Q(f)-\sum_{n=1}^{N}2\;\mathrm{Re}(Q(f,c_{n}\phi_{n}))+\sum_{m,n=1}^{N}Q(c_{m}\phi_{m},c_{n}\phi_{n})$
(44) $\displaystyle=$ $\displaystyle
Q(f)-\sum_{n=1}^{N}2\;\mathrm{Re}(c_{n}^{*}Q(f,\phi_{n}))+\sum_{m,n=1}^{N}c_{m}c^{*}_{n}Q(\phi_{m},\phi_{n})$
$\displaystyle=$ $\displaystyle
Q(f)-2\sum_{n=1}^{N}|c_{n}|^{2}+\sum_{n=1}^{N}|c_{n}|^{2}=Q(f)-\sum_{n=1}^{N}|c_{n}|^{2}.$

As the left side is always $\geq 0$, it follows that [for any $N$]

$\sum_{n=1}^{N}|c_{n}|^{2}\leq Q(f),$ (45)

thereby establishing the convergence of the series

$\sum_{n=1}^{\infty}|c_{n}|^{2}=\sum_{n=1}^{\infty}|Q(f,\phi_{n})|^{2},$ (46)

and also that the sum of the series is $\leq Q(f)$ (the second assertion).

The first assertion of the theorem follows from the obvious identitiesThe
first identity follows from

$|Q(\frac{f\pm g}{2},\phi_{n})|^{2}=\frac{1}{4}(|Q(f,\phi_{n})|^{2}\pm
2\mathrm{Re}(Q(f,\phi_{n})Q(g,\phi_{n})^{*})+|Q(g,\phi_{n})|^{2}),$ (47)

by subtracting the right hand sides with opposite choices of sign. Similarly
with the second identity.

$\displaystyle\mathrm{Re}(Q(f,\phi_{n})Q(g,\phi_{n})^{*})$ $\displaystyle=$
$\displaystyle|Q(\frac{f+g}{2},\phi_{n})|^{2}-|Q(\frac{f-g}{2},\phi_{n})|^{2},$
(48) $\displaystyle\mathrm{Im}(Q(f,\phi_{n})Q(g,\phi_{n})^{*})$
$\displaystyle=$
$\displaystyle|Q(\frac{f+ig}{2},\phi_{n})|^{2}-|Q(\frac{f-ig}{2},\phi_{n})|^{2},$
(49)

as the sum over each term on the right hand sides converges absolutely [by the
previous argument].∎

Theorem 3. Let $\phi_{1},\phi_{2},\ldots$ be an orthonormal system. The series
$\sum_{n=1}^{\infty}c_{n}\phi_{n}$ convergesNote, that this is convergence in
$\bar{\mathfrak{H}}$! Therefore, if one considers a continuous realization
$\mathfrak{H}$ of $\bar{\mathfrak{H}}$ (the space of all functions $f$ defined
in $\Omega$ with finite $\int_{\Omega}|f|^{2}dv$), then this refers not to
pointwise convergence, but convergence in the mean. if and only if
$\sum_{n=1}^{\infty}|c_{n}|^{2}$ is finite.
Proof: By axiom E [Cauchy-completeness of $\bar{\mathfrak{H}}$] the
convergence of $\sum_{n=1}^{\infty}c_{n}\phi_{n}$ is equivalent to the
requirement that for every $\epsilon>0$ there is a $N=N(\epsilon)$, so that
for $N\leq m\leq n$

$\sqrt{Q(\sum_{p=1}^{n}c_{p}\phi_{p}-\sum_{p=1}^{m}c_{p}\phi_{p})}\leq\epsilon.$
(50)

But it now follows that [for $m<n$]

$\displaystyle Q(\sum_{p=1}^{n}c_{p}\phi_{p}-\sum_{p=1}^{m}c_{p}\phi_{p})$
$\displaystyle=$ $\displaystyle Q(\sum_{p=m+1}^{n}c_{p}\phi_{p})$ (51)
$\displaystyle=$
$\displaystyle\sum_{p,q=m+1}^{n}c_{p}c^{*}_{q}Q(\phi_{p},\phi_{q})$
$\displaystyle=$ $\displaystyle\sum_{p=m+1}^{n}|c_{p}|^{2}$ $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}|c_{p}|^{2}-\sum_{p=1}^{m}|c_{p}|^{2},$

whereby we have exactly the convergence condition [this is the Cauchy
completeness property for the real numbers] for the series
$\sum_{p=1}^{\infty}|c_{p}|^{2}$.∎

Corollary. For this $f$ [$=\sum_{n=1}^{\infty}c_{n}\phi_{n}$], one has
$Q(f,\phi_{p})=c_{p}$.
Proof: Certainly we have, for $p\leq N$,

$Q(\sum_{n=1}^{N}c_{n}\phi_{n},\phi_{p})=\sum_{n=1}^{N}c_{n}Q(\phi_{n},\phi_{p})=c_{p}.$
(52)

Now $Q(f^{\prime},g^{\prime})$, on account of its bilinearity, and the
inequality $|Q(f^{\prime},g^{\prime})|\leq\sqrt{Q(f^{\prime})Q(g^{\prime})}$,
is a continuous function of $f^{\prime}$ and $g^{\prime}$.In other words, if
$f_{n}\rightarrow f,n\rightarrow\infty$ (which means $Q(f-f_{n})\rightarrow
0$), then $Q(f_{n},g)\rightarrow Q(f,g)$. And similarly for a sequence
$g_{n}\rightarrow g$. So we can let $N\rightarrow\infty$, and obtain
$Q(f,\phi_{p})=c_{p}$ [i.e. defining $f^{N}=\sum_{n=1}^{N}c_{n}\phi_{n}$, by
continuity
$c_{p}=\lim_{N\rightarrow\infty}Q(f^{N},\phi_{p})=Q(\lim_{N\rightarrow\infty}f^{N},\phi_{p})=Q(f,\phi_{p})$].∎

Theorem 4. Let $\phi_{1},\phi_{2},\ldots$ be an orthonormal system. The series

$f^{\prime}=\sum_{n=1}^{\infty}c_{n}\phi_{n},\;\;c_{n}=Q(f,\phi_{n}),$ (53)

is convergent for all $f$, and $f-f^{\prime}$ is orthogonal to all
$\phi_{1},\phi_{2},\ldots$
Proof: Follows directly from theorems 2 and 3.

Theorem 5. Any one of the following three conditions is both necessary and
sufficient for an orthonormal system $\phi_{1},\phi_{2},\ldots$ to be
complete:
$\alpha$. The linear space spanned by $\phi_{1},\phi_{2},\ldots$ is everywhere
dense.
$\beta$. For all $f$,
$f=\sum_{n=1}^{\infty}c_{n}\phi_{n},c_{n}=Q(f,\phi_{n})$.
$\gamma$. For all $f,g$,
$Q(f,g)=\sum_{n=1}^{\infty}Q(f,\phi_{n})Q(g,\phi_{n})^{*}$.

Proof: [In the following, “completeness” is short for “completeness of the
orthonormal $\phi_{1},\phi_{2},..$] First, completeness implies $\beta$, as
the $f-f^{\prime}$ of Theorem 4 must vanish [by definition, there is no
nonzero element orthogonal to all elements of a complete set].

Secondly, $\beta$ implies $\alpha$, as the sequence $f^{N},N=1,2,..$,
$f^{N}=\sum_{p=1}^{N}c_{p}\phi_{p}$ has $f$ as its accumulation point, and all
the $f^{N}$ belong to the linear space spanned by the
$\phi_{1},\phi_{2},\ldots$

Thirdly, $\beta$ implies $\gamma$, as (cf. Eq.(35))

$Q(\sum_{n=1}^{N}c_{n}\phi_{n}-f)=Q(f)-\sum_{n=1}^{N}|c_{n}|^{2},$ (54)

and for $N\rightarrow\infty$ (as $Q(f)$, as remarked above, is continuous),
the left hand side tends to zero, we have

$\sum_{n=1}^{\infty}|c_{n}|^{2}=Q(f),\;\;\sum_{n=1}^{\infty}|Q(f,\phi_{n})|^{2}=Q(f).$
(55)

If we now sequentially replace $f$ by $\frac{f+g}{2}$, $\frac{f-g}{2}$, and
$\frac{f+ig}{2}$, $\frac{f-ig}{2}$, and subtract twice, one obtains, as in the
proof of Theorem 2, the representation $\gamma$.[See note 47.]

Fourth, $\alpha$ implies completeness. For suppose a $f$ exists which is
orthogonal to all $\phi_{1},\phi_{2},\ldots$: then it is also orthogonal to
the entire space spanned by them, and as the latter is everywhere dense,
orthogonal to itself, i.e. $Q(f)=0,f=0$.

Fifth, $\gamma$ implies completeness: for if $f$ is orthogonal to all
$\phi_{1},\phi_{2},\ldots$, then $\gamma$ implies (putting $g=f$) that
$Q(f)=0,f=0$.

We therefore have the following logical scheme: completeness
$\Rightarrow\beta$, $\beta\Rightarrow\alpha\Rightarrow$ completeness,
$\beta\Rightarrow\gamma\Rightarrow$ completeness. All four assertions [i.e.
completeness, $\alpha$, $\beta$, $\gamma$] are therefore logically
equivalent.∎

Theorem 6. To every sequence $f_{1},f_{2},\ldots$ corresponds an orthonormal
system $\phi_{1},\phi_{2},\ldots$ (both sequences may terminate after a finite
number of elements) which span the same linear space.
Proof: Denote the first nonzero element of $f_{1},f_{2},..$ as $g_{1}$; the
first element of $f_{1},f_{2},..$ $\neq a_{1}g_{1}$ as $g_{2}$; the first
element of $f_{1},f_{2},..$ $\neq a_{1}g_{1}+a_{2}g_{2}$ as $g_{3}$; and so
on. The $g_{1},g_{2},\ldots$ are obviously linearly independent, and span the
same linear space as $f_{1},f_{2},..$ By the well-known procedure of
[Schmidt]-orthogonalizationThis procedure [introduced in [35, Ch. 1, Sec. 5]]
runs as follows:

$\displaystyle\gamma_{1}$ $\displaystyle=$ $\displaystyle
g_{1},\;\;\phi_{1}=\frac{1}{\sqrt{Q(\gamma_{1})}}\gamma_{1},$ (56)
$\displaystyle\gamma_{2}$ $\displaystyle=$ $\displaystyle
g_{2}-Q(g_{2},\phi_{1})\phi_{1},\;\;\phi_{2}=\frac{1}{\sqrt{Q(\gamma_{2})}}\gamma_{2},$
(57) $\displaystyle\gamma_{3}$ $\displaystyle=$ $\displaystyle
g_{3}-Q(g_{3},\phi_{1})\phi_{1}-Q(g_{3},\phi_{2})\phi_{2},\;\;\phi_{3}=\frac{1}{\sqrt{Q(\gamma_{3})}}\gamma_{3},\ldots$
(58)

The $\phi_{1},\phi_{2},\ldots$ clearly span the smae linear space as the
$g_{1},g_{2},\ldots$, and are normalized, and orthogonal to each other. one
can transform the $g_{1},g_{2},\ldots$ into an orthonormal system.∎

Corollary. There exist complete orthonormal systems.
Proof: By axiom D, choose an everywhere dense sequence $f_{1},f_{2},\ldots$
Replace it by an orthonormal system as in Theorem 6. This is complete, by
Theorem 5$\alpha$.∎

If we now take an arbitrary complete orthonormal system
$\phi_{1},\phi_{2},\ldots$, and to every $f$ in $\bar{\mathfrak{H}}$ assign
the sequence $c_{1},c_{2},\ldots$ ($c_{n}=Q(f,\phi_{n}),n=1,2,\ldots$), then
the sequence $c_{1},c_{2},\ldots$ belongs to $\mathfrak{H}_{0}$ by Theorem 2,
and determines in its turn $f$ by Theorem 5$\beta$. This one-to-one
representation of $\bar{\mathfrak{H}}$ covers all of $\mathfrak{H}_{0}$ by
Theorem 3. The invariance of the [vector] addition and [scalar] multiplication
in this mapping is trivial, the invariance of $Q$ follows from Theorem
5$\gamma$.

In this way we have indeed demonstrated that every space $\bar{\mathfrak{H}}$
obeying A.–E. must agree in all its properties with the usual Hilbert space
$\mathfrak{H}_{0}$. (This mapping from $\bar{\mathfrak{H}}$ to
$\mathfrak{H}_{0}$ is naturally analogous to that sketched in Section 3 from
[the function space] $\mathfrak{H}$ to $\mathfrak{H}_{0}$.)

###  7 Operator Calculus

Following the considerations of previous Section, we can use abstract
(complex) Hilbert space as the foundation for further investigations. Appeal
to its various representations (the discrete, as well as the various
continuous ones, cf. the beginning of Section 5) will be seen in the
forthcoming Sections (7-11, 13) to be absolutely unnecessary: all of our
general developments are independent of these. We shall only need to concern
ourselves with these aspects later, with the physical applications.

The first thing which we must develop in $\bar{\mathfrak{H}}$ is the operator
calculus—we already know (cf. the end of Section 2) the fundamental importance
it has for quantum mechanics.

A function in $\bar{\mathfrak{H}}$ which is defined in certain (perhaps all)
points of $\bar{\mathfrak{H}}$, and has points of $\bar{\mathfrak{H}}$ as
values, is called an operator. An operator $T$ is linear, if, first, it is
defined in a linear manifold (which by no means needs to be closed), and if,
second, it always satisfies

$T(a_{1}f_{1}+a_{2}f_{2}+\cdot\cdot\cdot+a_{k}f_{k})=a_{1}Tf_{1}+a_{2}Tf_{2}+\cdot\cdot\cdot+a_{k}Tf_{k},$
(59)

where $a_{1},a_{2},\ldots a_{k}$ are complex constants.

As we already observed in Section 4, the expressions “continuous” and “bounded
in the sphere of radius 1 centered on 0” are meaningful for operators, and for
linear operators obviously imply the same thing.The second condition means:
$Q(f)\leq 1$ implies $Q(Tf)\leq C$ ($C$ a constant). Given that $T(af)=aTf$,
this implies

$\displaystyle Q(Tf)$ $\displaystyle\leq$ $\displaystyle CQ(f),$ (60)
$\displaystyle Q(Tf-Tg)$ $\displaystyle\leq$ $\displaystyle CQ(f-g),$ (61)

in other words, continuity. Conversely: for continuous $T$ there exists an
$\epsilon>0$ such that $Q(f)<\epsilon$ implies $Q(Tf)\leq 1$, from which it
immediately follows that for $Q(f)\leq 1$ we have
$Q(Tf)\leq\frac{1}{\epsilon}$[$\equiv C$]. In this case the Hilbert expression
“bounded” is the common one. If we consider the discrete representation
$\mathfrak{H}_{0}$, then every bounded linear operator corresponds to an
infinite matrix, and the associated bilinear form (of infinitely many
variables) belong to that class of bounded bilinear forms, for which Hilbert
has completely solved the eigenvalue problem (cf. note 12)The 1912 treatise of
Hilbert summarizes this work, see, for example [22, p. 15]. The spectral
theory of bounded forms was first presented in [21].. The operators of
interest in quantum mechanics do not (cf. same note) however belong to this
class.

In the continuous realizations $\mathfrak{H}$ of Hilbert space one indeed, for
certain bounded operators $T$, has at hand the integral kernel representation
analogous to matrices [in the discrete case]:

$Tf(P)=\int_{\Omega}\Phi(P,Q)f(Q)dv_{Q},$ (62)

where the integration [in $\Omega$, with volume element $dv_{Q}$] is over $Q$;
however, such a representation fails already with the simplest of all
operators, the “identity operator” (which transforms $f$ to $f$). Our
considerations will be accomplished without employment of either a matrix or
integral kernel representation.

It is essential to recall some simple operators, and rules for calculation
with operators. One has

$0f=0,\;\;\;1f=f.$ (63)

(0 thus appears in three forms: as a number, as a point in
$\bar{\mathfrak{H}}$, and as the null operator; but there is hardly any danger
of confusing them.) Moreover we have:

$\displaystyle(aT)f$ $\displaystyle=$ $\displaystyle
a\;Tf\;\;(a\;\mathrm{a\;complex\;constant}),$ (64) $\displaystyle(R+T)f$
$\displaystyle=$ $\displaystyle Rf+Tf,$ (65) $\displaystyle(RT)f$
$\displaystyle=$ $\displaystyle R(Tf).$ (66)

It is generally known that addition of operators is commutative and
associative, both types of multiplication [scalar and operator] are
distributive and associative, but in general not commutative, and that 0 and 1
play the role of null and unity.

Finally we observe the following: if $T$ is a linear operator, and for a
particular element $f$ of $\bar{\mathfrak{H}}$ there exists a $\bar{f}$, so
that (given existence of $Tg$) we always have

$Q(\bar{f},g)=Q(f,Tg),$ (67)

then we denote this $\bar{f}$ by $T^{\dagger}f$. It is uniquely determined
provided that the $g$ for which $Tg$ exists are everywhere dense. Indeed, if
this were the case for two elements $\bar{f}_{1},\bar{f}_{2}$ we would have

$Q(\bar{f}_{1},g)=Q(\bar{f}_{2},g),\;\;Q(\bar{f}_{1}-\bar{f}_{2},g)=0,$ (68)

for an everywhere dense set of $g$, hence [by continuity of $Q$] for all $g$,
and therefore [setting $g=\bar{f}_{1}-\bar{f}_{2}$]

$Q(\bar{f}_{1}-\bar{f}_{2})=0,\;\bar{f}_{1}=\bar{f}_{2}.$ (69)

This requirement (of meaningful action on an everywhere dense set of $g$) will
now be imposed on all operators to be considered—it will indeed be satisfied
by all the operators of real interest to us.The meaning of this requirement
can be made clear with the following two examples:
Let $\Omega$ be the interval $(0,1)$, $T$ the derivative operation. $Tf$ does
not always exist (not all $f(x)$ are differentiable), but differentiable
functions on this interval are certainly dense, as any function can be
arbitrarily well approximated—in the mean—by polynomials (certainly
differentiable!).
Let $\Omega$ be the interval $(-\infty,\infty)$, $T$ multiplication by $x$.
$Tf$ does not always exist ($\int_{-\infty}^{\infty}|f(x)|^{2}dx$ can be
finite, while $\int_{-\infty}^{\infty}x^{2}|f(x)|^{2}dx$ is not). However,
these $f$ [for which $Tf$ exists] are everywhere dense: all $f$, which vanish
identically outside an (arbitrarily large, but finite) interval, belong to
this set.

