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\title{On the Stone-von Neumann Uniqueness Theorem and Its Ramifications}
\author{Stephen J. Summers \\ Department of Mathematics \\ University of
Florida \\ Gainesville, Florida 32611, U.S.A. \\ e-mail: sjs@math.ufl.edu
\thanks{ To be published in \emph{John von Neumann and the 
Foundations of Quantum Mechanics} (tentative title), edited by M. Redei and M. 
Stoelzner.}}

\date{\normalsize{July, 1998}}

\begin{document}

\maketitle
{\abstract A brief history of the Stone-von Neumann uniqueness theorem
and its ramifications is provided. The influence of this theorem on the 
development of quantum theory, which was its initial source of motivation, is 
emphasized. In addition, its impact upon mathematics itself is suggested by 
considering certain subsequent developments in originally unanticipated
directions.}
\\[24pt]
\Large
{\bf 1. Introduction}
\\[18pt]
\normalsize

     In the mid to late 1920's, the emerging theory of quantum mechanics
had two main competing (and, initially, mutually antagonistic) formalisms --- 
the wave mechanics of E. Schr\"odinger~\cite{Schr1} and the matrix mechanics 
of W. Heisenberg, M. Born and P. Jordan~\cite{Heis2}\cite{BJ}\cite{BHJ}.\footnote{Significant portions of \cite{BJ} were obtained independently by P.A.M.
Dirac~\cite{Dir1}.}  Though a connection between the two was quickly pointed 
out by Schr\"odinger himself --- see paper III in \cite{Schr1} --- among 
others, the folk-theoretic ``equivalence'' between wave and matrix mechanics 
continued to generate more detailed study, even into our times. One outgrowth 
of this was associated with the canonical commutation relations (CCR):
%
\begin{equation}  
PQ - QP = \frac{h}{2 \pi i} \idty \; ,
\end{equation}  
%
which had begun to play such an important role in quantum theory 
\cite{Dir1}\cite{Heis2}\cite{BJ}\cite{BHJ} and were particularly central in
the matrix mechanics approach. 

     Schr\"odinger found a representation of (1) in the context of his wave 
mechanics in paper III of ~\cite{Schr1}. Given in modern language, his $Q$ is 
the multiplication operator
%
\begin{displaymath}
(Q\Psi)(x) = x\Psi(x) \; , \; x \in \RR \; , 
\end{displaymath}
%
on $L^2(\RR)$ and $P$ is the differential operator
%
\begin{displaymath}
(P\Psi)(x) = -i\frac{h}{2\pi}\frac{d\Psi}{dx}(x) \; , \; x \in \RR \; ,
\end{displaymath}
%
on $L^2(\RR)$. Born and Jordan~\cite{BJ} had found another with $P$ and $Q$ 
formal matrices with infinitely many entries. Jordan~\cite{J1} subsequently 
made a heuristic argument to the effect that these two representations of (1) 
are, in fact, equivalent in the sense described below. If that were indeed the 
case, it would be a very powerful confirmation that the physical content of 
matrix mechanics and wave mechanics coincided, since all physically relevant 
quantities can be expressed in terms of $P$ and $Q$. And that, in turn, would 
enable physicists to employ with confidence whichever approach was most 
convenient.

     However, much work remained to be done before this assertion could be
mathematically well-formulated and then proven rigorously. First, quantum
theory needed to be formulated in Hilbert space, a crucial step begun by D.
Hilbert himself \cite{HvN},\footnote{In the winter term of 1926-27, Hilbert 
gave lectures on the new quantum mechanics which were prepared in 
collaboration with his assistants, L. Nordheim and J. von Neumann. The notes 
of the lectures were written out by Nordheim, with von Neumann's assistance, 
and were published in \cite{HvN}.} made explicit by von Neumann in \cite{vN0}, 
and reached culmination in von Neumann's book \cite{vN2}. 

     Then, because there is no realization of $P$ and $Q$ satisfying (1) as 
bounded operators on Hilbert space~\cite{Wie}\cite{Win}\footnote{Of course, 
the founders of quantum theory did not have these later results, but they had 
realized that all of their examples were unbounded.}, one needed to address 
the fact that (1) could not be understood as an operator equation on all of 
Hilbert space. This difficulty was side-stepped by reformulating the 
problem~\cite{Weyl}: formally, if $P$ and $Q$ satisfy (1), then, with $U(a)$ 
and $V(a)$ defined by
%
\begin{displaymath}
U(a) = e^{\frac{-i2 \pi a P}{h}} \; \textnormal{and} \;
V(a) = e^{\frac{-i2 \pi a Q}{h}} \; , 
\end{displaymath}
%
it follows that, for any $a,b \in \RR$,
%
\begin{equation}
U(a)V(b) = e^{\frac{i2 \pi ab}{h}}V(b)U(a) \; .
\end{equation}
%
This is the Weyl form of the CCR for one degree of freedom. $U(a)$ and
$V(a)$ are, formally, unitary operators and therefore bounded; hence,
(2) \emph{may} be understood as an operator equation on all of the Hilbert
space of states. In the Schr\"odinger representation we have
%
\begin{equation}
(U(a)\Psi)(x) = \Psi(x-a) \; \textnormal{and} \; 
(V(b)\Psi)(x) = e^{\frac{-i2 \pi bx}{h}}\Psi(x) \; ,
\end{equation}
%
for any $\Psi \in L^2(\RR)$, and these are, indeed, unitary operators on
$L^2(\RR)$.

