A[arabic] B[alph]

# Von Neumann’s 1927 Trilogy on the Foundations of Quantum Mechanics.
Annotated Translations

Anthony Duncan

(September 3, 2024)

###### Contents

  1. 0 Introduction
    1. 1 Early life: from Janos to Johann
    2. 2 Berlin-Zürich-Göttingen
    3. 3 Von Neumann becomes a physicist
    4. 4 Paper 1: The Mathematical Foundation of Quantum Mechanics
    5. 5 Paper 2: Probability-theoretic Construction of Quantum Mechanics
    6. 6 Paper 3: The thermodynamics of quantum mechanical ensembles
    7. 7 A note on the translation and annotations
  2. 1 Paper 1: Mathematical Foundation of Quantum Mechanics
    1. 1 Introduction
    2. 2 Hilbert space [motivation]
    3. 3 Hilbert space [introduction of $l^{2}$ and $L^{2}$]
    4. 4 [Critique of Dirac-Jordan transformation theory]
    5. 5 [Hilbert space: formal axiomatic definition]
    6. 6 [Basic properties of orthogonal systems in Hilbert space]
    7. 7 Operator Calculus
    8. 8 Projection Operators
    9. 9 The eigenvalue problem
    10. 10 [The eigenvalue problem: further discussion]
    11. 11 The absolute value of an operator
    12. 12 The statistical postulate of quantum mechanics.
    13. 13 The statistical postulate of quantum mechanics (contd.)
    14. 14 Applications
    15. 15 Conclusion
    16. 16 Appendix 1
    17. 17 Appendix 2
    18. 18 Appendix 3
  3. 2 Paper 2: Probability-theoretic Construction of Quantum Mechanics
    1. 1 Introduction
    2. 2 Basic assumptions
    3. 3 General form of the expectation values.
    4. 4 States.
    5. 5 Measurements and States.
    6. 6 Measurements and States (contd.)
    7. 7 Statistical connection of diverse measurements.
    8. 8 [Statistical consequences of measurement.]
    9. 9 Summary.
  4. 3 Paper 3: The thermodynamics of quantum-mechanical ensembles.
    1. 1 Introduction
    2. 2 [Introduction of thermodynamic concepts: statement of principal result.]
    3. 3 [Derivation of Boltzmann distribution.]
    4. 4 [Thermodynamic interpretation of the elementary random ensemble; abstract of remaining sections.]
    5. 5 General Preparatory Matters
    6. 6 [Reversible transformation of pure states]
    7. 7 Transmission of quantum systems through semipermeable walls
    8. 8 Determination of entropy changes.
    9. 9 Determination of entropy changes (contd.)
    10. 10 [Determination of the equilibrium (maximum entropy) ensemble.]
    11. 11 The temperature.

##  Chapter 0 Introduction

In his witty and insightful biography of John von Neumann (born December 28,
1903 in Budapest, as Neumann, Janos Lajos), Norman Macrae [30] suggests that
von Neumann had realized “nearly all his achievements while he was mainly
engaged in something else.” Perhaps the best illustration of this paradox can
be found in the year 1927, when von Neumann, while completing the requirements
for his habilitation (basically, approval to lecture officially at a German
University) at the Friedrich-Wilhelm University in Berlin by submitting _two_
separate theses, one on mathematical logic (“The axiomatization of set
theory”), the other in functional analysis (“General eigenvalue theory of
symmetric functional operators”), also managed to produce three papers laying
the foundations for a mathematically rigorous and conceptually coherent
formulation of quantum theory. The object of the present article is to present
new, fully annotated translations of these remarkable papers.111The article
will elaborate on the treatment of these papers in Duncan and Janssen [15, 16,
Vol. 2, Ch. 17]. We begin with some brief biographical information on von
Neumann; this will be followed by a short discussion of the scientific context
of von Neumann’s incursion into physics; and, finally, a (very condensed!)
description of the contents of each paper of the trilogy.

###  1 Early life: from Janos to Johann

Neumann Janos (in Hungarian the family name comes first) was born in Budapest
on December 28, 1903. His father, Neumann Miksa (Max), was a prominent and
successful banker with a law degree; on his mother’s (Kann Margit) side, the
family had access to the considerable resources of the Kann-Heller hardware
firm, whose sales offices occupied the ground floor of an imposing building,
while the Neumann’s lived in the 18-room apartment on the top floor. The
Neumanns were, like many other middle class Hungarian Jews, secular and
completely acculturated to the “k.u.k” (_kaiserliche und königliche_ ,
“imperial and royal”) society of turn-of-the-century Austria-Hungary.222For a
nontechnical biography of von Neumann, see Bhattacharya [2]. Also
nontechnical, but with somewhat more detail on the conceptual advances
initiated by von Neumann, Israel and Gasca[24].

The term “child prodigy” seems almost inadequate to describe the young Janos,
who possessed remarkable powers of retention: he apparently memorized huge
tracts of Oncken’s multivolume world history, and received tutoring—on the
insistence of his father, for the practical reason that it would allow his
children freedom of mobility in a still very fractured European continent
[2]—in several modern European languages (English, German, French, and
Italian) as well as the obligatory Latin and (Ancient) Greek. His future
mathematical aptitude [18, see] first appeared in his capacity (age 6) for
arithmetical mental gymnastics (of the “divide two eight digits numbers”
type), but soon matured into a mastery of calculus (age 8), and of
contemporary real and complex analysis (age 12), as presented in Borel’s [3]
treatise.

In 1913 Max Neumann was ennobled for his services to the Austro-Hungarian
state, and the family name became Margittai Neumann (Hungarian for “Neumann of
Margitta”, the latter being a place name chosen by Max in honor of his wife),
later, following German practice, replaced by “von Neumann” (omitting the
object of the “von”). Janos entered secondary school a year later at the
Lutheran gymnasium (of which most of the students were Jewish), and
fortunately found in László Rátz, the mathematics teacher, a mentor who
immediately recognized the extraordinary aptitude of his new pupil. Rátz
arranged for Janos to obtain additional tutoring directly from instructors of
mathematics at the University of Budapest: first, from Gábor Szegö, and later,
from Lipót Fejér (one of the foremost Hungarian mathematicians at the time),
and Fejér’s student Mihály Fekete. Other than in mathematics, Janos’ secondary
school education followed the standard Lutheran gymnasium syllabus. While
still at the gymnasium, Janos coauthored a paper with Fekete on the zeroes of
Chebyshef polynomials. He also began to think about the set-theoretical
foundations of mathematics—specifically, how to avoid the infamous paradoxes
discovered by Bertrand Russell, and the development of a completely rigorous
foundation for the natural numbers (first by defining ordinal numbers, and
from them, the associated cardinals). These all too abstract preoccupations
worried his father Max sufficiently that he insisted that his eldest son
transfer his attention to a more practical area of study: chemical
engineering, for which the ETH in Zürich was the obvious destination. At this
point, however, the aspiring engineer’s grasp of the prerequisite chemistry
was inadequate for admission to the prestigious Swiss school, and it was
decided that Janos should spend some time first at the University of Berlin to
acquire the required background—while simultaneously working on a doctorate in
the axiomatization of set theory at the University of Budapest, a concession
no doubt extracted by Janos from his father in exchange for his consent to
chemical studies. Janos arrived in Berlin in September 1921—and adopted for
his matriculation the German equivalent—Johann—for his Hungarian given name.

###  2 Berlin-Zürich-Göttingen

For the next eight years von Neumann moved back and forth between Berlin,
Zürich,333As pointed out by Macrae [30, Ch. 4], Zürich was probably a
practical necessity, given the rampant inflation in Germany: in Switzerland
money was stable and one could avoid the financial vertigo of hyperinflation.
Von Neumann came from a very well-to-do family in Budapest, and had no
intention of enduring the Bohemian poverty of many German students at the
time. and Göttingen. While fully completing the chemical engineering
prerequisites for admission to the ETH (1923), he mainly occupied himself with
mathematical matters. In particular, he interacted frequently with the
mathematician Erhard Schmidt at the (Friedrich Wilhelms) University of Berlin:
Schmidt was a student of Hilbert who had contributed substantially to the
early development of functional analysis. It was no doubt through the offices
of Schmidt that Hilbert (in Göttingen) was kept informed about the activities
of the new Hungarian prodigy, who was already making profound contributions to
the foundations of mathematics, a central area of interest for Hilbert.

In von Neumann’s Berlin student years, chemistry clearly played a secondary
role to his mathematical studies: with Erhard Schmidt, Issai Schur, Richard
von Mises, and Ludwig Bieberbach in senior positions, the University attracted
a large number of talented and engaged students in mathematics at all levels,
and the mathematics seminar was typically fully attended, with lively
discussions between the students following the lectures. Von Neumann’s fellow
students at this time have testified to his preternatural speed in mental
computation. He clearly impressed Schmidt—an expert in function theory, and in
fact the person who had first given a precise description of the class of
infinite-dimensional linear spaces, which were becoming known (somewhat
unfairly to Schmidt) as _Hilbert_ spaces. In particular, Schmidt was
interested in the foundational issues in the formal axiomatization of
mathematics (particularly set theory, and the Cantorian theory of transfinite
numbers), which von Neumann was in the process of revolutionizing. On the
side, von Neumann acquired the knowledge of chemistry needed to matriculate in
the ETH for a course in chemical engineering.

The move to Zürich for his course at the ETH (beginning in the Fall of 1923)
did not distract von Neumann from mathematics: as a mentor he simply replaced
Schmidt with Hermann Weyl (another student of Hilbert), who, in addition to an
interest in axiomatics, was deeply interested in a number of areas of
mathematical physics (most prominently, relativity theory). By the end of
summer 1926, von Neumann had passed his doctoral exam at the University of
Budapest (with a thesis on the axiomatization of set theory), as well as
graduating from the ETH with the long-planned degree in chemical engineering.

In Göttingen, David Hilbert, and his assistant Richard Courant, had by this
point (starting in January 1926) begun the process of applying to the
International Education Board (established by J. D. Rockefeller as a source of
fellowship grants to promising young scholars trying to survive the miserable
economic climate of the early 1920s in Europe) for a grant to bring von
Neumann to Göttingen. By the Fall of 1926 we find von Neumann installed in
Göttingen, where a new topic of interest—the quantum-mechanical theories of
Heisenberg and Schrödinger, so different in form, but apparently identical in
physical content—had, at least temporarily, displaced axiomatics in Hilbert’s
focus of activities.

Hilbert had long displayed a keen interest in developments in theoretical
physics. In 1915/16 he was involved in a somewhat acrimonious priority dispute
with Einstein on the discovery of generally covariant gravitational field
equations [25, a dispute now typically acknowledged to have been resolved in
Einstein’s favor; see, e.g.,]. He also followed closely developments in
quantum theory, giving lectures on the old quantum theory in the Winter
semester of 1922/23, which were written up by Lothar Nordheim, a doctoral
strudent of Max Born [33]. Nordheim had been explicitly recruited by Hilbert
from Sommerfeld’s group in Munich as a sort of liaison for Hilbert with the
theoretical physics community.

After the dust had at least partially settled following the explosive
developments in matrix mechanics (Fall of 1925) and wave mechanics
(Winter/Spring 1926), Hilbert enlisted Nordheim’s help in addressing the
mathematical formulation of the new theory (Winter semester, 1926/27). In
particular, in tune with Hilbert’s preference for axiomatic formulations, the
recent article of Pascual Jordan [28], entitled “On a new foundation of
quantum mechanics” (_Über eine neue Begründung der Quantenmechanik”_), which
purported to provide a single axiomatic framework encompassing the various
apparently distinct versions of the new theory, became the central focus of
Hilbert’s activity in this area at just the point when the recently arrived
(now, Johann) von Neumann joined the fray. Nordheim recalled later (AIP
interview of Lothar Nordheim by John Heilbron, July 30, 1962), when asked by
Heilbron if von Neumann had been concerned with quantum mechanics prior to
arriving in Göttingen,

> I think that he got his start in this field in Göttingen, and being so
> tremendously fast, he got hold of it very soon. Yes, he had one of the
> fastest minds I have ever met. His brains always worked on roller bearings
> [!].

###  3 Von Neumann becomes a physicist

When von Neumann arrived in Göttingen, the broad outlines of what would before
long become a consistent, conceptually sound theory of atomic phenomena were
already visible.444For a detailed account of the development of quantum theory
from about 1900 to 1927, see Duncan and Janssen [16]. Werner Heisenberg and
his collaborators Max Born and Pascual Jordan had developed the former’s
proposal to replace classical mechanical quantities with complex matrices
constrained by certain nonlinear conditions (the commutation relation of
classically conjugate variables, such as position and momentum) into a well-
defined theory of energy levels and quantum transitions, at least for systems
with a discrete spectrum (such as bound electrons in atoms). In a completely
separate development, Erwin Schrödinger, in Zürich, had developed an approach
to quantization in which classical quantities were replaced by linear
operations (including derivatives) on a complex function $\psi(q)$ (where $q$
represents all the coordinate variables of the particle(s) described by this
“wave function”). Schrödinger had also been able to show that his theory gave
identical results to the Göttingen matrix mechanics for systems with a purely
discrete spectrum. However, his theory clearly encompassed non-discrete
systems (such as free, or unbound, electrons) with ease, which could not be
said for the matrix approach.

At the point where von Neumann arrived in Göttingen (Fall, 1926), there were
two outstanding issues of great conceptual importance which the new theories
posed, to wit:

  1. 1.

The essentially stochastic nature of the emerging quantum theory was only
understood in an incomplete and sometimes contradictory way.

  2. 2.

The precise mathematical structure underlying the two (superficially
completely different) versions (matrix vs wave) of the theory was not yet
visible.

The remarkable achievement of von Neumann was to provide a completely rigorous
mathematical formulation of the theory in which its probabilistic
aspects—specifically, the Born rule for probabilities—would emerge naturally
once a few basic assumptions on the properties of measurements of quantum
systems were made.

Hilbert’s desire to keep abreast of the latest developments in quantum theory
provided an ideal gateway for von Neumann’s entry into the field. Presumably
Nordheim, Born’s doctoral student, was the natural conduit to von Neumann of
the activities of Born, Heisenberg, and, increasingly of relevance to our
story here, Pascual Jordan. Jordan’s attempt to axiomatize quantum theory, and
thereby to subsume its disparate versions under a single theoretical
roof,555For more detailed discussion of Jordan’s approach, see Duncan and
Janssen [15, 16, Vol. 2, Sec. 16.2]. would clearly have appealed to von
Neumann, a disciple of Hilbert for whom an exact formal foundation was the
ultimate desideratum of any mathematically based science.

In the summer of 1926, Max Born, in his analysis of scattering processes in
quantum mechanics, proposed an explicitly probabilistic interpretation of the
Schrödinger wave function: its absolute square would provide an indication of
the probability of finding the scattered particle in a given region (more
exactly, traveling in a given angular direction) after the scattering:

> One obtains _no_ answer to the question “what is the state [of the scattered
> particle] after the collision”, but rather, to the question “how probable is
> any prescribed effect of the collision?” [4, p. 866]

In the Fall of 1926, Pascual Jordan, stimulated by numerous communications
with Wolfgang Pauli, attempted to construct an axiomatic formulation of
quantum theory in which (a) the peculiarities of the new theory would be
subsumed in postulates of a probabilistic nature, but in which _probability
amplitudes_ (of which the Schrödinger wave function was the best known special
case) would obey combination laws which mimic those of classical probability
theory—despite the fact that actual probabilities were related nonlinearly
(via the absolute square, as emphasized by Born) to the underlying (in general
complex) amplitudes, and (b) these amplitudes would serve as a basis for a
completely general axiomatic formulation of the theory, capable of reproducing
all previous versions (matrix mechanics, wave mechanics, Born-Wiener theory,
and the Dirac q-number theory).