$T^{\dagger}$ is also, as one immediately sees, also a linear operator; also
for $T^{\dagger}$ we shall presume that it acts meaningfully on an everywhere
dense set of elements $f$. It is defined by the equation

$Q(f,Tg)=Q(T^{\dagger}f,g).$ (70)

On the basis of the properties of $Q$ (Section 4B) one can easily show (with
the necessary assumptions of meaningfulness):

$\displaystyle(aT))^{\dagger}$ $\displaystyle=$ $\displaystyle aT^{\dagger}$
(71) $\displaystyle(R+T)^{\dagger}$ $\displaystyle=$ $\displaystyle
R^{\dagger}+T^{\dagger}$ (72) $\displaystyle(RT)^{\dagger}$ $\displaystyle=$
$\displaystyle T^{\dagger}R^{\dagger}$ (73) $\displaystyle T^{\dagger\dagger}$
$\displaystyle=$ $\displaystyle T.$ (74)

A linear operator $T$ is called _symmetric_ (more accurately, complex
hermitian) if $T=T^{\dagger}$. Using the above equations one can verify
immediately the following assertions:
If $T$ is symmetric, so is $aT$ if and only if $a$ is real.
If $R,T$ are symmetric then so is $R+T$;
on the other hand, $RT$ is symmetric if and only if $R$ and $T$ commute
($RT=TR$).
0 and 1 [null and identity] are both symmetric.

###  8 Projection Operators

After these general considerations we now wish to consider an important
special class of operators: the projection operators,The word [German
_Einzeloperator_ (literally, “Identity-operator”)], is derived from Hilbert’s
word for the analogous bilinear forms: _Einzelform_. defined as follows: a
symmetric [modern: hermitian] linear operator $E$ is a projection operator
(abbrev. P. Op. [German, _Einzeloperator, kurz E. Op._]) if $E=E^{2}$. One
sees at once that 0 and 1 are P. Ops. Moreover, if $E$ is a P. Op., so is
$1-E$, as $(1-E)^{2}=1-2E+E^{2}=1-2E+E=1-E$.

We now wish to derive a few theorems concerning projection operators. While we
shall not need all of these results inthe following discussion, they are
nonetheless useful in throwing the right light on the essence of this
fundamental concept. In this respect the following two facts are of primary
interest:

  1. 1.

A P. Op. maps certain $f$ into $f$, and others into 0. The first set
constitutes the interior, the second set the exterior of the given P. Op.
Every $f$ in $\bar{\mathfrak{H}}$ can be decomposed in precisely one way as
$g+h$, with $g$ in the interior and $h$ in the exterior of $E$. Thus, $Ef$
arises from $f$ by discarding its “outer component” $h$.

  2. 2.

There is a certain [partial] ordering of size among projection operators:
namely, according to the extension of its interior (or, inversely, according
to the extension of its exterior).

Keeping these results in view one may skip the following discussion (in which
the definition of the interior and exterior of $E$ are used).

One has obviously

$\displaystyle Q(Ef,Eg)$ $\displaystyle=$ $\displaystyle
Q(f,E^{\dagger}Eg)=Q(f,E^{2}g)=Q(f,Eg),$ (75) $\displaystyle Q(Ef,Eg)$
$\displaystyle=$ $\displaystyle Q(E^{\dagger}Ef,g)=Q(E^{2}f,g)=Q(Ef,g),$ (76)

and in particular [setting $g=f$]

$Q(f,Ef)=Q(EF).$ (77)

From this follows

Theorem 1. One has always $0\leq Q(EF)\leq Q(f)$.
Proof: The first inequality is trivial, the second holds because $1-E$ is also
a P. Op.:

$\displaystyle Q(f,Ef)$ $\displaystyle=$ $\displaystyle Q(f)-Q(f,(1-E)f),$
(78) $\displaystyle Q(Ef)$ $\displaystyle=$ $\displaystyle Q(f)-Q((1-E)f)\leq
Q(f).\qed$ (79)

Therefore every P. Op. is continuous, indeed all P. Ops. are equally
continuous [i.e. have identical operator norm (equal to unity)].

Theorem 2. If $E,F$ are P. Ops. then $EF$ is a P. Op. if and only if they
commute ($EF=FE$). $E+F$ is then a P. Op. if and only if $EF=0$ (or
equivalently $FE=0$). $F-E$ is then a P. Op. if and only if $EF=E$ (or
equivalently $FE=E$).
Proof: In the case of the product $EF$, $EF=FE$ is already both necessary and
sufficient for symmetry [hermiticity] to hold [as
$(EF)^{\dagger}=F^{\dagger}E^{\dagger}=FE$]; but as from this also follows

$(EF)^{2}=EFEF=EEFF=E^{2}F^{2}=EF,$ (80)

the proof is completed in this case.

For $E+F,F-E$ the symmetric property is assured, one need only check the
relation $T^{2}=T$ [$T=E+F$ or $F-E$]. Take first $E+F$. If $E+F$ is a P. Op.,

$\displaystyle Q(f,Ef)+Q(f,Ff)$ $\displaystyle=$ $\displaystyle Q(f,(E+F)f),$
(81) $\displaystyle Q(Ef)+Q(Ff)$ $\displaystyle=$ $\displaystyle Q((E+F)f)\leq
Q(f);$ (82)

and from $Ef=f$ it then follows that

$Q(Ff)\leq 0,\;\;Ff=0.$ (83)

Now for any $g$ it is always the case that $EEg=Eg$, so $FEg=0$, whence
$FE=0$.

Conversely, if $FE=0$, so also $EF=0$ (as $FE$ is a P. Op., therefore $F,E$
commute), and

$(E+F)^{2}=E^{2}+EF+FE+F^{2}=E+0+0+F=E+F,$ (84)

and consequently $E+F$ is a P. Op. In other words, $FE=0$ is necessary and
sufficient [for $E+F$ to be a P. Op.]; and as the roles of $E$ and $F$ are
fully symmetric, the same holds for $EF=0$.

Next, consider $F-E$. It is then and only then a P. Op. if $1-(F-E)=E+(1-F)$
is a P. Op. For this to hold it is however both necessary and sufficient (as
$E$ and $1-F$ are P. Ops.) that

$E(1-F)=0,\;\;E=EF,$ (85)

and also

$(1-F)E=0,\;\;E=FE.\qed$ (86)

We now introduce the following definitions: if $E+F$ is a P. Op., we say that
$E$ and $F$ are _orthogonal_ [German: _fremd_]; if $F-E$ is a P. Op. we write
$E\leq F$. Theorem 2 thus provides simple necessary and sufficient conditions
for both situations. One sees right away that $\leq$ is an ordering relation
for the P. Ops.: i.e. that it satisfies $E\leq E$; $E\leq F$ and $F\leq E$
imply $E=F$; and $E\leq F$ and $F\leq G$ imply $E\leq G$. In this ordering 0
precedes, 1 follows all $E$. $E\leq F$ is equivalent to the statement that $E$
and $1-F$ are orthogonal, or to $1-F\leq 1-E$.

We should now investigate more closely the interior and exterior sets of $E$.
That both are closed linear manifolds follows from the linearity and
continuity of $E$. Further one sees immediately that they are simply
interchanged under the mapping $E$ to $1-E$. The interior of 1 and the
exterior of 0 contains all $f$ of $\bar{\mathfrak{H}}$; the exterior of 1 and
the interior of 0 consists on the other hand only of the null element 0.

Theorem 3. The interior of $E$ consists of elements of the form $Eg$ ($g$
arbitrary), the exterior of $E$ those of form $(1-E)g$.
Proof: The second assertion follows from the first, simply by replacing $E$
with $1-E$. The first is proven as follows: if $f$ isin the interior of $E$,
then $f=Ef$, which is of the desired form; conversely, if $f=Eg$, then
$Ef=EEg=Eg=f$, i.e. $f$ is in the interior of $E$.∎

Theorem 4. $f$ lies in the interior (resp. exterior) of $E$ if and only if
$Q(Ef)=Q(f)$ (resp. $=0$).
Proof: The second part is trivial, as $Q(Ef)=0$ if and only if $Ef=0$. The
first part follows from this by interchange of $E$ and $1-E$ [namely:
$Q(f-Ef)=0$ iff $f=Ef$; however, a short calculation shows that
$Q(f-Ef)=2(Q(f)-Q(Ef))=0$ iff $Q(Ef)=Q(f)$].∎

Theorem 5. Each $f$ can be decomposed in a unique fashion as $g+h$ where $g$
(resp. $h$) is in the interior (resp. exterior) of $E$. This unique
decomposition is in fact $Ef+(1-E)f$.
Proof: That $Ef+(1-E)f$ accomplishes the desired decomposition is clear; the
uniqueness follows from the fact that the interior and exterior of $E$ are
linear spaces which share only the null element 0.∎

Theorem 6. The interior (resp. exterior) of $E$ consists of those, and only
those, elements of $\bar{\mathfrak{H}}$ which are orthogonal to every element
of the exterior (resp. interior) of $E$.
Proof: If $f$ (resp. $g$) lie in the interior (resp. exterior) of $E$, they
must be orthogonal:

$Q(f,g)=Q(Ef,g)=Q(f,Eg)=Q(f,0)=0.$ (87)

Our assertion follows obviously from this, together with Theorem 5.∎

Theorem 7. $E,F$ are orthogonal P. Ops. if and only if each $f$ in the
interior of $E$ lies in the exterior of $F$.
Proof: The condition is necessary and sufficient, as it amounts to saying that
for all $f$ in the interior of $E$ one has $Ff=0$, i.e. we have always $FEg=0$
[for arbitrary $g$], i.e. $FE=0$, whence $E$ and $F$ are orthogonal.∎

Theorem 8. $E\leq F$ is equivalent to the statement that the interior of $E$
is a subset of the interior of $F$, or also, that the exterior of $F$ is a
subset of the exterior of $E$.
Proof: $E\leq F$ means: $E$ and $1-F$ are orthogonal [as $E=EF$]. One can then
apply Theorem 7 first to $E$ and $1-F$, then to $1-F$ and $E$.∎

Theorem 9. $E\leq F$ is equivalent to the statement that for all $f$,
$Q(Ef)\leq Q(Ff)$.
Proof: The condition is sufficient, for from it we deduce that if $f$ lies in
the exterior of $F$,

$Q(Ef)\leq Q(Ff)=0,\;\;Ef=0,$ (88)

i.e. $f$ is in the exterior of $E$, showing that $E\leq F$. It is necessary,
as from $E\leq F$ [which means $E=EF$] we deduce [cf. Theorem 1]:

$Q(Ef)=Q(EFf)\leq Q(Ff).\qed$ (89)

We are now in a position to formulate the eigenvalue problem of a general
symmetric operator with the help of projection operators: indeed, in contrast
to Section 3, in such a way that the continuous spectrum is correctly treated.

###  9 The eigenvalue problem

As we mentioned previously, we find the most obvious formulation of the
eigenvalue problem unsatisfying. It goes as follows:
Let $T$ be a linear symmetric [hermitian] operator. One seeks all real numbers
$l$ and all $f\neq 0$ in $\bar{\mathfrak{H}}$, for which

$Tf=lf.$ (90)

The $l$ are the eigenvalues, the $f$ the eigenfunctions.

This formulation has two essential defects:

  1. 1.

The eigenfunctions $f$ are not uniquely determined objects, even when one
normalizes them via $Q(f)=1$: a factor of absolute value 1 remains
undetermined. In fact, in the case of a degenerate eigenvalue (i.e. a $l$
associated with several linearly independent eigenfunctions) there is even the
freedom of an arbitrary orthogonal [unitary] transformation of the
eigenfunctions [associated with the given $l$].

  2. 2.

This formulation fails for the continuous spectrum. If we go back to the
$\mathfrak{H}_{0}$ (resp. $\mathfrak{H}$) realizations, it becomes apparent
that there are possibly eigensequences $x_{1},x_{2},\ldots$ (resp.
eigenfunctions $f(P)$) which however do not belong to $\mathfrak{H}_{0}$
(resp. $\mathfrak{H}$), i.e.

$\sum_{n=1}^{\infty}|x_{n}|^{2}\;\;(\mathrm{resp.}\;\;\int_{\Omega}|f(P)|^{2}dv)$
(91)

is infinite.This suggests the objection, that we have arbitrarily restricted
excessively admission to $\mathfrak{H}_{0}$ (resp. $\mathfrak{H}$—therefore of
$\bar{\mathfrak{H}}$); perhaps one should also admit functions to
$\mathfrak{H}$ with infinite $\int_{\Omega}|f(P)|^{2}dv$. But this objection
is not sound: we shall showt hat for quantum mechanics what is important for
the continuous spectrum is not the eigenfunctions, but quite different
features, for the description of which our $\bar{\mathfrak{H}}$ provides the
most suitable space. For the preceding cases the admission of all functions
$f(P)$ would not even have sufficed, which can be clarified with two examples
we now give. In one case, the indicated expansion of the function space helps,
but in the other it does not.
First: let $\Omega$ be the real line $(-\infty,\infty)$, $T$ the symmetric
[self-adjoint] operator, well known from quantum mechanics,

$p=\frac{h}{2\pi i}\frac{d}{dx}\cdot\cdot\cdot.$ (92)

As one sees immediately, the spectrum consists of the entire interval
$(-\infty,\infty)$, and the eigenvalue $l$ is associated with the
eigenfunction $\exp{(2\pi ilx/h)}$. As $\int_{-\infty}^{\infty}|\exp{(2\pi
ilx/h)}|^{2}dx=\int_{-\infty}^{\infty}dx=\infty$, this eigenfunction does not
belong to $\mathfrak{H}$.
Second: Let $\Omega$ be as before, $T$ the similarly very important symmetric
[self-adjoint] operator $q=x\cdot\cdot\cdot$. It now appears (and will be
rigorously established in Section 10) that again every $l$ between $-\infty$
and $\infty$ is an eigenvalue, but now the eigenfunction must vanish for all
$x\neq l$ [as $xf(x)=lf(x)$ implies $(x-l)f(x)=0$]. Such a function is
equivalent in any integral to zero, so must be regarded as [identically] zero:
in other words, there are no eigenfunctions. (Naturally one can consider the
Dirac $\delta(x-l)$ (cf. Section 1$\zeta.$) as an eigenfunction, but it is
“improper.”
One sees therefore that there is no way to delineate a mathematically
unobjectionable function space, in which the continuous spectrum can be
handled directly.

We are therefore forced to seek another formulation. We shall try to
approximate $\bar{\mathfrak{H}}$ by using spaces of known character! For this
purpose we shall use the discrete realization $\mathfrak{H}_{0}$: the space of
all sequences $x_{1},x_{2},\ldots$ with finite
$\sum_{n=1}^{\infty}|x_{n}|^{2}$. We can regard this space as the limiting
case of 1-, 2-, 3-,…dimensional complex Euclidean spaces, as the dimension
grows beyond bound.

We therefore denote $k$-dimensional (complex) Euclidean space by
$\mathfrak{R}^{(k)}$: the space of all [finite] sequences
$x_{1},x_{2},\ldots,x_{k}$. It is natural to define $Q(x,y)$ in
$\mathfrak{R}^{(k)}$ as $\sum_{n=1}^{k}x_{n}y^{*}_{n}$ (i.e. as the inner
product of the vectors $x,y$). Then every linear operator A in
$\mathfrak{R}^{(k)}$ can be specified through a matrix $\\{a_{\mu\nu}\\}$,
inasmuch as A transforms the vector $x$ into the vector $y$ as follows

$y_{\nu}=\sum_{\mu=1}^{k}a_{\mu\nu}x_{\mu},\;\;\;(\nu=1,2,\ldots,k),$ (93)

and symmetry [hermiticity (of A), namely $Q(x,\mathbf{A}y)=Q(mathbf{A}x,y)$]
is equivalent to the property

$a_{\mu\nu}=a^{*}_{\nu\mu}.$ (94)

As is well known, the hermitian form associated to A,

$\mathbf{A}(x|y)=Q(\mathbf{A}x|y)=Q(x|\mathbf{A}y)=\sum_{\mu,\nu=1}^{k}a_{\mu\nu}x_{\mu}y^{*}_{\nu},$
(95)

can always be brought into “principal axis form”:

$\mathbf{A}(x|y)=\sum_{p=1}^{k}l_{p}(\alpha_{p1}x_{1}+\cdot\cdot\cdot\alpha_{pk}x_{k})(\alpha_{p1}y_{1}+\cdot\cdot\cdot\alpha_{pk}y_{k})^{*},$
(96)

where the matrix $\\{\alpha_{\mu\nu}\\}$ is orthogonal (unitary), i.e. the
“identity form”

$Q(x|y)=\sum_{n=1}^{k}x_{n}y_{n}^{*}$ (97)

is transformed into itself [by the matrix $\\{\alpha_{\mu\nu}\\}$]:

$\sum_{n=1}^{k}x_{n}y_{n}^{*}=\sum_{p=1}^{k}(\alpha_{p1}x_{1}+\cdot\cdot\cdot+\alpha_{pk}x_{k})(\alpha_{p1}y_{1}+\cdot\cdot\cdot+\alpha_{pk}y_{k})^{*}.$
(98)

The “eigenvalues” $l_{1},l_{2},\ldots,l_{k}$ are known to be uniquely
specified, up to reordering; not so the “eigenvectors”
$\alpha_{1\nu},\alpha_{2\nu},\ldots,\alpha_{k\nu},\;(\nu=1,2,\ldots,k)$. For
each of them there is still the freedom of a factor of absolute value one (the
“phase”), and, if several eigenvalues coincide (“degeneracy”), even the
freedom of an orthogonal (unitary) transformation of the associated
eigenvectors [to a given eigenvalue] among themselves.

Nevertheless, it is easy to arrive at a unique standard form for
$\mathbf{A}(x|y)$. We shall label the distinct values among the
$l_{1},l_{2},\ldots,l_{k}$ (ordered by magnitude)
$L_{1},L_{2},\ldots,L_{q},(q\leq k)$. Further, define

$\mathbf{E}(l;x|y)=\sum_{L_{p}\leq
l}(\alpha_{p1}x_{1}+\cdot\cdot\cdot+\alpha_{pk}x_{k})(\alpha_{p1}y_{1}+\cdot\cdot\cdot+\alpha_{pk}y_{k})^{*}.$
(99)

One can easily establish that the $L_{p}$ and $\mathbf{E}(l;x|y)$ are uniquely
determined, despite the above mentioned indeterminacy of the $l_{p}$ and
$\alpha_{\mu\nu}$. The matrix associated with [the bilinear form]
$\mathbf{E}(l;x|y)$ we write as

$\mathbf{E}(l)=\\{e_{\mu\nu}(l)\\}.$ (100)

Evidently

$\mathbf{A}(x|y)=\sum_{p=1}^{q}L_{p}\\{\mathbf{E}(L_{p};x|y)-\mathbf{E}(L_{p-1};x|y)\\},$
(101)

where $L_{0}$ is chosen as an arbitrary number $<L_{1}$.As
$\mathbf{E}(l;x|y)$, viewed as a function of $l$, is constant on the intervals
$l<L_{1}$, $L_{p-1}\leq l<L_{p},(p=2,\ldots,q)$, $L_{q}\leq l$, this can also
be written

$\mathbf{A}(x|y)=\int_{-\infty}^{\infty}ld\mathbf{E}(l;x|y),\;\;Q(x|\mathbf{A}y)=\int_{-\infty}^{\infty}ldQ(x|\mathbf{E}(l)y).$
(102)

(The integral $\int_{-\infty}^{\infty}$ is the so called Stieltjes integral:
see Appendix 3 for further details.)