     In 1930, M.H. Stone~\cite{S1} stated\footnote{with an indication of 
the elements of a proof} and, in 1931, von Neumann~\cite{vN1} proved the 
following theorem. Note that a representation of the Weyl form of the CCR is 
said to be irreducible if the only subspaces of the Hilbert space $\Hs$ of 
states left invariant by the operators
$\{ U(a) \mid a \in \RR \} \cup \{ V(a) \mid a \in \RR \}$ are $\{ 0 \}$
and $\Hs$ itself.

\begin{theorem} If $\{ \tilde{U}(a) \mid a \in \RR \}$ and
$\{ \tilde{V}(a) \mid a \in \RR \}$ are (weakly continuous\footnote{As shown by
von Neumann~\cite{vN4}, this continuity assumption may be replaced by mere
weak Lebesgue-measurability as long as the Hilbert space is separable.}) 
families of unitary operators acting irreducibly on a (separable\footnote{Note 
that if the weakly continuous Weyl representation $\{\tilde{U},\tilde{V},\Hs\}$
is irreducible, then $\Hs$ must be separable.}) Hilbert space $\Hs$ such that
%
\begin{eqnarray*}
\tilde{U}(a)\tilde{U}(b) = \tilde{U}(a + b) \; & , & \;
\tilde{V}(a)\tilde{V}(b) = \tilde{V}(a + b) \; , \\
\tilde{U}(a)\tilde{V}(b) & = & e^{\frac{i2 \pi ab}{h}}\tilde{V}(b)\tilde{U}(a) 
\; , 
\end{eqnarray*}
%
then there exists a Hilbert space isomorphism\footnote{For Hilbert spaces, an 
isomorphism is a one-to-one linear norm-preserving transformation from one 
Hilbert space onto the other.} $W : \Hs \rightarrow L^2(\RR)$ such that 
%
\begin{displaymath}
W \tilde{U}(a) W^{-1} = U(a) \; \textnormal{and} \; 
W \tilde{V}(a) W^{-1} = V(a) \; , 
\end{displaymath}
%
for all $a \in \RR$, where $U(a)$ and $V(a)$ are the Weyl unitaries in the
Schr\"odinger representation defined in (3).\footnote{If $\Hs$ is not 
separable and $\tilde{U},\tilde{V}$ are not weakly continuous, then the 
conclusion of this portion of the theorem is false~\cite{Emch2}.} If 
$\{ \tilde{U}, \tilde{V}, \Hs \}$ is not irreducible but $\Hs$ is separable, 
then $\Hs$ decomposes into a direct sum of countably many closed subspaces, on 
each of which the restriction of $\{ \tilde{U}, \tilde{V} \}$ is once again 
unitarily equivalent to the Schr\"odinger representation $\{ U,V,L^2(\RR)\}$.
\end{theorem}

     Hence, every irreducible Weyl representation of the CCR for one degree of
freedom is unitarily equivalent to the Weyl form of the Schr\"odinger 
representation, and this is true, up to multiplicity, for reducible 
representations, as well. It therefore follows that the physical content of 
the irreducible representation $\{ \tilde{U}, \tilde{V}, \Hs \}$ is identical 
to that of the Schr\"odinger representation $\{ U, V, L^2(\RR) \}$. This 
theorem is usually referred to in the literature as the Stone-von Neumann 
uniqueness theorem.\footnote{Many authors refer to it simply as the 
von Neumann uniqueness theorem.}

     This discussion has been presented for one degree of freedom, but it 
may be reformulated for any finite number of degrees of freedom, and, there
again, any irreducible representation of the Weyl form of the CCR for $n$ 
degrees of freedom is unitarily equivalent to the corresponding Schr\"odinger 
representation (with analogous results in the reducible case) \cite{vN1}. 
Hence, if one is considering a quantum system with only finitely many degrees 
of freedom, then it matters not which representation one chooses to work in, 
and the ``equivalence'' of matrix mechanics and wave mechanics is even more 
tightly knit. This seemed to satisfy the founders of quantum mechanics, 
though, much later, mathematical physicists found some clouds in this 
apparently brilliant sky when they refocussed their attention on the dynamical 
variables $P$ and $Q$, as we shall see. 

     Returning to the unbounded operators $P$ and $Q$, it should be noted that 
F. Rellich~\cite{Rell}, followed by many authors (see \cite{Put}\cite{JorM} 
for references and recent results), provided sufficient conditions on the 
canonical conjugates $P$ and $Q$ in a representation of the CCR (1) which 
ensured that they are unitarily equivalent to the corresponding operators in 
the Schr\"odinger representation. The strategy ordinarily adopted was to find 
conditions on $P$ and $Q$ so that they may be exponentiated in such a way that
(2) holds, and then to appeal to Theorem 1. As a useful and representative 
result of this type, we mention J. Dixmier's theorem, once again stated here 
only for one degree of freedom.

\begin{theorem}[\cite{Dix}] Let $P$ and $Q$ be closed symmetric operators in a 
Hilbert space $\Hs$. Let $\Ds$ be a dense, linear subspace of $\Hs$ contained 
in the domains of both $P$ and  $Q$ such that $P\Ds \subset \Ds$ and
$Q\Ds \subset \Ds$. If (1) holds on $\Ds$ and the restriction of $P^2 + Q^2$ 
to $\Ds$ is essentially self-adjoint, then $\Hs$ decomposes into a direct sum 
of closed subspaces, on each of which the restrictions of $P$ and $Q$ are 
unitarily equivalent to the corresponding operators in the Schr\"odinger 
representation.
\end{theorem}

     On the other hand, there are many results (see \cite{Put} and \cite{Schm}
for references) to the effect that even if $P$ and $Q$ are essentially 
self-adjoint on a common invariant dense domain $\Ds$, on which they satisfy 
(1), they need \emph{not} be unitarily equivalent to the Schr\"odinger 
representation. In fact, K. Schm\"udgen~\cite{Schm} has produced an 
uncountable set of pairwise inequivalent representations of this type! Of 
course, by Theorem 1, when these operators are exponentiated, the resulting 
unitaries do \emph{not} satisfy (2). Are all of these examples physically 
pathological? And even if so, could there be others which are not? The answer 
to this latter question is positive. H. Reeh~\cite{Reeh} has provided such an 
example arising in the description of a charged particle in the exterior of an 
infinitely long cylinder with a magnetic flux running through it. This is 
therefore a physically meaningful representation of the CCR with finitely many 
degrees of freedom (two, after the idealization of letting the radius of the
cylinder go to zero) which is not unitarily equivalent to the corresponding
Schr\"odinger representation. Seventy years ago, this example would have been
a bombshell; however, now that the developments described in the next
section have accustomed us to the nonequivalence of physically relevant
representations, Reeh's example\footnote{There may well be other such
physically motivated examples in the literature; we apologize in advance
for not being aware of them.} was hardly noticed. Nonetheless, even 
physicists should be a bit more careful when they proclaim the equivalence of
the Heisenberg and Schr\"odinger representations in their quantum mechanics 
lectures. 