We will briefly recall here the axiomatic framework proposed by Jordan [26].
in the first installment of his “New Foundation of Quantum Mechanics.”
Jordan’s notation is extraordinarily clumsy so we shall abandon it in favor of
a cleaner formulation.666We also eliminate the “supplementary amplitude”
introduced by Jordan in his first paper to allow for non-hermitian quantities.
The treatment throughout is for a system of a single degree of freedom, and
for observables with a continuous spectrum. The framework erected by Jordan
posits for the given quantum mechanical system (say, a particle on a line
subject to an imposed potential energy function) a set of mechanical
quantities $A,B,C,\ldots$ which stand in a definite kinematical relation to
one another. For example, classically such quantities would be definite
functions $A(Q,P),B(Q,P),C(Q,P)\ldots$ of the position/momentum canonical pair
$Q/P$ for the particle. For simplicity, Jordan assumes that these quantities
can only assume continuous values: in particular, the observable $A$ can be
found to have the (real number) value $\alpha$, the observable $B$ a value
$\beta$, and so on. Jordan postulates

  1. 1.

_Postulate I._ If the system is known to have a value $\beta$ of quantity $B$,
then the probability that the [measured] value of quantity $A$ lies in the
interval $(\alpha,\alpha+d\alpha)$ is
$|\varphi_{AB}(\alpha,\beta)|^{2}d\alpha$. Here, the functional dependence of
the probability amplitude $\varphi_{AB}i$ on its two arguments is completely
determined by the kinematical relation between the mechanical quantities $A$
and $B$: _it is completely indifferent to the particular dynamics of the
system, for example, the particular form of the Hamiltonian energy function
$H(Q,P)$, which for Jordan is just one more of the accessible mechanical
quantities of the system._

  2. 2.

_Postulate II._ The probability amplitude connecting two quantities is complex
symmetric:

$\varphi_{AB}(\alpha,\beta)=\varphi_{BA}(\beta,\alpha)^{*}.$ (1)

  3. 3.

_Postulate III._ The probability amplitudes (and not their squares!) obey
standard composition rules of the probability calculus. If two outcomes
[German _Tatsachen_], $F_{1}$ and $F_{2}$, associated with probability
amplitudes $\varphi_{1}$ and $\varphi_{2}$ are mutually exclusive [_sich
ausschliessen_] then the amplitude for “$F_{1}$ or $F_{2}$” is
$\varphi_{1}+\varphi_{2}$. If the two outcomes are independent [_unabhängig_],
then the amplitude for “$F_{1}$ and $F_{2}$” is $\varphi_{1}\varphi_{2}$.

  4. 4.

_Postulate IV._ For every mechanical quantity $q$ there exists a canonically
conjugate momentum $p$.

An immediate consequence of Postulate III is the following composition rule
for amplitudes:

$\varphi_{AC}(\alpha,\gamma)=\int\varphi_{AB}(\alpha,\beta)\varphi_{BC}(\beta,\gamma)d\beta.$
(2)

As an illustration, imagine that the observable $C$ (with measured value
$\gamma$) corresponds to the position of a particle emitted from a diffuse
source at time $t_{0}$, observable $B$ to the (now discrete) location $\beta$
of a hole (or slit) in a screen through which the particle passes at a time
$t_{1}>t_{0}$, and $A$ to the location $\alpha$ at which the particle stops on
a fluorescent screen at $t_{2}>t_{1}>t_{0}$. The independent passage from
source to the perforated screen, then from the perforated screen to the final
fluorescent one, requires the multiplication of the $\varphi_{AB}$ and
$\varphi_{BC}$ amplitudes. The mutually exclusive choice of which hole (or
slit) to pass through then requires a sum over such holes, which would of
course mean that a discrete sum would replace the integral over $\beta$ in Eq.
(2). In the more general case, in which the intermediate state requires a
continuous specification, the integral form given above applies.777More
generally still, the $B$ observable might contain both a discrete and
continuous spectrum—say, the position on a screen perforated with slits as
well as opeinings of finite size—in which case the right hand side of Eq. (2)
would involve both an integral and a sum. The familiar two-slit example
illustrating the interference phenomena peculiar to quantum mechanics
furnishes thus just a very simple special case of Jordan’s composition rule.

There are however two serious problems with Jordan’s postulated axioms for
probability amplitudes. The first is a deep conceptual one, the second of a
more technical nature. Namely:

  1. 1.

First, the exact operational status of the first line of Postulate I—“If the
system is known …then the probability that …”—is never clearly specified in
_New Foundation_. Von Neumann would, in the second paper of his trilogy, lay
out the precise meaning of this superficially straightforward assertion by
applying a rigorous version of the frequentist definition of probabilities
that he had absorbed from Richard von Mises in his Berlin studies. Values (or,
more precisely, _ranges of values_) of observable quantities were to be
assigned to ensembles of physically identical quantum systems subjected to
identical measurement interventions. Conditional probabilities would emerge as
the limit of frequencies of results obtained by subsequent measurements on
such suitably prepared ensembles. Once the association is made between the
values obtained by measurements and the spectrum of associated symmetric
[i.e., hermitian] operators, a few almost self-evident assumptions on the
statistical properties of these ensembles would lead von Neumann to an
essentially unique construction of the formalism of quantum mechanics,
including the Born rule encapsulated in Postulate I.

  2. 2.

The mathematical status of Jordan’s amplitudes $\varphi_{AB}(\alpha,\beta)$
was extremely obscure. The problem becomes acute whenever a continuous
spectrum is present. Consider again Eq. (2), but assume that the observables
$A$, $C$ are the same, with $B$ a different observable (all with continuous
spectra). Using Postulate II, this becomes

$\displaystyle\varphi_{AA}(\alpha,\alpha^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\int\varphi_{AB}(\alpha,\beta)\varphi_{AB}^{*}(\alpha^{\prime},\beta)d\beta=0\;\;\mathrm{if}\;\alpha\neq\alpha^{\prime}$
(3) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\infty\;(?)\;\;\mathrm{if}\;\alpha=\alpha^{\prime}.$

The modern physicist will recognize the peculiar behavior of the
$\varphi_{AA}$ amplitude as associated with the “delta function”
$\delta(\alpha-\alpha^{\prime})$ introduced heuristically by Dirac, but it
must be understood that the appropriate mathematical framework for dealing
with an “improper function” of this kind lay approximately 25 years in the
future, with Laurent Schwarz’ distribution theory. In particular, the
appearance of such an improper function was particularly embarassing given the
probability base of the theory, as attempting to square it gives nonsense. As
we shall see, von Neumann was at pains, especially in the first paper of the
trilogy, to indicate the mathematical problems introduced by admitting such
improper functions, while emphasizing that their use _was neither physically
sensible, or mathematically necessary._ He does this by taking the examples of
quintessentially continuous observables of quantum mechanics—position and
momentum—and showing how the essence of Postulate I can be maintained for
measurements of position and momentum without the appearance of a single
“improper” function.

Hilbert, in keeping with his long-standing desire to keep abreast of the
latest developments in fundamental theoretical physics, gave a lecture course
on quantum theory in the Winter semester of 1926/1927 in which he presented
matrix mechanics and wave mechanics. This was just the time when Jordan was
formulating and submitting for publication his “New Foundation …” paper with
the four postulates outlined above. No doubt Nordheim, Hilbert’s “spy” in the
Göttingen physics department, was keeping a close eye on these developments,
which would naturally appeal to Hilbert’s formalist outlook. The idea of
building up a theory from formal elements, initially given no interpretation
or “meaning”, but specified solely by posited interrelations and structural
connections—the “axioms” of the theory—appealed greatly to Hilbert. Only after
the nexus of “theorems” describing more complex properties derivable from the
axioms had been worked out would one then choose a specific mathematical (or
physical) interpretation to the basic formal elements. Hilbert had followed
precisely this approach in his _Grundlagen der Geometrie_ [Foundations of
Geometry, 1899], in which abstract concepts such as “point, line, plane, etc”
are introduced in a purely axiomatic fashion and their aggregates shown to
satisfy a rich set of properties, without ever needing to identify these words
with their commonplace equivalents (in two-dimensional Euclidean geometry).
Hilbert was enthusiastic about the possibility of doing the same with quantum
mechanics: a complete structure could be erected using Jordan’s complex
probablilty amplitudes, axiomatically prescribed to have certain properties.
Any analytic realization of these properties, supplemented with a physical
interpretation of the amplitudes, would then constitute an acceptable quantum-
mechanical theory.

In the Spring of 1927, Hilbert enlisted both Nordheim, and the recently
arrived prodigy Johann von Neumann (whose formalist credentials were hardly in
doubt, given his work on the axiomatization of set theory) in realizing a
reworking of Jordan’s _Neue Begründung_ which would both clarify the axiomatic
structure proposed by Jordan, and then show the consistency of these axioms by
exhibiting an analytic realization of the axioms in terms of mathematically
well-defined operations. The resultant paper would not be published for
another year, and was by the time of its publication completely superseded by
the trilogy of von Neumann. Nevertheless, it is a valuable index to von
Neumann’s state of knowledge at the outset of his career in physics. The
second paragraph of the paper gives an indication of the relative roles of the
participants in the writing of this paper [23]:

> The work presented below arose from a seminar given by D. Hilbert in the
> Winter semester 1926/27 on the recent development of quantum mechanics, the
> preparation of which profited from the essential assistance of L. Nordheim.
> Important parts of the mathematical execution are due to J. v. Neumann.

In the Hilbert-Nordheim-von Neumann paper, the theory is built up from a set
of six axioms, three of which (I, II, and IV) correspond to Postulates I-III
of Jordan (the missing one being Jordan’s assertion on the existence of
conjugate coordinate/momentum pairs). The “extra” three are really already
explicitly present in Jordan’s _Neue Begründung_ paper, if not listed as
fundamental postulates: namely (a) (III) the “sharp” (i.e., delta-function)
behavior of the amplitudes for an observable with respect to itself (cf. Eq.
(3)), (b) (V) the requirement that the form of the amplitudes depend only on
the kinematical relation between the respective observables, not on the
particular dynamics of the theory (e.g., the Hamiltonian), and (c) (VI) the
assertion that the probabilities do not depend on the particular choice of
coordinate system in which we choose to work.

Despite the implications of the quote above specifically attributing aspects
of the mathematical execution to von Neumann, the analytic apparatus employed
in this paper to concretely realize the Jordan axiomatic framework owes much
more to Hilbert than to his young acolyte. In an attempt to find a unified
representation of Jordan’s amplitudes, Hilbert resorts to the use of integral
kernels, a completely natural choice given his central role in working out a
complete spectral theory of linear integral equations (with bounded kernels)
in the two decades preceding this paper. Unfortunately, this approach
immediately necessitates the introduction of the improper “functions” (such as
the delta function) to which von Neumann was strongly allergic [47, as
emphasized in]. If probability amplitudes $\varphi_{AB}(\alpha,\beta)$ were to
play a dual role as kernels of linear transformation operators on spaces of
functions,888We have slightly altered the notation used by Hilbert et al. to
conform to our earlier usage.

$f(\beta)\rightarrow g(\alpha)\equiv
T_{AB}f(\alpha)=\int\varphi_{AB}(\alpha,\beta)f(\beta)d\beta,$ (4)

then the simplest possible linear operator, the identity operation
$f\rightarrow f)$, must be represented by an integral kernel which is just the
Dirac delta function:

$T_{AA}f(\alpha)=f(\alpha)=\int\delta(\alpha-\beta)f(\beta)d\beta.$ (5)

Unfortunately, as von Neumann undoubtedly realized from the outset, this could
not serve as the basis for developing a rigorous spectral theory. While the
formal manipulations of Hilbert et al., which we recognize today as perfectly
justified by the distribution theory developed in the late-1940s (and by the
use of Gelfand triplets, the so-called “rigged” Hilbert spaces), the rigorous
mathematical status of these maneuvers was still a complete mystery in the
mid-1920s. So the highly formal manipulations of the Hilbert-Nordheim-von
Neumann paper bear only a superficial similarity to the (completely rigorous!)
spectral theory of integral operators developed by the Hilbert school. This
latter theory was in fact explicitly—and correctly—deployed in the culminating
“Three Man Paper” of matrix mechanics [8], in a section written by Born [16,
for discussion, see]. What Born realized at the time was that the Hilbert
theory (amplified by Hellinger to include a rigorous theory of operators with
a partly continuous spectrum) only applied to _bounded_ operators—those,
roughly speaking—whose eigenvalues were limited in absolute magnitude. Almost
all the operators appearing in quantum mechanics (position, momentum, energy,
etc) are _unbounded_ operators. Born, in a footnote, expressed the hope that
an appropriate extension of the Hilbert theory would allow treatment of such
quantities as well.

Born did not, however (in 1925), fully appreciate the structure of the
function space underlying the spectral theories of Hilbert and Hellinger, for
which he can hardly be blamed, as neither did Hilbert! The critical concept,
that of a infinite-dimensional vector space, equipped with an (in general
complex-valued) inner product, and with a complete denumerable orthogonal
basis set dense in the space, had already been introduced by Erhard Schmidt,
whom von Neumann had come to know very well in Berlin, more than a decade
earlier. This space, redubbed by von Neumann a “Hilbert space”, would be
placed center stage by von Neumann in his first paper (as sole author) on the
mathematical foundations of quantum mechanics. We turn now to a brief review
of the contents of this work.

###  4 Paper 1: The Mathematical Foundation of Quantum Mechanics

The article of Hilbert, Nordheim and von Neumann, although prepared for
publication by April of 1927, was not published until the following year, long
past the point where von Neumann’s three seminal papers had completely upended
and permanently reframed the formal structure of quantum theory. It has
therefore become something of an historical oddity, which, despite the
eminence of its provenance, essentially vanished from the mainstream
development of the theory. In his first solo paper on quantum mechanics, von
Neumann [47] would essentially reject the methodology of the Jordan
axiomatization, replacing probability amplitudes as the central building
blocks of the theory with vectors (and, more generally, projection operators
onto linear subspaces) in an infinite dimensional vector space of a special
sort.

The paper begins with an itemized summary of the present status of “quantum
mechanics”, a theory—or better, nexus of ideas
[_Begriffsbildungen_]—attributed explicitly to Heisenberg, Dirac, Born,
Schrödinger and Jordan.

  1. 1.

The behavior of atomic systems is connected to a particular eigenvalue
problem, with the spectrum of eigenvalues coinciding with the possible
measured values [of what quantity, is not stated—but we may presume that von
Neumann is talking about the energy here].

  2. 2.

The peculiar mixture of continuous and discrete aspects appearing in quantum
mechanics is therefore attributable to the simple fact that eigenvalue
problems can easily lead to the coexistence of continuous and discrete
spectra.

  3. 3.

The quantum laws of Nature are in general not strictly causal, although in
certain restricted cases the behavior of the system can be reliably predicted;
thus the basic laws give probability distributions.

  4. 4.

The eigenvalue problem can appear in quite distinct forms: as the
diagonalization problem of an infinite matrix, or of a differential equation.
These two formulations can be related by expanding the “wave function” in the
differential operator case in a (discrete) infinite series of orthogonal
functions, in which case the action of the differential operator can be
rewritten as the action of an infinite matrix on the expansion coefficients of
the wave function.

  5. 5.

Both approaches (matrix/differential equation) have their own difficulties. In
the matrix case, the transformation to diagonal form is only possible if there
is no continuous spectrum—this obviously is not the case even in the simple
case of a hydrogen atom. The use of such constructs as “continuous matrices”,
in which one operates simultaneously with discrete matrices and integral
equation kernels, is very difficult to accomplish with full mathematical
rigor.