The $\mathbf{E}(l;x|y)$ are the so called projection forms
[German:_Einzelformen_ , or “identity forms”], namely, they are symmetric
[hermitian] $e_{\mu\nu}(l)=e_{\nu\mu}(l)^{*}$, and their matrices satisfyThat
$\mathbf{E}(l;x|y)$ is an identity form can be readily verified by direct
calculation, as it corresponds to the bilinear form

$\sum L(x)L^{*}(y),$ (103)

where the $L(x)$ are a set of mutually orthogonal and normalized linear
combinations of $x_{1},x_{2},\ldots,x_{k}$.

$\mathbf{E}(l)^{2}=\mathbf{E}(l).$ (104)

We can therefore regard the corresponding operators

$\displaystyle\mathbf{E}(l)x$ $\displaystyle=$ $\displaystyle y,$ (105)
$\displaystyle y_{\nu}$ $\displaystyle=$
$\displaystyle\sum_{\mu=1}^{k}e_{\mu\nu}(l)x_{\mu}$ (106)

as P. Ops. in the space $\mathfrak{R}^{k}$.

Further, one can immediately see that for $l<L_{1}$ (resp. $\geq L_{q}$),

$\mathbf{E}(l;x|y)=0,\;\;\mathrm{resp.}\;=\sum_{p=1}^{k}x_{p}y_{p}^{*},$ (107)

i.e. $\mathbf{E}(l)=$ the null (resp. the identity) operator; and that it
follows from $l\leq l^{\prime}$ that

$\mathbf{E}(l;x|x)\leq\mathbf{E}(l^{\prime};x|x).$ (108)

The matrix $\mathbf{E}(l)$ is therefore everywhere constant, save at the
eigenvalues, where it makes discontinuous jumps: the same holds for
$\mathbf{E}(l;x|x)$, which is moreover monotonically increasing [for fixed
$x$, as a function of $l$].

This formulation can be taken over without difficulty to the space
$\mathfrak{H}_{0}$, and indeed to [the abstract Hilbert space]
$\bar{\mathfrak{H}}$. Here again we seek, for each symmetric [hermitian]
linear operator $T$ a family of P. Ops. $E(l)$, so that

$Q(f,Tg)=\int_{-\infty}^{\infty}ldQ(f,E(l)g)$ (109)

always holds. This state of affairs can more concisely be expressed as
follows:

$Tg=\int_{-\infty}^{\infty}ld(E(l)g),\;\;T=\int_{-\infty}^{\infty}ldE(l).$
(110)

In this new situation we require always that, for $l\leq l^{\prime}$,
$Q(E(l)f)\leq Q(E(l^{\prime})f)$, i.e. $E(l)\leq E(l^{\prime})$ (analogously
to the monotonicity of $\mathbf{E}(l;x|x)$!). $\mathbf{E}(l)$ had at most [a
finite number] $k$ of points of discontinuity, but as we now must let $k$
become infinity, so that $\mathfrak{R}^{k}$ approximates $\mathfrak{H}_{0}$,
we cannot demand the same of $E(l)$. On the other hand, the fact that
$\mathbf{E}(l)$ begins as zero (for small $l$) and ends as unity (for large
$l$) can be interpreted [in the infinite dimensional case] by analogy as
follows: for $l\rightarrow+\infty$ (resp. $-\infty$), $E(l)f\rightarrow f$
(resp. 0).

Finally, the jumps of $\mathbf{E}(l)$ are always on the left side of the
points of discontinuity (i.e. it is upper semi-continuous, namely a sum of the
form $\sum_{L_{p}\leq l}$, not $\sum_{L_{p}<l}$). We will require the
corresponding property for $E(l)$: for $l^{\prime}\rightarrow l,l^{\prime}>l$
[$l^{\prime}\searrow l$], one also has $E(l^{\prime})f\rightarrow E(l)f$.

Let us summarize our requirements:
1. $\quad Q(f,Tg)=\int_{-\infty}^{\infty}ldQ(f,E(l)g)$,
or, concisely,

$Tg=\int_{-\infty}^{\infty}ld\\{E(l)g\\},\;\;T=\int_{-\infty}^{\infty}ldE(l).$
(111)

2a. $l\leq l^{\prime}$ implies $E(l)\leq E(l^{\prime})$.
2b. For $l\rightarrow+\infty$ (resp. $-\infty$), $E(l)f\rightarrow f$ (resp.
0).
2c. For $l^{\prime}\searrow l$, $E(l^{\prime})f\rightarrow f$.
Under these circumstances, we call $E(l)$ the “partition of unity associated
with $T$.”In essence this is the form in which the eigenvalue problem for
bounded bilinear forms has been solved by Hilbert. To be sure, we consistently
suppress the separation of the point and continuous spectrum usual in the
mathematical literature. Moreover, the operator
$T=\int_{-\infty}^{\infty}ldE(l)$ does not always act sensibly; one can show
that $Tf$ exists (in $\bar{\mathfrak{H}}$), if and only if the integral
$\int_{-\infty}^{\infty}l^{2}dQ(E(l)f)$ is finite.

###  10 [The eigenvalue problem: further discussion]

The definition just given of the eigenvalue representation [now: spectral
representation] of a symmetric linear operator requires naturally some
additional critical considerations.

First, it is not immediately clear whether every symmetric linear operator
possesses an eigenvalue representation,and whether it is uniquely determined.
For bounded operators the theorems of Hilbert establish that one and only one
such representation exists (see note 12); for unbounded operators only the
uniqueness has been established.For the case of real symmetric operators the
author could show that one, and only one, solution exists; for complex
symmetric (hermitian) operators one conjectures the same, but certain
difficulties stand in the way of the proof. Cf. the article mentioned in note
12, to appear in _Mathematische Annalen_.Two years later, von Neumann
published [46], a comprehensive treatment of the eigenvalue problem for a
large class of densely defined operators (hypermaximal hermitian operators
$R$), not necessarily bounded, in which a spectral decomposition
$R=\int\lambda dE(\lambda)$ can be rigorously established. The basic tool of
von Neumann’s proof is the use of a Cayley transform to map such operators to
unitary (hence, bounded) ones to which the Hilbert spectral theory apples.

Further, it is certainly desirable to arrive at a direct interpretation of the
P. Ops. $E(l)$. To this end, let us consider the simple case in which $T$ has
a point spectrum [only]: eigenvalues $l_{1},l_{2},\ldots$ with associated
eigenfunctions $\phi_{1},\phi_{2}\ldots$ These are well known to constitute
(e.g. in a continuous realization, and therefore, also in
$\bar{\mathfrak{H}}$) a complete orthonormal system, whence

$f=\sum_{n=1}^{\infty}Q(f,\phi_{n})\phi_{n},$ (112)

and, as they are eigenfunctions [of $T$],

$Tf=\sum_{n=1}^{\infty}Q(f,\phi_{n})l_{n}\phi_{n}.$ (113)

(As we are concerned with a merely preliminary orientation, we shall not
consider further here any questions of convergence.) From this it further
follows that

$Tf=\int_{-\infty}^{+\infty}ld[\sum_{l_{n}\leq l}Q(f,\phi_{n})\phi_{n}].$
(114)

This is however just the desired eigenvalue representation [i.e. spectral
representation], with

$E(l)f=\sum_{l_{n}\leq l}Q(f,\phi_{n})\phi_{n};$ (115)

$E(l)$ clearly has the properties 1.-2. of Section 9.

The [projection] operator $E(l)$ therefore serves the following function: one
expands $f$ in the eigenfunctions of $T$ ($\phi_{1},\phi_{2},..$) and omits
all terms with eigenvalues greater than $l$. The interior of $E(l)$ therefore
consists of all functions $f$ in whose expansion only eigenfunctions with
eigenvalues $\leq l$ appear, i.e. it is the set of linear combinations of all
eigenfucntions with eigenvalues $\leq l$.

Now this is also the meaning of the $E(l)$ if a continuous spectrum is
present: the interior of $E(l)$ (which in fact determines $E(l)$, as follows
from the results of Section 6) consists of all linear combinations (which can
also include integrals int eh case of a continuous spectrum) of eigenfunctions
of $T$ with eigenvalues $\leq l$.The first example of note 24 shows how linear
combinations of eigenfunctions which do not belong to $\mathfrak{H}$ (i.e.
those with infinite $\int_{\Omega}|\phi|^{2}dv$) can be constructed which do
lie in $\mathfrak{H}$:
The eigenfunctions $\exp{(2\pi ilx/h)}$ are not square-integrable, but linear
combinations of them are indeed so:

$\int_{l_{1}}^{l_{2}}\exp{(2\pi ilx/h)}dl=\frac{h}{2\pi i}\left(e^{2\pi
il_{2}x/h}-e^{2\pi il_{1}x/h}\right)\cdot\frac{1}{x},$ (116)

where the last factor $\frac{1}{x}$ guarantees the square-integrability.

This is clearly not yet rigorous, but in many cases it provides a clue to the
determination of the $E(l)$, as one may know or have an indication of the
eigenfunctions of the continuous spectrum: the exact definition at the
conclusion of Section 9 then allows a verification, whether the $E(l)$ so
obtained are the correct ones. To clarify this we would like to give two
examples. We consider a continuous realization, with $\Omega$ the interval
$(-\infty,+\infty)$. Let $T$ be the operator

$p=\frac{h}{2\pi
i}\frac{d}{dx}\cdot\cdot\cdot\mathrm{(resp.}\;\;q=x\cdot\cdot\cdot),$ (117)

as in note 24.

In the first case we have the eigenfunctions $e^{2\pi ilx/h}$, the
eigenfunction-expansion of a function $f$ is therefore the Fourier integral

$f(x)=\int_{-\infty}^{+\infty}\left[\int_{-\infty}^{+\infty}e^{-2\pi
ilz/h}f(z)dz\right]e^{2\pi ilx/h}dl.$ (118)

$E(l)$ effects the “excision of the remainder”:

$E(l)f(x)=\int_{-\infty}^{l}\left[\int_{-\infty}^{+\infty}e^{-2\pi
il^{\prime}z/h}f(z)dz\right]e^{2\pi il^{\prime}x/h}dl^{\prime}.$ (119)

Indeed, we have:

$\displaystyle\int_{-\infty}^{\infty}ld\left[E(l)f(x)\right]$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}ld\left(\int_{-\infty}^{l}\left[\int_{-\infty}^{\infty}e^{-2\pi
il^{\prime}z/h}f(z)dz\right]e^{2\pi il^{\prime}x/h}\right)$ (120)
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}l\;\left[\int_{-\infty}^{\infty}e^{-2\pi
ilz/h}f(z)dz\right]e^{2\pi ilx/h}dl.$ (121) $\displaystyle=$
$\displaystyle\frac{h}{2\pi i}f^{\prime}(x).$ (122)

In the second case [coordinate operator $q$] we would look for eigenfunctions
as those functions that are non-vanishing only for $x=l$ [as
$qf(x0:=xf(x)=lf(x)$ implies $(x-l)f(x)=0$, so either $x=l$ or $f(x)=0$],
which is impossible. However, we should still expect that linear combinations
of all eigenfunctions belonging to eigenvalues $\leq l$ should simply be those
functions which can only be nonzero for $x\leq l$. $E(l)$ can therefore be
defined as follows:

$\displaystyle E(l)f(x)$ $\displaystyle=$ $\displaystyle
f(x),\;\;\mathrm{if}\;\;x\leq l,$ (123) $\displaystyle=$ $\displaystyle
0,\;\;\mathrm{if}\;\;x>l.$ (124)

In this case one easily verifies that

$\int_{-\infty}^{\infty}ld\left[E(l)f(x)\right]=xf(x).$ (125)

Finally, let us consider yet another aspect of the eigenvalue representation
[spectral representation]. It allows a very simple representation of powers of
$T$. Indeed, one has

$T^{n}=\int_{-\infty}^{\infty}l^{n}dE(l).$ (126)

For $n=0$ this is trivial ($T^{0}=1$), for $n=1$ true by definition. We shall
demonstrate it for $n=2$, the proof is analogous for all higher $n$ (i.e. one
deduces the identity for $n+1$ given $n$ exactly as we do below for $n=2$
given $n=1$). One has:

$\displaystyle Q(f,T^{2}g)$ $\displaystyle=$ $\displaystyle Q(Tf,Tg)$ (127)
$\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}ldQ(Tf,E(l)g)$ (128)
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}ld\left[\int_{-\infty}^{\infty}l^{\prime}dQ(E(l^{\prime})f,E(l)g)\right]$
(129) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}ld\left[\int_{-\infty}^{\infty}l^{\prime}dQ(f,E(l^{\prime})E(l)g)\right]$
(130) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}ld\left[\int_{-\infty}^{\infty}l^{\prime}dQ(f,E(\mathrm{min}(l,l^{\prime}))g)\right].$
(131)

For $l^{\prime}>l$ we have $\mathrm{min}(l,l^{\prime})$ is constant (=$l$), so
it suffices to run the inner integral over the interval $(-\infty,l)$:For
Stieltjes integrals one has the relation

$\int_{A}^{B}u(l)d\left[\int_{A}^{l}v(l^{\prime})dw(l^{\prime})\right]=\int_{A}^{B}u(l)v(l)dw(l),$
(132)

corresponding to the relation

$\int_{A}^{B}u(l)\frac{d}{dl}\left[\int_{A}^{l}v(l^{\prime})dl^{\prime}\right]dl=\int_{A}^{B}u(l)v(l)dl$
(133)

for conventional [Riemann] integrals.

$\displaystyle Q(f,T^{2}g)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}ld\left[\int_{-\infty}^{l}l^{\prime}dQ(f,E(l^{\prime})g)\right]$
(134) $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}l\cdot
ldQ(f,E(l)g)$ (135) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}l^{2}dQ(f,E(l)g),$ (136)

or, in our abbreviated notation,

$T^{2}=\int_{-\infty}^{\infty}l^{2}dE(l),$ (137)

as was to be proved.

We will also make use of the following notation: since $l\leq l^{\prime}$
implies $E(l)\leq E(l^{\prime})$, and therefore that $E(l^{\prime})-E(l)$ is a
P. Op., which we shall denote $E(l,l^{\prime})\equiv E(I)$, where $I$ is the
interval $(l,l^{\prime})$.

###  11 The absolute value of an operator

Before proceeding to physical applications, we still need to develop one last
set of concepts which we shall be using. Let $A$ be a linear operator (which,
as well as [its adjoint] $A^{\dagger}$, acts meaningfully on a dense subset of
$\bar{\mathfrak{H}}$, cf. Section 7). We take two complete orthonormal sets
$\phi_{1},\phi_{2},\ldots$ and $\psi_{1},\psi_{2},\ldots$, where
$A\psi_{1},A\psi_{2},..$ exist,By selecting an everywhere dense sequence
$f_{1},f_{2},\ldots$ from the everywhere dense set of $f$ where $Af$ is
defined, and then applying Section 6, Theorem 6, we obtain the desired
orthonormal system $\psi_{1},\psi_{2},\ldots$ and set

$[A;\phi_{\mu},\psi_{\nu}]=\sum_{\nu,\nu=1}^{\infty}|Q(\phi_{\mu},A\psi_{\nu})|^{2}.$
(138)

The sum may be finite or infinite, but is always meaningful, as only non-
negative terms appear in it.This quantity is now referred to as the _Hilbert-
Schmidt norm_ (squared) $||A||^{2}_{HS}$ of the operator $A$. It was exploited
in Schmidt’s reformulation [34] of Hilbert’s integral equation theory, in the
integral form 153. The finiteness of the Hilbert-Schmidt norm identifies the
special class of “compact” operators, which have a purely point spectrum.

One then has

$\sum_{\mu=1}^{\infty}|Q(\phi_{\mu},A\psi_{\nu})|^{2}=Q(A\psi_{\nu}),$ (139)
$[A;\phi_{\mu},\psi_{\nu}]=\sum_{\nu=1}^{\infty}Q(A\psi_{\nu}),$ (140)

i.e. the dependence of the $[A;\phi_{\mu},\psi_{\nu}]$ on the $\phi_{\mu}$ is
only an apparent one. As a consequence of

$\sum_{\mu,\nu}^{\infty}|Q(\phi_{\mu},A\psi_{\nu})|^{2}=\sum_{\mu,\nu}^{\infty}|Q(A^{\dagger}\phi_{\mu},\psi_{\nu})|^{2}=\sum_{\mu,\nu}^{\infty}|Q(\psi_{\nu},A^{\dagger}\phi_{\mu})|^{2},$
(141)

we have (provided all $A^{\dagger}\phi_{\mu}$ exist, cf. note 30)

$[A;\phi,\psi]=[A^{\dagger};\psi,\phi].$ (142)

The same [independence] holds for the $\psi_{\nu}$, so that $[A;\phi,\psi]$
depends solely on $A$ [and not on the choice of orthonormal systems
$\phi_{\mu},\psi_{\nu}$]:

$[A;\phi,\psi]=[A].$ (143)

The equation (142) consequently implies

$[A]=[A^{\dagger}].$ (144)

The quantity $\sqrt{[A]}$ will be called the absolute value of the operator
$A$. We will now study this quantity in greater detail. Our first task will be
to establish the meaning of $[A]$ in the various representations of [the
abstract space] $\bar{\mathfrak{H}}$.

In the discontinuous realization $\mathfrak{H}_{0}$, $A$ can be represented by
a matrix

$\\{a_{\mu\nu}\\},\;\;a_{\mu\nu}=a_{\nu\mu}^{*}\;\;\;\;(\mu,\nu=1,2,\ldots).$
(145)

The points [i.e. vectors, represented as row sequences] (1,0,0,…), (0,1,0,…),
(0,0,1,…),…form, as one easily sees, a complete orthonormal system. [The
operator] $A$ correspondingly transforms these to

$(a_{11},a_{21},a_{31},\ldots),(a_{12},a_{22},a_{32},\ldots),(a_{13},a_{23},a_{33},\ldots),\ldots$
(146)

Denoting these vectors by $\psi_{1},\psi_{2},..$, we find

$[A]=\sum_{\nu=1}^{\infty}Q(A\psi_{\nu})=\sum_{\nu=1}^{\infty}\left[\sum_{\mu=1}^{\infty}|a_{\mu\nu}|^{2}\right]=\sum_{\mu,\nu=1}^{\infty}|a_{\mu\nu}|^{2}.$
(147)

Therefore: $[A]$ is the sum of the absolute squares of all elements of the
matrix [of $A$ relative to this orthonormal basis].