     We have been led to representations of the Weyl form of the CCR through 
the physically motivated interest in representations of conjugate $P$'s and 
$Q$'s. However, physically interesting applications have been found for 
representations $\{ U,V,\Hs\}$ in nonseparable Hilbert spaces which 
have no connection with unbounded operators satisfying (1) at all  --- see the 
recent preprint~\cite{CMS} for references. In such representations the 
functions $a \mapsto U(a)$ and $a \mapsto V(a)$ are not weakly continuous; 
these representations are called nonregular. In \cite{CMS} is given a 
generalization of Theorem 1 to the case of weakly measurable nonregular 
representations, which is sufficient to subsume the known physical models. We 
shall say no more about this interesting line of development here.  

     In this introduction, the mathematical level of the discussion has been
deliberately held low. This will not be possible in the
balance of the paper. We shall first consider the consequences of the fact that
the analogue of Theorem 1 for infinitely many degrees of freedom is false; 
indeed, in that case, there is an enormously infinite number of unitarily 
inequivalent representations of the CCR in the Weyl form and, therefore, also
of the original CCR. This fact was only 
slowly and painfully realized, because physicists chose to ignore the 
restriction in the hypothesis of the Stone-von Neumann uniqueness theorem. We 
shall indicate how this obduracy was overcome and what mathematical 
physicists have discovered in their exploration of this rich set of 
inequivalent representations in both its mathematical and physical aspects. 
We shall also discuss the correct generalization of Theorem 1 to infinitely
many degrees of freedom. Finally, to trace another line of influence of the 
Stone-von Neumann uniqueness theorem, we shall briefly describe certain 
generalizations and their role in the harmonic analysis of locally compact 
groups, which has found particular application in such diverse fields as 
number theory, imaging science, communication theory and data/signal analysis.
However, given the limitations of space imposed upon us, we have here no 
ambitions of completeness. 
\\[24pt]
\Large
{\bf 2. Infinitely Many Degrees of Freedom}
\\[18pt]
\normalsize

     Though the hypothesis of Theorem 1 clearly restricts its import to 
finitely many degrees of freedom and close examination of von Neumann's 
proof makes it evident that the argument loses its mathematical validity
when extended to infinitely many degrees of freedom, physicists have always
trusted their physical ``intuition'' more than mathematical proof. Indeed,
that which a physicist calls a proof is often viewed by a mathematician as a 
plausibility argument, at best. Physicists are, however, often justified in 
not waiting for the mathematicians, whose concern for rigor they regard with
impatience, to firmly bolster the physicists' ideas. If they were to do so, 
the natural sciences would not have advanced as rapidly as they have. 
Significantly, physicists have a source of conviction which mathematicians do 
not: mathematically unconstrained speculations can be checked, to a certain 
extent, in the laboratory. Nonetheless, important aspects of physicists' 
theories of nature --- their attempts to formalize the physical intuitions 
gleaned from the complex feedback loop between theory and experiment --- have 
often enough either remained vague or revealed themselves to be incorrect, if 
not nonsensical. 

     An example of this is the physicists' long-lived belief, based upon their
experience with systems having finitely many degrees of freedom and the
Stone-von Neumann uniqueness theorem, that the choice of representation of the 
CCR was merely a matter of convenience --- one only needed to keep track of the
number of degrees of freedom. It was realized quite early that quantum field 
theory necessitated infinitely many degrees of freedom in its canonical 
variables (see already \cite{Dirac}). When dealing with infinitely many 
degrees of freedom, they worked exclusively in the representation of the CCR
associated with a Hilbert space containing a dense set of states describing 
only finitely many particles. This representation emerged heuristically in the 
first papers on quantum field theory by Heisenberg and W. Pauli~\cite{HP} and 
was later formalized more completely by V. Fock~\cite{Fock} (see \cite{Cook} 
for the first mathematically rigorous and Poincar\'e covariant presentation of 
this representation, now usually called the Fock\footnote{or Fock-Cook, among 
mathematicians} representation). Since the Fock representation, using 
annihilation and creation operators and a distinguished vacuum vector, is so 
well-known, and it is equally well-known that the Schr\"odinger representation 
can be re-written as a Fock representation with only finitely many 
annihilation and creation operators, we shall not interrupt the flow of our 
story with the details (but see \cite{BratRob} or \cite{Emch}, if necessary). 

     The Fock representation was therefore viewed as the natural 
generalization of the Schr\"odinger representation to infinitely many degrees 
of freedom and inherited its royal mantle of distinction. Hence, quantum 
field models were written in the Fock representation by theoretical physicists,
insofar as a representation was actually specified, with the firm belief
that it was the only representation they needed.           

     It is an interesting aside that von Neumann apparently did not appreciate
systems of infinitely many degrees of freedom. He wrote in his treatment of
radiation in \cite{vN2}:
%
\begin{quote}
Nun ist es formal unbequem und bedenklich, Systeme mit unendlich vielen
Freiheitsgraden bzw. Wellenfunktionen mit unendlich vielen Argumenten
zuzulassen.\footnote{Now it is inconvenient formally and of doubtful validity 
to admit systems with infinitely many degrees of freedom, or wave functions 
with infinitely many arguments. --- See page 141, resp. page 265, of 
\cite{vN2}.}
\end{quote}
%
In what is effectively the Fock representation, he therefore considered $N$ 
degrees of freedom, computed energy spectrum, and then let 
$N \rightarrow \infty$. In order to compute this spectrum, von Neumann 
performed a canonical transformation\footnote{in mathematical terms, a 
symplectic transformation} to obtain a second representation of the CCR in 
which the transformed Hamiltonian has a simpler form. For finite $N$ this 
transformation is, by Theorem 1, unitarily implementable. However, in the 
limit $N \rightarrow \infty$ the transformation is not unitarily 
implementable and the representations are unitarily inequivalent. In other 
words, without realizing it, von Neumann himself worked with unitarily 
inequivalent representations of the CCR. His argument about the energy 
spectrum is therefore suspect.\footnote{We return to this point below.}     

     From the very beginning of the subject, quantum field theory was
plagued by divergences; when one source of infinity was heuristically
taken care of, yet another was stumbled upon. This became such an apparently
insurmountable problem, that some of the founders became quite pessimistic
(particularly Bohr and Dirac) and decided that yet another conceptual 
revolution would be required to transcend quantum field theory and avoid its 
apparently inherent problems. However, some researchers had not yet given up 
on the possibility of getting sensible answers from quantum field theory and 
were trying to discern and then engage the various sources of these infinities 
from increasingly profound starting points. 