  6. 6.

Only recently, with the work of Born and Jordan, has the connection to
probabilities, visible at the outset in the matrix approach [the square of
matrix elements of the coordinate operator gave atomic transition
probabilities], emerged. This connection was expanded to a complete and closed
system by Jordan, but only at the expense of introducing mathematically
inadmissible objects. In particular, one obtains eigenfunctions that are
“improper” functions, such as the one introduced by Dirac, “which is supposed
to have the following (absurd) properties:”

$\delta(x)=0,\;\;\mathrm{for}\;x\neq
0,\quad\int_{-\infty}^{+\infty}\delta(x)dx=1.$ (6)

  7. 7.

A common defect of all these methods is the introduction of elements in the
calculation which are in principle unobservable and physically meaningless.
While the final probabilities are unique and meaningful, the eigenfunctions
which appear at intermediate stages of the calculation are undetermined up to
a phase (in cases of degeneracy, up to an entire unfixed unitary rotation).

Following this terse (and incisive) review of the state of affairs, von
Neumann alerts the reader that the rest of the paper would present a detailed
mathematical preparation (sections 2–11), followed by the application of the
formalism to quantum theory in the final sections (12–15). The title of the
first part (preceding section 2), “The Hilbert space” (_Der Hilbertsche
Raum_), requires some comment. What precisely von Neumann means is first
explained in section 3, following a brief reference to Hilbert’s 1906 theory
of bilinear forms:

> In this case there are good grounds to focus on the following manifold
> [space]: all sequences $x_{1},x_{2},\ldots$ of complex numbers with finite
> $\sum_{n=1}^{\infty}|x_{n}|^{2}$. For this space it is customary to use the
> name (complex) Hilbert space [47, p. 7].

The term “customary” (_üblich_) is a little puzzling here, as the term
“Hilbert space” does not appear in any of Hilbert’s numerous works on integral
equations and bilinear forms in the relevant period (mainly 1906–1912), nor,
to the author’s knowledge, in the papers of other workers (pre-1927) in the
field of function spaces (later, functional analysis). The particular Hilbert
space cited in the quote above (now referred to by mathematicians as $l^{2}$)
was in fact first defined explicitly by Schmidt [35]. All of the defining
properties of a Hilbert space (linear space, existence of bilinear inner
product, Cauchy completeness) were shown by Erhard Schmidt in his 1908 paper
to apply to this space, the geometrical interpretation of which as an
infinite-dimensional vector space he attributes in a footnote to Gerhard
Kowalewski. So a more historically accurate term should perhaps be “Schmidt-
Kowalewksi” spaces. It is however quite possible that in conversations between
mathematicians (even, perhaps, between von Neumann and Schmidt himself) the
term “Hilbert space” was already in informal use. This usage became universal
once it was sanctified by von Neumann in his mathematical papers form 1927 on.

A brief summary of the contents of this groundbreaking paper follows:

Section 2.

The formulation of the eigenvalue problem (cited in his introduction as _the
basic problem_ in quantum theory) is reviewed in both matrix and wave
mechanics. In both cases one has to deal with the determination of the
possible values of $w$ where $Hf=wf$, $H$ is a linear operator, and $f$ is
either a sequence (with $H$ a matrix) or a function (with $H$ a differential
operator).

Section 3.

The deep analogy between the space of square-summable sequences ($l^{2}$ now,
in von Neumann’s notation $\mathfrak{H}_{0}$) and square-integrable functions
$\varphi(x)$ over some space $\Omega$ (thus:
$\int_{\Omega}|\varphi(x)|^{2}dv<\infty$) is discussed ($L^{2}$ now, in von
Neumann’s notation $\mathfrak{H}$), with clear reference to the matrix and
continuous (wave) formulations of quantum mechanics. (The rigorous isomorphism
of the spaces will be explained in Section 4). The notation $Q(f,g)$ for the
complex inner product of two vectors $f$ and $g$ is introduced (in modern
Dirac notation, this would be written $\langle g|f\rangle$—note the inverted
order).

Section 4.

The isomorphism of $\mathfrak{H}_{0}$ ($l^{2}$) with $\mathfrak{H}$ ($L^{2}$)
is shown, citing the Riesz-Fischer theorem.

Section 5.

The axioms defining a (separable) Hilbert space are laid out: linear vector
space, complex inner product with associated metric, infinite-dimensional with
a countable basis (separable), and Cauchy-complete.

Section 6.

A plethora of basic theorems, propositions and lemmas following directly from
the axioms (Bessel’s inequality, Parseval identity, etc.).

Section 7,

Operator calculus: the importance of densely defined operators (that can act
sensibly on vectors arbitrarily close to any element of the Hilbert space),
definition of the hermitian adjoint and symmetric operators (those equal to
their hermitian adjoints).

Section 8.

Projection operators (_Einzeloperator_) are introduced: $E$ is a projection
operator if it is symmetric ($E=E^{\dagger}$) and idempotent, $E=E^{2}$.
Several important definitions and theorems are presented involving projection
operators.

Section 9.

A universally valid formulation of the eigenvalue problem can now be given:
the operator $A$ in question is to be written as an integral over
(differential) spectral projection operators. If $E(l)$ projects onto the
subspace spanned by all eigenvectors of $A$ with eigenvalue less than or equal
to $l$ (so $E(-\infty)=0$ and $E(+\infty)=1$), then

$A=\int_{-\infty}^{+\infty}ldE(l),\quad 1=\int_{-\infty}^{+\infty}dE(l).$ (7)

How this works is illustrated for operators in a finite-dimensional Euclidean
space.

Section 10.

The Hilbert-Hellinger theory establishes the validity of Eq. (7) for all
symmetric bounded operators. For the unbounded operators predominant in
quantum mechanics, the existence of such a spectral representation is at this
point not known in general. However, for the fundamental (unbounded!)
operators $q$ and $p$, the spectral representation is explicitly constructed,
making it clear that there is no fundamental obstruction to this desirable
result for unbounded operators.

Section 11.

A norm $||A||$ is introduced for arbitrary operators $A$: one evaluates the
matrix of the operator $A_{\mu\nu}$ with respect to a complete orthonormal
basis in the Hilbert space, then computes the sum of absolute squares:
$||A||:=\sqrt{\sum_{\mu,\nu}|A_{\mu\nu}|^{2}}\equiv\sqrt{[A]}$. Typical
properties attached to a norm are shown to hold, e.g., the triangle inequality
$||A+B||\leq||A||+||B||$.

Section 12.

The precisely defined operator norms of section 12 are deployed to put
Jordan’s conditional probabilities on a completely rigourous footing. The
necessary ingredients are just the spectral operators associated with any
observables whose value (ranges) are prescribed/measured. As measurements have
necessarily limited precision, it suffices to specify, for any given interval
$I=(l_{1},l_{2})$ (with $l_{1}<l_{2}$), the associated spectral projection
operator, for (say) operator $A$, $E(I)=E(l_{2})-E(l_{1})$. Then the
conditional probability that another observable, associated with operator $B$
(spectral operator $F$), is found to have a value in the interval $J$, given
that $A$ is known to be in the interval $I$, is just
$[E(I)F(J)]=||E(I)F(J)||^{2}$. _This is the Born rule in its most precise and
mathematically rigorous form._

Section 13.

The probability rule of section 12 is extended to simultaneous measurements of
sets of compatible (i.e., commuting) observables. The postulates forming the
basis of Jordan’s formulation of quantum mechanics in _Neue Begründung_ are
discussed in the context of the new, mathematically rigorous framework. The
symmetry of amplitudes (Jordan Postulate 2) becomes a symmetry under
“replacement of all assertions (_Behauptungen_) with all conditions
(_Voraussetzungen_).” This symmetry follows from the obvious identity
$[AB]=[BA]$.

Following Jordan, von Neumann asserts that the law of multiplication of
probabilities (for independent events) in not generally valid—rather a weaker
law, of “superposition of probability amplitudes”, as stated by Jordan, holds.
That this is not unexpected, claims von Neumann, is because“the dependency
relations of our probabilities can be arbitrarily complicated.” What does this
mean? It should be remembered that at the time of writing this paper, von
Neumann had not yet read Heisenberg’s [20] uncertainty paper (written in
Copenhagen, and not published till more than a week after von Neumann’s paper
was submitted on May 20). The “dependency relations” somewhat hazily alluded
to by von Neumann were exposed with brilliant clarity in Heisenberg’s
discussion of the passage of atoms through multiple Stern-Gerlach devices. In
this thought experiment it is shown that the establishment of the “assertions”
concerning the intermediate state of the atoms needed to apply the classical
law for multiplication of probabilities would require measurements of this
intermediate state, which would destroy the phase relations of the atomic
states. This intimate interconnection of the atomic phases which requires use
of products of complex amplitudes (rather than of positive real probabilities)
can, with some charity, be equated with von Neumann’s vague “dependency
relations.” In any event, von Neumann would soon absorb the lesson of
Heisenberg’s paper, which is prominently cited in his next paper [48], and, in
agreement with Heisenberg, maintain that the classical probability laws (both
multiplication and addition) would always be perfectly valid, _provided the
appropriate measurements needed to establish the claimed conditions and
assertions were carried out._

The law of addition for probabilities of exclusive outcomes was far less
problematic, and holds without further ado. Taking for simplicity assertions
and conditions associated with a single observable each, with spectral
projections $E$, $F$ respectively, then, if $J^{\prime}$ and
$J^{\prime\prime}$ are two non-overlapping intervals, with $J=J^{\prime}\cup
J^{\prime\prime}$, then

$[E(I)F(J)]=[E(I)F(J^{\prime}]+[E(I)F(J^{\prime\prime})].$ (8)

Section 14.

In this section (entitled “Applications”— _Anwendungen_) von Neumann discusses
the spectrum of the Hamiltonian operator, which is of course of central
physical importance. In general, the spectrum contains both point eigenvalues,
some of which may be degenerate, and continuum eigenvalues. The thorny issue
of the a-priori probability of a quantum state can now be settled with
complete precision. Von Neumann applies the term “quantized state” for the
eigenfunctions associated with the point spectrum, and “unquantized state” for
the continuum spectrum. The a-priori probability associated with an interval
$I$ is $[E(I)]$, which is just the dimension of the subspace projected onto by
$E(I)$, to wit, the number of independent eigenfunctions
$\varphi_{\nu_{1}},\varphi_{\nu_{2}},\ldots$ with eigenvalues in the interval
$I$. For an unquantized state, the a-priori probability is zero: the
probability of finding the system in a state with any preassigned _exact_
value of the energy in the continuum part of the spectrum is zero. Note that
in this entire discussion, the probabilities referred to are typically
relative ones: in the hydrogen atom, for example, there are infinitely many
discrete eigenstates, so the assertion is that each individual state has the
same a-priori probability as any other, while an exactly prescribed continuum
state has zero probability relative to any of the discrete ones. There is a
short discussion of the calculation of conditional probabilities when two
observables are “causally” connected (by which von Neumann means, functionally
connected, $A=f(B)$). Finally, he derives the original Born rule, in the
context of the time evolution of eigenstates of a time-dependent Hamiltonian
$H(t)$: if the system is placed in eigenstate $\varphi_{\mu}^{(0)}$ of $H(0)$
at time 0, the probability that it will be found to be in the $\nu$’th
eigenstate $\varphi_{\nu}^{(t)}$ of $H(t)$ at time $t$ is
$|Q(\varphi_{\mu}^{(0)},\varphi_{\nu}^{(t)})|^{2}$; i.e., just the absolute
square of the inner product of the two eigenstates.

Section 15.

The conclusion reiterates the basic result for conditional probabilities in
quantum mechanics, in the most general case, where an arbitrary number of
compatible observables ($R_{1},R_{2},\ldots$, spectral functions
$E_{1},E_{2},\ldots$) are presumed to lie in the intervals
$I_{1},I_{2},\ldots$, and one then wishes to find the probability than another
set of mutually compatible observables ($S_{1},S_{2},\ldots$, spectral
functions $F_{1},F_{2},\ldots$) will be found to have values in the intervals
$J_{1},J_{2},\ldots$. The desired conditional probability is given by the
beautifully compact and intuitive formula

$[E_{1}(I_{1})\cdot E_{2}(I_{2})\cdot\ldots\cdot F_{1}(J_{1})\cdot
F_{2}(J_{2})\cdot\ldots].$ (9)

The paper concludes with the admonition that mutually incompatible observables
(corresponding to noncommuting operators) not be allowed among either the
conditions (the $R_{i}$ above) or the assertions (the $S_{j}$ above), although
the $R_{i}$ and $S_{j}$ may of course not be compatible. As von Neumann puts
it, “quantum mechanics does not in any way allow the positing of such a
question!”

The enormous wealth of new insights and beautifully precise (and powerful!)
formalism brought to bear by von Neumann on the conceptual makeup of quantum
theory should not blind us to the two really central, and for the future
development of the theory, seminal contributions of this paper:

  1. 1.

The paper gives a mathematically unassailable demonstration of the complete
identity of matrix and wave mechanics, which is nothing more or less than the
isomorphism between the Hilbert spaces of square-summable series ($l^{2}$ now;
$\mathfrak{H}_{0}$ for von Neumann) and of (Lebesgue) square-integrable
functions ($L^{2}$ now; $\mathfrak{H}$ for von Neumann). The underlying
theorem (Riesz-Fischer) was not due to von Neumann, but he understood clearly
its relevance to establishing the underlying unity of the two regnant forms of
the theory. _Any problem of wave (resp. matrix) mechanics, can be rephrased,
with no loss of information, as a problem of matrix (resp. wave) mechanics!_
Even the hydrogen atom, with its annoying mix of pointlike and continuous
spectrum, can be completely described by an infinite discrete matrix
Hamiltonian, _which contains all the information concerning the normalizable
discrete bound states as well as the non-normalizable continuous eigenstates
of the system._ 999One could, for example, use the square-integrable Hermite
eigenfunctions of a three-dimensional oscillator as a complete orthonormal
basis, and evaluate the matrix elements of the Coulomb Hamiltonian of the
hydrogen atom in this basis. Such a matrix is of necessity non-diagonal, if a
continuous spectrum is present, but it nonetheless contains _all_ the physics
of the system.

  2. 2.

Equally importantly, the mathematical expression of the theory correctly
identifies its meaningful physical content. The states that are physically
accessible to an electron are the ones with a normalizable (i.e., square-
integrable) wave function, corresponding to a ray (unique direction) in
Hilbert space. Electrons are never found localized to exactly a single point
(with a wave function proportional to a delta function), or infinitely
extended through an infinite volume universe, with exactly well-defined
momentum (the plane waves of collision theory). These are idealizations which,
it should not be surprising, are not accommodated in the physical Hilbert
space, where every vector has finite norm. They may be convenient
idealizations for certain calculations, but the states corresponding to real
particles always have some dispersion in position and momentum: they are wave
packets of ultimately finite extent, when viewed in coordinate space. Thus,
the importance of von Neumann’s projection operators, which are bounded and
always produce states of finite norm; they also correspond to the fact that
measurements are never infinitely precise, but determine the desired
quantities up to finite error.101010One may for convenience introduce a
broader formalism (e.g., rigged Hilbert space) to accommodate continuum
eigenstates, but the $L^{2}$ space of von Neumann still remains the only
physically meaningful setting for all processes in the quantum world.
Moreover, by phrasing the empirical content of the theory in terms of
projection operators rather than states, the (for von Neumann) annoying phase
ambiguity in the state vectors disappears: the arbitrary phase $\vartheta$ in
a state vector $e^{i\vartheta}|\varphi\rangle$ (helping ourself to Dirac
notation) disappears once we construct the projection operator onto that
state: $P_{\varphi}=e^{i\vartheta}|\varphi\rangle\cdot
e^{-i\vartheta}\langle\varphi|=|\varphi\rangle\langle\varphi|$.

