# Why do we need Hilbert spaces?

Curcuraci Luca
Department of Physics, University of Trieste,
Strada Costiera 11 34151, Trieste, Italy
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
(March 2016)

###### Abstract

These are the notes written for the talk given at the workshop “ _Rethinking
foundations of physics 2016_ ". In section 2, a derivation of the the quantum
formalism starting from propositional calculus (quantum logic) is reviewed,
pointing out which are the basic requirements that lead to the use of Hilbert
spaces. In section 3, a similar analysis is done following for the
reconstruction of quantum theory using an operational approach. In both cases,
non-commutativity plays a crucial role. Finally, in section 4 a toy model
which try to motivate non-commutativity is proposed. Despite this last section
is interesting to read, the analysis performed there is not complete. This toy
model will be re-formulated in a rigorous way (and extended) in future works.

###### Contents

  1. 1 Introduction
  2. 2 Propositions about a quantum system: QM from QL
    1. 2.1 Propositions for quantum systems
    2. 2.2 From propositions to lattice
    3. 2.3 From lattice to Hilbert spaces
    4. 2.4 From Hilbert space to quantum mechanics
  3. 3 Modelling a lab: operational reconstruction of QM
    1. 3.1 A bit of math: some notions of algebra
    2. 3.2 Operational approach for a physical theory: description of a single physical quantity
    3. 3.3 Another bit of math: represent an algebra
    4. 3.4 Operational approach for a physical theory: description of a physical system
    5. 3.5 Algebraic formulation of quantum mechanics
  4. 4 What is the meaning of an Hilbert space? A toy-model proposal
    1. 4.1 The toy model: random jumps over a random space
    2. 4.2 Entropy for $V_{N}$ and entropy for $X_{N}$
    3. 4.3 Algebraic description of a point-like particle
    4. 4.4 Conclusions, weak points of the model and possible further development

##  1 Introduction

Despite all its success, quantum mechanics after more that one hundred years,
is still under debate. All the problems due to its interpretation are
originated from the choice to formulate the theory using the Hilbert space
formalism. Here we will discuss how this choice is unavoidable and suggests an
interesting motivation for this unavoidability.

In general, two are the main basic observations that can be done,
experimentally, when one deal with non-relativistic quantum systems

  1. O1

the outcomes of an experiment about a quantum system is probabilistic;

  2. O2

there are physical quantities that can be measured _simultaneously_ and other
that do not.

The first observation is a common feature of all the physical system, once one
take seriously into account the fact that, even the best experimental
physicist in the word, can perform measurement with a finite resolution. The
second observation is the distintive feature of a quantum system and, as we
shall see, it is the origin of all the differences between classical and
quantum system. Starting from these two observations, one can derive (as
logical consequence and with few further assumptions) almost all the
postulates of quantum mechanics. In what follow we will present two approaches
for such derivation, pointing out where we need to do an assumption in order
to continue the reconstruction. The first approach is based on (quantum)
logical arguments, while the second make use of operational considerations.
Finally we will present a toy model to motivate the second observation
(deriving it from the first, in some sense) for the description of a point-
like particle, moving over a random space.

##  2 Propositions about a quantum system: QM from QL

In this section, the quantum logic derivation of the Hilbert space structure
of quantum mechanics will be briefly reviewed. The main idea behind this
approach is to find the theoretical foundations of the postulates for a
quantum theory starting from the proposition that one can formulate about a
quantum system. This is the so called quantum logic (QL) approach to the
foundations of quantum mechanics (QM). For a more detailed treatment, we refer
to [4],[2], [3] for the quantum logic and [1] for the mathematical formulation
of the quantum mechanics’ postulates.

###  2.1 Propositions for quantum systems

In everyday life, it is a common fact to formulate _propositions_ to describe
something and this, of course, holds also in science. Propositions are the
basic outcomes of any experiment, and so it is reasonable to expect that some
basic feature of the physical system under study, can be deduced from the
propositions we may formulate from the experiments. If a quantity $A$ can be
measured (assign an objective numerical value), it is common to formulate
proposition like

_‟ $A$ takes the value $v_{A}$˝_

or better, taking into account the finite resolution of any measurement device
(hence O1)

_‟ $A$ takes value in $[a,b]$˝_

These proposition can be considered as the simplest possible proposition.
Introducing the very natural logical connectivities AND/OR and considering a
second measurable quantity $B$, one may also formulate composite propositions.
Two basic examples are