In the continuous realization $\mathfrak{H}$ [i.e. $L^{2}$] we shall evaluate
$[A]$ only in the special case that the operator $A$ can be represented by an
integral kernel $\Phi$:

$Af(P)=\int_{\Omega}\Phi(P,Q)f(Q)dv_{Q}.$ (148)

Then one finds:

$\displaystyle[A]$ $\displaystyle=$
$\displaystyle\sum_{\nu=1}^{\infty}Q(A\psi_{\nu})$ (149) $\displaystyle=$
$\displaystyle\sum_{\nu=1}^{\infty}\int_{\Omega}|A\psi_{\nu}(P)|^{2}dv_{P}$
(150) $\displaystyle=$
$\displaystyle\sum_{\nu=1}^{\infty}\int_{\Omega}\left|\int_{\Omega}\Phi(P,Q)\psi_{\nu}(Q)dv_{Q}\right|^{2}dv_{P}$
(151) $\displaystyle=$
$\displaystyle\int_{\Omega}\left[\sum_{\nu=1}^{\infty}\left|\int_{\Omega}\Phi(P,Q)\psi_{\nu}(Q)dv_{Q}\right|^{2}\right]dv_{P}$
(152) $\displaystyle=$
$\displaystyle\int_{\Omega}\int_{\Omega}|\Phi(P,Q)|^{2}dv_{P}dv_{Q}.$ (153)

So in this case $[A]$ is the integral of the absolute square of the integral
kernel [see note r].

Next, we shall derive the most important properties of $[A]$.

Theorem 1. $[A]\geq 0$ always, and $[A]=0$ only if $A=0$. (The same holds
consequently for $\sqrt{[A]}$.)
Proof: $[A]\geq 0$ is obvious. Let $f$ be an arbitrary element of
$\bar{\mathfrak{H}}$ for which $Af$ exists [i.e. $f$ in domain of $A$: $A$ is
assumed to be densely defined in this proof]. Either $f$ is zero, in which
case $Af=0$, or we may define $\phi=f/\sqrt{Q(f)}$, with $Q(\phi)=1$. Setting
$\psi_{1}=\phi$, we may complete $\phi$ to a complete orthonormal set
$\psi_{1},\psi_{2},\ldots$ (where all $A\psi_{\nu}$ exist!). As $[A]=0$,

$\sum_{\nu=1}^{\infty}Q(A\psi_{\nu})=0,\;\;Q(A\psi_{1})\leq 0\Rightarrow
A\psi_{1}=0$ (154)

whence $A\phi=0$, $Af=0$. Accordingly $A$ must = 0. ∎

Theorem 2. The following relations hold:

$\displaystyle\sqrt{[aA]}$ $\displaystyle=$ $\displaystyle|a|\sqrt{[A]},$
(155) $\displaystyle\sqrt{[A+B]}$ $\displaystyle\leq$
$\displaystyle\sqrt{[A]}+\sqrt{[B]},$ (156) $\displaystyle\sqrt{[AB]}$
$\displaystyle\leq$ $\displaystyle\sqrt{[A]}\;\sqrt{[B]}.$ (157)

Proof: The first formula is trivial. The second follows from the identity

$Q((A+B)\psi_{\nu})-Q(A\psi_{\nu})-Q(B\psi_{\nu})=2\mathrm{Re}\;Q(A\psi_{\nu},B\psi_{\nu}).$
(158)

Taking absolute values, we have, using the inequality

$|\mathrm{2Re}\;Q(A\psi_{\nu},B\psi_{\nu})|\leq
2\sqrt{Q(A\psi_{\nu})Q(B\psi_{\nu})},$ (159)

and after summation $\sum_{\nu=1}^{\infty}$,For all non-negative $a_{n},b_{n}$
one has the inequality
$\sqrt{a_{1}b_{1}}+\cdot\cdot\cdot\sqrt{a_{n}b_{n}}\leq\sqrt{(a_{1}+\cdot\cdot\cdot+a_{n})(b_{1}+\cdot\cdot\cdot+b_{n})}$.
[Squaring both sides, the assertion follows from the inequality
$2\sqrt{a_{i}b_{i}a_{j}b_{j}}\leq a_{i}b_{j}+a_{j}b_{i}$, which itself
follows, after squaring, from $(a_{i}b_{j}-a_{j}b_{i})^{2}\geq 0$.]

$\displaystyle[A+B]-[A]-[B]$ $\displaystyle\leq$
$\displaystyle\left|\sum_{\nu=1}^{\infty}2\mathrm{Re}\;Q(A\psi_{\nu},B\psi_{\nu})\right|$
(160) $\displaystyle\leq$ $\displaystyle
2\sum_{\nu=1}^{\infty}\sqrt{Q(A\psi_{\nu})Q(B\psi_{\nu})}$ (161)
$\displaystyle\leq$ $\displaystyle
2\sqrt{\sum_{\nu=1}^{\infty}Q(A\psi_{\nu})\sum_{\nu=1}^{\infty}Q(B\psi_{\nu})}$
(162) $\displaystyle=$ $\displaystyle 2\sqrt{[A][B]}.$ (163)

Thus

$\displaystyle[A+B]$ $\displaystyle\leq$
$\displaystyle[A]+[B]+2\sqrt{[A][B]}=(\sqrt{[A]}+\sqrt{[B]})^{2}$ (164)
$\displaystyle\sqrt{[A+B]}$ $\displaystyle\leq$
$\displaystyle\sqrt{[A]}+\sqrt{[B]}.$ (165)

The third relation can be proved as follows:

$\displaystyle[AB]$ $\displaystyle=$
$\displaystyle\sum_{\mu,\nu=1}^{\infty}\left|Q(\phi_{\mu},AB\psi_{\nu})\right|^{2}$
(166) $\displaystyle=$
$\displaystyle\sum_{\mu,\nu=1}^{\infty}\left|Q(A^{\dagger}\phi_{\mu},B\psi_{\nu})\right|^{2}$
(167) $\displaystyle\leq$
$\displaystyle\sum_{\mu,\nu=1}^{\infty}Q(A^{\dagger}\phi_{\mu})Q(B\psi_{\nu})$
(168) $\displaystyle=$
$\displaystyle\sum_{\mu=1}^{\infty}Q(A^{\dagger}\phi_{\mu})\cdot\sum_{\nu=1}^{\infty}Q(B\psi_{\nu})$
(169) $\displaystyle=$ $\displaystyle[A^{\dagger}][B]=[A][B],$ (170)
$\displaystyle\sqrt{[AB]}$ $\displaystyle\leq$
$\displaystyle\sqrt{[A]}\sqrt{[B]}.\qed$ (171)

Theorem 3. The equation

$[A+B]=[A]+[B]$ (172)

holds, as long as one of the four equations below is satisfied:

$AB^{\dagger}=0,\;\;A^{\dagger}B=0,\;\;BA^{\dagger}=0,\;\;B^{\dagger}A=0.$
(173)

Proof: For $A^{\dagger}B=0$ we have

$Q((A+B)\psi_{\nu})-Q(A\psi_{\nu})-Q(B\psi_{\nu})=2\mathrm{Re}\;Q(A\psi_{\nu},B\psi_{\nu})=2\mathrm{Re}\;Q(\psi_{\nu},A^{\dagger}B\psi_{\nu})=0,$
(174)

which, after summation $\sum_{\nu=1}^{\infty}$ yields

$[A+B]-[A]-[B]=0\Rightarrow[A+B]=[A]+[B].$ (175)

We can replace $A,B$ with $A^{\dagger},B^{\dagger}$, as then from
$A^{\dagger}B=0$ we obtain the sufficient condition
$A^{\dagger\dagger}B^{\dagger}=AB^{\dagger}=0$; further, we can interchange
$A$ and $B$, obtaining [the sufficient conditions] $B^{\dagger}A=0$ and
$BA^{\dagger}=0$.∎

It is clear that

$\sum_{\mu,\nu=1}^{\infty}\left|Q(A\phi_{\mu}),B\psi_{\nu})\right|^{2}=\sum_{\mu,\nu=1}^{\infty}\left|Q(\phi_{\mu}),A^{\dagger}B\psi_{\nu})\right|^{2}=[A^{\dagger}B].$
(176)

The expression on the left (manifestly dependent on $A,B$, but independent of
the choice of the $\phi$ and $\psi$) depends only on $A^{\dagger}B$. It will
later play a significant role, and we will therefore introduce an independent
notation for it, to wit:Von Neumann uses the notation $[A,B]$, which invites
an almost unavoidable identification with the commutator for modern readers.
We have therefore elected to use angle brackets instead. This makes it
reminiscent of an inner product, and indeed, we shall shortly see that it
satisfies many of the expected properties for such.

$\langle A,B\rangle\equiv[A^{\dagger}B].$ (177)

In particular, from
$(A^{\dagger}B)^{\dagger}=B^{\dagger}A^{\dagger\dagger}=B^{\dagger}A$ it
follows that

$\langle A,B\rangle=\langle B,A\rangle.$ (178)

From Theorem 2 it follows that:

$\langle aA,bB\rangle=|ab|^{2}\langle A,B\rangle.$ (179)

Likewise, from Theorem 3 we have

$\langle A,B+C\rangle=\langle A,B\rangle+\langle A,C\rangle,$ (180)

when any of the following four conditions is satisfied:

$\displaystyle A^{\dagger}B(A^{\dagger}C)^{\dagger}$ $\displaystyle=$
$\displaystyle A^{\dagger}BC^{\dagger}A=0,$ (181)
$\displaystyle(A^{\dagger}B)^{\dagger}A^{\dagger}C$ $\displaystyle=$
$\displaystyle B^{\dagger}AA^{\dagger}C=0,$ (182) $\displaystyle
A^{\dagger}C(A^{\dagger}B)^{\dagger}$ $\displaystyle=$ $\displaystyle
A^{\dagger}CB^{\dagger}A=0,$ (183)
$\displaystyle(A^{\dagger}C)^{\dagger}A^{\dagger}B$ $\displaystyle=$
$\displaystyle C^{\dagger}AA^{\dagger}B=0.$ (184)

In particular, $BC^{\dagger}=0$ or $CB^{\dagger}=0$ are sufficient [for (180)
to hold].

We will primarily be concerned with expressions of the type $\langle
E,F\rangle$, where $E,F$ are P. Ops. If $E,F$ commute, then $EF$ is also a P.
Op., in which case $\langle E,F\rangle$ has a simple geometrical
interpretation. For in that case

$\langle E,F\rangle=[E^{\dagger}F]=[EF],$ (185)

and we are therefore just concerned with the meaning of $[E]$, where $E$ is a
P. Op. It is easy to construct a complete orthonormal system of $\psi_{\nu}$
which lie exclusively in the interior or exterior of $E$,Let
$f_{1},f_{2},\ldots$ (resp. $g_{1},g_{2},\ldots$) be an everywhere dense
sequence in the interior (resp. exterior) of $E$. By application of Section 6,
Theorem 6 we can construct from each orthonormal systems
$\rho_{1},\rho_{2},\ldots$ (resp. $\sigma_{1},\sigma_{2},\ldots$). From
Section 8, Theorem 6, $\rho_{1},\sigma_{1},\rho_{2},\sigma_{2},\ldots$ is also
an orthonormal system, and from Section 8, Theorem 5 it spans an everywhere
[in $\bar{\mathfrak{H}}$] dense space, i.e it is complete. in which case

$\displaystyle[E]$ $\displaystyle=$
$\displaystyle\sum_{\nu=1}^{\infty}Q(E\psi_{\nu})$ (186) $\displaystyle=$
$\displaystyle\sum_{\psi_{\nu}\;\mathrm{in\;interior\;of\;}E}Q(\psi_{\nu})+\sum_{\psi_{\nu}\;\mathrm{in\;exterior\;of\;}E}Q(0)$
(187) $\displaystyle=$
$\displaystyle\sum_{\psi_{\nu}\;\mathrm{in\;interior\;of\;}E}1$ (188)
$\displaystyle=$
$\displaystyle\mathrm{number\;of}\;\psi_{\nu}\;\mathrm{in\;interior\;of\;}E.$
(189)

But this is just the dimension of the interior of $E$. Consequently, we have,
for example, $[1]=\infty$ (as the interior of the identity ! encompasses the
whole space $\bar{\mathfrak{H}}$), and $[0]=0$.

Theorems 1 and 2 show that $\sqrt{[A]}$ can indeed be treated as the absolute
value of operators. Nevertheless, it is clearly only useful in the immediate
neighborhood of 0: $\sqrt{[1]}$ is already infinite.

###  12 The statistical postulate of quantum mechanics.

We are now in a position to undertake the program of particular interest here:
the mathematically unobjectionable unification of statistical quantum
mechanics. To this end we shall start by considering the simplest possible
case, one with unambiguously clear results, and then translate these (known)
results into our [mathematical] language: this will then provide the clue to
the method which should be employed in the general case.

We consider therefore the following extremely simple case arising in the
continuous realization [i.e. in $\mathfrak{H}$, or $L^{2}$]: let $\Omega$ be
$k-$dimensional Euclidean space, and consider a quantum-mechanical system for
which the Schrödinger equation reads

$H\psi-l\psi=0.$ (190)

As is well known,Cf. for example note 3. the symmetric linear operator $H$
arises in the following manner: one takes the classical-mechanical expression
for the energy as a function of the coordinates $q_{1},q_{2},\ldots,q_{k}$ and
the momenta $p_{1},p_{2},\ldots,p_{k}$, and replaces each $q_{\mu}$ with the
operator $q_{\mu}\cdot\ldots$ [i.e. multiplication by $q_{\mu}$], and each
$p_{\mu}$ by the operator $\frac{h}{2\pi i}\frac{\partial}{\partial
q_{\mu}}\cdot\ldots$. (Here there arises some ambiguity, as the
$q_{\mu},p_{\nu}$ commute multiplicatively with one another as normal numbers,
in contrast to the operators $q_{\mu}\cdot\ldots$ and $\frac{h}{2\pi
i}\frac{\partial}{\partial q_{\mu}}\cdot\ldots$ Clearly one has the constraint
on $H$ that it be symmetric [self-adjoint]; however, this does not suffice to
determine it uniquely. This constitutes one of the essential deficiencies in
quantum theory.)

Now let us assume that only a non-degenerate (i.e. consisting of only simple
eigenvalues) point spectrum is present, and call the eigenvalues
$l_{1},l_{2},\ldots,$ and the corresponding (normalized) eigenfunctions
$\phi_{1},\phi_{2},\ldots$ We have already determined in Section 10 the
partition of unity corresponding to this $H$. It is:

$E(l)f=\sum_{l_{n}\leq l}Q(f,\phi_{n})\cdot\phi_{n}.$ (191)

Further, we need the partition of unity $F_{\mu}(l)$ corresponding to the
operator $q_{\mu}$. This is a trivial generalization of the final example
considered in Section 10 (where $k$ was simply 1), namely

$\displaystyle F_{\mu}(l)f(q_{1},\ldots,q_{k})$ $\displaystyle=$
$\displaystyle f(q_{1},\ldots,q_{k})\;\;\mathrm{for}\;q_{\mu}\leq l,$ (192)
$\displaystyle=$ $\displaystyle 0\;\;\mathrm{for}\;q_{\mu}>l.$ (193)

(This can be shown by exactly the same qualitative consideration as
previously; once again, one easily establishes the result

$q_{\mu}f(q_{1},\ldots,q_{k})=\int_{-\infty}^{\infty}ld\\{F_{\mu}(l)f(q_{1},\ldots,q_{k})\\},$
(194)

which holds by definition [of the $F_{\mu}(l)$].

The probability postulate of Pauli and DiracCf. for example the work of Jordan
cited in note 7.The identification of (195) with a probability for observing a
particle in a given region of space goes back to Born’s examination of
collision phenomena in [4],[5]. It is now universally referred to as the Born
rule. Further elaborations of the probabilistic interpretation of wave
functions by Pauli (primarily in private correspondence with Heisenberg) were
acknowledged by Jordan [27]; Dirac’s development of statistical transformation
theory [14] occurred more or less simultaneously with, and independently of,
Jordan’s. applicable in this case now takes the form: if the system is in the
$n$-th quantum state $(l_{n},\phi_{n})$, then the probability that the
coordinates $q=(q_{1},\ldots,q_{k})$ lie in the $k$-dimensional region $K$ is
(writing $dq$ for $dq_{1}\cdot\cdot\cdot dq_{k}$)

$\int_{K}|\phi_{n}(q)|^{2}dq.$ (195)

We can further generalize this Ansatz. If we only know that the energy lies in
the interval $I$, then the probability (apart from normalization factors) is
given by

$\sum_{l_{n}\in I}\int_{K}|\phi_{n}(q)|^{2}dq.$ (196)

Indeed: if only a single eigenvalue lies in $I$, this follows from our earlier
assertion; while if several are present, it follows from this case if we
assume (as is usual) the single non-degenerate quantum states ar to be
regarded as a-priori equally probable.