###  5 Paper 2: Probability-theoretic Construction of Quantum Mechanics

In the second paper of the trilogy presented here, von Neumann [48] sets out
to provide an inductive construction of the kinematic structure of quantum
mechanics, starting from some foundational assumptions of the simplest
possible character. “Kinematic”, because the paper is not concerned with the
dynamics of a specific physical system (specifying the degrees of freedom,
Hamiltonian, time development of states, etc.); instead, the focus is on the
statistical properties of the theory, and how these properties constrain the
mathematical framing of the theory. What was lacking in the previous deductive
approaches (calculation of the expansion coefficients of the wave function
from the Schrödinger equation, followed by a “dogmatic” identification of
absolute squares thereof with probabilities), “a systematic derivation of
quantum mechanics starting from empirical facts, or probability-theoretical
axioms”, would be provided by this paper. Von Neumann is at pains to emphasize
his confidence in the absolute validity of conventional probability theory—a
confidence inspired by his absorption of the arguments of Heisenberg’s [20]
uncertainty paper—specifically, the validity of _both_ the addition and
multiplication laws for probabilities of mutually exclusive or independent
events. Jordan’s assertion that the laws applied for probability amplitudes,
and not their squares (the probabilities themselves), would be shown to arise
from an incomplete analysis of the connection between the processes of
measurement and the underlying quantum mechanical structures. We provide here
a brief summary of the paper.

Section 2.

What are the foundational assumptions from which von Neumann would manage to
reconstruct, almost magically, the full kinematic structure of an arbitrary
quantum system? They turn out to be of two kinds. First, the notion of
expectation value (_of an observable_ , _in an ensemble of exemplars of a
given physical system_) is introduced, and certain intuitively plausible
formal properties of the expectation value are imposed (axioms A and B below).
Second, each observable is associated with a linear, symmetric (i.e., self-
adjoint) operator, with the spectrum of this operator covering the values of
the associated observable which can be obtained by the measurement of the
observable on any given exemplar of the system. The mapping from observables
to operators is then subjected to two requirements, again of an intuitively
plausible character (axioms C and D below). The entire
statistical/transformation theory of Jordan-Dirac is then shown to be an
inevitable consequence of the stated axioms. In particular, the Born rule
emerges in an extremely natural manner.

The probability theory lying in the background of the entire paper is the
frequentist theory championed by Richard von Mises [45], with which von
Neumann was certainly familiar (von Mises was professor of applied mathematics
at the University of Berlin during von Neumann’s time in Berlin).
Intriguingly, von Neumann also uses language more reminiscent of the
information theory approach that would later emerge with the work of Shannon
in the 1940s. He speaks, for example, of degrees of “knowledge” (_Kenntnis_)
of an ensemble, corresponding to the measurement of the expectation values of
sets of observables.

The basic objects studied by von Neumann in this paper are ensembles
${\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots.}$ consisting of the concatenation
of (in general, infinitely many) exemplars of a definite physical system
(e.g., a hydrogen atom) $\mathfrak{S}$. Each $\mathfrak{S}_{i}$ thus
represents the physical system $\mathfrak{S}$ in a particular quantum state.
There is no obstruction to the same state appearing several times in the
ensemble. However, the ensembles envisaged are _very large_ , in the sense
that any randomly selected subensemble will retain the same statistical
properties as the original ensemble. One ensemble plays a particularly central
role in the argument, as it can serve as the source of all others, via well-
defined “pruning” processes. This is the “elementary disordered” ensemble, in
which “every imaginable state of the system $\mathfrak{S}$ occurs with equal
frequency.” Such an ensemble corresponds to the situation in which “one knows
nothing about the system.” All other ensembles $\mathfrak{S}^{\prime}$ can be
obtained from the elementary one by a process of pruning. The technical
machinery to accomplish this exactly corresponds to the algebra of projection
operators developed in paper I. For example, the process of measuring some
kinematic quantity $\mathfrak{a}$, performed on each of the (infinitely many!)
members of the elementary ensemble results in a value in the spectrum of
$\mathfrak{a}$: by discarding all systems except those where this value lies
in some interval $I$, we obtain a new ensemble which can be associated in an
obvious way with the projection operator $E(I)$ (where $E$ is the spectral
operator associated with $\mathfrak{a}$). The resulting ensemble is “sharp”
with respect to the observable $E(I)$—a measurement of $E(I)$ gives the value
1 on every system in the ensemble.

Von Neumann defines the knowledge (or better, information) we have about an
ensemble
$\mathfrak{S}^{\prime}={\mathfrak{S}_{1}^{\prime},\mathfrak{S}_{2}^{\prime},\ldots}$
as the collection of all possible expectation values
$\mathbf{E}(\mathfrak{a})$ (i.e., the average value obtained when the quantity
$\mathfrak{a}$ is measured on every system $\mathfrak{S}_{i}^{\prime}$ with
$i=1,2,\ldots$ in the ensemble). This includes the possibility of measuring
all powers of a quantity, and therefore, via knowledge of all the moments,
reconstruction of the full distribution of the quantity in the ensemble.

The inductive reconstruction of quantum kinematics in this paper is based on
the following four axioms, the first two of which constrain the expectation
values $\mathbf{E}(\mathfrak{a})$, while the second two assert basic
properties of the operators associated with each observable.

A.

If $\mathfrak{a},\mathfrak{b},\mathfrak{c},\ldots$ form a finite or infinite
set of quantities, and $\alpha,\beta,\gamma,\ldots$ are real numbers (with
convergent sum), then

$\mathbf{E}(\alpha\mathfrak{a}+\beta\mathfrak{b}+\gamma\mathfrak{c}+\ldots)=\alpha\mathbf{E}(\mathfrak{a})+\beta\mathbf{E}(\mathfrak{b})+\gamma\mathbf{E}(\mathfrak{c})+\ldots$
(10)

B.

If $\mathfrak{a}$ is an intrinsically non-negative quantity, then

$\mathbf{E}(\mathfrak{a})\geq 0.$ (11)

C.

Let $S,T,\ldots$ (finite or infinite in number) be normal operators,111111A
normal operator is one that commutes with its adjoint. This class, which is
the most expansive for which the spectral theory of von Neumann applies,
includes self-adjoint and unitary operators. and $\alpha S+\beta T+\ldots$
likewise normal. If $S,T,\ldots$ are associated to the quantities
$\mathfrak{a},\mathfrak{b},\ldots$, then $\alpha S+\beta T+\ldots$ is
associated to $\alpha\mathfrak{a}+\beta\mathfrak{b}+\ldots$

D.

Let $S$ be a normal operator, $f(x)$ a real-valued function defined for all
real $x$, and let $S$ correspond to the quantity $\mathfrak{a}$. Then $f(S)$
corresponds to the quantity $f(\mathfrak{a})$.

These innocent assumptions, almost self-evident in appearance, will quite
remarkably allow von Neumann to reconstruct the entire measure-theoretic
structure of quantum mechanics. In particular, the Born rule will emerge as
just one very special case of the general theory.

Section 3.

Working in the discrete realization of Hilbert space (von Neumann’s
$\mathfrak{H}_{0}$, modern $l^{2}$, the space of square-summable sequences),
von Neumann shows that the complete statistical information contained in an
ensemble of states of an arbitrary quantum system can be encapsulated in a
single matrix $U$. The operator represented by this matrix is symmetric (i.e.,
self-adjoint) and positive (i.e., no negative eigenvalues).121212This is the
famous “density operator”, or “density matrix” at the center of modern quantum
statistical mechanics. For _any_ observable represented by an operator $S$,
the mean value for the observable when measured on all elements of the
ensemble—in other words, the expectation value of the observable in the
ensemble— is given by131313For a detailed reconstruction of von Neumann’s
derivation of this result, see Duncan and Janssen [15, pp. 248–251] or Duncan
and Janssen [16, Vol. 2, Sec. 17.2].

$\mathbf{E}(S)=\mathrm{Tr}(SU).$ (12)

Section 4.

Moreover, if the ensemble is “pure”, or “uniform” (von Neumann uses both
denotations), with the ensemble consisting of identical systems, there exists
a unit vector $\varphi$ in the Hilbert space such that $U=P_{\varphi}$, the
projection operator onto that state vector. For such a pure ensemble, the
expectation value of any observable $S$ is just $Q(\varphi,S\varphi)$ (in
modern notation: $\langle\varphi|S|\varphi\rangle$). This is essentially the
content of the “Born rule.”

Section 5.

In this section von Neumann explores the formal implications of simultaneous
measurability of several quantities $\mathfrak{a}_{1},\ldots,\mathfrak{a}_{m}$
(associated with operators $S_{1},\ldots,S_{m}$). The normal interpretation of
this property implies the commutativity of the operators $S_{i}$, but von
Neumann insists on more: the mutual commutativity of the spectral projectors
$E_{\mu}(\lambda)$ associated with $S_{\mu}$ (where $\mu=1,2,\ldots,m$). The
proof relies on the existence of a single observable, with operator $S$, from
which the values of all the $S_{\mu}$ follow. This can be shown explicitly if
the $\mathfrak{a}_{\mu}$ have purely discrete spectra [15, see], but is not at
all clear when we are dealing with operators—such as the various components of
coordinates or momenta in a multi-dimensional problem—with a partly or
completely continous spectrum. Of course, it is perfectly straightforward to
show by explicit construction that the spectral projectors for the various
components of the momentum vector (say) commute: this is self evident from the
discussion of the spectral theorem for these operators by von Neumann [47].
The commutativity of the spectral projectors (which is named “complete
commutativity”),

$[E_{i}(\lambda),E_{j}(\lambda^{\prime}]=0,\quad(i,j=1,2,\ldots,m),$ (13)

is therefore the appropriate expression, according to von Neumann, of the
simultaneous measurability of the corresponding quantities. In fact, this is
both mathematically and physically sensible. For unbounded operators, the
products $S_{i}S_{j}$ may encounter domain difficulties ($S_{j}$ may produce a
vector not in the domain of $S_{i}$), while the spectral projectors $E_{i}$
and $E_{j}$ are bounded and everywhere defied, so the calculation of products
and commutators is troublefree. And physically, measurements in the real
worlds only establish the value of measured quantities within finite bounds
determined by the limitations of the experimental setup: formally this
corresponds to measurement of the projection operator $E_{i}(I)$ (with $I$ an
interval).

Section 6.

The exploration of the properties of the statistical operator $U$ associated
with an ensemble ${\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots}$ continues, in
particular, the properties of ensembles specified by the simultaneous
measurement of $m$ compatible (commuting) quantities
$\mathfrak{a}_{1},\mathfrak{a}_{2},\ldots,\mathfrak{a}_{m}$. If, for example,
it is known that in every system in the ensemble, a measurement of
$\mathfrak{a}_{1}$ yields a value in the interval $I_{1}$, of
$\mathfrak{a}_{2}$ yields a value in the interval $I_{2}$, etc., then, von
Neumann shows, the statistical operator $U$ of the ensemble is constrained to
commute with the projection operator

$E=E_{1}(I_{1})E_{2}(I_{2})\cdot\cdot\cdot E_{m}(I_{m}).$ (14)

In certain cases the simultaneous projections may result in $E$ projecting
onto a one dimensional subspace, spanned by a vector $\varphi$. In that case
we have simply $E=U=P_{\varphi}$, and the ensemble is a pure one in which
every system is in exactly the state $\varphi$.

Section 7.

The arguments of section 6 are extended to provide a complete specification of
the density operator $U$ of an ensemble of which it is known that each of $m$
compatible observables lie in specified intervals $I_{1},I_{2},\ldots,I_{m}$
for each system in the ensemble. The previous arguments had only established
$UE=EU=U$, with $E$ given in Eq. (14). Now, von Neumann is able to show that,
up to an unimportant overall real positive normalization, $U=E$.

Section 8.

The rigorous and fully general extension of Jordan’s specification of
conditional probabilities in quantum mechanics is derived: the result turns
out to be a remarkably simple formula involving the trace of products of
spectral projection operators. Specifically, the conditional probability that
an ensemble corresponding to the projector $E$ of Eq. (14) (i.e.,
$\mathfrak{a}_{1}$ known to be in interval $I_{1}$ etc.) will, if another set
of observables $\mathfrak{b}_{1},\ldots$ (spectral projectors
$F_{1},F_{2},\ldots$) is measured, be found to have these observables in
intervals $J_{1},J_{2},\ldots$ is given by $\mathrm{Tr}(EF)$, where
$F=F_{1}(J_{1})F_{2}(J_{2})\ldots$ In this section von Neumann points out an
extremely important feature of the statistical operator $U$ of a ensemble: it
does not, in general, uniquely specify the actual individual states appearing
in the ensemble. Different mixtures of different states can combine to give an
ensemble with exactly the same statistical operator $U$—an ensemble, in other
words, statistically indistinguishable with respect to arbitrary measurements,
from the first. This simple observation would turn out to have striking
implications for quantum measurement theory. This ambiguity is only avoided in
the case that the ensemble is pure: every system in the ensemble in the same
state $\varphi$.

Section 9

Conclusions: (a) Measurements inevitably disturb the measured object; in the
case of compatible measurements, simultaneous measurement is equivalent to a
single measurement, (b) the immediate repetition of a measurement will given
the same result, and (c) physical quantities are to be represented by
functional operators (i.e., linear operators in a Hilbert space). Further, the
a-causality inherent in quantum theory is not to be attributed to quantum
dynamics, which is fully causal, but rather to limitations in the possibility
of simultaneous measurement of incompatible observables: if an experiment is
performed to ascertain the value of an observable in an “incompatible” state
(i.e., one which is not an eigenvector of the associated operator) the
preexistent state is “destroyed” (_zertrümmert_), and a new one produced in an
unpredictable way, governed only by statistical laws. This is the first clear
expression of the notion of “collapse of the wave function” which would emerge
as a central component of the Copenhagen school of quantum measurement theory.

###  6 Paper 3: The thermodynamics of quantum mechanical ensembles

The main objective of the third paper [49] is the derivation of a general
formula for the entropy $S$ of an ensemble of $N$ quantum-mechanical systems,
the statistical properties of which are specified by a statistical operator
$U$, as defined and studied in the second paper [48]. The resulting formula
for the “von Neumann entropy” plays a fundamental role in quantum statistical
mechanics:

$S=-Nk\;\mathrm{Tr}(U\ln{U}).$ (15)

The remarkable thing about von Neumann’s derivation of this formula is that it
is shown to be a direct consequence of the volume dependence of the entropy of
a _classical ideal gas_! Most of the paper is devoted to establishing some
ancillary results extending the classical properties of semipermeable
membranes to the quantum case. The argument leading to Eq. (15), when it
appears, is very short and appears almost miraculous.

Section 1.

In the introduction, the essential results of von Neumann [48] are summarized:
the notion of an ensemble of quantum systems $\mathfrak{S}$, with an
associated statistical operator $U$ such that the expectation value of an
arbitrary observable associated to operator $R$ is given by
$E(R)=\mathrm{Tr}(UR)$. Two special ensembles (in a sense, extreme opposites
of each other) are identified: (a) pure ensembles, in which all systems are in
the same quantum state $\varphi$, with statistical operator $U=P_{\varphi}$,
and (b) the “elementary random” ensemble in which all possible states with
equal a-priori weight appear with the same statistical weight, corresponding
to statistical operator $U=1$. For purely mathematical questions, the reader
is referred to Paper 1 [47].

Section 2.