     Of direct relevance to our story, L. van Hove 
examined a simple model and argued that the origin of the divergences of 
perturbation theory (which is always carried out in Fock space) could be 
located in the fact that the state vectors of the interacting model were 
``orthogonal'' to the state vectors in Fock space. In modern terms, what he 
argued was that the folium of states\footnote{\emph{i.e.} the set of states 
determined by the density matrices on the Hilbert space of the given 
representation} of the interacting model was disjoint from the folium of states
of the Fock representation. An immediate consequence of this observation would 
have been that the interacting representation for his model was unitarily 
inequivalent to the Fock representation. He did not quite get to this 
point.\footnote{It is of interest to note that van Hove perceived a connection 
between his model and von Neumann's infinite product spaces~\cite{vN3}. With 
the benefit of hindsight, we see that he was anticipating the theory of 
infinite product representations of the CCR~\cite{KMW}.}

     Also in the early 1950's, K.O. Friedrichs~\cite{Fried} undertook an 
influential attempt to reduce the hand-waving typical of quantum field theory
up to that time. For our purposes here, the result of greatest interest was
his construction of some representations of the CCR for infinitely many
degrees of freedom which were \emph{not} unitarily equivalent to the Fock
representation. As he wrote:
%
\begin{quote}
Accordingly, there are different --- non-equivalent --- realizations of
the basic field operators, and consequently different --- non-equivalent ---
kinds of fields, a fact which seems worth noticing.\footnote{See p. 3 of 
\cite{Fried}.}
\end{quote}
%
In point of fact, he constructed representations in which the number operator 
does not exist (cf. the discussion further below).\footnote{It would seem
that Friedrichs was not aware of either van Hove's example nor von Neumann's 
paper \cite{vN3} when he did this work; they were mentioned only in the 
Comments and Corrections at the very end of the book \cite{Fried}, added after 
the work had been completed.}
 
     Though it would appear that not many theoretical physicists did take 
notice of Friedrichs' results, at least a handful of mathematical physicists 
and mathematicians were paying attention. In particular, in the following year
L. G\aa rding and A.S. Wightman~\cite{GaWi}, taking their cue 
from Friedrichs and trying to classify representations of the CCR using 
properties of a number operator,\footnote{They also made use of von Neumann's
paper on infinite products.} proved that there exists a large class of 
inequivalent representations of the CCR for infinitely many degrees of 
freedom.\footnote{As straightforward an operation as multiplying all the 
$P$'s by 2 and all the $Q$'s by $\frac{1}{2}$ produces a representation of
the CCR which is unitarily inequivalent to the initial representation.} 
Indeed, it slowly emerged that there exists an unimaginably 
infinite number of inequivalent representations --- the space of unitary
equivalence classes of such representations cannot even admit a separable
Borel structure~\cite{MacB}\cite{Glimm}. The task of classifying these
representations would thus appear to be hopeless.

     Another researcher who reacted to the examples of van Hove and Friedrichs 
was R. Haag. Aware of these preceding works, he presented an argument to the 
effect that the interaction representation, widely in use in quantum field 
theory on the basis of its prior success in quantum mechanical scattering 
theory, did not exist unless there was no interaction at all! This important 
assertion found a number of mathematically rigorous formulations and proofs, 
which can, perhaps, be summarized into two types, represented in \cite{SW} and 
\cite{Emch}. We state Haag's theorem in a somewhat restricted form along the 
lines of \cite{SW}. 

\begin{theorem}[Haag's Theorem] Let $\phi(x)$ be a free hermitian scalar
field\footnote{and therefore irreducible, local and Poincar\'e-covariant} 
of mass $m > 0$, and let $\psi(x)$ be an irreducible local
Poincar\'e-covariant field. If $\phi(x)$ and $\psi(x)$, resp. the canonical
conjugates $\dot{\phi}(x)$ and $\dot{\psi}(x)$, are unitarily equivalent at 
some time $t$, then $\psi(x)$ is also a free field of mass $m$.
\end{theorem}
  
     Of course, the indicated hypothesis holds for the ``free'' field and the
``interacting'' field in the interacting representation. This was extremely
inconvenient for the then-standard scattering theory for quantum fields.
But it is clear that Haag's theorem also implies that the representations 
which are of physical interest, precisely \emph{because} they involve 
interaction, are to be found among those \emph{inequivalent} to the Fock 
representation. Therefore, by 1955, both the existence and the necessity of 
using representations inequivalent to the Fock representation had been firmly 
established --- though not established in all theorists' minds: as late as  
1961, a standard text on quantum field theory~\cite{Schw} could present the
old scattering theory in the interaction representation with no mention of 
Haag's theorem\footnote{and yet still cite \cite{Haag} for other purposes!}. 

     Before we turn to a recounting of the progress made in constructing
representations inequivalent to the Fock representation, we answer the
natural question: which representations \emph{are} equivalent to the
Fock representation? It was evident to Friedrichs that a necessary
condition for this equivalence is the existence of a number operator in
the representation. A series of papers followed Friedrichs' lead and gave 
successively more general, rigorously proven content to the assertion 
``a representation of the CCR is unitarily equivalent to the Fock 
representation if and only if the number operator exists as a 
densely-defined self-adjoint positive operator in the representation.''
However, as was emphasized by J.M. Chaiken~\cite{Chai1}, this result is
very sensitive to the definition of ``number operator''.\footnote{See
\cite{BratRob} for further references and \cite{Schaf} for a recent paper
on those representations which have a ``generalized number operator''.}

     The work of G\aa rding and Wightman did not provide an explicit 
construction of inequivalent representations. Wightman and S.S. 
Schweber~\cite{WS} later constructed some classes of inequivalent 
representations of the CCR, as did I.E. Segal (see a later 
account~\cite{Seg} and the references given there). Many further classes of 
inequivalent representations have been constructed and brought under 
mathematical control since then. We mention the infinite product 
representations~\cite{KMW}, coherent representations \cite{Klau}, quasi-free 
representations \cite{Rob}, quadratic representations~\cite{PRS} and 
higher-order representations~\cite{FS}. These various classes of 
representations have found physical application and will surely prove to be of 
further use in the future. 