This expression, in the event that $K$ (resp. $I$) is defined by the
inequalities $q_{1}^{\prime}<q_{1}\leq
q_{1}^{\prime\prime},q_{2}^{\prime}<q_{2}\leq
q_{2}^{\prime\prime},\ldots,q_{k}^{\prime}<q_{k}\leq q_{k}^{\prime\prime}$
(resp. $l^{\prime}<l\leq l^{\prime\prime}$), can however be written also in
the following fashion:

$\displaystyle\sum_{l_{n}\in I}\int_{K}|\phi_{n}(q)|^{2}dq$ $\displaystyle=$
$\displaystyle\sum_{l_{n}\in
I}\int_{q_{1}^{\prime}}^{q_{1}^{\prime\prime}}\cdot\cdot\cdot\int_{q_{k}^{\prime}}^{q_{k}^{\prime\prime}}|\phi_{n}(q_{1},\ldots,q_{k})|^{2}dq_{1}\cdot\cdot\cdot
dq_{k}$ (197) $\displaystyle=$ $\displaystyle\sum_{l_{n}\in
I}\int_{-\infty}^{\infty}\cdot\cdot\cdot\int_{-\infty}^{\infty}|F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime})*\ldots*F_{k}(q_{k}^{\prime},q_{k}^{\prime\prime})\phi_{n}(q_{1},\ldots,q_{k})|^{2}dq_{1}\cdot\cdot\cdot
dq_{k}$ $\displaystyle=$ $\displaystyle\sum_{l_{n}\in
I}\int_{\Omega}|F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime})*\ldots*F_{k}(q_{k}^{\prime},q_{k}^{\prime\prime})\phi_{n}(q)|^{2}dq$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\int_{\Omega}|F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime}*\ldots*F_{k}(q_{k}^{\prime}q_{k}^{\prime\prime})\cdot
E(l^{\prime},l^{\prime\prime})\phi_{n}(q)|^{2}dq$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}Q(F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime})*\ldots*F_{k}(q_{k}^{\prime},q_{k}^{\prime\prime})\cdot
E(l^{\prime},l^{\prime\prime})\phi_{n}(q))$ $\displaystyle=$
$\displaystyle[F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime})*\ldots*F_{k}(q_{k}^{\prime},q_{k}^{\prime\prime})\cdot
E(l^{\prime},l^{\prime\prime})].$

Now one may easily check that the
$F_{\mu}(q_{\mu}^{\prime},q_{\mu}^{\prime\prime})$ commute with one another,
from which one further concludes that

$\displaystyle\sum_{l_{n}\in I}\int_{K}|\phi_{n}(q)|^{2}dq$ $\displaystyle=$
$\displaystyle\langle(F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime})*\ldots*F_{k}(q_{k}^{\prime},q_{k}^{\prime\prime})^{\dagger},E(l^{\prime},l^{\prime\prime})\rangle$
(198) $\displaystyle=$ $\displaystyle\langle
F_{k}(q_{k}^{\prime},q_{k}^{\prime\prime})*\ldots*F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime}),E(l^{\prime},l^{\prime\prime})\rangle$
$\displaystyle=$ $\displaystyle\langle
F_{1}(q_{1}^{\prime},q_{1}^{\prime\prime})*\ldots*F_{k}(q_{k}^{\prime},q_{k}^{\prime\prime}),E(l^{\prime},l^{\prime\prime})\rangle.$

In other words: if the energy lies in $I$, the (relative) probability that
$q_{1}$ lies in $J_{1}$, $q_{2}$ lies in $J_{2}$, etc. is given by

$\langle E(I),F_{1}(J_{1})*\ldots*F_{k}(J_{k})\rangle.$ (199)

But inversely, by a postulate of Jordan, the quantity

$\sum_{l_{n}\in I}\int_{K}|\phi_{n}(q)|^{2}dq.$ (200)

may also be regarded as the probability that the $l_{n}$, i.e. the energy,
lies in $I$, if $q$ lies in $K$ (i.e. $q_{1}$ in $J_{1}$,…,$q_{k}$ in
$J_{k}$—cf. note 7). Or more exactly: the Pauli-Jordan assertion is relevant
only in the limit of infinitely small $K$ (where, after dividing by the
proportionality factor of the “volume of $K$” one is left with the probability
[density] $\sum_{l_{n}\in I}|\phi_{n}(q)|^{2}$). Nevertheless, a correct
result also emerges in the opposite limit $K=\Omega$ (i.e. the entire space):

$\sum_{l_{n}\in I}\int_{\Omega}|\phi_{n}(q)|^{2}dq=\sum_{l_{n}\in
I}1=\mathrm{number\;of}\;l_{n}\;\mathrm{in}\;I,$ (201)

i.e. all quantized (non-degenerate) states are a-priori equiprobable, and
unquantized states are impossible—both of which assertions belong to the basic
assumptions of quantum theory.

We therefore conclude: if $q_{1}$ lies in $J_{1}$,…, $q_{k}$ in $J_{k}$, then
the (relative) probability that the $l_{n}$ (the energy) lies in $I$ is given
by the same expression as previously, which we will in this case write as
follows:

$\langle F_{1}(J_{1})*\ldots*F_{k}(J_{k}),E(I)\rangle.$ (202)

###  13 The statistical postulate of quantum mechanics (contd.)

The results just obtained suggest the following postulate: let
$R_{1},R_{2},\ldots,R_{i}$ and $S_{1},S_{2},\ldots,S_{j}$ be symmetric linear
operators, representing particular sensible physical quantities. (We cannot
more closely examine here the exact meaning of this last concept of
“representation of a quantity by an operator”, which at the moment is
absolutely fundamental in quantum mechanics; cf. the discussions at the
beginning of preceding sections on the relation of the classical-mechanical
Hamiltonian function and the “energy operator” $H$.) The partitions of unity
associated with the $R_{\mu},\;\mu=1,2,\ldots,i$ (resp.
$S_{\nu},\;\nu=1,2,\ldots,j$) will be denoted $E_{\mu}(l)$ (resp.
$F_{\nu}(l)$).The mathematical apparatus developed at great length in this
paper finally is brought to bear on the central interpretational problem of
quantum mechanics: how are the formal elements of the theory linked to
phenomenologically observable quantities? The key is the use of projection
operators. On the formal side , they possess the following desirable
properties, identified by von Neumann from the outset as absent in the Dirac-
Jordan theory: (a) they are unambiguous (no phase ambiguities), (b) they are
_bounded operators_ , defined on all elements of the Hilbert space, and for
which multiplication operations can be rigorously defined—in particular, there
is no appeal to “improper functions”, such as the delta function. On the
phenomenological side, they correspond naturally to the inescapable
limitations of actual physical measurements, which establish the value of
observed quantities only in finite ranges, and not with infinitesimal
exactitude. Von Neumann’s elegant rephrasing of the Born rule in the language
of projection operators is the direct precursor to [48], where the entire
statistical formalism outlined here is built up inductively from a few,
apparently innocuous, basic assumptions on measurements carried out on
ensembles of quantum systems.

We assume that all the $E_{\mu}(l)$ commute with each other, as do all the
$F_{\nu}(l)$.Here, the notion of “simultaneous measurement of compatible
observables”, which had first been exhibited clearly in Heisenberg’s
“uncertainty paper” [20], is introduced: von Neumann wants to derive the
conditional probabilities which apply for the most general measurement which
are (at least in principle) possible given the unavoidable interference of
measurements in quantum mechanics with the values of conjugate quantities in
the measured system. This property will be called the “complete commutability”
of the $R_{1},R_{2},\ldots,R_{i}$ (likewise, of the
$S_{1},S_{2},\ldots,S_{j}$). (For complete commutability of two operators
$T^{\prime},T^{\prime\prime}$, the conventional commutability property, is
clearly necessary; it is also sufficient, provided at least one of the
operators is bounded. If both are unbounded, certain difficulties of a formal
nature arise, which we shall not discuss here. For all operators appearing in
quantum mechanics this is the case.)

We now make the following physical asssumption:

> If the quantities represented by $R_{1},\ldots,R_{i}$ have assumed values
> which lie in the intervals $I_{1},\ldots,I_{i}$, then the (relative)
> probability that quantities represented by $S_{1},\ldots,S_{j}$ lie in the
> intervals $J_{1},\ldots,J_{j}$ is given by
>
> $\langle
> E_{I_{1}}*\ldots*E_{I_{i}},F_{J_{1}}*\ldots*F_{J_{j}}\rangle=\langle
> E_{I_{1}}*\ldots*E_{I_{i}}*F_{J_{1}}*\ldots*F_{J_{j}}\rangle.$ (203)
>
> The equality holds because the $E_{I_{\nu}}$ all commute with one another.

We shall now draw some conclusions from this assumption, in order to indicate
its usefulness:
$\alpha.$ Both the premises ($(R_{1},\ldots,R_{i}$) as well as the conclusions
$(S_{1},\ldots,S_{j}$) may be freely interchanged, without altering the
probability distribution. This follows from the commutability with each other
of the $E_{\mu}(l)$ (resp. the $F_{\nu}(l)$), whence also of differences of
the $E_{\mu}(l)$ (resp. the $F_{\nu}(l)$).
$\beta.$ Interchange of all premises with all conclusions changes nothing
(i.e. the probability distribution behaves as if it arises from a priori
probabilities). This follows from the generally valid formula $\langle
A,B\rangle=\langle B,A\rangle$.
$\gamma.$ Tautologous premises or conclusions may be freely introduced or
removed (i.e. those for which the interval is $(-\infty,\infty)$). For they
occasion just the appearance or removal of a factor of

$E_{\mu}(-\infty,\infty)\;\mathrm{or}\;F_{\nu}(-\infty,\infty)=1-0=1$ (204)

$\delta.$ The multiplication law for probabilities does not in general hold
(rather, a weaker law holds, corresponding to the “superposition of
probability amplitudes” of Jordan—see note 7—which we will not discuss further
here).In [48], this assertion (of the invalidity of the multiplication law in
quantum mechanics) is withdrawn: the rules of probability are mathematical
certainties, and their application to quantum mechanics, if carried out with
sufficient care, is unobjectionable. This is hardly surprising, given that the
dependency relations of our probabilities can be arbitrarily complicated;
furthermore, one is dealing with relative probabilities.
$\epsilon.$ The addition law for probabilities is valid. (Indeed, this is
valid in the conventional probability calculus without regard to
dependencies.) We must show that from
$J_{j}^{\prime}+J_{j}^{\prime\prime}=J_{j}$ follows

$\displaystyle[E_{1}(I_{1})*\ldots*E_{i}(I_{i})*F_{1}(J_{1})*\ldots*F_{j}(J_{j}^{\prime})]+[E_{1}(I_{1})*\ldots*E_{i}(I_{i})*F_{1}(J_{1})*\ldots*F_{j}(J_{j}^{\prime\prime})]$
(205) $\displaystyle=$
$\displaystyle[E_{1}(I_{1})*\ldots*E_{i}(I_{i})*F_{1}(J_{1})*\ldots*F_{j}(J_{j})].$

(From $\alpha.$ and $\beta.$ we can certainly confine ourselves to the case
where the last interval, $J_{j}$, can be split [into two subintervals].) This
can also be written

$[AF_{j}(J_{j}^{\prime})]+[AF_{j}(J_{j}^{\prime\prime})]=[AF_{j}(J_{j})].$
(206)

Using Theorem 3, Section 11, this is certainly the case provided

$\displaystyle AF_{j}(J_{j}^{\prime})+AF_{j}(J_{j}^{\prime\prime})$
$\displaystyle=$ $\displaystyle AF_{j}(J_{j}),$ (207) $\displaystyle
AF_{j}(J_{j}^{\prime})(AF_{j}(J_{j}^{\prime\prime}))^{\dagger}$
$\displaystyle=$ $\displaystyle
AF_{j}(J_{j}^{\prime})F_{j}(J_{j}^{\prime\prime})A^{\dagger}=0.$ (208)

The first equation follows from
$F_{j}(J_{j}^{\prime})+F_{j}(J_{j}^{\prime\prime})=F_{j}(J_{j})$, but the
second follows as well, as then we must have that $F_{j}(J_{j}^{\prime})$ and
$F_{j}(J_{j}^{\prime\prime})$ are orthogonal [projection operators] (cf.
Theorem 2, Section 8). However, if $J_{j}^{\prime},J_{j}^{\prime\prime},J_{j}$
are the intervals
$(l^{\prime},l^{\prime\prime}),(l^{\prime\prime},l^{\prime\prime\prime}),(l^{\prime},l^{\prime\prime\prime})$
[with $l^{\prime}\leq l^{\prime\prime}\leq l^{\prime\prime\prime}$], then it
is clear that

$\displaystyle F_{j}(J_{j}^{\prime})$ $\displaystyle=$ $\displaystyle
F_{j}(l^{\prime\prime})-F_{j}(l^{\prime}),$ (209) $\displaystyle
F_{j}(J_{j}^{\prime\prime})$ $\displaystyle=$ $\displaystyle
F_{j}(l^{\prime\prime\prime})-F_{j}(l^{\prime\prime}),$ (210) $\displaystyle
F_{j}(J_{j})$ $\displaystyle=$ $\displaystyle
F_{j}(l^{\prime\prime\prime})-F_{j}(l^{\prime}).$ (211)

$\theta.$ Our expression for probabilities is invariant with respect to
canonical transformations. By a canonical transformation we mean the following
(cf. references in note 7): let $U$ be a linear operator with the property
$UU^{\dagger}=U^{\dagger}U=1$: in this case we call $U$ “orthogonal” [modern
term: unitary]. A canonical transformation consists of replacing every linear
operator $R$ by $URU^{\dagger}$. With regard to such a transformation, the
operations $aR,R+S,RS,R^{\dagger}$ are invariant, and therefore also the
properties of being symmetric [self-adjoint], of being a projection operator,
and, between projection operators, the relations $\leq$ and of orthogonality.
Further, $Q$ [the inner product] is invariant if $\bar{\mathfrak{H}}$ is
mapped onto itself by $U$:

$Q(Uf,Ug)=Q(U^{\dagger}Uf,g)=Q(f,g),$ (212)

so that $U\phi_{1},U\phi_{2},\ldots$ is a complete orthonormal system if
$\phi_{1},\phi_{2},\ldots$ is one, from which the invariance of [the operator
absolute value] $[A]$ follows. Additionally, the invariance of the eigenvalue
[spectral] representation is manifest (cf. Section 9).

As none of the concepts used by us is changed [by a canonical transformation],
the same holds for the (relative) probabilities derived on the basis of these.

###  14 Applications

We now consider some physical applications. First of all, [take] the case of
Schrödinger’s equations. These were already discussed in the case of non-
degenerate systems in Section 12, and if we also admit degeneracies—that is,
multiple eigenvalues—nothing in the results there obtained is changed. The a
priori probabilities of the single [i.e. non-degenerate] eigenvalues, for
example, are (cf. the characterization given in Section 12)

$\displaystyle\langle
F_{1}(-\infty,\infty)*\ldots*F_{k}(-\infty,\infty),E(I)\rangle$
$\displaystyle=$ $\displaystyle\langle 1,E(I)\rangle=[E(I)]$ $\displaystyle=$
$\displaystyle\mathrm{dimension\;of\;the}\;f(q)\;\mathrm{lying\;in\;the\;interior\;of}\;E(I).$

In the interior of $E(I)$ however, as we know, are found all linear
combinations

$a_{1}\phi_{\nu_{1}}(q)+a_{2}\phi_{\nu_{2}}(q)+\cdot\cdot\cdot,$ (214)

where $l_{\nu_{1}},l_{\nu_{2}},\ldots$ are the eigenvalues of $H$ lying in the
interval $I$ (multiple eigenvalues counted separately), and
$\phi_{\nu_{1}},\phi_{\nu_{2}},\ldots$ are the associated eigenfunctions. The
dimension indicated is therefore just the number of eigenvalues in $I$.

This result also implies that the a priori probability of a quantized state is
the multiplicity of the eigenvalues for that state, and unquantized states are
impossible.

Secondly, let us consider the case of sharp and causal dependencies.[Von
Neumann’s use of the word “causal” in the following is somewhat jarring. Here
there is no implication of of temporal sequence: von Neumann is simply
referring to a situation in which the determination of the value of one
quantity immediately determines, through a unique functional relationship, the
value of the other. Thus, a measurement of the distance $r$ of an electron
from the nucleus of a hydrogen atom would immediately determine the
electrostatic potential $-e^{2}/r$, and vice-versa.] We set $i=j=1$ and assume
that a certain quantity is given, while we seek another, which is a function
of the first—i.e. is causally determined by it. The corresponding operators we
can denote $R$ (resp. $S$), with $S=f(R)$, where $f(x)$ is a real function. We
assume, that $f(x)$ is monotone increasing. (The restrictions $i=j=1$, and the
monotonicity of $f(x)$ are not essential, but are assumed, only for the sake
of orientation, to simplify the calculation).

Let the partition of unity associated with $R$ be $E(l)$:

$R=\int_{-\infty}^{\infty}ldE(l).$ (215)

As we showed in Section 10, one then has

$R^{n}=\int_{-\infty}^{\infty}l^{n}dE(l),$ (216)

whence

$S=f(R)=\int_{-\infty}^{\infty}f(l)dE(l)=\int_{-\infty}^{\infty}l^{\prime}dE(g(l^{\prime})),$
(217)

where $g$ is the inverse of $f$ [i.e. $f(g(l))=l$]. Accordingly, $E(g(l))$ is
the partition of unity associated with $S$.

If $J$ is the interval $(l^{\prime},l^{\prime\prime})$, then we will denote
the interval $(g(l^{\prime}),g(l^{\prime\prime})$ by $g(J)$. One thus has [for
the partition of unity associated with $S$]:

$F(J)=E(g(J)),$ (218)

and consequently,

$\langle E(I),F(J)\rangle=[E(I)F(J)]=[E(I)E(g(J)]=E(I\cap g(J)).$ (219)

(Here $I\cap g(J)$ is the intersection of the intervals $I$ and $g(J)$; one
easily verifies the identity which ensures always $E(I)E(J^{\prime})=E(I\cap
J^{\prime})$.)

If the intervals $I$, $g(J)$ have no intersection, this is equal to zero; but
$I\cap g(J)$ is empty just when there exists no $x$ in $I$ for which $f(x)$ is
in $J$. Otherwise said: values in contradiction to a causal connection do not
appear.

If $I\cap g(J)$ is not empty, then it corresponds to the interval $I^{\prime}$
of all $x$ in $I$, for which $f(x)$ lies in $J$, i.e. those values allowed on
the basis of the causal connection. We therefore have the probability
relation:

$\displaystyle[E(I^{\prime})]$ $\displaystyle=$
$\displaystyle\mathrm{dimension\;of\;the}\;f\;\mathrm{lying\;in}\;E(I^{\prime}),\;\mathrm{or,\;as\;we\;know,}$
(220) $\displaystyle=$
$\displaystyle\mathrm{number\;of\;eigenvalues\;of\;R\;lying\;in}\;I^{\prime}.$

In summary: in the context of causal connections one obtains, if $R$ is
quantized, the usual quantum-theoretical result. (Besides, it is clear that
this probability becomes infinite if $R$ has a continuous spectrum which
intrudes in $I^{\prime}$.)

With these two examples we believe that we have demonstrated that statistical
quantum mechanics, despite its probabilistic character, is very well capable
of making sharp and binding assertions, as long as the occasion for such
arises, e.g. in the case of absolute quantum prohibitions [e.g. selection
rules] and causal conditions.

Finally, we consider the quantum-mechanical treatment of time-dependent
systems as given by Born.This example I owe to an observation of L. Nordheim.
The Hamiltonian function of the system depends explicitly on time, as does the
associated operator $H(t)$ (time is handled as a number, not as an operator
“quantity”). Let us exclude degeneracies, and denote the eigenfunctions at
time $t_{0}$ by

$\phi_{1}^{(0)},\phi_{2}^{(0)},\ldots$ (221)

and at time $t$ by

$\phi_{1}^{(t)},\phi_{2}^{(t)},\ldots$ (222)

Then, according to Born,_Zeitschrift für Physik_ , Vol. 38, p. 803; Vol. 40,
p. 167 (1926). the probability that the system is in the $\nu$-th state at
time $t$, given that it was in the $\mu$-th state at time $t_{0}$, is
$|c_{\mu,\nu}(t)|^{2}$, where $c_{\mu,\nu}(t)$ is the $\mu$-th expansion
coefficient of $\phi_{\nu}^{(t)}$ with respect to the
$\phi_{1}^{(0)},\phi_{2}^{(0)},\ldots$:

$\phi_{\mu}^{(t)}=\sum_{\nu=1}^{\infty}c_{\mu,\nu}(t)\phi_{\nu}^{(0)}.$ (223)

Indeed, suppose that $E_{0}(l)$ is the partition of unity for $H(t_{0})$, and
$E_{t}(l)$ that for $H(t)$. Ass we saw in Section 10, the interior of
$E_{0}(l)$ (resp. $E_{t}(l)$) consists of the linear combination of all
$\phi_{\mu}^{(0)}$ (resp. $\phi_{\mu}^{(t)}$) with eigenvalues $\leq l$.
Consequently the interior of $E_{0}(I)$ (resp. $E_{t}(I)$) consists of the
linear combination of all $\phi_{\mu}^{(0)}$ (resp. $\phi_{\mu}^{(t)}$) with
eigenvalues lying in the interval $I$.