The central thermodynamics concepts—heat ($Q$), temperature ($T$), and entropy
($S$)—are introduced, now in the context of ensembles of quantum systems. Two
ensembles $\mathfrak{S}_{1}$ and $\mathfrak{S}_{2}$ are defined to have a
difference of entropy $\Delta S=-Q/T$ if a reversible transformation of
$\mathfrak{S}_{1}$ into $\mathfrak{S}_{2}$ can be accomplished via a transfer
of heat $Q$ to an ambient (infinite) heat reservoir at temperature $T$. The
additive ambiguity in the entropy can be removed by defining the entropy of
all pure ensembles to be zero, whence the principal result of the paper for
the entropy of an ensemble with statistical operator $U$ (stated here without
proof), Eq. (15). The stratagem to be employed derives from a paper by
Einstein [17], in which individual quantum-mechanical systems are enclosed in
“containers”, which are then given the dynamics of a _classical_ ideal gas.

Section 3.

Assuming the basic result for quantum entropy, Eq. (15), von Neumann shows
that the ensembles with maximum entropy for a given number of systems $N$ and
prescribed energy $E$ (thus, with $\mathrm{Tr}(U)=1$ and $\mathrm{Tr}(UH)=E$)
are those satisfying the Boltzmann distribution, i.e.

$U=\alpha\,e^{\beta H},$ (16)

with $\beta=-1/kT$, where $k$ is Boltzmann’s constant. The usual (canonical-
ensemble) thermodynamics formulas for $E$ and $S$ (as partition sums) are then
derived.

Section 4.

In this short transitional section, the normalization constant $\alpha$ in Eq.
(16) is set to unity, and the final result obtained: $U=e^{-H/kT}$. This is
seen to tend to $U=1$ in the limit of infinite temperature, as expected for
the elementary random ensemble where all states (of equal a-priori weight) are
equally represented in the ensemble.

Section 5.

The quantum mechanical analog of the semipermeable wall of classical
thermodynamics is introduced. One imagines a partition with “doors” at which
one stations measurement devices for some quantity $\mathfrak{a}$, associated
with some measurable property of the quantum systems contained in each of the
“containers/molecules” constituting the ideal gas of Section 2. If the
orthogonal eigenstates of $\mathfrak{a}$ are segregated into two non-
overlapping sets $\varphi_{\mu}$ and $\psi_{\nu}$, then a molecule arriving at
one of the doors and subjected to a measurement of $\mathfrak{a}$ will be
allowed to pass through (resp. reflected back from) the door according to the
measured state being found to be a $\varphi_{\mu}$ (resp. $\psi_{\nu}$).

Section 6.

Here, von Neumann derives another important preliminary result: the existence
of a reversible quasi-static transformation of one pure ensemble
$\mathfrak{S}_{1}$ (with statistical operator $P_{\varphi}$) into another pure
ensemble $\mathfrak{S}_{2}$ (statistical operator $P_{\psi}$). This
transformation is shown to involve (in general) performance of work, but no
exchange of heat, and is hence isentropic.

Section 7.

Here, the converse of the result of section 5 is demonstrated: if a
semipermeable wall perfectly transmits (i.e., allows to pass through) systems
in a state $\varphi$, but perfectly reflects systems in a state $\psi$, the
$\varphi$ and $\psi$ must be orthogonal. Here the basic result (not yet
proven) Eq. (15) is assumed.

Section 8.

The central result of the paper, the von Neumann entropy formula Eq. (15) is
derived, using an argument which can only be described as a remarkable feat of
mathematical legerdemain. The net effect is to transform the entropy formula
for an ideal classical gas (more exactly, the volume-dependent part of this
formula) into the formula for the entropy of an arbitrary ensemble of $N$
states of a given quantum system. What von Neumann actually calculates is the
change in entropy $S$ entailed by the reversible transformation of a pure
ensemble $\mathfrak{S}_{1}=\\{\varphi,\varphi,\varphi,\ldots.\\}$ into a
general mixed ensemble $\mathfrak{S}_{2}$ in which orthogonal states
$\psi_{1},\psi_{2},\psi_{3},\ldots,\psi_{N}$ occur with weights
$w_{1},w_{2},w_{3},\ldots,w_{N}$, where $w_{1}+w_{2}+w_{3}+\ldots+w_{N}=1$.
Thus, the statistical operator for the ensemble is

$U=\sum_{\mu}w_{\mu}P_{\psi_{\mu}},$ (17)

and

$\mathrm{Tr}(U\ln{U})=\sum_{\mu}w_{\mu}\ln{w_{\mu}}.$ (18)

The argument is simplicity itself. Recall that we are employing the
Einsteinian artifice of enclosing each quantum state in the ensemble in a
separate massive box. The collection of these boxes/molecules are then
supposed to form an ideal classical gas at some temperature $T$, bouncing
around in a containing volume $V$. The entropy of the entire system is thus
the sum of the quantum entropy of the systems enclosed by the boxes, plus the
classical formula for the entropy of an ideal gas. One starts (in the pure
ensemble) by inserting walls randomly closing off fractions $w_{1}V$,
$w_{2}V,\ldots$ of the entire volume, thereby enclosing $w_{1}N$,
$w_{2}N$,…molecules in each subvolume. The systems (all $\varphi$) in the
first box can be isentropically transformed into all $\psi_{1}$, those in the
second to all $\psi_{2}$, and so on. The first enclosure is then expanded
isothermally (this affects only the “molecules”, not the quantum systems
contained in them) from volume $w_{1}V$ back to $V$, entailing an increase in
entropy $(w_{1}N)k\ln{\\{V/(w_{1}V)\\}}=-Nkw_{1}\ln{w_{1}}$, and so on for all
the other enclosures. Finally, all the enclosures (once again of volume $V$)
are isentropically merged by clever use of semipermeable walls of the type
explained in previous sections, yielding the ensemble $\mathfrak{S}_{2}$. The
only change in entropy occurs in the isothermal expansion stage. The net
result, with the help of Eq. (18), is the desired entropy formula Eq. (15).

Section 9.

As already demonstrated in section 6, all pure ensembles have the same
entropy, which can conveniently be normalized to zero. As Eq. (18) displays
the change in entropy required in going from a pure ensemble
$\mathfrak{S}_{1}$ to a general mixed ensemble $\mathfrak{S}_{2}$, it must
give directly the entropy of the latter. This is the famous von Neumann
entropy formula. With this, the result previously announced in section 2 is
therefore established.

Section 10.

The Boltzmann distribution for general quantum ensembles is now derived. The
problem amounts to finding the particular statistical operator $U$ (normalized
to $\mathrm{Tr}(U)=1$) which maximizes the entropy Eq. (15), subject to the
constraint that the expectation value of the energy $E$, given by
$\mathrm{Tr}(UH)$, is held fixed. The result, after some simple algebra, is,
as expected $U=\alpha\exp{(\beta H)}$, with $\alpha=1/\mathrm{Tr}(\exp{(\beta
H)})$. The interpretation of the Lagrange multiplier $\beta$ requires
thermodynamic reasoning.

Section 11.

To fix the interpretation of $\beta$ one calculates the derivative of energy
with respect to entropy, finding $\beta=-1/kT$. The thermodynamics of the
ensemble is thus fully determined. To conclude the paper, von Neumann points
out that the results obtained depend on the underlying quantum system having a
completely discrete spectrum, with eigenvalues tending to infinity (if they
are bounded there must be an accumulation point, and the partition sums would
diverge). This is guaranteed if the system is enclosed in a finite volume,
which is a perfectly natural restriction for thermodynamic systems.

The concept of von Neumann entropy would be further deepened by Claude Shannon
in his seminal work on information and communication theory in 1948. The
density operator was also introduced independently and at about the same time
as von Neumann by Lev Landau [29], but in a much more limited context
(radiative damping in wave mechanics).

###  7 A note on the translation and annotations

A literal word-for-word translation of academic scientific German from the
1920s will inevitably appear extremely stilted, and sometimes confusing, when
rendered in modern English. I have therefore allowed myself a certain degree
of freedom in rephrasing von Neumann’s original words, while, of course,
trying to maintain as far as possible the original sense. Von Neumann’s
original footnotes (which begin from 1 on each new page) have been reproduced
as the numerically ordered footnotes in the translation, which (as
conventional nowadays) do not restart on each page. Short comments of the
annotator are enclosed inline in square brackets. More extended annotations
and commentary are relegated to alphabetically labelled footnotes (a, b, c,
etc.).

##  Chapter 1 Paper 1: Mathematical Foundation of Quantum Mechanics

### J. v. Neumann

### Submitted by M. Born in the session of May 20, 1927

###  1 Introduction

The formulations of quantum mechanics given by Heisenberg, Dirac, Born,
Schrödinger and Jordan See various works of the given authors in 1926/27 in
_Annalen der Physik, Zeitschrift für Physik, Proceedings of the Royal
Society_. Most importantly, [19], [9], [8], [10], [11], [12], [36], [37],
[38], [39], [40], [6], [7], [27], [28]. have introduced many quite novel
concepts and questions, among which we would like to emphasize the following:

$\alpha.$ It has become apparent that the behavior of an atomic system is
somehow connected with a certain eigenvalue problem (the formulation of which
will be treated later, in Sec. 12), in particular, the values of the
characteristic quantities describing the system are the eigenvalues
themselves.
$\beta.$ In this fashion one has achieved in a satisfying way the long awaited
amalgamation of continuous (classical-mechanical) and discontinuous
(quantized) phenomena in the world of atoms: for an eigenvalue spectrum can
indeed just as easily possess continuous as well as discontinuous parts.The
eigenvalue problem as such, in the context of infinite bilinear forms (rather
than matrices), and for linear integral equations (rather than differential
equations), was treated exhaustively by Hilbert in a series of papers from
1904-1910; the resulting theory (presented in full in [22]), which was
restricted to bounded forms—and a further subcategory of these, the completely
continuous (now, _compact_) forms—also showed the way to deal rigorously with
situations in which both a point and continuous spectrum arose. The use by
Hilbert of the term “spectrum” to delineate the set of eigenvalues seems in
hindsight remarkably prescient.
$\gamma.$ Moreover, indications have emerged in the new quantum mechanics that
the laws of Nature (or at least the quantum laws known to us) do not determine
atomic processes in an unambiguously causal way, but rather that the
elementary laws merely provide probability distributions, which only
degenerate in exceptional cases to causally sharp ones.
$\delta.$ The eigenvalue problem appears in various manifestations: as the
problem of determining eigenvalues of an infinite matrix (equivalently, the
transformation of the matrix to diagonal form), or as the eigenvalue problem
of a differential equation. Nonetheless, the two formulations are equivalent
to one another, as the matrix (viewed as a linear transformation) arises from
the differential operator (which gives the left side of the differential
equationThe typical (Schrödinger) differential equation of quantum mechanics
has the form $H\psi=\lambda\psi$, where $H$ is a differential operator,
$\lambda$ is the eigenvalue parameter, and $\psi$ is the “wave function”, of
which certain regularity and boundary conditions are required, leading to the
eigenvalue problem. when applied to the “wave function”), once one goes over
from the “wave function” to its expansion coefficients in a complete
orthogonal set.Schrödinger, _Annalen der Physik_ , Vol. 79/8, p. 734
(1926).[Schrödinger’s “equivalence” paper.] (The matrix then provides the
corresponding [to the action of the differential operator] transformation of
the expansion coefficients).A rigorous proof of the equivalence of wave and
matrix mechanics will, as von Neumann shows in Section 4, rely on theorems of
Riesz and Fischer going back to the first decade of the century (note 15).
$\epsilon.$ Both methods have their problems. In the matrix approach one has
almost always to deal with an unsolvable problem: the transformation of the
energy matrix to diagonal form. This is indeed only possible when there is no
continuous spectrumHellinger, inaugural dissertation, Section 4 (Göttingen,
1907). It is immediately apparent that this condition is necessary: the
spectrum is invariant under a transformation, and a diagonal matrix has only a
discontinuous spectrum (with the diagonal elements as the eigenvalues). in
which case the treatment of the problem is uniform (even though in the
opposite sense [i.e. discrete rather than continuous] of classical mechanics):
only discontinuous (quantized) quantities appear therein. (The hydrogen
atom—which also contains a continuous spectrumSchrödinger, _Annalen der
Physik_ , Vol. 79/4, p. 367 (1926).[Schrödinger’s first paper on wave
mechanics.]—cannot therefore be correctly treated [by matrix methods]). One
can of course help oneself to the use of “continuous matrices”,Cf. primarily
Dirac, _Proceedings of the Royal Society_ , Vol. 113, p. 621 (1927). but this
procedure (which is actually a simultaneous operation with matrices and
integral kernels) can only be carried through with great difficulty in a
mathematically rigorous fashion, as one would then have to introduce concepts
such as infinitely large matrix elements or infinitely close diagonal
elements.See note 24.
$\zeta.$ Initially the probability postulates of the matrix method were not
available in the differential equation approach. (These will be discussed
later in detail). This lacuna was repaired first by Born and then later by
Pauli and Jordan; however, the complete procedure, as it was built up by
Jordan to a closed system,_Zeitschrift für Physik_ , Vol. 40, 11/12, p. 809
(1927). Cf. also a paper shortly to appear in _Mathematische Annalen_ of
Hilbert, Nordheim, and the present author.[[23]] is subject to difficult
mathematical problems.In particular one cannot avoid the admission of so
called “improper eigenfunctions”, for example, the [delta] function
$\delta(x)$ employed first by Dirac, which is supposed to have the following
(absurd) properties:

$\displaystyle\delta(x)$ $\displaystyle=$ $\displaystyle
0\;\;\mathrm{for\;\;}x\neq 0,$
$\displaystyle\int_{-\infty}^{+\infty}\delta(x)dx$ $\displaystyle=$
$\displaystyle 1.$ (1)

A particular difficulty in Jordan’s approach is the need to calculate not only
the transformation operators (whose integral kernels are “probability
amplitudes”), but also the variable region [range] onto which the operators
map, i.e. the eigenvalue spectrum.
$\theta.$ A common deficiency of all of these methods, however, is that they
introduce elements into the calculation that are in principle unobservable and
without physical content: eigenfunctions must be calculated, which as a
consequence of their normalization remain undetermined up to a constant of
absolute value 1 (the “phase” $e^{i\phi}$).The unavoidable presence of complex
phases, some, but not all, of which were physically irrelevant, was a subject
of confusion in the early days of matrix mechanics: see, for example [19, p.
883], [9, p. 875]. Indeed, in the case of a $k$-fold degeneracy (i.e. an
eigenvalue appearing $k$ times) they remain undetermined up to $k$-dimensional
orthogonal transformation.Here we mean the complex orthogonality of the
transformation matrix $\\{\alpha_{\mu\nu}\\}$:

$\displaystyle\sum_{\rho=1}^{k}\alpha_{\mu\rho}\alpha^{*}_{\nu\rho}$
$\displaystyle=$ $\displaystyle 1,\;\mathrm{for}\;\mu=\nu,$ (2)
$\displaystyle=$ $\displaystyle 0,\;\mathrm{for}\;\mu\neq\nu,$

which leaves the “hermitian identity form”
$\sum_{\rho=1}^{k}x_{\rho}y^{*}_{\rho}$ invariant. In the mathematical
literature the usual term for this is “unitary.” The probabilities appearing
at the end of the calculation are indeed invariant [i.e. unambiguous], but it
is unsatisfying and unclear why this detour through unobservable and non-
invariant quantities is necessary.

In the present work we shall present an approach which remedies these defects,
and which, we believe, summarizes the statistical interpretation of quantum
mechanics as it now stands in a unified and consistent way.

A detailed investigation of purely mathematical questions has been avoided
throughout, wherever calculational aspects are not essential; nonetheless, our
exposition should be regarded as mathematically rigorous in all essential
respects. It could not be avoided devoting the greater part of the work to
developing and explaining formal concepts prior to showing their application.
Accordingly, sections 2-11 have a preparatory character, while the actual
theory is presented in sections 12-14.