_‟ $A$ takes value $v_{A}$OR $B$ takes value $v_{B}$˝_

_‟ $A$ takes value $v_{A}$ AND $B$ takes value $v_{B}$˝_

In everyday life, both the propositions make sense. Nevertheless, for a
quantum system, the second proposition cannot be formulated in general: if $A$
and $B$ cannot be measured simultaneously, the measurement to formulate this
proposition cannot be performed in general because O2. We can see that the
effect of O2 is to reduce the number of propositions that we may formulate
using AND. Let us also observe the following fact: the proposition

_‟A take value $v_{A}$ IMPLIES THAT $B$ take value $v_{B}$˝_

make sense only for quantities that can be measured at the same time, which is
again a consequence of O2. In order to explore better the consequences of
this, we will adopt the following conventions: a proposition like _‟ $A$ takes
value $v_{A}$˝_ or _‟ $A$ takes value $[a,b]$˝_ will be labeled simply by $a$,
the logical connectors AND by $\wedge$, $OR$ by $\vee$, the implication by
$\Rightarrow$ and finally the logical negation by $\neg$. We say that two
propositions are equal $a=b$ when $a\Rightarrow b$ and $b\Rightarrow a$. With
this notation, the observation O2 restrict the number of propositions about a
physical system having form $a\wedge b$ which make sense. Before to go on,
suppose $X$ may take only two values $v_{X}$ and $u_{Y}$. Then consider the
following proposition

_‟ $Y$ takes value in $v_{Y}$ AND, $X$ take value $v_{X}$ OR $X$ take value
$u_{X}$˝_

Apparently it seems equivalent to

_‟ $Y$ takes value in $v_{Y}$ AND $X$ takes value $v_{X}$, OR $Y$ takes value
in $v_{Y}$ AND $X$ takes value $u_{X}$˝_

Using the conventions introduced above $a\wedge(b\vee c)=(a\wedge
b)\vee(a\wedge c)$. It is not difficult to understand that if $X$ and $Y$
cannot be measured at the same time, the second proposition do not make sense:
this means that for the possible propositions that we can formulate about a
quantum system, in general

$a\wedge(b\vee c)\neq(a\wedge b)\vee(a\wedge c)$ (1)

where $\neq$ simply means that it is not true that are equivalent. Consider
now a different situation. Let $X,Y,Z$ be three measurable quantities. If

_‟ $X$ takes value $v_{X}$ IMPLIES THAT $Y$ takes value $v_{Y}$˝_

is true, namely the value assumed by $X$ determine the value of $Y$, then

_‟ $X$ takes value $v_{X}$, OR $Y$ takes value $u_{Y}$ AND $Z$ takes value
$w_{Z}$ IMPLIES THAT $Y$ takes value $u_{Y}$, AND $X$ takes value $v_{X}$ OR
$Z$ takes value $w_{Z}$˝_

which in symbols can be written as: if $a\Rightarrow b$ then $a\vee(b\wedge
c)\Rightarrow b\wedge(a\vee c)$. Again we can see that, if $a$ and $c$ cannot
be measured at the same time (and so also $b$ cannot be measured at the same
time of $c$), the first part of this proposition in general doesn’t make
sense. This means that for the possible propositions we can formulate about a
quantum system, in general

$\mbox{if }a\Rightarrow b\mbox{ then }a\vee(b\wedge c)\nRightarrow
b\wedge(a\vee c)$ (2)

This is again a consequence of O2. Finally we observe that in general when

_‟ $X$ takes value $v_{X}$ IMPLIES THAT $Y$ takes value $v_{Y}$˝_

is true, for $u_{Y}$ arbitrary, the proposition

_‟ $X$ takes value $v_{X}$, OR $X$ DOES NOT take value $v_{X}$ AND $Y$ takes
value $u_{Y}$ IS EQUIVALENT TO $Y$ takes value $u_{Y}$˝_

always make sense for a quantum system, since the two physical quantities can
always be measured at the same time by assumption. In symbols we can write
that, for the set of propositions about a quantum system

$\mbox{if }a\Rightarrow b\mbox{ then }a\vee(\neg a\wedge b)=b$ (3)

holds. From this discussion we can understand the the logical connectors OR,
AND, NOT, IS EQUIVALENT TO and IMPLIES THAT cannot be used in a
straightforward manner for a quantum system: thus the usual logic is not
suitable in this case. As we will see, (1),(2) and (3) will help us to select
the right structure to describe mathematically the set of all the propositions
we may formulate about a quantum system, namely to implement O2.