Now suppose that [the interval] $I_{\mu}$ contains a single eigenvalue of
$H(0)$, the $\mu$-th, and $J_{\nu}$ a single eigenvalue of $H(t)$, the
$\nu$-th. Then the interior of $E_{0}(I_{\mu})$ (resp. $E_{t}(J_{\nu})$)
consists of [the one dimensional space consisting of] multiples of
$\phi_{\mu}^{(0)}$ (resp. $\phi_{\nu}^{(t)}$), so that

$\displaystyle E_{0}(I_{\mu})f$ $\displaystyle=$ $\displaystyle
Q(f,\phi_{\mu}^{(0)})\cdot\phi_{\mu}^{(0)},$ (224) $\displaystyle
E_{t}(J_{\nu})f$ $\displaystyle=$ $\displaystyle
Q(f,\phi_{\nu}^{(t)})\cdot\phi_{\nu}^{(t)}.$ (225)

We now calculate $\langle E_{0}(I_{\mu}),E_{t}(J_{\nu})\rangle$, using as a
complete orthonormal system $\phi_{1}^{(0)},\phi_{2}^{(0)},\ldots$:

$\displaystyle\langle E_{0}(I_{\mu}),E_{t}(J_{\nu})\rangle$ $\displaystyle=$
$\displaystyle[E_{t}(J_{\nu})E_{0}(I_{\mu})]$ (226) $\displaystyle=$
$\displaystyle\sum_{\rho=1}^{\infty}Q(E_{t}(J_{\nu})E_{0}(I_{\mu})\phi_{\rho}^{(0)})$
(227) $\displaystyle=$ $\displaystyle Q(E_{t}(J_{\nu})\phi_{\mu}^{(0)})$ (228)
$\displaystyle=$ $\displaystyle
Q(Q(\phi_{\mu}^{(0)},\phi_{\nu}^{(t)})\cdot\phi_{\nu}^{(t)})$ (229)
$\displaystyle=$ $\displaystyle|Q(\phi_{\mu}^{(0)},\phi_{\nu}^{(t)})|^{2},$
(230)

which is precisely Born’s expression (given that
$c_{\mu,\nu}(t)=Q(\phi_{\nu}^{(t)},\phi_{\mu}^{(0)})=Q^{*}(\phi_{\mu}^{(0)},\phi_{\nu}^{(t)})$).

###  15 Conclusion

It is apparent that our postulate of the double nature of dynamics
(continuous–discontinuous) is justified by the association of a partition of
unity into projection operators,
$E(l^{\prime},l^{\prime\prime})=E(l^{\prime\prime})-E(l^{\prime}),l^{\prime}\leq
l^{\prime\prime}$, to each physical quantity (and to the system from which it
derives).

The function $E(l)$ is monotone nondecreasing (in the sense appropriate for
projection operators), and this monotonicity displays all the characteristics
familiar from the study of atoms: its growth from 0 (for $l=-\infty$) to 1
(for $l=\infty$) can proceed in discrete jumps (quantized states), or
continuously (unquantized states), and in between there can also be intervals
where it is constant (forbidden states). The assertion “the quantity
associated with $R$ has a value lying in the interval $l^{\prime}<x\leq
l^{\prime\prime}$” is represented in out calculus by the projection operator
$E(l^{\prime},l^{\prime\prime})$.

In the event that several assertions are made, e.g. if the values of the
quantities $R_{1},R_{2},\ldots,R_{i}$ lie in the intervals
$I_{1},I_{2},\ldots,I_{i}$, while the values of the quantities
$S_{1},S_{2},\ldots,S_{j}$ lie in the intervals $J_{1},J_{2},\ldots,J_{j}$,
one must form the product

$E_{1}(I_{1}*\ldots*E_{i}(I_{i})*F_{1}(J_{1})*\ldots*F_{j}(J_{j}),$ (231)

(which is the form in which the multiplication law for probabilities is to be
taken over [in quantum mechanics]); the absolute square [of this product]

$[E_{1}(I_{1}*\ldots*E_{i}(I_{i})*F_{1}(J_{1})*\ldots*F_{j}(J_{j})]=\langle
E_{1}(I_{1})*\ldots*E_{i}(I_{i}),F_{1}(J_{1})*\ldots*F_{j}(J_{j})\rangle$
(232)

is then the (relative) probability of the simultaneous validity [of these
assertions]. Note that in this calculation it is not at all necessary to
introduce a cause/effect distinction. Such a distinction will subsequently
appear automatically as a consequence of the commutation relations: if all the
$E_{\mu}(I_{\mu})$ on the one hand, and the $F_{\nu}(J_{\nu})$ on the other,
all commute with each other, then the product decomposes automatically into
these two groups. Within each of these groups the order of factors is
insignificant ($\alpha.$ in Section 13), as is the order of the two groups
taken as a whole ($\beta.$ in Section 13), i.e. what one regards as the cause
and what one regards as effect.

Of course the commutation relations do not on their own necessarily determine
the cause–effect distinction uniquely: if, for example, a $E_{\rho}$ commutes
with all the $F_{\nu}$, it can be freely assigned to either one of the groups.

On the other hand, certain assignments are excluded from the outset: it is not
possible in all cases to enforce the absolutely sharp connection which as a
rule applies for mutually commuting quantities (cf. the second example in
Section 14), in particular to observe (as one does in classical mechanics) all
coordinates and their conjugate momenta, and to make them individually and
collectively “causes”, i.e. $R_{\mu}$s. (Dirac was the first to point out the
impossibility of such a procedure.) For it is well known that the operators of
a coordinate and its conjugated momentum do not commute ($q_{\mu}\cdot\ldots$
and $\frac{h}{2\pi i}\frac{\partial}{\partial q_{\mu}}\cdot\ldots$), so that
they must unavoidably be separated: one must be the cause, the other the
effect (resp. the observed and the prescribed). Even observing everything does
not help: quantum mechanics (which encompasses practically everything that we
now know with exactitude about atoms) simply affords no way to frame the
question!

It should be mentioned in conclusion that the material presented here does not
by any means exhaust the applicability of our method. We will return to this
on another occasion, as well as to certain formal mathematical questions which
remain unresolved here.

###  16 Appendix 1

Let $w$ be an eigenvalue [of $H$], and $(x_{1},x_{2},\ldots)$ a sequence which
is transformed by $H$ into itself, multiplied by $w$. By application of
$S^{-1}$ let this sequence be transformed to $(y_{1},y_{2},\ldots)$. Under
these circumstances one has that $(y_{1},y_{2},\ldots)$ is transformed by
$S^{-1}HS$ into itself, multiplied by $w$. Now $S^{-1}HS=W$ [=diagonal matrix]
transforms $(y_{1},y_{2},\ldots)$ into $(w_{1}y_{1},w_{2}y_{2},\ldots)$, so
all $y_{\mu}$ must =0 unless $w_{\mu}=w$. Consequently, if $w$ differs from
all $w_{1},w_{2},\ldots$, then all $y_{\mu}$, and therefore all $x_{\mu}$,
must be zero—hence, $w$ cannot be an eigenvalue.

Conversely, if $w$ is equal to some of the $w_{\mu}$, let’s say
$w_{\mu^{\prime}},w_{\mu^{\prime\prime}},\ldots,$ then
$y_{\mu^{\prime}},y_{\mu^{\prime\prime}},\ldots,$ can take arbitrary values
(all other $y_{\mu}=0$). The $(x_{1},x_{2},\ldots)$ arise from
$(y_{1},y_{2},\ldots)$ by application of $S$, so

$x_{\mu}=\sum_{\mu=1}^{\infty}s_{\mu\nu}y_{\nu}=s_{\mu\mu^{\prime}}y_{\mu^{\prime}}+s_{\mu\mu^{\prime\prime}}y_{\mu^{\prime\prime}}+\cdot\cdot\cdot.$
(233)

Therefore $(x_{1},x_{2},\ldots.)$ can in fact be an arbitrary linear
combination of the columns $(s_{1\mu^{\prime}},s_{2\mu^{\prime}},\ldots)$,
$(s_{1\mu^{\prime\prime}},s_{2\mu^{\prime\prime}},\ldots)$,…, and of no
others.

###  17 Appendix 2

We wish to show that the space $\mathfrak{H}$ of all functions $f(P)$ defined
on $\Omega$ ($\Omega$ an arbitrary $k$-dimensional surface in $l$-dimensional
space, $P$ an arbitrary point in $\Omega$, $dv$ the differential volume
element in $\Omega$) with finite $\int_{\Omega}|f(P)|^{2}dv$ satisfies the
conditions A.–E. of Section 4. For the standard Hilbert space (all sequences
$x_{1},x_{2},\ldots$ with finite $\sum_{n=1}^{\infty}|x_{n}|^{2}$) the proof
is superfluous, as, according to Section 6 [last paragraph], every space with
the properties A.–E. has the properties of $\mathfrak{H}_{0}$, whence
$\mathfrak{H}_{0}$ itself must satisfy A.–E.See the final paragraph of Section
6 (not Section 5, as referenced by von Neumann). The complete isomorphism of
$\mathfrak{H}$ with $\mathfrak{H}_{0}$ is the rigorous mathematical framing of
the complete equivalence of wave and matrix mechanics.

The definition of the operations $af$ and $f+g$ is clear; that $f$ belonging
to $\mathfrak{H}$ implies $af$ belonging to $\mathfrak{H}$ is obvious, while
$f,g$ belonging to $\mathfrak{H}$ implies that $f+g$ belongs to $\mathfrak{H}$
as a consequence of the inequality given in note 16.[A. is now established.]
We must define $Q(f,g)$ as $\int_{\Omega}f(P)g^{*}(P)dv$, the motivation for
which was given in Section 3. This gives a finite number for all $f,g$ of
$\mathfrak{H}$ as

$|f(P)g^{*}(P)|\leq\frac{1}{2}|f(P)|^{2}+\frac{1}{2}|g(P)|^{2},$ (234)

and $\int_{\Omega}|f(P)|^{2}dv$ and $\int_{\Omega}|g(P)|^{2}dv$ are
finite.[The second axiom, B., now follows directly.]

Let us now proceed to the verification of the requirements A.–E.! A. and B.
are manifestly satisfied.
C.: [Infinite-dimensionality] We shall choose $k$ non-overlapping regions
$\mathfrak{M}_{1},\mathfrak{M}_{2},\ldots,\mathfrak{M}_{k}$ in $\Omega$. Let
$f_{p}(P)$ be [the characteristic function of $\mathfrak{M}_{p}$] 1 if $P$ is
in $\mathfrak{M}_{p}$, 0 otherwise (for $p=1,2,\ldots,k$. The $f_{p}(P)$
($p=1,2,\ldots,k$) are linearly independent, as if

$a_{1}f_{1}+\cdot\cdot\cdot+a_{k}f_{k}=0,$ (235)

then, in particular, for points $P$ in $\mathfrak{M}_{p}$,

$a_{1}f_{1}(P)+\cdot\cdot\cdot+a_{k}f_{k}(P)=0$ (236)

implies $a_{p}=0$, for any $p=1,2,\ldots,k$. [As $k$ can be chosen arbitrarily
large, the space $\mathfrak{H}$ must be infinite dimensional.]
D.: [Separable property] A general verification of this property for all
spaces $\Omega$ is not really possible without going further into the precise
definition of general [i.e. measurable] spaces and the so-called Lebesgue
measure.Cf. for example, Carathéodory, _Vorlesungen über reele Funktionen_ ,
Berlin-Leipzig 1918, Chaps. 5-9. We shall not do so here, but will give
examples in two characteristic cases of everywhere dense sequences
$f_{1},f_{2},\ldots$ in $\mathfrak{H}$. First, let $\Omega$ be the
“$n$-dimensional cube” of all points $x_{1},x_{2},\ldots,x_{n}$ satisfying

$0\leq x_{\nu}\leq 1,\;\;(\nu=1,2,\ldots,n).$ (237)

$\mathfrak{H}$ is then the space of all (complex) functions $f$ defined on
$\Omega$ with finitely integrable absolute square. We can expand all such
functions $f(P)=f(x_{1},x_{2},\ldots,x_{n})$, following Fourier, as follows

$f(x_{1},x_{2},..,x_{n})=\sum_{r_{1},r_{2},..,r_{n}=-\infty}^{\infty}c_{r_{1},r_{2},\ldots,r_{n}}e^{2\pi
i(r_{1}x_{1}+r_{2}x_{2}+\cdot\cdot\cdot+r_{n}x_{n})}.$ (238)

Now, the partial sum

$\sum_{r_{1},r_{2},..,r_{n}=-N}^{N}c_{r_{1},r_{2},\ldots,r_{n}}e^{2\pi
i(r_{1}x_{1}+r_{2}x_{2}+\cdot\cdot\cdot+r_{n}x_{n})}$ (239)

converges in the mean to $f$ for $N\rightarrow\infty$.A general and direct
construction of an everywhere dense sequence in [a general Hilbert space]
$\mathfrak{H}$ will be given in the work cited in note 12. There therefore
exists, arbitrarily close to an arbitrary element $f$ of $\mathfrak{H}$, a
function of the form (239), with rational $c_{r_{1},r_{2},\ldots,r_{n}}$. But
it is well known that these [coefficients] can be written in the form of a
sequence.One can, as is well known, write any assembly of finitely many
rational numbers (which is the case here) in the form of a sequence.

Second: let $\Omega$ be the $n$-dimensional (real) space of all points
$x_{1},x_{2},\ldots,x_{n}$ [$-\infty<x_{i}<\infty$], $\mathfrak{H}$ the
corresponding [function-space]. We consider an arbitrary (differentiable)
function $\phi(x)$ which maps the interval $(0,1)$ to the real line
$(-\infty,\infty)$, and let $\psi(y)$ be its inverse (for example,
$\phi(x)=\ln{\frac{x}{1-x}},\;\psi(y)=\frac{e^{y}}{e^{y}+1}$). Then one has
generally that

$\int_{-\infty}^{\infty}\cdot\cdot\cdot\int_{-\infty}^{\infty}f(x_{1},\ldots,x_{n})dx_{1}\ldots
dx_{n}=\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}f(\phi(u_{1}),\ldots,\phi(u_{n}))\phi^{\prime}(u_{1})*\ldots*\phi^{\prime}(u_{n})du_{1}*\ldots*du_{n}.$
(240)

Accordingly, we only need to choose $f_{1},f_{2},\ldots$ so that they are
everywhere dense in the $\mathfrak{H}$ of the first example, whence the
$g_{1},g_{2},\ldots$, defined as

$g_{\mu}(x_{1},\ldots,x_{n})=\frac{f_{\mu}(\psi(x_{1}),\ldots,\psi(x_{n}))}{\phi^{\prime}(\psi(x_{1}))*\ldots*\phi^{\prime}(\psi(x_{n}))}$
(241)

are then everywhere dense in our [second] example.There is a small subtlety in
this argument: denseness requires closeness with respect to the metric in each
space, which involves the integral of the absolute square, possibly multiplied
by a positive measure factor. One needs to define a mapping between square
integrable functions $g(x_{i})\in L^{2}[(-\infty,\infty)^{n}]$ and
$f(u_{i})\in L^{2}[(0,1)^{n}]$, which can be done by a small modification of
von Neumann’s formula (241). The correct association is given by setting

$g_{\mu}(x_{1},\ldots,x_{n})=\frac{f_{\mu}(\psi(x_{1}),\ldots,\psi(x_{n}))}{\sqrt{\phi^{\prime}(\psi(x_{1}))*\ldots*\phi^{\prime}(\psi(x_{n}))}}$
(242)

Note that the function $\phi(u)$ (e.g. $\ln{u/(1-u)}$) is monotonic so
$\phi^{\prime}$ is strictly positive, allowing the square-root. Alternatively,
if we stick with von Neumann’s mapping, we arrive at the Hilbert space for
square-integrable functions on $(-\infty,\infty)^{n}$ with a non-trivial
measure $dv=\prod_{i}\phi^{\prime}(\psi(x_{i}))dx_{i}$.
E.: Let $f_{1},f_{2},\ldots$ be a sequence of functions in $\mathfrak{H}$, and
for every $\epsilon>0$ let there be a $N=N(\epsilon)$, so that $N\leq m\leq n$
implies

$\int_{\Omega}|f_{m}(P)-f_{n}(P)|^{2}dv\leq\epsilon.$ (243)

Choose a monotone increasing sequence of integers $N_{1},N_{2},\ldots$ such
that

$N_{\nu}\geq N(\frac{1}{8^{\nu}}).$ (244)

We then have

$\int_{\Omega}|f_{N_{\nu+1}}(P)-f_{N_{\nu}}(P)|^{2}dv\leq\frac{1}{8^{\nu}}.$
(245)

The set of points $P$ for which

$|f_{N_{\nu+1}}(P)-f_{N_{\nu}}(P)|\geq\frac{1}{2^{\nu}}$ (246)

therefore forms a region of measure (surface-extensionThis is to be understood
as the Lebesgue measure, see note 37.) $\leq\frac{1}{2^{\nu}}$.

[Proceeding in this way,] the set of points for which we do _not_ have

$\displaystyle|f_{N_{\nu+1}}(P)-f_{N_{\nu}}(P)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2^{\nu}}$ (247)
$\displaystyle|f_{N_{\nu+2}}(P)-f_{N_{\nu+1}}(P)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2^{\nu+1}}$ (248)
$\displaystyle|f_{N_{\nu+3}}(P)-f_{N_{\nu+2}}(P)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2^{\nu+2}}$ (249)
$\displaystyle\ldots\ldots\ldots\ldots$ (250)

must have a total measure

$\leq\frac{1}{2^{\nu}}+\frac{1}{2^{\nu+1}}+\frac{1}{2^{\nu+2}}+\cdot\cdot\cdot=\frac{1}{2^{\nu-1}}.$
(251)

But everywhere, where the above inequalities hold, the sequence
$f_{N_{1}}(P),f_{N_{2}}(P),\ldots$ must converge [being a Cauchy sequence of
complex numbers]: the points of non-convergence therefore form a region of
measure $\leq 1/2^{\nu-1}$. This holds for all $\nu$, hence the region of non-
convergence must have measure zero. The limit of $f_{N_{\nu}}(P)$ therefore
exists everywhere (with the exceptiojn of a set of measure zero): let us call
it $f(P)$.

If $m\geq N=N(\epsilon)$, then for $N_{\nu}\geq m$, (i.e. for all but at most
a finite number of the $N_{\nu}$),

$\int_{\Omega}|f_{m}(P)-f_{N_{\nu}}(p)|^{2}dv\leq\epsilon;$ (252)

so we can let $\nu\rightarrow\infty$: thus,

$\int_{\Omega}|f_{m}(P)-f(P)|^{2}dv\leq\epsilon.$ (253)

Thus: $f$ belongs to $\mathfrak{H}$, and the sequence $f_{1},f_{2},\ldots$
converges to $f$.