###  2 Hilbert space [motivation]

As we saw in section 1.$\delta$, eigenvalue problems in quantum mechanics
appear in two main forms: as eigenvalue problems of infinite matrices (or,
what amounts to the same thing, bilinear forms), and as eigenvalue problems of
differential equations.

We would like to first examine each of these problems on its own, and then
emphasize the common features. Throughout we will have to pay attention to
long established mathematical facts, which are also not physically novel,
inasmuch as they have already in essence appeared in the work of Schrödinger
cited above (note 3)[the “equivalence” paper] and in numerous works of
Dirac.[10], [11],[12],[13],[14]. Nonetheless it is perhaps appropriate to
develop everything in a connected way, and it will also be useful to first
present results which will motivate the abstract conceptual developments of
the following sections.

Let us first consider the matrix formulation. Here we encounter an infinite
matrix representing the energy (how one arrives at it will be explained
later), and our task is to transform it to diagonal form, as the diagonal
elements are just the energy levels.Cf. for example Born, _Probleme der
Atomdynamik_ , p. 86 (Berlin 1926). Let us assume that this proceeds without
complications—i.e. that only a point spectrum is present (cf. 1.$\epsilon$ and
note 4).

The energy matrix will be written

$H=\\{h_{\mu\nu}\\}\quad(\mu,\nu=1,2,\ldots);$ (3)

it is assumed to be hermitian, i.e.

$h_{\mu\nu}=h^{*}_{\nu\mu}.$ (4)

One seeks a transformation matrix

$S=\\{s_{\mu\nu}\\}\quad(\mu,\nu=1,2,\ldots)$ (5)

with the following properties: it is orthogonal [unitary], i.e.

$\sum_{\rho=1}^{\infty}s_{\mu\rho}s^{*}_{\nu\rho}=1\;(\mathrm{for}\;\mu=\nu),\;\;0\;(\mathrm{for}\;\mu\neq\nu),$
(6)

and $S^{-1}HS$ is in diagonal form.

We will denote by $W$ the matrix $S^{-1}HS$, and the diagonal elements of this
(diagonal) matrix will be denoted $w_{1},w_{2},\ldots$ Thus we have

$\displaystyle HS$ $\displaystyle=$ $\displaystyle SW,$ (7)
$\displaystyle\sum_{\rho=1}^{\infty}h_{\mu\rho}s_{\rho\nu}$ $\displaystyle=$
$\displaystyle s_{\mu\nu}w_{\nu}.$ (8)

In other words, the $\nu$’th column of $S$, ($s_{1\nu},s_{2\nu},\ldots$), is
multiplied by $w_{\nu}$ through the application of $H$. Each column of $S$ is
consequently a solution of the eigenvalue problem, which consists of finding
just those sequences $x_{1},x_{2},\ldots$ which are transformed by $H$ into a
multiple of themselves (by some constant $w$). ($x_{1},x_{2},\ldots$ is then
an eigenvector, the proportionality factor $w$ an eigenvalue, excluding
naturally the trivial solution $0,0,\ldots$. The eigenvalue associated with
$s_{1\nu},s_{2\nu},\ldots$ is thus $w_{\nu}$.)Here, von Neumann is extending
the standard procedure for diagonalizing a finite-dimensional hermitian matrix
to the infinite-dimensional case, under the assumption that the associated
linear operator has only a discrete point spectrum. It is _not_ assumed that
the operator is bounded—for example, the harmonic oscillator can be handled in
this way, despite having an unbounded energy spectrum.

One can now show that these [sequences] are the only solutions: more exactly,
there are no eigenvalues different from the $w_{1},w_{2},..$, and if $w$ is an
eigenvalue, the eigenvectors associated with it are linear combinations of all
the columns $s_{1\nu},s_{2\nu},\ldots$ for which $w_{\nu}=w$ (see Appendix 1).

Consequently, the determination of the matrix $S$ is essentially accomplished
as soon as the eigenvalue problem, as formulated above, is completely solved.

In the formulation in terms of differential equations the situation is even
clearer: it is from the outset given in terms of an eigenvalue problem. One is
given a differential operator (for example, for a rotating rigid body, an
oscillator, hydrogen atom, one has, respectively

$\displaystyle(\frac{1}{\sin{\theta}}\frac{\partial}{\partial\theta}\sin{\theta}\frac{\partial}{\partial\theta}+\frac{1}{\sin^{2}{\theta}}\frac{\partial}{\partial\phi^{2}})\cdot\cdot\cdot$
(9)
$\displaystyle(\frac{d}{dq^{2}}-\frac{16\pi^{4}}{h^{2}}\nu_{0}^{2}q^{2})\cdot\cdot\cdot$
(10) $\displaystyle(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial
z^{2}}+\frac{8\pi^{2}me^{2}}{h^{2}}\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}})\cdot\cdot\cdot)$
(11)

and one seeks a function $\psi$ (in our examples $\psi$ is a function of
$\theta,\phi;q;x,y,z$ respectively) for which

$H\psi=w\psi,$ (12)

i.e. which is transformed into a multiple of itself by the action of $H$, and
for which the eigenvalues $w$ are again, up to a factor of $8\pi^{2}m/h^{2}$,
the energy levels. Naturally, $\psi$ must satisfy certain regularity
conditions, and not vanish identically.Schrödinger, _Annalen der Physik_ ,
Vol. 79/4, p. 361, and 79/6, p. 489 (1926).

What is the common feature of all these cases? Apparently this: in each case
one is given a space of certain quantities (namely, the manifold of all
sequences $x_{1},x_{2},\ldots$; or that of all functions of two angles
$\theta,\phi$, or of a single coordinate $q$, or of three coordinates
$x,y,z$), and a linear operator $H$ in this space. In each case one seeks all
solutions of the eigenvalue problem associated with the operator $H$, namely
all (real) numbers $w$ for which there exists a non-vanishing element $f$ of
this space satisfying

$Hf=wf.$ (13)

These eigenvalues $w$ then represent the energy levels [of the system].

It is now our task to proceed from a unified formulation to a unique problem.
This will be achieved once we show that all spaces introduced in this way—in
particular, absolutely all the spaces to which one is led in the usual
formulations of quantum mechanics at the present time—are in essence identical
to one another; i.e. that they can all be obtained from a single space (which
will be described in the subsequent sections) by a mere redefinition.

To this end we will need however to specify more exactly which sequences
(resp. functions) can be accepted in the spaces mentioned above—in other
words, we will need to indicate the regularity- and boundary-conditions, which
turn out to be of decisive importance in the eigenvalue problem.

###  3 Hilbert space [introduction of $l^{2}$ and $L^{2}$]

We begin with the space of numerical [complex] sequences $x_{1},x_{2},\ldots.$
It seems reasonable to require the convergence of the squared sum
$\sum_{n=1}^{\infty}|x_{n}|^{2}$. In fact, this holds for the solutions of the
eigenvalue problem, as long as there is only a point spectrum, which are just
the columns $s_{1\nu},s_{2\nu},\ldots$ of the matrix $S$ (cf. Section 2);
indeed, as we required above, one has

$\sum_{\rho=1}^{\infty}s_{\mu\rho}s^{*}_{\mu\rho}=\sum_{\rho=1}^{\infty}|s_{\mu\rho}|^{2}=1$
(14)

(If there is a continuous spectrum, it is no longer possible to solve the
eigenvalue problem only employing eigen-sequences with finite
$\sum_{n=1}^{\infty}|x_{n}|^{2}$. We shall see below that our procedure
nevertheless will allow a comprehensive treatment of the continuous
spectrum).Von Neumann is here emphasizing in fact, not often fully
appreciated, that a complete and rigorous treatment of bounded linear
operators is available even in the case where a _continuous_ spectrum is
present, in the framework of the _discrete_ space of square-summable
sequences.

Even the simplest linear transformations $H$ (i.e. corresponding to matrices
$h_{\mu\nu},\mu,\nu=1,2,\ldots$) fail us when acting on sequences with
arbitrary $\sum_{n=1}^{\infty}|x_{n}|^{2}$, inasmuch as the series
$\sum_{\nu=1}^{\infty}h_{\mu\nu}x_{\nu}$ are not guaranteed to converge. (It
will become clear that the matrices of quantum mechanics have the property
that the squared row sums $\sum_{\nu=1}^{\infty}|h_{\mu\nu}|^{2}$ are all
finite, in which case the convergence of
$\sum_{\nu=1}^{\infty}h_{\mu\nu}x_{\nu}$ follows from the finiteness of
$\sum_{n=1}^{\infty}|x_{n}|^{2}$.The series
$\sum_{\nu=1}^{\infty}h_{\mu\nu}x_{\nu}$ in fact converges absolutely as a
consequence of the inequality
$|h_{\mu\nu}x_{\nu}|\leq\frac{1}{2}|h_{\mu\nu}|^{2}+\frac{1}{2}|x_{\nu}|^{2}$.)Quantum
mechanics is rife with operators which are _not_ bounded, e.g. momentum,
position, energy, etc. What von Neumann is saying here is that the theory can
be formulated in such a way that its statistical phenomenological content is
completely expressed in terms of _bounded_ operators—specifically, the bounded
projection operators associated with the spectral resolution of all the
operators—usually, unbounded—representing physical quantities of interest. How
this can be done is shown in Sections 12-14 of the paper.

Finally, it is precisely this limitation of the range of the
$x_{1},x_{2},\ldots$ [namely, $\sum_{n=1}^{\infty}|x_{n}|^{2}<\infty$] which
has proven most suitable in the theory of infinite matricesThe theory of
infinite matrices (resp. bilinear forms) is in essence due to Hilbert, based
on the complete clarification of the relations (and in particular, on the
solution of the eigenvalue problem) between the important classes of the so-
called completely continuous and the (originating from these) bounded bilinear
forms (cf. _Göttingen Nachrichten, Mathematische-Physikalische Klasse_ , 1906,
pp. 159-227). A large part of the mathematical confusions and difficulties of
quantum mechanics derive from the fact that even the simplest operators (or
matrix bilinear forms) appearing therein do not belong to the class of bounded
operators treated by Hilbert. The present author has recently shown how the
eigenvalue problem of arbitrary, including therefore unbounded, operators can
be unambiguously solved (to appear shortly in _Mathematische Annalen_). In
this work we will naturally need to investigate the precise formulation of an
eigenvalue problem for unbounded operators. (and on the appropriate
generalization of these which must serve as the foundation for the
mathematical construction of quantum mechanics).

There is therefore a good motivation for distinguishing the following space:
_all sequences of complex numbers $x_{1},x_{2},\ldots$ with finite
$\sum_{n=1}^{\infty}|x_{n}|^{2}$_. The name _(complex) Hilbert space_ is usual
for this space.The space of square-summable sequences, now referred to as
“$l^{2}$”, was first studied in detail in a seminal paper by Erhard Schmidt,
[35]. In this paper he introduces the standard geometrical concepts associated
with “Hilbert spaces” ( a term never used by Hilbert)—e.g. inner product,
norm, orthogonal projections, etc. Many of the results obtained earlier by
Hilbert for bounded bilinear forms and integral kernels were rederived in a
simpler form in this language.

At this point, let us move on to the so-called _function spaces_ considered in
Section 2. Here the situation with regard to supplementary conditions (i.e.
regularity and boundary conditions) is more complicated. It is commonplace to
require double differentiability, as well as uniqueness, vanishing at infinity
or at the boundary of the domain of definition, and the like. How can we
arrive at a unified standpoint in such a situation?

Let the domain of definition of the functions in question (in our examples:
the $\theta,\phi$ space, i.e. the sphere; the $q$-space, i.e. the real line;
the $x,y,z-$space, i.e. the usual [3-]space) be called $\Omega$. Of course
only uniquely defined functions in $\Omega$ are to be considered, to which the
operator $H$ can be applied. Consequently, if $H$ is a differential operator
of second order (which is generally the case in quantum mechanical problems),
double differentiability of $\psi$ is essential. With the reservation that
this need only hold at those points of $\Omega$ allowed by the application of
$H$—for example, at points where factors in $H$ display singularities
($x=y=z=0$, say, in the hydrogen atom [where the Coulomb potential diverges]),
this [double-differentiability] is not required.For example, the ground state
solution for the hydrogen atom (in Schrödinger’s notation $n=1,l=0$)
$\psi(x,y,z)=\exp{(-\sqrt{x^{2}+y^{2}+z^{2}})}$, has a point singularity at
$x=y=z=0$. Indeed, it is well known that it is precisely this imposition of
regular behavior at such points that results in the appearance of an
eigenvalue problem: still, one can (and must) achieve this by the introduction
of considerably weaker conditions.

We will therefore require, that $\int_{\Omega}|\psi|^{2}dv$ (denoting with
$dv$ respectively the line, surface, and volume element of $\Omega$ [in the
examples cited previously]) remain finite: this precludes a too rapid
divergence of the singularities in $H$, and also guarantees a part of the
boundary conditions, namely, the vanishing [of $\psi$] at infinity. Success
will show that we have made the right choice.

Certain boundary conditions (e.g. vanishing at the boundary of a finite region
[cf. particle in a box]) are still not encompassed in the previous
requirements. In order to clarify the role of these, consider the following:
in order that an eigenvalue problem be at all possible, we must require that
the matrices $H$ possess _hermitian symmetry_

$H=\\{h_{\mu\nu}\\},\;\;h_{\mu\nu}=h^{*}_{\nu\mu};$ (15)

and in the case of differential equations, _self-adjointness_ is essential,
i.e. thatThe importance of enforcing appropriate boundary conditions in order
to obtain the desired self-adjoint property of differential operators was
already well-known in the mathematical literature, see, e.g. [22, pp. 39-41].

$\int_{\Omega}\\{\psi_{1}\cdot(H\psi_{2})^{*}-(H\psi_{1})\cdot\psi_{2}^{*}\\}dv$
(16)

vanish for all functions $\psi_{1},\psi_{2}$ that vanish sufficiently fast at
the boundary of their domain of definition. (Equivalently: the quantity
$\\{\psi_{1}\cdot(H\psi_{2})^{*}-(H\psi_{1})\cdot\psi_{2}^{*}\\}dv$ should be
an [exact] differential of some expression built from $\psi_{1},\psi^{*}_{2}$
and their derivatives.) For this to hold the boundary conditions are in
certain circumstances absolutely important. For example, let $\Omega$ be the
line segment [0,1] and $H$ the operator $i\frac{d}{dx}$. We then have
[$dv=dx$]

$\displaystyle\int_{\Omega}\\{\psi_{1}\cdot(H\psi_{2})^{*}-(H\psi_{1})\cdot\psi_{2}^{*}\\}dv$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\\{\psi_{1}(x)\cdot[-i(\psi_{2}^{\prime}(x))^{*}]-i\psi_{1}^{\prime}\cdot\psi_{2}^{*}(x)\\}dv$
(17) $\displaystyle=$
$\displaystyle-i\int_{0}^{1}\\{\psi_{1}(x)(\psi_{2}^{\prime}(x))^{*}+\psi_{1}^{\prime}(x)\psi_{2}^{*}(x)\\}dv$
$\displaystyle=$ $\displaystyle-i[\psi_{1}(x)\psi_{2}^{*}(x)]_{0}^{1}.$

This certainly vanishes, provided $\psi_{1}$ and $\psi_{2}$ vanish at both
end-points of $\Omega$, but, lacking imposition of all boundary conditions,
could very well be nonzero.

We therefore further require (and this will prove to contain the last
essential boundary conditions) that $\psi$ behaves in such a way as to
preserve the self-adjointness of $H$.