###  2.2 From propositions to lattice

Let us now try to formalise mathematically the discussion done before. In
order to do that, we need to state some technical definitions.

######  Definition 1.

Let $X$ be a set. A relation $\preccurlyeq$ on $X$ is said _partial order_ if
it is reflexive ($x\preccurlyeq x$, $\forall x\in X$), transitive
($x\preccurlyeq y$ and $y\preccurlyeq z$ implies $x\preccurlyeq z$, $\forall
x,y,z\in X$) and skew-symmetric ($x\preccurlyeq y\preccurlyeq x$ implies
$x=y$, $\forall x,y\in X$). The couple $(X,\preccurlyeq)$ is said _poset_.

Using the ordering relation of the poset, one may define the following

######  Definition 2.

Let $(X,\preccurlyeq)$ be a poset and consider a subset $Y\subset X$. The
_lower bound of $Y$_ (_upper bound of $Y$_) is an element $a\in X$ such that
$a\preccurlyeq x$ ($x\preccurlyeq a$) for any $x\in X$. The _greatest lower
bound, GLB_ (_least upper bound, LUB_) of $Y$ is a lower bound (upper bound)
of $Y$ $b$ such that $b\preccurlyeq a$ ($a\preccurlyeq b$) for every lower
bound (upper bound) $a$ of $Y$.

It is not difficult to see that if the GLB (LUB) exists it is unique. Now we
are ready to introduce the central mathematical concept of this paragraph.

######  Definition 3.

Given a poset $(X,\preccurlyeq)$, it is a _lattice_ if for any $x,y\in X$, the
GLB and LUB always exist (denoted $x\wedge y$ and $x\vee y$, respectively).

Not all the poset are lattice. The symbols $\wedge$ and $\vee$ used in the
definition above, can be defined as the following maps

  1. a)

$\wedge:X\times X\rightarrow X$, such that for any $x,y,z\in X$, then $x\wedge
y\preccurlyeq x$, $x\wedge y\preccurlyeq y$ and, if $z\preccurlyeq x$ and
$z\preccurlyeq y$ then $z\preccurlyeq x\wedge y$.

  2. a)

$\vee:X\times X\rightarrow X$, such that for any $x,y,z\in X$, then
$x\preccurlyeq x\vee y$, $y\preccurlyeq x\vee y$ and, if $x\preccurlyeq z$ and
$y\preccurlyeq z$ then $x\wedge y\preccurlyeq z$.

and it is not difficult to see that the writing $x\wedge y=x$, $x\vee y=y$ and
$x\preccurlyeq y$ are equivalent. Lattices are classified according to the
following

######  Definition 4.

A lattice $(X,\preccurlyeq)$ is said

  1. a)

_distributive_ if $x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)$, $\forall
x,y,z\in X$;

  2. b)

_modular_ if $x\preccurlyeq y$ implies $x\vee(y\wedge z)=y\wedge(x\vee z)$,
$\forall x,y,z\in X$;

  3. c)

_bounded_ if there exist two elements $\mathbf{0}\in X$ and $\mathbf{1}\in X$
such that $\mathbf{0}\preccurlyeq x\preccurlyeq\mathbf{1}$, $\forall x\in X$;

  4. d)

_orthocomplemented_ if it is bounded and equipped with an operation
$x\mapsto\neg x$ (called _orthocomplementation_) such that

    1. i)

$x\vee\neg x=\mathbf{1}$, $\forall x\in X$;

    2. ii)

$x\wedge\neg x=\mathbf{0}$, $\forall x\in X$;

    3. iii)

$\neg(\neg x)=x$, $\forall x\in X$;

    4. iv)

$x\preccurlyeq y$ implies $\neg y\preccurlyeq\neg x$, $\forall x,y\in X$;

  5. e)

_orthomodular_ if orthocomplemented and $x\preccurlyeq y$ implies that
$x\vee(\neg x\wedge y)=y$, $\forall x,y\in X$;

One can prove that: distributivity implies modularity which implies
orthomodularity, but the converse is not true. The last notions we need to
reach the goal of this paragraph, are about the elements of a lattice

######  Definition 5.

Let $(X,\preccurlyeq)$ be a bounded lattice then

  1. a)

an element $x\in X$ _covers_ $y\in X$ if $y\prec x$ (namely, $y\preccurlyeq x$
but $x\neq y$) and doesn’t exist $z\in X$, such that $y\prec z\prec x$;