     But the most ambitious and difficult constructions of representations
of the CCR have been carried out under the rubric ``constructive quantum
field theory.''  This work was motivated by the desire to mathematically 
construct the sort of representations the quantum field theorists were
tacitly referring to; in other words, to give some mathematical meaning
to the quantum field models at the center of the theorists' discourse. This 
latter goal has been approached from two different directions --- on the one 
hand, various axiom systems have been erected which hope to subsume basic 
principles common to large classes of quantum fields: then theorems are proven 
to establish physically interesting properties of all quantum fields 
satisfying the given axioms; and on the other hand, concrete models have been 
constructed to show that the axiom systems are not vacuous: of course, in this 
connection, valiant efforts have been made to construct the standard models of 
the quantum field theorists. The axiomatic approach will not be further 
discussed here.\footnote{We refer the interested reader to 
\cite{SW}\cite{HaagB} and also, for a historical overview, \cite{Wight}.} 
Instead, we shall briefly indicate those results of constructive quantum field 
theory which are of direct relevance to the topic at hand. 

     We first discuss J. Glimm and A. Jaffe's construction of the 
$(\phi^4)_2$-model \cite{GJ1}. Let $\Hs_0$ be the Fock space for a scalar 
hermitian Bose field $\phi(x,t)$ of mass $m > 0$. Let 
$\pi(x,t) = \partial \phi(x,t)/\partial t$ and $\Ds\subset\Hs_0$ be the dense 
set of finite-particle vectors in $\Hs_0$. Then, for every $f$ in a dense 
subspace $\Ss(\RR)$ of $L^2(\RR)$, the operator 
$\phi(f) \equiv \phi(f,0) = \int\phi(x,0)f(x)dx$ is essentially self-adjoint 
on $\Ds$ and $\phi(f)\Ds \subset \Ds$ (similarly for $\pi(f)$). Then one has
on $\Ds$ the CCR\footnote{As is customary in quantum field theory, we adopt
physical units in which $c = h/2\pi = 1$.}
%
\begin{eqnarray}
\phi(f)\pi(g) - \pi(g)\phi(f) & = & i<f,g>\idty \; , \\
\phi(f)\phi(g) - \phi(g)\phi(f) = & 0 & = \pi(f)\pi(g) - \pi(g)\pi(f) \; ,
\end{eqnarray}
%
for all $f,g \in \Ss(\RR)$. When exponentiated, these operators provide a Weyl 
representation of the CCR. For each bounded open subset $\bf{O}\subset\RR$,
denote by $\As(\bf{O})$ the von Neumann algebra\footnote{so-called because 
they were introduced in \cite{MvN}} generated by the Weyl unitaries
%
\begin{displaymath}
\{e^{i\phi(f)},e^{i\pi(f)} \mid f \in \Ss(\RR) \; , \; 
\textnormal{supp}(f) \subset \bf{O}\} \; .
\end{displaymath}
%
Note that, though there are many $C^*$-algebras associated with the CCR in the
Fock representation,\footnote{As opposed to the case of the canonical 
anticommutation relations, the $C^*$-algebra obtained here depends upon the 
choice of the dense subspace $\Ss(\RR)$ of test functions --- see 
\cite{BratRob} for a detailed discussion of this point. However, once the
dense subspace of test functions has been fixed, the closure $\As_0$ in the
operator norm of the algebra generated by the set 
$\{e^{i\phi(f)},e^{i\pi(f)} \mid f \in \Ss(\RR) \}$ has the property that to
each, not necessarily continuous representation of the CCR, (4)-(5), there 
corresponds a representation of $\As_0$ \cite{Slaw}. Moreover, $\As_0$ is
simple \cite{Man}\cite{Slaw}, so it is representation-independent --- see the
discussion further below. } they all have the same weak closure. 

     The total energy 
%
\begin{displaymath}
H_0 = \frac{1}{2}\int \ :(\pi(x,0)^2 + \nabla\phi(x,0)^2 + m^2\phi(x,0)^2): dx
\end{displaymath}
%
of this field is a positive quadratic form on $\Ds \times \Ds$ and 
therefore determines uniquely a self-adjoint operator, which we also denote by 
$H_0$. With $g \in L^2(\RR)$ nonnegative of compact support,  
Glimm and Jaffe showed that, for each $\lambda > 0$, the cut-off interacting 
Hamilton operator
%
\begin{displaymath}
H(g) \equiv H_0 + \lambda \int :\phi(x,0)^4: g(x) dx
\end{displaymath}
%
is essentially self-adjoint on $\Ds$,\footnote{Without the cutoff $g$, 
the interacting Hamilton operator is \emph{not} densely defined in Fock space.}
and its self-adjoint closure, also denoted by $H(g)$, is bounded from below. 
By adding a suitable multiple of the identity we may take $0$ to be the 
minimum of its spectrum. Then, $0$ is a simple eigenvalue of $H(g)$ with 
normalized eigenvector $\Omega(g) \in \Hs_0$. 