Altogether, we therefore require: $\psi$ is uniquely defined in $\Omega$, has
the property that $H$ can be applied to it and has the property of self-
adjointness, and $\int_{\Omega}|\psi|^{2}dv$ is finite. The only nontrivial
condition—the one responsible for the “regularity requirement” giving rise to
the eigenvalue problem—is, as one sees, the last. (One could raise the
objection that this requirement excludes access to the continuous spectrum,
where the eigenfunctions are never square-integrable. We also encountered an
analogous situation in the matrix case. It will however become apparent, that
precisely this constraint allows a particularly fruitful—and mathematically
fully rigorous—conception of the continuous spectrum, completely faithful to
its physical meaning, without having to admit improper forms as
eigenfunctions. Cf. Section 10.)

To summarize: we have a space $\mathfrak{H}$ (in the matrix formulation the
space of all sequences $x_{1},x_{2},\ldots$ with finite
$\sum_{n=1}^{\infty}|x_{n}|^{2}$; in the differential equation formulation the
space of all functions $\psi$ defined in $\Omega$ with finite
$\int_{\Omega}|\psi|^{2}dv$, where $\Omega$ can be a surface of arbitrary
dimension, with or without boundary, and bounded or unbounded) the elements of
which we denote $f,g,\ldots$ to maintain a unified notation. In addition, a
linear operator $H$ is given (providing for each $f$ in $\mathfrak{H}$ another
element $Hf$, and for which $H(af)=aHf$ ($a$ complex), and $H(f+g)=Hf+Hg$),
which is also symmetric (resp. self-adjoint). This last requirement can also
be formulated in a unified way. For by defining [the inner product]
$Q(f,g)=\sum_{n=1}^{\infty}x_{n}y^{*}_{n}$ (resp.
$\int_{\Omega}\phi\psi^{*}dv$), where
$f=x_{1},x_{2},\ldots,g=y_{1},y_{2},\ldots$ in the matrix case, and
$f=\phi,g=\psi$ in the differential equation case, the symmetry/self-
adjointness condition takes the form, for allowed $f,g$,

$Q(f,Hg)=Q(Hf,g)$ (18)

We now seek the solutions of the eigenvalue problem ($Hf=wf,f\neq 0,w$ a real
constant) for the operator $H$. In this way we obviously in the first instance
only obtain the point spectrum, but the general formulation achieved in this
fashion will subsequently enable us to get control also of the continuous
spectrum.

It is apparent that these formulations differ only in the underlying space
$\mathfrak{H}$. Is it not possible also to interpret these spaces somehow in a
unified fashion?

###  4 [Critique of Dirac-Jordan transformation theory]

A suggestive approach would be to exploit the analogous construction of the
expressions $\sum_{n=1}^{\infty}|x_{n}|^{2}$ and $\int_{\Omega}|\psi|^{2}dv$
(similarly $\sum_{n=1}^{\infty}x_{n}y^{*}_{n}$ and
$\int_{\Omega}\phi\psi^{*}dv$) to say: given a space $R$, which can be
discrete or continuous (the “space” 1,2,…, or a $\Omega$), we consider all
functions on $R$ (all sequences $x_{1},x_{2},\ldots$, or, all functions on
$\Omega$). There exists an “integration over $R$” ($\sum_{n=1}^{\infty}x_{n}$,
or $\int_{\Omega}\psi dv$), and we admit only such “proper” functions into
consideration the absolute square of which possesses a finite “integral over
$R$” ($\sum_{n=1}^{\infty}|x_{n}|^{2}$ finite [the space $l^{2}$, in modern
notation] or $\int_{\Omega}|\psi|^{2}dv$ finite [$L^{2}[\Omega]$, in modern
notation]. Naturally, in certain cases $R$ might be of mixed type, containing
for example both discrete points and line intervals.

In essence this is the approach followed by Dirac with great persistence and
unquestioned success in a number of ground breaking works in quantum
mechanics.In several articles of 1926/27 in the _Proceedings of the Royal
Society_. If we nevertheless attempt to find a different solution, it is just
for the reason that the analogy sketched above necessarily remains a
superficial one, as long as one adheres to normal standards of mathematical
rigor.

For example, in the case where $R$ is the “space” 1,2,…, so that
$\mathfrak{H}$ contains the sequences $x_{1},x_{2},\ldots$, all linear
operators $H$ are representable by matrices $\\{h_{\mu\nu}\\}$:

$\displaystyle H(x_{1},x_{2},\ldots)$ $\displaystyle=$
$\displaystyle(y_{1},y_{2},\ldots),$ (19) $\displaystyle y_{n}$
$\displaystyle=$ $\displaystyle\sum_{\nu=1}^{\infty}h_{\mu\nu}x_{\nu}.$ (20)

If one tries to perform an analogous representation in the case where $\Omega$
is the interval [0,1] (so $\mathfrak{H}$ contains all functions defined on
this interval), then one must demand that every linear operator $H$ be
represented by an integral kernel $\Phi(x,y)$:

$H\phi(x)=\int_{0}^{1}\Phi(x,y)\phi(y)dy.$ (21)

This fails already for the simplest operators (e.g. the “identity” operator,
which takes every function to itself). If one still wishes to “fake” the
correctness of this false assertion, one is forced, with Dirac, to consider
“improper” integral kernels [e.g. the Dirac delta “function”].Von Neumann is
at pains throughout his work on quantum mechanics to appeal to functional
analytic methods and results which were on a firm mathematical footing in the
mid-1920s. A rigorous theory of “distributions”, including Dirac’s
$\delta(x)$, would emerge, starting in the 1930s with the work of Sobolev, and
reaching maturity in the period 1945-1950 with Laurent Schwartz’s work
[41],[42]. Schwartz was initially motivated to extend the solution space of
differential equations by allowing arbitrary derivatives of continuous
functions. A fully developed theory based on continuous linear functionals
over suitable topological function spaces was presented in his 1950 treatise.
As von Neumann shows in the present paper, Hilbert space provides a completely
adequate arena to construct a rigorous operator theory containing the full
statistical content of quantum mechanics—the use of “improper functions”
(later, distributions), is not necessary.

We shall embark on a different route, which at heart lies at the foundation of
Schrödinger’s “equivalence proof” of the matrix [e.g. matrix mechanics] and
differential equation [e.g. wave mechanics] formulations, and can be derived
from long known mathematical material. In order to describe this approach,
some preparatory remarks are required.

Let $\Omega$ once again be a surface (1,2,…dimensional; with or without
boundary; finite or infinite), $dv$ the differential element of integration on
$\Omega$. We consider all complex-valued functions $f,g,\ldots$ on $\Omega$
with finite $\int_{\Omega}|f|^{2}dv,..$ as constituting the “space”
$\mathfrak{H}$. The integral $\int_{\Omega}fg^{*}dv$ (which provides the only
context in which integration over $\Omega$ will appear in the following) will
be written for brevity $Q(f,g)$ [modern notation: $(g,f)$ or (Dirac) $\langle
g|f\rangle$, with the anti-linear element on the left]. For $Q(f,f)$ (the
finiteness of which specifies $\mathfrak{H}$!) we write simply $Q(f)$ [modern
notation $(f,f)$, $\langle f|f\rangle$, or $|f|^{2}$].

Two functions $f,g$ of $\mathfrak{H}$ are called orthogonal if

$Q(f,g)=0,$ (22)

and a system $f_{1},f_{2},\ldots$ is a normalized orthogonal [orthonormal]
system if

$\displaystyle Q(f_{\mu},f_{\nu})$ $\displaystyle=$ $\displaystyle
1,\;\;\mathrm{for\;}\mu=\nu,$ (23) $\displaystyle=$ $\displaystyle
0,\;\;\mathrm{for\;}\mu\neq\nu.$

Finally, a complete orthonormal system is one to which no further $f$ can be
added while preserving the orthonormality conditions.(This is clearly
equivalent to the statement that there is no function $f$, not identically
zero, orthogonal to all $f_{n}$.)

One can show that there exist complete orthonormal systems in $\mathfrak{H}$
(see Section 5 for a more exact discussion of this and all further unproven
assertions [in the remainder of this Section]). Let $\phi_{1},\phi_{2},\ldots$
be one such. If $f$ is a function in $\mathfrak{H}$, we shall call

$c_{\mu}=Q(f,\phi_{\mu}),(\mu=1,2,\ldots)$ (24)

the expansion coefficients of $f$ (with regards to
$\phi_{1},\phi_{2},\ldots$). The sum $\sum_{\mu=1}^{\infty}|c_{\mu}|^{2}$ is
always finite (indeed, it equals $Q(f)$, which is the so-called Parseval
formula), and the series $\sum_{\mu=1}^{\infty}c_{\mu}\phi_{\mu}$ converges in
a certain sense to $f$. The series in fact need not converge in any particular
point of $\Omega$; but when one follows the deviation of the $N$’th partial
sum of $f$ (namely, $\sum_{n=1}^{N}c_{\mu}\phi_{\mu}-f$) over all of $\Omega$,
and characterizes the net deviation through the integral of its absolute
square (thus, $Q(\sum_{n=1}^{N}c_{\mu}\phi_{\mu}-f)$, which, the smaller it
is, the closer $\sum c_{\mu}\phi_{\mu}$ approximates $f$ over all of $\Omega$,
and not just at special points of $\Omega$), then this integral converges for
$N\rightarrow\infty$ to 0, i.e
$\lim_{N\rightarrow\infty}Q(\sum_{n=1}^{N}c_{\mu}\phi_{\mu}-f)=0$. (This form
of convergence is denoted “convergence in the mean”. It is less stringent than
point-wise convergence, but is the more appropriate concept when dealing with
orthogonal expansions.)

The converse statement is also correct (Fischer-Riesz theorem):Göttingen
Nachrichten, Mathematische-Physikalische Klasse, 1907, pp. 116-122.
If $c_{1},c_{2},\ldots$ is an arbitrary sequence of complex numbers with
finite $\sum_{\mu=1}^{\infty}|c_{\mu}|^{2}$, then (function)-sum
$\sum_{\mu=1}^{\infty}c_{\mu}\phi_{\mu}$ converges in the mean (i.e. there
exists a $f$ in $\mathfrak{H}$ such that
$Q(f-\sum_{\mu=1}^{N}c_{\mu}\phi_{\mu})\rightarrow 0$ as
$N\rightarrow\infty$), and the sum $f$ has the expansion coefficients
$c_{1},c_{2},\ldots.$

We have therefore a one-to-one mapping between functions $f$ in $\mathfrak{H}$
and sequences $c_{1},c_{2},\ldots$ with finite
$\sum_{\mu=1}^{\infty}|c_{\mu}|^{2}$. This mapping is manifestly linear, i.e.
if $f$ corresponds to $c_{1},c_{2},..$, then $af$ ($a$ a complex constant)
corresponds to $ac_{1},ac_{2},\ldots$; and if $f$ (resp. $g$) corresponds to
$c_{1},c_{2},..$ (resp. $d_{1},d_{2},\ldots$), then $f+g$ corresponds to
$c_{1}+d_{1},c_{2}+d_{2},\ldots$From the relation
$|u+v|^{2}\leq|u+v|^{2}+|u-v|^{2}=2|u|^{2}+2|v|^{2}$ one sees directly that
the finiteness of
$\sum_{\mu=1}^{\infty}|c_{\mu}|^{2},\sum_{\mu=1}^{\infty}|d_{\mu}|^{2}$ (resp.
$\int_{\Omega}|f|^{2}dv,\int_{\Omega}|g|^{2}dv$) implies the finiteness of
$\sum_{\mu=1}^{\infty}|c_{\mu}+d_{\mu}|^{2}$ (resp.
$\int_{\Omega}|f+g|^{2}dv$). Moreover $Q(f,g)$ is given by
$\sum_{\mu=1}^{\infty}c_{\mu}d^{*}_{\mu}$, which is the generalized Parseval
formula

$\displaystyle c_{\mu}$ $\displaystyle=$ $\displaystyle
Q(f,\phi_{\mu}),\;\;d_{\mu}=Q(g,\phi_{\mu}),\;\;(\mu=1,2,\ldots)$ (25)
$\displaystyle Q(f,g)$ $\displaystyle=$
$\displaystyle\sum_{\mu=1}^{\infty}c_{\mu}d^{*}_{\mu},$ (26)

to be demonstrated in the next Section.

Precisely since we denoted $\sum_{\mu=1}^{\infty}x_{\mu}y^{*}_{\mu}$ (in the
space of sequences $x_{1},x_{2},\ldots$) $Q(x,y)$ ($x$ for
$x_{1},x_{2},\ldots$, $y$ for $y_{1},y_{2},\ldots$), we can assert:
$\mathfrak{H}$ can be mapped in a one-to-one fashion to the space of all
sequences $x_{1},x_{2},\ldots$ with finite
$\sum_{\mu=1}^{\infty}|x_{\mu}|^{2}$ in such a way that the operations $af$
($a$ a complex constant), $f+g$, $Q(f,g)$—i.e. all operations previously used
in the description of quantum mechanics—are transformed into themselves (i.e.
to their analogs in the space of sequences). Consequently, all function spaces
$\mathfrak{H}$ (whatever the associated $\Omega$) are distinguished from the
space of sequences $x_{1},x_{2},\ldots$ with finite
$\sum_{\mu=1}^{\infty}|x_{\mu}|^{2}$ (and likewise, with one another) solely
by the denotation of elements and operations, while agreeing completely with
each other in all their properties.

In other words: even without the introduction of “continuous matrices” and
“improper forms”, the various sequence- and function-spaces which underly
quantum mechanics are in essence, and with absolute mathematical rigor,
identical.This is, in essence, the desired proof of equivalence of matrix and
wave mechanics, satisfying “usual levels of mathematical rigor” whcih von
Neumann found wanting in the Dirac-Jordan approach.

###  5 [Hilbert space: formal axiomatic definition]

The space of all complex valued number sequences $x_{1},x_{2},\ldots$ with
finite $\sum_{\mu=1}^{\infty}|x_{\mu}|^{2}$ will be called complex infinite-
dimensional Euclidean space or _complex Hilbert space_ ; we will denote this
space $\mathfrak{H}_{0}$ [our emphasis; in modern notation $\mathfrak{H}_{0}$
is written $l^{2}$].

As we saw in previous sections, all function spaces $\mathfrak{H}$ have formal
properties which coincide with each other and with the space
$\mathfrak{H}_{0}$: the differences lie solely in the notation used for, or
the interpretation of, the elements of the space. This suggests that we can
characterize all thses spaces by their common properties, and define any space
possessing all the specified characteristic properties as an abstract Hilbert
space. One can therefore imagine all the particular sequence- and function-
spaces as arising from the particular notation attached to its elements (by
interpreting them as sequences $x_{1},x_{2},..$ or functions $\psi$ on a space
$\Omega$), rather as one can arrive at a space-time interpretation in
relativity theory of a metrically homogeneous 4-dimensional “world” by choice
of a special coordinate system (namely, by specifying world points as number
quadruplets).