  2. b)

an element $x\in X$ is said _atom_ if it covers $\mathbf{0}$;

  3. c)

two elements $x,y\in X$ are said _orthogonal_ , written $x\perp y$, if
$x\preccurlyeq\neg y$

A bounded lattice $(X,\preccurlyeq)$ is said _atomic_ if for any $y\in
X/\\{\mathbf{0}\\}$ there exist an atom $x\in X$ such that $x\preccurlyeq y$.
A bounded lattice is said _atomistic_ if any element of the lattice can be
seen as the join of atoms. An atomic lattice $(X,\preccurlyeq)$ is said _with
the covering property_ if for any $x\in X$ and every atom $a\in X$ such that
$a\wedge x=\mathbf{0}$, the element $a\vee x$ covers $x$.

For an orthomodular lattice, one can prove that if it is atomic, it is also
atomistic. The discussion done in the previous paragraph seems to suggest the
following: if the set of all the propositions about a quantum system with the
operations $\Rightarrow,\wedge$ and $\vee$ is a lattice, then it must be an
orthomodular lattice because of O2. Nevertheless, in order to say this we need
to find a way to define the partial ordering, namely the $\Rightarrow$ that in
the previous paragraph played the role of logical implication. To define this
ordering relation, the observation O1, suggests that the following
mathematical definition is physically reasonable

######  Definition 6.

Let $(X,\preccurlyeq)$ be an orthomodular lattice, a _probability-like
measure_ on $X$ is a function $p:X\rightarrow[0,1]$ such that

  1. a)

$p(\mathbf{1})=1$ and $p(\mathbf{0})=0$;

  2. b)

for every sequence $\\{x_{i}\\}_{i\in I}$ of orthogonal elements of $X$,
$p\left(\bigvee_{i}x_{i}\right)=\sum_{i}p(x_{i})$

This probability-like measure induces an ordering relation on $X$, in
particular $x\preccurlyeq y$ if and only if $p(x)\leqslant p(y)$ for every
possible $p$: but notice that the ordering relation exists independently to
the existence of $p$. In any case, the observation O1 tells us that when we
study a quantum system (and in general any physical system) this notion is at
disposal: the measure $p$ can be interpreted as a ‟degree of belief˝(or ‟truth
value˝) of a certain proposition, namely, if $a$ is a proposition, $p(a)$ tell
us how much we are sure that $a$ happens in real word. But one must be careful
about O2: the degree of belief of a proposition can be tested and compared
with the one of another proposition, only if these propositions are associated
to observables that are measurable at the same time. Keeping this fact in
mind, we can say that, if we are agree on O1, we have an ordering relation at
disposal over the set of all the physical propositions. This partial order,
allows us to define the met and join between all the propositions (actually
this is an assumption, despite it is reasonable), thus we can conclude that it
is a lattice. In what follow, the lattice of the physical proposition about a
quantum system $Q$ will be denoted by $\mathcal{L}_{Q}(\Rightarrow)$, where
$\Rightarrow$ denotes the partial ordering relation described before. The
observation O2 suggests that is an orthomodular lattice because we expect (3)
to hold. Nevertheless we should also check if it is bounded and define an
orthocomplementation on it. We need to define $\mathbf{1}$ and $\mathbf{0}$.
The first can be though as the proposition

_‟The measurement of some quantity is a real number ˝_

which is clearly always true for a physical system. $\mathbf{0}$ can be
thought as the proposition

_‟We are not measuring anything ˝_

which is always false if we assume that we formulate propositions only after
that at least one experiment was performed. It is not difficult to see that
any other proposition in between these two, or more formally,
$\mathbf{0}\Rightarrow a\Rightarrow\mathbf{1}$. Thus $\mathbf{0}$ and
$\mathbf{1}$ belongs to the lattice of all the propositions about the physical
system $\mathcal{L}_{Q}(\Rightarrow)$ and this lattice is bounded. Once we
have this, the orthocomplementation of a proposition
$a\in\mathcal{L}_{Q}(\Rightarrow)$ is the unique proposition $\neg a$ with
truth value $p(\neg a)=1-p(a)$, and it is not difficult to understand that it
is the negation (in common language sense) of the initial proposition. Hence
it is reasonable to think $\mathcal{L}_{Q}(\Rightarrow)$ as an _orthomodular
lattice_.

