The
frame functions considered by Bell correspond precisely to
dispersion-free states. But these frame functions are not continuous.
Gleason’s theorem implies that no such frame functions exist.
Therefore there are no dispersion-free states. But Gleason’s theorem
uses Mackey’s postulate of additivity of expectations for *commuting*
projections. Bell’s argument based on Gleason’s theorem avoids the
unjustified assumption of additivity of expectation values for
noncommuting operators.

Bell writes: “That so much follows from such apparently innocent
assumptions leads one to question their innocence.” He points out
that if __\( P \)__, __\( Q \)__, and __\( R \)__ are projections with __\( P \)__ and __\( Q \)__ orthogonal
to __\( R \)__ but not to each other, we might be able to measure __\( R \)__ and __\( P \)__, or
__\( R \)__ and __\( Q \)__, but not necessarily both, because __\( P \)__ and __\( Q \)__ do not commute.
Concretely, the two sets of measurements may well require different
experimental arrangements. (This point was often made by Niels Bohr.)
Bell expresses this fundamental fact emphatically: “The danger in fact
was not in the explicit but in the implicit assumptions. It was
tacitly assumed that measurement of an observable must yield the same
value independently of what other measurements are made
simultaneously.” In other words, the measurement may depend on its
context. This amounts to saying that Gleason’s frame functions may not
be well defined from the point of view of actual experiments.
Accordingly, one should examine Mackey’s apparently plausible
derivation that projection-valued measures truly provide part of a
valid axiomatization of quantum mechanics.

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.