We thus have the task of describing abstract Hilbert space on the basis of its
“inner” properties: i.e. those that can be formulated independently of the
interpretation of the elements of the space as sequences or functions. This
will be accomplished by specifying 5 properties, from which all others follow,
as we shall show. These 5 properties refer to an “abstract Hilbert space
$\bar{\mathfrak{H}}$”, with elements $f,g,\ldots,$ in which the operations
$af$ ($a$ a complex constant), $f+g$, $Q(f,g)$ are defined ($af,f+g$ are again
elements of $\bar{\mathfrak{H}}$, while $Q(f,g)$ is a complex number). These
operations (in particular $Q(f,g)$) should now however be viewed independently
of their definition in Section 3: we require of them just the 5 properties to
be specified. Of course, these properties will be satisfied (cf. Appendix 2)
if we replace $\bar{\mathfrak{H}}$ with either its “discrete realization”
$\mathfrak{H}_{0}$ [$l^{2}$] (i.e. space of all sequences $x_{1},x_{2},\ldots$
with finite $\sum_{n=1}^{\infty}|x_{n}|^{2}$,
$Q(x,y)=\sum_{n=1}^{\infty}x_{n}y^{*}_{n}$), or with its “continuous
realization” $\mathfrak{H}$ (space of all functions $\phi$ defined on $\Omega$
with finite
$\int_{\Omega}|\phi|^{2}dv,Q(\phi,\psi)=\int_{\Omega}\phi\psi^{*}dv$ [modern
$L^{2}[\Omega]$]). To visualize the concrete meaning of our axioms, one has
just to replace the abstract $\bar{\mathfrak{H}}$ with one of the special
realizations cited above.

We shall now give the 5 characteristic properties A.-E. of abstract (complex)
Hilbert space $\bar{\mathfrak{H}}$ mentioned above. In each of them we shall
immediately derive the simplest consequences; at the end shall follow the
proof that these properties fully determine $\bar{\mathfrak{H}}$, in the sense
that $\bar{\mathfrak{H}}$ can be uniquely constructed—given invariance of the
operations $af,f+g,Q(f,g)$—from $\mathfrak{H}_{0}$ (the idea of the proof was
already sketched in Section 4 for the function-spaces $\mathfrak{H}$). That
the spaces $\mathfrak{H}_{0},\mathfrak{H}$ can in fact be regarded as
$\bar{\mathfrak{H}}$’s (i.e. possess the properties A.-E.) will be
demonstrated in Appendix 2.

Our five axioms A.–E. for the abstract complex Hilbert space
$\bar{\mathfrak{H}}$ read as follows:Schmidt’s treatment of the fundamental
Hilbert space $l^{2}$ (not referred to as such!) in [35] includes discussion
of the properties (not identified as “axioms”) A.-C. and E. (Cauchy-
completeness). The property D. (separability) is not discussed explicitly.
Schmidt refers throughout to a “function space”, i.e. the space of complex
functions defined on the integers, thus complex sequences $c_{1},c_{2},\ldots$
The inner product of two such functions is written $(f;g^{*})$ (instead of von
Neumann’s $Q(f,g)$), and the norm as $||f||$ (instead of von Neumann’s
$\sqrt{Q(f)}$)—in both cases, somewhat closer to modern usage!. However, both
authors place the antilinear (complex conjugated) element in the inner product
on the right, rather than the left, as done at present. All the standard
inequalities, including the triangle inequality (35), Bessel’s inequality
(Section 6, Theorem 2), etc. are derived by Schmidt. He credits (note 8,
op.cit.) Kowalewski for the geometric interpretation of the concepts and
theorems discussed: specifically, for the treatment of his “functions” as
“vectors in a space of infinitely many dimensions.”

A. $\bar{\mathfrak{H}}$ is a linear space. Namely: $\bar{\mathfrak{H}}$
possesses an addition $f+g$ and a multiplication $af$ ($f,g$ in
$\bar{\mathfrak{H}}$, $a$ a complex number, $f+g,af$ in $\bar{\mathfrak{H}}$).
These operations satisfy the known rules followed by the analogous operations
with vectors (in particular: existence of a [null vector] 0, commutativity and
associativity of addition, distributivity and associativity of [scalar]
multiplication).

A fundamental concept which can be grounded solely on the basis of A. is that
of linear independence: any [set of] elements $f_{1},f_{2},\ldots,f_{k}$ of
$\bar{\mathfrak{H}}$ are said to be linearly independent if

$a_{1}f_{1}+a_{2}f_{2}+\cdot\cdot\cdot+a_{k}f_{k}=0$ (27)

implies $a_{1}=a_{2}=\cdot\cdot\cdot=a_{k}=0$. A further concept so grounded
is that of a subspace of $\bar{\mathfrak{H}}$ generated by a subset
$\mathfrak{M}$: this is the set of all
$a_{1}f_{1}+a_{2}f_{2}+\cdot\cdot\cdot+a_{k}f_{k}$, where $a_{1},a_{2},\ldots
a_{k}$ are arbitrary complex numbers, and $f_{1},f_{2},\ldots,f_{k}$ are
arbitrary members of $\bar{\mathfrak{H}}$.

B. $\bar{\mathfrak{H}}$ is a metric space, with a metric derived from a
bilinear form $Q(f,g)$. Namely: there exists a function $Q(f,g)$ (defined for
all $f,g$ in $\bar{\mathfrak{H}}$, and complex numbers as values) with the
following properties:

  1. 1.

$Q(af,g)=aQ(f,g)$, ($a$ a complex constant).

  2. 2.

$Q(f_{1}+f_{2},g)=Q(f_{1},g)+Q(f_{2},g)$.

  3. 3.

$Q(f,g)=(Q(g,f))^{*}$.
From 1, 2 (given 3) we can conclude:
1′. $Q(f,ag)=a^{*}Q(f,g)$.
2′. $Q(f,g_{1}+g_{2})=Q(f,g_{1})+Q(f,g_{2})$.
1, 2, 1′, 2′ express the bilinear hermiticity of $Q$, 3 the symmetry. From 3
it follows that $Q(f,f)$ is always real, but we shall further require (writing
again $Q(f)$ for $Q(f,f)$):

  4. 4.

$Q(f)\geq 0$, and only -0, if $f=0$. From 1-4 one concludes without difficulty
that

$|Q(f,g)|\leq\sqrt{Q(f)Q(g)}$ (28)

always holds, and that alsoOne easily calculates (with $a,b$ real constants;
Re($z$), Im($z$) the real and imaginary parts of the complex number $z$)

$Q(af+bg)=a^{2}Q(f)+ab(Q(f,g)+Q(g,f))+b^{2}Q(g)=a^{2}Q(f)+2ab\mathrm{Re}(Q(f,g))+b^{2}Q(g).$
(29)

The left side is always $\geq 0$, the right is a quadratic form in $a,b$, so
must have its discriminant $\leq 0$, whence

$\mathrm{Re}(Q(f,g))^{2}-Q(f)Q(g)\leq
0,\;\;|\mathrm{Re}(Q(f,g))|\leq\sqrt{Q(f)Q(g)}.$ (30)

If we now replace $f$ with $e^{i\phi}f$ ($\phi$ real), the right side is
unchanged, while the left side becomes

$|\mathrm{Re}(e^{i\phi}Q(f,g))|=|\cos{(\phi)}\mathrm{Re}(Q(f,g))-\sin{(\phi)}\mathrm{Im}(Q(f,g))|.$
(31)

As the maximum of this quantity is
$\sqrt{\mathrm{Re}(Q(f,g))^{2}+\mathrm{Im}(Q(f,g))^{2}}=|Q(f,g)|$ [the right
hand side of (31) can be regarded as the inner product of the two-vectors
$(\cos{\phi},-\sin{\phi})$ and $(\mathrm{Re}(Q(f,g)),\mathrm{Im}(Q(f,g))$,
which reaches its maximum value when these two vectors are aligned—as the
first is a unit vector, the inner product then just becomes the magnitude of
the second vector] we indeed have

$|Q(f,g)|\leq\sqrt{Q(f)Q(g)}.$ (32)

Further, one has [$a$ complex]
$Q(af)=aa^{*}Q(f)=|a|^{2}Q(f),\;\sqrt{Q(af)}=|a|Q(f)$, and

$\displaystyle Q(f+g)$ $\displaystyle=$ $\displaystyle
Q(f)+Q(f,g)+Q(g,f)+Q(g)=Q(f)+2\mathrm{Re}(Q(f,g))+Q(g)$ (33)
$\displaystyle\leq$ $\displaystyle
Q(f)+2\sqrt{Q(f)Q(g)}+Q(g)=(\sqrt{Q(f)}+\sqrt{Q(g)})^{2},$ (34)
$\displaystyle\sqrt{Q(f+g)}$ $\displaystyle\leq$
$\displaystyle\sqrt{Q(f)}+\sqrt{Q(g)}.$ (35) $\displaystyle\sqrt{Q(af)}$
$\displaystyle=$ $\displaystyle|a|\sqrt{Q(f)}$ (36)
$\displaystyle\sqrt{Q(f+g)}$ $\displaystyle\leq$
$\displaystyle\sqrt{Q(f)}+\sqrt{Q(g)}.$ (37)

The last two relations motivate the consideration of $\sqrt{Q(f)}$ as the
absolute value of $f$, and $\sqrt{Q(f-g)}$ as the distance between $f$ and
$g$.From relation (35) follows the fundamental postulate [triangle inequality]
for every distance relation:

$\mathrm{distance}(f,h)\leq\mathrm{distance}(f,g)+\mathrm{distance}(g,h).$
(38)

Our distance function $\sqrt{Q(f-g)}$ is therefore given in $\mathfrak{H}_{0}$
(resp. $\mathfrak{H}$, where we indeed know the [inner product] $Q$) by
$\sqrt{\sum_{n=1}^{\infty}|x_{n}-y_{n}|^{2}}$ (resp.
$\sqrt{\int_{\Omega}|f-g|^{2}dv}$); namely, the reasonable generalization of
the distance concept in the usual euclidean spaces.

In this way $Q$ indeed provides $\bar{\mathfrak{H}}$ with a metric, a concept
of separation. Consequently expressions such as “continuous”, “bounded”,
“arbitrarily close” etc. acquire a meaning in $\bar{\mathfrak{H}}$.

C. $\bar{\mathfrak{H}}$ has infinitely many dimensions: i.e. there are an
arbitrarily large number of linearly independent elements of
$\bar{\mathfrak{H}}$.

D. There exists an everywhere dense sequence in $\bar{\mathfrak{H}}$. Namely:
there exists a sequence $f_{1},f_{2},\ldots$ such that elements of the
sequence appear in an arbitrary neighborhood of any element $f$ of
$\bar{\mathfrak{H}}$.

E. The Cauchy convergence requirement holds in $\bar{\mathfrak{H}}$. Namely:
Every sequence $f_{1},f_{2},\ldots$ in $\bar{\mathfrak{H}}$ satisfying the
Cauchy convergence condition (i.e. for every $\epsilon>0$ there is a
$N=N(\epsilon)$ such that $N\leq m\leq n$ implies
$\sqrt{Q(f_{m}-f_{n})}\leq\epsilon$) is convergent (i.e. there exists a $f$ in
$\bar{\mathfrak{H}}$ so that for every $\epsilon>0$ there is a $N=N(\epsilon)$
such that $N\leq m$ implies $\sqrt{Q(f_{m}-f)}\leq\epsilon$).

Temporarily postponing to Appendix 2 the question of whether
$\mathfrak{H}_{0},\mathfrak{H}$ [i.e. $l^{2}$ and $L^{2}[\Omega]$] satisfy
these conditions, we will instead proceed to derive the most immediate
consequences of A.–E., specifically the often repeated claim of
constructibility of $\bar{\mathfrak{H}}$ from $\mathfrak{H}_{0}$, in the
course of which we shall also acquire an insight into the structure of
orthogonal systems in $\bar{\mathfrak{H}}$ which will be critical for our
later considerations.

In the next Section we shall give (for the sake of completeness) a rigorous
discussion of these ideas using known mathematical procedures—one can skip the
details, paying attention only to the main results.

###  6 [Basic properties of orthogonal systems in Hilbert space]

We must first give some simple definitions of a general nature. $f$ is an
accumulation point of a subset $\mathfrak{M}$ of $\bar{\mathfrak{H}}$, if
points of $\bar{\mathfrak{H}}$ [–>$\mathfrak{M}$] can be found arbitrarily
close (cf. the remark following axiom B) to $f$. A subset $\mathfrak{M}$ of
$\bar{\mathfrak{H}}$ is closed if it contains all its accumulation points; it
is everywhere dense, if every $f$ in $\bar{\mathfrak{H}}$ is an accumulation
point [of $\mathfrak{M}$]; it is dense in [another subset] $\mathfrak{N}$, if
every point of $\mathfrak{N}$ is an accumulation point [of $\mathfrak{M}$].

$\mathfrak{M}$ is a linear subspace if it is mapped to itself by linear
operations: thus, if $f_{1},f_{2},\ldots,f_{k}$ is in $\mathfrak{M}$, so is
$a_{1}f_{1}+a_{2}+f_{2}+\ldots a_{k}f_{k}$.

Two elements $f,g$ are orthogonal, if $Q(f,g)=0$. $\mathfrak{M}$ is an
orthonormal system, if for all $f,g$ in $\mathfrak{M}$

$\displaystyle Q(f,g)$ $\displaystyle=$ $\displaystyle
1,\;\;\mathrm{for}\;\;f=g,$ (39) $\displaystyle=$ $\displaystyle
0,\;\;\mathrm{for}\;\;f\neq g.$ (40)

$\mathfrak{M}$ is a complete orthonormal system if no further $f$ can be added
to it while preserving the orthogonal character of the set. Equivalently: if
no $f$ (other than $f=0$) is orthogonal to all $g$ in $\mathfrak{M}$.

We can now proceed to the proof of the previously announced theorems:

Theorem 1. Every orthonormal system $\mathfrak{M}$ is either finite or a
sequence; every complete orthonormal system is a sequence.
Proof: Let $\mathfrak{M}$ be an orthonormal system, $f_{1},f_{2},\ldots$ the
everywhere dense sequence [cf. axiom D]. For every pair $f,g$ of
$\mathfrak{M}$ we have

$Q(f-g)=Q(f)-2\mathrm{Re}(Q(f,g))+Q(g)=1-0+1=2,$ (41)

so their separation is $\sqrt{2}$. We can associate to each element $f$ of
$\mathfrak{M}$ some element of the [everywhere dense] sequence
$f_{1},f_{2},\ldots$ which is nearer to it than $\sqrt{2}/2$, from which it
follows that two separate elements of the dense sequence are associated to two
different elements of $\mathfrak{M}$. Consequently, $\mathfrak{M}$ has at most
as many elements as the given dense sequence, which was to be proven.

If on the other hand $\mathfrak{M}$ is complete, we wish to show that
$\mathfrak{M}$ cannot be finite, i.e. that to any finite set
$\phi_{1},\phi_{2},\ldots,\phi_{k}$ there is an orthogonal $f\neq 0$. But we
cannot find $k+1$ linearly independent elements in the linear subspace spanned
by $\phi_{1},\phi_{2},\ldots,\phi_{k}$, so $\bar{\mathfrak{H}}$ must contain
[axiom C] an element $f$ outside of this space. Accordingly

$f-c_{1}\phi_{1}-c_{2}\phi_{2}-\cdot\cdot\cdot c_{k}\phi_{k}$ (42)

is never $=0$: indeed, by choosing $c_{n}=Q(f,\phi_{n})$ we can make it
orthogonal to all $\phi_{1},\ldots\phi_{k}$, [thus, any $\mathfrak{M}$ with a
finite basis cannot be complete], which was to be proven.∎

Theorem 2. Let $\phi_{1},\phi_{2},\ldots$ be an orthonormal system [not
necessarily complete!]. Then every series

$\sum_{n=1}^{\infty}Q(f,\phi_{n})Q(g,\phi_{n})^{*}$ (43)

is absolutely convergent, and in particular [for $f=g$], one has
$\sum_{n=1}^{\infty}|Q(f,\phi_{n})|^{2}\leq Q(f)$.
Proof: Choosing $c_{n}=Q(f,\phi_{n}),n=1,2,\ldots$, we have

$\displaystyle Q(\sum_{n=1}^{N}c_{n}\phi_{n}-f)$ $\displaystyle=$
$\displaystyle