     For any $t \in \RR$, let ${\bf O}_t$ denote the subset of $\RR$ consisting
of all points with distance less than $\vert t \vert$ to $\bf{O}$. By choosing
the cutoff function $g$ to be equal to $1$ on ${\bf O}_t$, then for any 
$A \in \As({\bf O})$ the operator
%
\begin{displaymath}
\sigma_t(A) \equiv e^{itH(g)}Ae^{-itH(g)}
\end{displaymath}
%
is independent of $g$ and is contained in $\As({\bf O}_t)$. For any bounded 
open $\Os \subset \RR^2$ and $t \in \RR$, let
${\bf O}(t) = \{ x \in \RR \mid (x,t) \in \Os \}$ be the time $t$ slice of 
$\Os$. We define $\As(\Os)$ to be the von Neumann algebra generated by
$\bigcup_s \sigma_s(\As({\bf O}(s)))$.\footnote{One can 
then show that the algebra $\As(\Os)$ coincides with the von Neumann algebra 
generated by bounded functions of the self-adjoint field operators 
$\int\phi(x,t)f(x,t)dxdt$, with test functions $f(x,t)$ having support in 
$\Os$.}  Finally, we let $\As$ denote the
closure in the operator norm of the union $\bigcup \As(\Os)$ over all
open bounded $\Os\subset\RR^2$. Hence, $\sigma_t$ is an automorphism on $\As$
and implements the time evolution associated with the interacting field. 
Similarly, ``locally correct'' generators for the Lorentz boosts and the 
spatial translations can be defined, resulting in an automorphic action on 
$\As$ of the entire Poincar\'e group in two spacetime dimensions.   

     For each $A \in \As$, we set $\omega_g(A) = <\Omega(g),A\Omega(g)>$
to define the locally correct vacuum state $\omega_g$ of the interacting 
field. Taking a limit as the cutoff function $g$ approaches the constant 
function $1$, Glimm and Jaffe showed that $\omega_g(A) \rightarrow \omega(A)$, 
for each $A \in \As$, defines a new (locally normal) state $\omega$ on $\As$ 
which is Poincar\'e invariant. By the GNS construction one then obtains a new 
Hilbert space $\Hs$, a representation $\rho$ of $\As$ as a $C^*$-algebra 
acting on $\Hs$, and a vector $\Omega \in \Hs$ such that $\rho(\As)\Omega$ is 
dense in $\Hs$ and
%
\begin{displaymath}
\omega(A) = <\Omega,\rho(A)\Omega> \; , \; \textnormal{for all} \; A \in \As
\; . 
\end{displaymath}
%
In addition, one obtains a strongly continuous unitary representation of the 
Poincar\'e group in two spacetime dimensions under which the algebras 
$\rho(\As(\Os))$ transform covariantly. The axioms of both the 
algebraic~\cite{HaagB} and the field approach~\cite{SW} have been verified for 
this model. 

     It is in this representation $(\rho,\Hs)$ that the field equations for 
this model find a mathematically satisfactory interpretation~\cite{Schr}. And 
it is to the physically significant quantities in this representation that 
perturbation theory in $\lambda$ is asymptotic --- see the discussion in 
\cite{GJ}. For this and other reasons, $\omega$ is interpreted as the exact 
vacuum state in the interacting theory, and its folium of states contains the 
physically admissible states of the interacting theory.

     The generators of the strongly continuous Abelian unitary groups \newline
$\{ \rho(e^{it\phi(f)}) \mid t \in \RR \}$ and 
$\{ \rho(e^{it\pi(f)}) \mid t \in \RR \}$ satisfy the CCR (4). However, this 
representation of the CCR in $\Hs$ is not unitarily equivalent to the initial 
representation in Fock space. Indeed, by taking different values of the 
coupling constant $\lambda$ in the above construction, one obtains an 
uncountably infinite family of mutually inequivalent representations of the 
CCR (4) (see \cite{Froh})!\footnote{Of relevance 
to quantum field theory, but particularly to quantum statistical mechanics, 
which also must face systems with infinitely many degrees of freedom 
\cite{BratRob}, is the observation that also equilibrium states at different 
temperatures are associated with mutually inequivalent representations 
\cite{Tak}. The Fock representation is associated with temperature zero.}

     Similar constructions with similar results have been carried out for
general polynomial interactions $P(\phi)$ and for the Yukawa model, both in two
spacetime dimensions. For the sake of technical convenience, these 
constructions were redone in the Euclidean approach, and many additional
models were constructed in that manner.\footnote{For an introduction to this
work, as well as further references, see \cite{GJ}.} Although the program of 
constructing the standard models of quantum field theory in four spacetime 
dimensions is not completed, the lessons taught by the constructions of 
interacting field models in lower dimensions cannot be overlooked. In
particular, quantum fields with different interactions are associated with 
mutually inequivalent representations of the CCR, which are in turn 
inequivalent to the Fock representation. The choice of representation would
thus appear to be quite significant. Tersely summarized, one could say that
the kinematics of the physical system fixes the CCR-algebra and the dynamics
determines the representation.\footnote{In fact, in a certain sense, the
representation also determines the dynamics --- see \cite{Ar}.}

     Though the original problem was stated in terms of unitary equivalence,
there are, in fact, weaker notions of equivalence which are also of physical
relevance. A notion of physical equivalence introduced by Haag and 
D. Kastler~\cite{HK} will be discussed next.

     Since one can carry out only finitely many experiments which themselves 
have only a finite accuracy, the experimental situation strictly limits our
ability to test the many idealizations which are implicit in any physical
theory and which are particularly strongly present in quantum mechanics
and quantum field theory. These limitations on measurement and the statistical 
interpretation of the basic objects in the theory induce a natural topology on 
the set of states on the algebra of observables $\As$. Let 
$\{A_i\}_{i=1}^n \subset \As$ be a set of observables of a system which has 
been prepared in the state $\omega$. Let $\{a_i\}_{i=1}^n$ be their measured 
average values to within the (respective) errors $\{\epsilon _i\}_{i=1}^n$. 
Hence one has the $n$ inequalities
%
\begin{displaymath}
\vert \omega (A_i) - a_i \vert < \epsilon_i \; , \; i = 1, \ldots ,n \; .  
\end{displaymath}
%
On the other hand, recall that the $\sigma (\As ^*,\As)$-topology on the set 
$\As ^*$ of all continuous, linear complex-valued functions on $\As$ is 
generated by the seminorms $N_A (\omega) \equiv \vert \omega (A) \vert$,
for each $A \in \As$. In other words, the $\sigma (\As ^*,\As)$-topology is 
the locally convex topology with basis of neighborhoods at the origin given by
%
\begin{displaymath}
\{ \Ns_{\{A_i\}_{i=1}^n , \{\epsilon _i\}_{i=1}^n} \mid n \in \IN,
\{ A_i \}_{i=1}^n \subset \As , \{\epsilon _i\}_{i=1}^n \subset 
(0,\infty)\} \; ,
\end{displaymath}
%
where 
%
\begin{displaymath}
\Ns_{\{ A_i \}_{i=1}^n , \{\epsilon _i\}_{i=1}^n} = 
\{\omega \in \As ^* \mid \vert \omega (A_i)\vert < \epsilon_i, i = 1, \ldots 
,n\} \; .
\end{displaymath}
%
Thus we see that any experiment (or set of experiments, necessarily 
finite) determines the state of the system only up to a neighborhood in the
$\sigma (\As ^*,\As)$-topology. 