Now, we will try to motivate other two properties that
$\mathcal{L}_{Q}(\Rightarrow)$ should have: atomiticy and the covering
property. When we deal with a physical system we have at disposal a set of
elementary propositions, like

_‟The physical quantity $A$ takes exactly the value $v_{A}\in\mathbb{R}$˝_

and from them we may construct more complex propositions (like ‟$A$ takes
value in $[a,b]$˝). In principle for a physical system, we have at disposal an
infinte number of this kind of propositions. Moreover, if $A$ is an elementary
physical quantity, in the sense that cannot be expressed in terms of other
physical quantities, then these propositions may cover only $\mathbf{0}$. This
means that propositions of this kind correspond to the atoms of
$\mathcal{L}_{Q}(\Rightarrow)$. It is also physically reasonable to say that,
any proposition about a physical system or is an atom or it cover an atom
(excluding the trivial case of the $\mathbf{0}$ proposition). Hence this
suggest that $\mathcal{L}_{Q}(\Rightarrow)$ is an _atomistic lattice_. The
covering property is more subtle but still reasonable. Suppose we have two
elementary propositions $a$ and $b$ (hence two atoms) and consider also a
third proposition $c$ (which may not be in general an atom). Now, suppose that
we know $a\Rightarrow b\vee c$. This means that $p(a)\leqslant p(b\vee c)$,
and so that at the same time $a,b$ and $c$ should hold. Because of this
simultaneous truth of these propositions, and because $a$ cannot implies $b$
and viceversa (they are atoms), it is also reasonable to assume that
$p(b)\leqslant p(a\vee c)$ (which means $b\Rightarrow a\vee c$) simply
because, if it is not so, for a physical system the previous requirement
($a\Rightarrow b\vee c$) doesn’t make sense anymore. Thus more rigorously we
can write, if $a,b$ are atoms, then $a\Rightarrow b\vee c$ implies
$b\Rightarrow a\vee c$ for any $c$. This is another possible characterisation
of the covering property for the case of orthomodular lattice. Hence
$\mathcal{L}_{Q}(\Rightarrow)$ can also be considered as a _lattice with the
covering property_.

Finally we conclude this paragraph with the last lattice-theoretical concept
which, by the way, can always be assumed: _irreducibility_.

######  Definition 7.

Let $(X,\preccurlyeq)$ be an orthocomplemented lattice. Consider two elements
$a,b\in X$, we say that $a$ _commute with_ $b$ if

$a=(a\wedge b)\vee(a\wedge\neg b)$

The set of all the elements of the lattice commuting with any other element of
the lattice is called _center_. A lattice is said _irreducible_ if its center
is just $\\{\mathbf{0},\mathbf{1}\\}$.

We can easily see that two propositions commute if and only if they are
testable at the same time (hence they are associated to two simultaneously
measurable quantities or to the same quantity). Thus the commutativity can be
interpreted as simultaneous testability. This means that O2 implies the loss
of commutativity between propositions, in lattice-theoretical terms, and so it
determines the impossibility to use the usual interpretation of the logical
connectors, as discussed in the beginning. It can be proved that any reducible
(i.e. not irreducible) lattice can be seen as the direct sum (in set-
theoretical sense) of lattices that are irreducible. Hence, even if the set of
propositions about a quantum system is not irreducible, we may always recast
the problem in lattices that are irreducible. For this reason we will always
consider irreducible lattices.
Thus we may conclude the following fact: _the lattice of propositions we can
formulate on a quantum system $Q$, ${\mathcal{L}_{Q}(\Rightarrow)}$, is an
orthomodular, irreducible, atomistic lattice with the covering property whose
ordering relation is represented by the truth value of a proposition_.

##### Remark.

The arguments presented here doesn’t prove rigorously that a quantum system,
and the set of propositions about it, are described by the lattice
$\mathcal{L}_{Q}(\Rightarrow)$ with the properties mentioned above. The aim of
this paragraph is to convince the reader that it is physically well motivated
to assume this structure as starting point, and that the motivations lie at
the heart of all the experimental observations about a quantum system.

###  2.3 From lattice to Hilbert spaces

We have seen that an orthomodular, irreducible, atomistic lattice with the
covering property can be used to model the set of proposition we can formulate
about a quantum system. In this paragraph we will see how it is possible to
map this rather abstract mathematical structure to the usual Hilbert spaces in
which quantum mechanics is typically formulated. Two are the main results that
we need, but before to state them, we need to introduce some technical
definitions.