###  18 Appendix 3

The Stieltjes integral

$\int_{a}^{b}f(x)d\phi(x)$ (254)

is defined, whether $a,b$ is a finite or infinite interval, if $f(x)$ is a
continuous function in this interval (and also at the endpoints $a,b$!), and
$\phi(x)$ is a monotone non-decreasing function in the same interval (finite
at $a,b$)—or the difference of two such functions (i.e. a function of bounded
variation). It is defined,Stieltjes, _Recherches sur les fractions continues_
, _Annales de la Faculté des Sciences de Toulouse_ , 1894/95, Chapter 6. A
short treatment can be found, for example, in Carlemann, _Équations Intégrales
à noyeau réel singulier_ , Uppsala 1923, pp. 7-9. in analogy to the Riemann
integral, as the limit under arbitrarily fine divisions
$x_{0},x_{1},\ldots,x_{n}$ of the interval $(a,b)$ ($a=x_{0}\leq
x_{1}^{\prime}\leq x_{1}\leq x_{2}^{\prime}\leq\cdot\cdot\cdot\leq x_{n-1}\leq
x_{n-1}^{\prime}\leq x_{n}=b$) of

$\sum_{n=1}^{N}f(x_{n}^{\prime})(\phi(x_{n}-\phi(x_{n-1})).$ (255)

Without going into this more closely here, let us indicate as follows a
geometrical realization of this quantity (for monotone $\phi$): draw in the
$x-y$ plane the curve

$x=\phi(u),\;\;y=f(u),\;\;(a\leq u\leq b).$ (256)

If $x$ falls in a gap, because $\phi(u)$ discontinuously jumps over it, the
neighboring $[\phi(u),]f(u)$ points are connected horizontally. The integral

$\int_{a}^{b}f(u)d\phi(u)$ (257)

is then the area of the plane between this curve (256), the $x-$axis, and the
abscissas $x=\phi(a)$ and $x=\phi(b)$ (see Figures 1–3).

Figure 1: von Neumann’s Figure a., showing the Stieltjes measure function
$\varphi(x)$ as a function of $x$. The function is monotone non-decreasing,
and continuous save for a finite number (two) of jump discontinuities. Figure
2: von Neumann’s Figures b,c., showing (b) the function $f(x)$ in (254) as a
function of $x$, and (c) the function of (256) defining the area (= German
_Fläche_) represented by the Stieltjes integral.

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

### J. v. Neumann

### Submitted by M. Born in the session of November 11, 1927

###  1 Introduction

The recent development of quantum mechanics has led, as is well known, to two
versions which differ in principle in the conceptual formulation of the
results, designated the “wave theory”, and the “transformation-” or
“statistical-theory.” It is the second of these which will be of primary
concern to us here.

The statistical theory was initiated by Born, Pauli and London, and has been
completed by Dirac and Jordan.In particular, see several works of P. A. M.
Dirac in the 1926/27 issues of the _Proceedings of the Royal Society_
(particularly Vol. 113, 1927), as well as P. Jordan, _Zeitschrift für Physik_
, Vol. 40, 11/12 (1927), and Vol. 44, 1/2 (1927). See also J. v. Neumann,
_Göttingen Nachrichten_ , session of May 20, 1927.[14],[27],[28],[47].
Primarily this theory allows one to answer questions of the following form:
Let us specify a particular physical quantity in a definite physical system.
What values can it assume? What are the a priori probabilities of these
values? How do these probabilities change, in the event that the values of
certain other quantities (previously measured) are specified?

As such, such questions are in no way unusual in classical mechanics, but
there it is always possible to make the statistical question a “sharp” one,
that is to say, to reach the point where a particular physical quantity
assumes a particular value with probability 1, and all other values with
probability 0. To achieve this, one has only to measure sufficiently many
quantities: namely, if the system has $f$ degrees of freedom, $2f$ independent
quantities (e.g. $f$ coordinates and their conjugate momenta).

That this is _not_ the case in quantum mechanics constitutes one of its most
characteristic features, for it is impossible to measure certain quantities
simultaneouslyThat is to say, the measurement of one influences the other, and
vitiates the validity of a previous measurement of the latter. Cf. for example
Dirac, _Proceedings of the Royal Society_ , Vol. 112 (1926), and Heisenberg,
_Zeitschrift für Physik_ , Vol. 43, 3/4 (1927).[12],[20].: for example,
measurements of a quantity and of its conjugated momentum are _always_
incompatible.

The usual approach employed up to now in statistical quantum mechanics was in
essence deductive: the absolute square of certain expansion coefficients of
the wave function,The designation “wave function” stems from the wave theory
of Schrödinger; Dirac views it as a row of a certain transformation matrix,
while Jordan calls it the “probability amplitude.” or of the wave function
itself, was more or less dogmatically equated to a probability—and agreement
with experience ascertained afterwards. A systematic derivation of quantum
mechanics from the phenomenological facts or basic probability-theoretic
assumptions, in other words, an inductive derivation, has not been given.
Additionally, its relation to the usual probability calculus was
insufficiently clarified: the validity of the basic laws [of probability
calculus] (the addition and multiplication laws for calculating probabilities)
was not adequately discussed.Thus, for example, according to Jordan
(_Zeitschrift für Physik_ , Vol. 40, 11/12) both the addition and the
multiplication rule apply to the “probability amplitudes”, and not to their
absolute squares, which are the probabilities themselves. In opposition to
this, cf. Heisenberg, _Zeitschrift für Physik_ , Vol. 43, 3/4.

In the following work an inductive construction of this type will be sought.
We assume throughout the absolute validity of the usual probability calculus.
It will become clear that this [calculus] is not only compatible with quantum
mechanics, but even (in combination with some not very deep factual and formal
assumptions—cf. the summary in Section 9, 1–3) suffices for its unique
derivation. Indeed, we shall be in a position to erect the entire “time-
independent” quantum mechanics on this foundation.

It will turn out to be useful to posit certain mathematical concepts as given
from the outset—e.g. those collected in Sections 1-11 of the author’s work
entitled “Mathematical Foundation of Quantum Mechanics.” _Göttingen
Nachrichten_ , session of May 20, 1927.[47] We shall therefore make use of the
contents of this work, all the more because it contains (Section 13) a
formulation of the statistical predictions [of quantum mechanics] which
deviates from (and is more general than) the usual one, which we will need in
any case. This work will be referred to as “M.B.Q.”

###  2 Basic assumptions

In the following, we denote by $\mathfrak{S}$ a permanently fixed physical
system. (In the following, many so designated systems, such as
$\mathfrak{S}^{\prime},\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots,\mathfrak{S}^{\prime}_{1},\mathfrak{S}^{\prime}_{2},\ldots$,
will appear.The formulation of probability theory followed by von Neumann here
adheres closely to the methodology propounded by [45], which summarizes the
frequentist approach developed over many years by Richard von Mises. The
terminology of von Neumann differs slightly from von Mises: the latter uses
the term “ _Kollektiv_ ” for an ensemble (von Neumann’s _Gesamtheit_), “
_Merkmal_ ” in von Mises becomes a “ _physikalische Grösse_ ” (physical
quantity) in von Neumann. All of these will have the physical structure of
$\mathfrak{S}$, only their states will vary—cf. note 6.) As we are concerned
with the application of statistical methods, we imagine this system to be
absolutely isolated from the environment (an isolation only occasionally
interrupted by measurements, i.e. external interventions), and presented in
very (perhaps infinitely) many exemplars
$\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots$ The physical quantities defined in
$\mathfrak{S}$ we shall denote $\\{\mathfrak{a},\mathfrak{b},\ldots\\}$; in
order to obtain the statistics of the quantity $\mathfrak{a}$ in the ensemble
$\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots$, one subjects each of the systems
$\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots$ to the experiment “measurement of
$\mathfrak{a}$”, and makes a note of the resulting distribution of values.
(Note that, in accordance with the fundamental principles of probability
theory,cf. ibid. [45, pp. 24-25]: “the limiting value [for a probability] must
remain unchanged, if one arbitrarily selects any subensemble of the full
ensemble and considers just these.” one obtains the same statistics from
examining, instead of $\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots$, an
arbitrarily chosen subset of the ensemble, as long as it has very many
members, even though it may be arbitrarily small in comparison to
$\\{\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots\\}$; the ensemble
$\\{\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots\\}$ is altered as little as we
please by such a “statistical levy.”) In particular, the statistics will be
termed “sharp” if the same value is measured in all the systems
$\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots$—i.e. the entire distribution
consists of a single number.

In any event, the foundation of any statistical investigation consists in
having at one’s disposal an “elementary random” ensemble
$\\{\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots\\}$, in which “all conceivable
states of the system $\mathfrak{S}$ appear equally frequently”; the
distribution of values of this ensemble must then be that ascribed to a system
$\mathfrak{S}$, of whose state one knows absolutely nothing.Naturally, one
knows the physical structure of $\mathfrak{S}$, which should be specified from
the outset—$\mathfrak{S}$ is perhaps a harmonic oscillator, or a hydrogen
atom, etc.; what one does not know is the present state of same, Even the
statistical properties of such systems $\mathfrak{S}^{\prime}$, of which one
knows something, are essentially dependent on [the elementary random ensemble]
$\\{\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots\\}$: if, for example, the only
thing that one knows about $\mathfrak{S}^{\prime}$ is that the value of the
quantity $\mathfrak{a}$ lies in the interval $I$, then one must take the
ensemble $\\{\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots\\}$, carry out an
experiment on each member $\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots$, which
decides if $\mathfrak{a}$ lies in the interval $I$, or not; those
$\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots$ for which the first situation holds
form an ensemble
$\\{\mathfrak{S}_{1}^{\prime},\mathfrak{S}_{2}^{\prime},\ldots\\}$, which is
to be taken as determining the statistics of our system
$\mathfrak{S}^{\prime}$. A system $\mathfrak{S}^{\prime}$, of which we have
some information, therefore always represents a particular ensemble
$\\{\mathfrak{S}_{1}^{\prime},\mathfrak{S}_{2}^{\prime},\ldots\\}$ which
arises from the [fundamental] ensemble
$\\{\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots\\}$ in a well-defined manner.

All “knowledge” about $\mathfrak{S}^{\prime}$, i.e. of an ensemble
$\mathfrak{S}_{1}^{\prime},\mathfrak{S}_{2}^{\prime},\ldots$, corresponds to
an association of an expectation value to each quantity $\mathfrak{a}$—namely,
the average value of the distribution of values [for $\mathfrak{a}$] that
occur in $\\{\mathfrak{S}_{1},\mathfrak{S}_{2},\ldots\\}$. Conversely,
“knowledge” of $\mathfrak{S}^{\prime}$ (i.e. the statistical composition of
$\\{\mathfrak{S}_{1}^{\prime},\mathfrak{S}_{2}^{\prime},\ldots\\}$) is
completely described once the association

$\mathfrak{a}\leftrightarrow\mathrm{Expectation\;value\;of}\;\mathfrak{a}=\mathbf{E}(\mathfrak{a})$
(1)

is given for _all_ quantities $\mathfrak{a}$.Naturally, the distribution of
values of $\mathfrak{a}$ is not determined solely by the specification of the
average value. Inasmuch however as we also know the average value of all
powers of $\mathfrak{a}$, i.e. all “moments” of the distribution, the latter
is fully determined. Consequently, it is conceptually equivalent, and formally
more convenient, to examine this association. Before we investigate it
further, however, we must study somewhat more closely the quantities
$\mathfrak{a},\mathfrak{b},\ldots$ in a system $\mathfrak{S}$.

In quantum mechanics, when one considers simultaneously two quantities
$\mathfrak{a},\mathfrak{b}$, there is a fundamental distinction between the
cases where they can be simultaneously observed, or not. In the case where
they can, there must be an experiment measuring $\mathfrak{a}$ and an
experiment measuring $\mathfrak{b}$, of such a kind that the experiments do
not disturb one another (cf. the remarks at the beginning of Section 1).
Alternatively, if we combine both experiments into a single one, there should
exist a single experiment which measures $\mathfrak{a}$ as well as
$\mathfrak{b}$. Given such an experiment we may very well presume that a third
quantity $\mathfrak{d}$ is being measured, the knowledge of which involves
that of both $\mathfrak{a}$ and $\mathfrak{b}$, i.e. one of which both
$\mathfrak{a}$ and $\mathfrak{b}$ are functions.At this point we deviate for
the first time in an essential way from classical mechanics. There, the
reduction of two, perhaps independent, quantities (which is very easily
compatible with simultaneous measurability) to a third, of which they should
both be functions, is completely unnatural. This reduction is also completely
incompatible with the concept of “degrees of freedom”, as
$\mathfrak{a},\mathfrak{b}$ (if independent) correspond to two degrees of
freedom, $\mathfrak{d}$ to just one.

It is perhaps also appropriate here to point to the fact, that quantum
mechanics is also lacking a concept of “degrees of freedom.” For example, a
hydrogen atom perturbed by an inhomogeneous electric field (with the nucleus
held fixed at the origin!) can be described either by three independent
quantities, the coordinates of the electron, but also by just one: the total
energy (as in this case there are no degeneracies!).

The concept “function of a quantity $\mathfrak{a}$”, $f(\mathfrak{a})$ (where
$f(x)$ is a function defined for all real numbers, and only for these) is
always meaningful: $f(\mathfrak{a})$ is a quantity which is measured whenever
$\mathfrak{a}$ is, with the value $f(w)$ ascribed to it whenever the value $w$
is observed for $\mathfrak{a}$. The situation is different for two (or more)
quantities, as these may possibly not be simultaneously measurable.
Nevertheless one may still in this case define the sum [of quantities not
simultaneously measurable], and for the following reason. In order to learn
the expectation value of a sum, one need only add the expectation values of
the two summands, without regard to the detailed properties of the individual
distributions (in particular, to questions of independence). Now, as only the
expectation values of quantities played a role in the preceding description of
quantum-mechanical systems, we can therefore regard
$\mathfrak{a}+\mathfrak{b}$ as a quantity, even if $\mathfrak{a},\mathfrak{b}$
are not simultaneously measurable.The simplest example being the total energy
$H=T+V$ for a mechanical system possessing both kinetic and potential energy.
In an analogous fashion we can indeed define the sum of arbitrarily many (even
infinitely) quantities $\mathfrak{a},\mathfrak{b},\mathfrak{c},..$: for the
expectation value of the sum (in agreement with basic principles of the
probability calculus) is defined as the sum of the expectation values. A
further (trivial) generalization is that the components of the sum can be
multiplied by arbitrary real constants.

Functions other than the sumHere, by “sum”, one should more broadly admit
“linear combinations,” as becomes clear in postulate A. below. Von Neumann is
pointing to the difficulties which arise when one considers kinematical
quantities involving _products_ of non-commuting quantities. of two (possibly
not simultaneously observable!) quantities have however absolutely no sense,
inasmuch as the expectation value of another function of
$\mathfrak{a},\mathfrak{b}$ cannot be expressed in terms of the expectation
values of $\mathfrak{a}$ and $\mathfrak{b}$.

We are now in a position to formulate completely what we shall demand of the
expectation value functions $\mathbf{E}(\mathfrak{a})$, which determine a
“knowledge” of $\mathfrak{S}^{\prime}$, i.e. of the ensemble
$\\{\mathfrak{S}_{1}^{\prime},\mathfrak{S}_{2}^{\prime},\ldots\\}$. Namely:
A. If $\mathfrak{a},\mathfrak{b},\mathfrak{c},\ldots$ are a finite or infinite
set of quantities, and $\alpha,\beta,\gamma,\ldots$ are real numbers,
thenObserve the deep implications of this property of quantum statistics:
suppose we have an electron with coordinates $q_{1},q_{2},q_{3}$, momenta
$p_{1},p_{2},p_{3}$, and mass $m$. The three quantities

$\alpha(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}),\;\;\frac{1}{2m}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}),\;\;\alpha(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})+\frac{1}{2m}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}),$
(2)

(i.e. the squared distance from the origin in appropriate units of the free
electron, the kinetic energy of the free electron, and the energy of the
electron trapped in a harmonic oscillator [potential]) have completely
different spectra: the first two each have a continuous spectrum, the third a
discrete spectrum; moreover, no two of them are simultaneously measurable.
Nonetheless, the sum of the expectation values of the first two is equal to
the expectation value of the third. (Additionally, it is convenient, in order
to find finite expectation values, to consider fixed “states” of the system,
cf. Section 4.)

$\mathbf{E}(\alpha\mathfrak{a}+\beta\mathfrak{b}+\gamma\mathfrak{c}+\cdots)=\alpha\mathbf{E}(\mathfrak{a})+\beta\mathbf{E}(\mathfrak{b})+\gamma\mathbf{E}(\mathfrak{c})+\cdots$
(3)

B. If $\mathfrak{a}$ is a quantity which is intrinsically non-negative, then

$\mathbf{E}(\mathfrak{a})\geq 0.$ (4)

The motivation for A. has already been discussed; B. is also an obvious
property of all expectation values. A topic so far omitted needs further
commentary, namely, the normalization requirement: if the quantity
$\mathfrak{a}$ always has the value 1, then we must also have
$\mathbf{E}(\mathfrak{a})=1$. Absent this we are confined to considering only
relative expectation values, given up to a proportionality factor. The reasons
for this generalization can best be illustrated with the following example.

Let $\mathfrak{a}$ be a quantity, $I$ an interval on the real line, and $f(x)$
the function defined to equal 1 in $I$ and 0 otherwise. The expectation value
for $f(\mathfrak{a})$ is then manifestly the probability that the value of
$\mathfrak{a}$ lies in the interval $I$. If now, for example, $\mathfrak{a}$
can assume all real values, and all with the same probability, then if we were
to normalize the total probability (i.e. that of the interval
$-\infty,+\infty$) to 1, the probability that the value of $\mathfrak{a}$ lies
in $I$, for any finite interval $I$, is equal to zero. On the other hand, we
have no such problem with the relative probabilities of two finite intervals.
Conclusions of this type are naturally much more important than the
requirement that the total probability be equal to 1: it is therefore useful
to give up on the latter (and allow the total probability to be perhaps
$+\infty$), and satisfy ourselves with relative expectation values and
relative probabilities—which can then in interesting (and _not_ impossible)
situations vanish!Indeed, in systems in which both the available space and the
potential energy are finite, this complication usually will not appear. Still,
it is appropriate to deal with it given that it becomes apparent in the most
typical systems in quantum mechanics: a free electron, hydrogen atom, etc.