     For purely mathematical reasons, J.M.G. Fell introduced the following
notion of equivalence of representations. If $(\rho,\Hs)$ is a representation
of $\As$, then its kernel is given by 
$\textnormal{Ker}(\rho) = \{ A \in \As \mid \rho(A) = 0 \}$.

\medskip

\noindent {\bf Definition}({\bf\cite{Fell}}) \emph{Two representations} 
$(\rho_1,\Hs_1)$ \emph{and} $(\rho_2,\Hs_2)$ of $\As$ \emph{are said to be 
weakly equivalent if} $\textnormal{Ker}(\rho_1) = \textnormal{Ker}(\rho_2)$.

\medskip

Unitary equivalence implies weak equivalence, but the converse is false. 
Fell showed that two representations $(\rho_1,\Hs_1)$ and $(\rho_2,\Hs_2)$ of 
$\As$ are weakly equivalent if and only if given every state $\omega_1$ on 
$\As$ determined by a density matrix on $\Hs_1$ and given any 
$\sigma (\As ^*,\As)$-neighborhood $\Ns$ of $\omega_1$, there exists a state 
$\omega_2 \in \Ns$ determined by a density matrix on $\Hs_2$. In other words, 
the $\sigma(\As^*,\As)$-closure of the folium of states associated with 
$(\rho_1,\Hs_1)$ coincides with the $\sigma(\As^*,\As)$-closure of the folium 
associated with $(\rho_2,\Hs_2)$.

     Therefore, if the kernels of two representations of $\As$ coincide,
then it is physically impossible to determine which representation one is
in (and conversely)! But if $(\rho, \Hs)$ is a representation of $\As$, then 
Ker($\rho$) is a norm-closed two-sided ideal of $\As$. Thus it follows that 
whenever $\As$ is simple, every representation of $\As$ must be faithful, and 
hence all representations of a simple algebra $\As$ are physically equivalent. 
What is more, in quantum field theory the quasilocal algebras $\As$ 
are typically simple!

     So, have we returned to the physicists' original point of view - the 
choice of representation is just a matter of convenience, even in systems
with infinitely many degrees of freedom? Not exactly! Let us posit, once 
again, that we have chosen observables $\{A_i\}_{i=1}^n \subset \As$ and made 
measurements with results $\{a_i\}_{i=1}^n$ to within errors 
$\{\epsilon _i\}_{i=1}^n$, thereby determining a 
$\sigma (\As ^*,\As)$-neighborhood $\Ns$ of the actual state $\omega$, normal 
in the true physical representation $(\rho,\Hs)$. If $\As$ is simple, then 
Fell's theorem entails that we can find a state $\rho_{\Ns}$, normal in any 
other fixed representation of $\As$, which is contained in $\Ns$ and therefore 
yields predictions conforming with the results of this experiment. But the 
moment we improve the experiment, \emph{i.e.} reduce the errors, or we change 
the experiment to include another set of observables (but still preparing the 
system in the same original state), then the neighborhood $\Ns$ changes 
(though $\omega$ does not change), and we must find \emph{another} approximate 
state in the given ``wrong'' representation to reproduce the results of the 
new experiment. In other words, in order to have (correct) predictive power 
beyond the particular experiments to which one fitted the approximate state, 
one \emph{needs} the correct state in the correct representation. This, surely,
is not merely a matter of convenience!\footnote{It is evident from this 
discussion that it is impossible to prove experimentally that a putative
exact state is the correct one (and, thus, that the correct representation
has been chosen). But at least it is logically possible to establish 
experimentally that it is \emph{not} the correct one (if, in fact, it is 
not).} 

     J. Manuceau's~\cite{Man} and J. Slawny's~\cite{Slaw} observation that the 
minimal $C^*$-algebra $\As_0$ associated with the CCR is simple and hence
all representations of $\As_0$ are isomorphic can be seen as the \emph{correct}
generalization of Theorem 1 to infinitely many degrees of freedom. The
existence of this algebraic isomorphism implies unitary equivalence in the
finite case but \emph{not} in the infinite case. This independence of
representation enables the rigorous study of the canonical transformations
commonly employed in theoretical physics (by von Neumann himself ---
see further above) as automorphisms on the $C^*$-algebra $\As_0$. In a 
given representation of $\As_0$, the given automorphism may or may not
be unitarily implementable. If it is, then the original Hamiltonian operator
will have the same spectrum as the transformed (diagonalized) one; if not,
then there need be no relation between the spectra of these operators.  

     To close this section, we remark that the significance of the 
Stone-von Neumann uniqueness theorem is further emphasized by the fact that
for the other important types of algebraic relations --- such as the 
canonical anticommutation relations and, more recently, supersymmetric
commutation relations, $p$-adic commutation relations, and the deformed
commutation relations of quantum groups --- one of the first questions
addressed is the validity of the counterpart of Theorem 1 in the given 
setting. For further reading, we mention the papers 
\cite{Schm2}\cite{Koch}\cite{GP}\cite{Ior}.
\\[24pt]
\Large
{\bf 3. Generalizations to the Harmonic Analysis of Locally Compact Groups}
\\[18pt]
\normalsize

     In 1949, G.W. Mackey~\cite{Mac1} provided a generalization of the
Stone-von Neumann uniqueness theorem to the setting of locally compact groups,
which itself found many applications in mathematics and elsewhere and which
may justifiably be seen as yet another impact of von Neumann's work. For 
simplicity, we shall restrict our attention to Abelian groups, though Mackey 
formulated and proved an analogous result for arbitrary locally compact 
groups. With $G$ a locally compact Abelian group and $\mu$ a Haar measure on 
$G$, one can naturally define the Hilbert space $L^2(G,d\mu)$. If $G^*$ is the 
topological character group of $G$, \emph{i.e.} each $\tau \in G^*$ is a 
continuous homomorphism from $G$ into the multiplicative group of complex 
numbers of modulus 1, and $G^*$ is endowed with the natural induced topology 
(so that it, too, becomes a locally compact Abelian group), then the analogue 
of the Weyl form of the Schr\"odinger representation is described by the 
following unitary operators on $L^2(G,d\mu)$:
%
\begin{equation}
(U_S(g)\Psi)(x) = \Psi(g^{-1}x) \; \textnormal{and} \; 
(V_S(\tau)\Psi)(x) = \tau(x)\Psi(x) \; ,
\end{equation}
%
for any $\Psi \in L^2(G,d\mu)$. (Compare with (3).)
   

\begin{theorem}[\cite{Mac1}] Let $G$ be an arbitrary 
separable\footnote{Loomis~\cite{Loo} 
later showed that the assumption of separability of $G$ could be dropped.} 
locally compact Abelian group, and let $G^*$ be its topological 
character group. Let $U$ be a weakly continuous representation of $G$ in the 
(separable) Hilbert space $\Hs$. If $V$ is a weakly continuous representation 
of $G^*$ on $\Hs$ such that $U(g)V(\tau) = \tau(g)V(\tau)U(g)$, for all 
$g \in G$ and $\tau \in G^*$, then $\Hs$ decomposes into a direct sum 
of at most countably many closed subspaces $\Hs_n$, each invariant under
$\{ U(g) \mid g \in G \} \cup \{ V(\tau) \mid \tau \in G^* \}$. 
Moreover, letting $U_n$, resp. $V_n$, denote the restriction of $U$, resp. $V$,
to $\Hs_n$, there exists a Hilbert space isomorphism 
$W_n : \Hs_n \rightarrow L^2(G,d\mu)$ with
%
\begin{displaymath}
W_n U_n(g) W_n^{-1} = U_S(g) \; \textnormal{and} \; 
W_n V_n(\tau) W_n^{-1} = V_S(\tau) \; , 
\end{displaymath}
%
for all $g \in G$ and $\tau \in G^*$.
\end{theorem}

\nind Subsequently, arguments which were more elementary than Mackey's 
original proof were found, as well as some additional reformulations - see, 
\emph{e.g.} \cite{SK}\cite{Rie1}. Theorem 4 is often called the 
Stone-von Neumann-Mackey theorem.\footnote{It may be of interest to note that 
Theorem 4 was evoked in J. Slawny's proof \cite{Slaw} of the existence and 
properties of the minimal $C^*$-algebra associated with the CCR, which was 
mentioned in the previous section.} It has been placed by Mackey into the 
context of his theory of induced representations and there was seen to be a 
consequence of his imprimitivity theorem. The interested reader is referred to 
\cite{MacBo} for an introduction to this circle of ideas.

     Theorem 1 is obtained as a special case of Theorem 4 by choosing $G$ to 
be the additive group of reals (for more than one degree of freedom, $G$ is 
chosen to be the additive group of vectors $\RR^n$). Note that, in that
case, $G^*$ is isomorphic to $G$ itself.

     From Theorem 4 follows one of the most useful theorems in Abelian
harmonic analysis, which is in turn a generalization of the crucial
Plancherel theorem in Fourier analysis (a special instance of Abelian harmonic
analysis). (See \cite{SK} for a proof.)

\begin{theorem} Let $G$ be a locally compact Abelian group. Given any
element $f \in L^1(G,d\mu) \cap L^2(G,d\mu)$, its Fourier transform $\hat{f}$,
defined by
%
\begin{displaymath}
\hat{f}(\tau) = \int \tau(g) f(g) d\mu(g) \; , 
\end{displaymath}
%
is in $L^2(G^*,d\mu^*)$, and the mapping $f \mapsto \hat{f}$
extends uniquely to a Hilbert space isomorphism from $L^2(G,d\mu)$ onto 
$L^2(G^*,d\mu^*)$ (with suitable normalization of the Haar measure $\mu^*$).
\end{theorem}

     From this then follows the generalized Riemann-Lebesgue lemma: the
Fourier transform of an integrable function on a locally compact Abelian group
$G$ vanishes at infinity on $G^*$. It is surely evident by now how central
a result the Stone-von Neumann-Mackey theorem is in Abelian harmonic analysis.

     To take yet another perspective on this topic, consider the 
$n$-dimensional Heisenberg group, which is the universal covering 
group of the non-Abelian group of unitary operators on $L^2(\RR^n)$ generated 
by the translations
%
\begin{displaymath}       
T_p f(x) = f(x + p) \; , \; p \in \RR^n \; ,
\end{displaymath}       
%
and the multiplications
%
\begin{displaymath}       
M_q f(x) = e^{iq \cdot x} f(x) \; , \; q \in \RR^n \; .
\end{displaymath}       
%
It is evident from the discussion in the introduction that the 
Stone-von Neumann uniqueness theorem may be used to classify the irreducible
representations of the Heisenberg group. This again permits the proof of 
a corresponding Plancherel theorem, \emph{etc.}. We refer the reader to
\cite{Tay} for a development of this theory, as well as indications of the
many sorts of applications which have arisen. Here we only mention one
buzzword: wavelets.     

     Finally, just to hint at further realms, we mention that Theorem 1 has 
also found applications to number theory (see, for example, \cite{Cart}),
function theory (see \cite{Schm3}) and invariant subspace theory(see 
\cite{Hel}).
 
\bigskip

\noindent {\bf Acknowledgements}: We have found \cite{Emch}
and \cite{Wight} particularly useful during the preparation of this paper
and further recommend them to the reader's attention.\\



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