We must add to the preceding considerations of a conceptual and foundational
nature the following remark of a formal character: a theory is impossible, as
long as we have not found a formal (i.e. accessible to calculation) equivalent
for the quantities $\mathfrak{a},\mathfrak{b},\mathfrak{c},\ldots$ of the
system $\mathfrak{S}$. But we know very well from quantum mechanics which
mathematical objects are to be associated with the physical quantities: they
are in fact the so-called linear symmetric [i.e. hermitian] functional
operators, which act on complex-valued functions (the wave functions) defined
in the state space of the system $\mathfrak{S}$, transforming them into other
such functions. For the time being we shall not discuss why this is so; later,
in the course of our considerations, we shall find a direct interpretation of
the operator assigned to a [physical] quantity (cf. Section 4, $\gamma$).

Now, the complex valued functions defined in the state space of $\mathfrak{S}$
form a realization of the abstract Hilbert space $\mathfrak{H}$;From here on,
we will make essential use of the concepts and terminology of M.B.Q. In
Section 4 of M.B.Q. it is stated, and in Section 6 and Appendix 2 proved
(Theorem of Fischer and F. Riesz), that the stated function space is a
realization of $\mathfrak{H}$ (i.e. the operations of addition, multiplication
by a constant and the so-called inner product—in the notation of
M.B.Q.:$f+g,a.g,Q(f,g)$—are mapped uniquely and invertibly to $\mathfrak{H}$).
Abstract Hilbert space is introduced in Sections 5,6. we can therefore replace
the linear symmetric functional operators acting on these with linear
symmetric operators of $\mathfrak{H}$. But an operator is only useful for
quantum mechanics if its “eigenvalue problem” is solved, or, at least,
solvable.For the possible values of the associated quantity are indeed the
eigenvalues, and to calculate the probabilities one must also know the
eigenfunctions! In addition, the representation in M.B.Q. (Section 14)
presumes knowledge of the partition of unity (defined there in Section 9)
associated to the operator, i.e. the solution of the eigenvalue problem.
Consequently, we must assign to every physical quantity a linear symmetric
operator which admits an “eigenvalue representation” [i.e. spectral
resolution].For further information on operators in $\mathfrak{H}$, see
Sections 7,8 in M.B.Q. The eigenvalue problem is discussed in Sections 9,10.
We call such operators, for the sake of brevity, normal: it is highly likely,
that every linear symmetric operator is normal.The present state of affairs is
as follows: A linear symmetric operator, which allows at all an eigenvalue
representation, allows only one such (for such an operator there is an exact
partition of unity, cf. M.B.Q., Section 9); all bounded (i.e. continuous)
operators, further, all differential operators of second order with
sufficiently regular coefficients, indeed absolutely all operators which have
ever appeared in quantum mechanics, admit an eigenvalue representation. The
existence of an eigenvalue representation for real symmetric linear operators
was shown in general by the author (to appear shortly in _Mathematische
Annalen_). Certain mathematical difficulties appear however in the general
proof for the complex linear symmetric operators which concern us here. Cf.
also M.B.Q., notes 12 and 27.The general spectral theory for self-adjoint—more
generally, normal—operators was presented in [46],[50]. In these works, von
Neumann identified the appropriate subclass of linear operators with “nice”
spectral properties as those (designated “normal”) which commute with their
adjoint, $AA^{\dagger}=A^{\dagger}A$,[46, p. 115]. This includes, in addition
to the self-adjoint operators, which are trivially normal (as
$A=A^{\dagger}$), the important class of unitary (and thus, automatically
bounded) operators $U$ for which $UU^{\dagger}=U^{\dagger}U=1$. We do not
however need to concern ourselves with this question here. With regard to all
this we shall therefore assume:
Every physical quantity $\mathfrak{a}$ of the system $\mathfrak{S}$
corresponds to a linear symmetric operator, and conversely.We will make rather
little use of the converse case, which is surely uncontroversial. How this
association is accomplished in detail is immaterial for us,Also, until quite
recently there was no prescription that determined it _in general_! we will
only employ two properties, in particular:
C. Let $S,T,\ldots$ (in finite or infinite number) as well as $\alpha S+\beta
T+\cdots$ be normal operators. If $S,T,\ldots$ are associated with the
quantities $\mathfrak{a},\mathfrak{b},\ldots$, then $\alpha S+\beta T+\cdots$
is associated with the quantity
$\alpha\mathfrak{a}+\beta\mathfrak{b}+\cdots$.In the highly unlikely, but at
present uncontradicted, circumstance (see note 14) that $S,T,\ldots$ are
normal, but $S+T+\cdots$ not, this will prevent us from forming the quantity
$\mathfrak{a}+\mathfrak{b}+\cdots$. This restriction will however be
unimportant for us.
D. Let $S$ be a normal operator, $f(x)$ a real-valued function defined for all
real $x$, and let $S$ be associated with the quantity $\mathfrak{a}$. Then
$f(S)$If $f(x)=x^{n}$ then the meaning of $f(S)$ is clear: it is just the
$n$-th iteration of $S$, $S^{n}$; and from this one obtains also the meaning
of $f(S)$ for polynomial $f(x)$. For arbitrary $f(x)$ one gets [$f(S)$] most
conveniently following M.B.Q., Section 14, second example: if $S$ has the
partition of unity $E(\lambda)$ (i.e. $S=\int_{-\infty}^{+\infty}\lambda
dE(\lambda)$ [see note h]) then

$f(S)=\int_{-\infty}^{+\infty}f(\lambda)dE(\lambda).$ (5)

The normal property of $S$ is therefore presumed; the normal property of
$f(S)$ can be easily demonstrated.For the reader familiar with Dirac notation,
the spectral representation $S=\int\lambda dE(\lambda)$ (more generally,
$f(S)=\int f(\lambda)dE(\lambda)$) is more familiar in the form
$S=\int\lambda|\lambda\rangle\langle\lambda|d\lambda$ (resp. $f(S)=\int
f(\lambda)|\lambda\rangle\langle\lambda|d\lambda$) , where the states are
continuum normalized
$\langle\lambda|\lambda^{\prime}\rangle=\delta(\lambda-\lambda^{\prime})$. The
“partition of unity” referred to by von Neumann is the completeness property
associated with the spectral representation of any normal operator, $\int
dE(\lambda)=\int|\lambda\rangle\langle\lambda|=1$. If there is a point
spectrum then a discrete sum is present in addition to the integral,
$\sum_{n}\lambda_{n}|\lambda_{n}\rangle\langle\lambda_{n}|$ (with
$\langle\lambda_{n}|\lambda_{m}\rangle=\delta_{nm}$). One easily establishes
in this version Eq. (5) for any power or polynomial function of $S$. Below, we
encounter projection operators $E(I)$ which project onto the subspace spanned
by eigenvectors $|\lambda\rangle$ of an observable with eigenvalues $\lambda$
in some interval $I$: in Dirac notation this would be
$E(I)=\int_{I}|\lambda\rangle\langle\lambda|d\lambda$. is associated with the
quantity $f(\mathfrak{a})$.

Both of these requirements are very plausible, and correspond to common
practice, making a more detailed discussion superfluous.

###  3 General form of the expectation values.

On the basis of A.–D. in Section 2 we can construct all possible expectation-
value functions $\mathfrak{a}\leftrightarrow\mathbf{E}(\mathfrak{a})$, i.e.
all possible “knowledge” about $\mathfrak{S}^{\prime}$ or statistical
ensembles $\\{\mathfrak{S}^{\prime}_{1},\mathfrak{S}_{2}^{\prime},\ldots\\}$.
For the purpose of calculation one associates a normal operator to the
expectation value of the physical quantity corresponding to the operator: if
$S$ is the operator, then we call this (relative) expectation value
$\mathbf{E}(S)$. $\mathbf{E}(S)$ must satisfy the conditions A. and B.,
whereby also C. and D. must also be taken into consideration.

In order to facilitate calculation, we will replace [the function space]
$\mathfrak{H}$ by its realization $\mathfrak{H}_{0}$ [$l^{2}$], the
conventional Hilbert space,i.e. the space of all sequences
$x_{1},x_{2},\ldots$ with finite $\sum_{n=1}^{\infty}|x_{n}|^{2}$, cf. M.B.Q.,
Section 5. in other words we map [$\mathfrak{H}_{0}$] one-to-one to
[$\mathfrak{H}$] (which is obviously possible in many ways). A linear operator
$S$ is then described by a matrix $\\{s_{\mu\nu}\\}(\mu,\nu=1,2,\ldots)$,
whereby generally

$\displaystyle S(x_{1},x_{2},\ldots)$ $\displaystyle=$
$\displaystyle(y_{1},y_{2},\ldots),$ (6) $\displaystyle y_{\mu}$
$\displaystyle=$ $\displaystyle\sum_{\nu=1}^{\infty}s_{\mu\nu}x_{\nu}.$ (7)

The symmetry [hermitian] property means, as one sees right away,
$s_{\mu\nu}=s_{\nu\mu}^{*}$. We now introduce the following operators:
$A_{\mu}$, with the matrix $s_{\mu\mu}=1$, otherwise $s_{\rho\sigma}=0$;
$B_{\mu\nu},(\mu<\nu)$, with the matrix $s_{\mu\nu}=s_{\nu\mu}=1$, otherwise
$s_{\rho\sigma}=0$; $C_{\mu\nu},(\mu<\nu)$, with the matrix $s_{\mu\nu}=i$,
$s_{\nu\mu}=-i$, otherwise $s_{\rho\sigma}=0$. They are all clearly linear
symmetric [hermitian], and also normal (the $A$ have the simple eigenvalue 1,
the $B$ and $C$ have the simple eigenvalues 1 and -1; otherwise only the
infinitely [degenerate eigenvalue] 0). In general one has (we use the notation
$\mathrm{Re}(z)$ (resp. $\mathrm{Im}(z)$) for $u$ (resp. $v$) when $z=u+iv$):

$S=\sum_{\mu}s_{\mu\mu}\cdot
A_{\mu}+\sum_{\mu<\nu}\mathrm{Re}(s_{\mu\nu})\cdot
B_{\mu\nu}+\sum_{\mu<\nu}\mathrm{Im}(s_{\mu\nu})\cdot C_{\mu\nu},$ (8)

and consequently one has

$\mathbf{E}(S)=\sum_{\mu}s_{\mu\mu}\cdot\mathbf{E}(A_{\mu})+\sum_{\mu<\nu}\mathrm{Re}(s_{\mu\nu})\cdot\mathbf{E}(B_{\mu\nu})+\sum_{\mu<\nu}\mathrm{Im}(s_{\mu\nu})\cdot\mathbf{E}(C_{\mu\nu}).$
(9)

If we now set

$\displaystyle u_{\mu\mu}$ $\displaystyle=$
$\displaystyle\mathbf{E}(A_{\mu}),$ (10) $\displaystyle u_{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{E}(B_{\mu\nu})+\frac{1}{2}i\mathbf{E}(C_{\mu\nu}),\;\;\mu<\nu,$
(11) $\displaystyle u_{\nu\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{E}(B_{\mu\nu})-\frac{1}{2}i\mathbf{E}(C_{\mu\nu}),\;\;\mu<\nu,$
(12)

then $u_{\mu\nu}=u_{\nu\mu}^{*}$, and one has also

$\mathbf{E}(S)=\sum_{\mu\nu}s_{\mu\nu}u_{\mu\nu}^{*}.$ (13)

The matrix $\\{u_{\mu\nu}\\}$ clearly corresponds to a linear symmetric
[hermitian] operator, which we shall denote $U$.Here, the “density operator”,
nowadays usually denoted $\rho$, is introduced into quantum theory. It would
become an indispensable tool in quantum statistical mechanics, in particular,
in the decoherence approach to quantum measurement theory. The result Eq. (13)
is now written $\mathbf{E}(S)={\mathrm{T}r}(\rho S)$ (by hermiticity
$u_{\mu\nu}^{*}=u_{\nu\mu}=\rho_{\nu\mu}$, so the right hand side of (13)
equals $\sum_{\mu}(S\rho)_{\mu\mu}=\mathrm{Tr}(S\rho)=\mathrm{Tr}(\rho S)$).

We should further consider the implications of B. If $\varphi$ is an arbitrary
point in the abstract Hilbert space $\mathfrak{H}$, with $Q(\varphi)$=1 (i.e.
on the unit sphere), then we define the operator $P_{\varphi}$ (“projection in
the direction of $\varphi$”) by

$P_{\varphi}f=Q(f,\varphi)\cdot\varphi;$ (14)

it is clearly linear, symmetric and normal (a simple eigenvalue 1, otherwise
infinitely degenerate 0). $P_{\varphi}$ is a projection operator, i.e.
$P_{\varphi}^{2}=P_{\varphi}$. In the $\mathfrak{H}_{0}$ [$l^{2}$]
realization, should $\varphi$ correspond to $(x_{1},x_{2},\ldots)$
($\sum_{n=1}^{\infty}|x_{n}|^{2}=1$), then $P_{\varphi}$ clearly has the
matrix $\\{x_{\mu}x_{\nu}^{*}\\}$.Thus, if $f=(y_{\mu}),\varphi=(x_{\mu})$,
then
$(P_{\varphi}f)_{\mu}=\sum_{\nu}x_{\mu}x_{\nu}^{*}y_{\nu}=Q(f,\varphi)\varphi_{\mu}$,
which is Eq. (14). Recall that in von Neumann’s notation for the inner
product, $Q(y,x)=\sum_{\nu}y_{\nu}x_{\nu}^{*}$ is antilinear in the _second_
argument, in contrast to modern usage. Thus $Q(f,g)$ (von Neumann)
$\rightarrow(g,f)$ or $\langle g|f\rangle$ (modern).

If $P_{\varphi}$ corresponds to the quantity $\mathfrak{a}$, then
$P_{\varphi}=P_{\varphi}^{2}$ corresponds to quantity $\mathfrak{a}^{2}$, i.e.
to an intrinsically non-negative one. Therefore $\mathbf{E}(P_{\varphi})\geq
0$, i.e.

$\sum_{\mu\nu}x_{\mu}^{*}x_{\nu}u_{\mu\nu}^{*}=\sum_{\mu\nu}u_{\nu\mu}x_{\nu}x_{\mu}^{*}\geq
0,$ (15)

where the left hand side is, as one easily verifies, equal to
$Q(\varphi,U\varphi)$. For all $\varphi$ with $Q(\varphi)=1$ we must therefore
have $Q(\varphi,U\varphi)\geq 0$, i.e.

$Q(f,Uf)\geq 0$ (16)

for any $f$ in $\mathfrak{H}$ whatsoever. An operator $U$ with this property
is said to be non-negative definite.If $f$ corresponds to the point
$x_{1},x_{2},\ldots$ in $\mathfrak{H}_{0}$, this means

$\sum_{\mu\nu}u_{\mu\nu}x_{\mu}x_{\nu}^{*}\geq 0,$ (17)

i.e. the hermitian form associated with the matrix $\\{u_{\mu\nu}\\}$ must be
non-negative definite; this characterization we carry over to the operator
$U$.

We now show the converse: every [non-negative] definite linear symmetric
operator $U$ determines (via the formula
$\mathbf{E}(S)=\sum_{\mu\nu}s_{\mu\nu}u^{*}_{\mu\nu}$) a statistics complying
with the requirements A.,B. That A. is satisfied is evident (after
consideration of C.); it remains to establish B.

If the quantity $\mathfrak{a}$ is intrinsically non-negative, we can form the
quantity $\mathfrak{b}=\sqrt{\mathfrak{a}}$, to which belongs the operator
$T$. As $\mathfrak{a}=\mathfrak{b}^{2}$, we have the operator $S=T^{2}$,
therefore one which is [non-negative] definite.For
$Q(f,Sf)=Q(f,T^{2}f)=Q(Tf,Tf)\geq 0$. We must show the following: if the [non-
negative] definite operators $S$, $U$ have the matrices
$\\{s_{\mu\nu}\\},\\{u_{\mu\nu}\\}$, then

$\sum_{\mu\nu}s_{\mu\nu}u_{\mu\nu}^{*}\geq 0.$ (18)

It suffices to show that for all $N=1,2,\ldots$

$\sum_{\mu,\nu=1}^{N}s_{\mu\nu}u_{\mu\nu}^{*}\geq 0,$ (19)

and then to let $N\rightarrow\infty$, considering that the (finite
dimensional!) hermitian forms

$\sum_{\mu,\nu=1}^{N}s_{\mu\nu}x_{\mu}x_{\nu}^{*},\;\;\sum_{\mu,\nu=1}^{N}s_{\mu\nu}x_{\mu}x_{\nu}^{*},$
(20)

are non-negative definite, given that the same holds for the corresponding
infinite-dimensional ones. One easily convinces oneself that
$\sum_{\mu\nu=1}^{N}s_{\mu\nu}u_{\mu\nu}^{*}$ is an orthogonal invariant (in
$N$-dimensional space)—consequently we can bring
$\sum_{\mu,\nu=1}^{N}s_{\mu\nu}x_{\mu}x_{\nu}^{*}$ into diagonal form. All
$s_{\mu\nu},\mu\neq\nu$ are then 0, so
$\sum_{\mu,\nu=1}^{N}s_{\mu\nu}u_{\mu\nu}^{*}$ is a sum of products of
diagonal elements. But diagonal elements of non-negative definite hermitian
forms are never negative, so our sum must always be $\geq 0$. This concludes
the proof.

We have therefore achieved the following result: all possible “knowledge” of a
system $\mathfrak{S}^{\prime}$, i.e. all possible statistical ensembles
$\\{\mathfrak{S}_{1}^{\prime},\mathfrak{S}_{2}^{\prime},\ldots\\}$, correspond
in one-to-one fashion to the [non-negative] definite linear symmetric
operators $U$. The correspondence is described by the expectation value
formula

$\mathbf{E}(S)=\sum_{\mu\nu}s_{\mu\nu}u_{\mu\nu}^{*},$ (21)

where $S,U$ have the matrices $\\{s_{\mu\nu}\\},\\{u_{\mu\nu}\\}$.

One sees from our formula that the modification of $U$ by a [multiplicative]
factor only changes $\mathbf{E}(s)$ by the same, constant factor [necessarily
real and positive, to preserve the non-negative definite property], and
therefore, as only relative expectation values concern us, is of no
importance. Any other [type of] alteration of $U$ is however significant.
Moreover, we cannot have $\mathbf{E}(S)=0$ identically, i.e. $U=0$, as then
all (relative) expectation values and probabilities vanish, and there would be
no [definable] statistics.The connection between $\mathbf{E}(S)$ and $U$ is
obviously dependent on the nature of the realization of $\mathfrak{H}$ in
$\mathfrak{H}_{0}$, i.e. the mapping of $\mathfrak{H}$ to $\mathfrak{H}_{0}$,
and this is possible in (infinitely) many ways. One can easily show [see note
$l$] that this dependence is only an apparent one, i.e. that $U$ is
invariant—but this will not concern us here.The freedom of choice of
orthonormal basis in $\mathfrak{H}_{0}$ [$l^{2}$] is reflected by a unitary
operator $V$ satisfying $V^{\dagger}V=1$: thus, if $\\{\varphi_{n}\\}$ is an
orthonormal basis, so is $\\{V\varphi_{n}\\}$. If $s,u$ are the matrices of