######  Definition 8.

Let $\mathbb{K}$ be a division ring and consider a vector space $\mathcal{H}$
on it. Then

  1. a)

an _involution_ is a map ${}^{*}:\mathbb{K}\rightarrow\mathbb{K}$ such that
$(a+b)^{*}=a^{*}+b^{*}$, $(ab)^{*}=b^{*}a^{*}$ and $(a^{*})^{*}=a$, $\forall
a,b\in\mathbb{K}$;

  2. b)

an _hermitian form_ is a map
$\langle\cdot,\cdot\rangle:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{K}$
such that

    1. i)

$\langle x,y\rangle=(\langle y,x\rangle)^{*}$, $\forall x,y\in\mathcal{H}$;

    2. ii)

$\langle x,ay+bz\rangle=a\langle x,y\rangle+b\langle x,z\rangle$, $\forall
x,y,z\in\mathcal{H}$ and $\forall a,b\in\mathbb{K}$;

    3. iii)

$\langle x,x\rangle=0$ if and only if $x=0$, $\forall x\in\mathcal{H}$

The couple $(\mathcal{H},\langle\cdot,\cdot\rangle)$ is called _hermitian
inner space_.

As usual, a subset where the vector space operation of $\mathcal{H}$ are
preserved is called _subspace_. As usual, two elements $x,y\in\mathcal{H}$ are
said orthogonal if $\langle x,y\rangle=0$. This allows us to define the
orthogonal complement of a subspace $N\subset\mathcal{H}$, which is the set

$N^{\perp}:=\\{y\in\mathcal{H}\quad|\quad\langle x,y\rangle=0,\forall x\in
N\\}$

Then the following definition holds

######  Definition 9.

Given an hermitian inner space $(\mathcal{H},\langle\cdot,\cdot\rangle)$, if
for any closed subspace $N\in\mathcal{H}$ one can write that
$\mathcal{H}=N\oplus N^{\perp}$, then
$(\mathcal{H},\langle\cdot,\cdot\rangle)$ is said _generalised Hilbert space_
(or _orthomodular space_).

We observe that, the class of Hilbert spaces is a particular class of
generalised Hilbert spaces, in fact in this definition the involution and the
field $\mathbb{K}$ are arbitrary. Now, we are ready to state the first
important theorem, due to Piron, that allows us to recover the Hilbert space
formulation.

######  Theorem 1 (Piron theorem).

Any (complete) irreducible, atomistic orthomodular lattice $(X,\preccurlyeq)$
with the covering property having at least four orthogonal atoms, is
isomorphic to the set of closed subspaces of some generalised Hilbert space
$(\mathcal{H},\langle\cdot,\cdot\rangle)$.

$\mathcal{L}_{Q}(\Rightarrow)$ fulfil all the requirements of this theorem
(the _completeness_ for a lattice was assumed when we declared that the met
and join always exist.) except for the number of atoms which are orthogonal.
Orthogonal atoms, means elementary propositions that are mutually exclusive.
This means that, if we try to evaluate the first proposition, then we know
that the second proposition is not true: this does not seem to be an
unphysical requirement since it seems reasonable that the number of mutually
exclusive propositions is infinite (because in general a physical quantity
assume value over $\mathbb{R}$). So, accepting that we have at least 4
orthogonal atoms, the Piron theorem guarantees that our lattice can be
represented using some generalised Hilbert space $\mathcal{H}$, and in
particular the propositions are in one-to-one correspondence with the closed
subspace of $\mathcal{H}$ (i.e. in one-to-one correspondence with projectors
over these subspaces).

We have not jet reached our goal to motivate the Hilbert space structure of
quantum mechanics with the quantum logic approach. In order tho do so we need
a second theorem, due to Sóler, which is able to select between all the
generalised Hilbert spaces exactly the three class of Hilbert spaces.

######  Theorem 2 (Sóler theorem).

If a generalised Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle)$ over
a field $\mathbb{K}$, admits a sequence of elements
$\\{e_{i}\\}_{i\in\mathbb{N}}$ such that

$\langle e_{i},e_{j}\rangle=\delta_{i,j}\lambda$

for some $\lambda\in\mathbb{K}$, then $\mathbb{K}$ must be the field of reals,
complex or quaternionic numbers and $(\mathcal{H},\langle\cdot,\cdot\rangle)$
is an infinite dimensional Hilbert space over one of these fields.

Thus we only need to find a sequence $\\{e_{i}\\}_{i\in\mathbb{N}}$ of
pairwise orthogonal element of $\mathcal{H}$. The existence of such sequence
can be motivated from the physical assumption of an infinite number of
orthogonal atoms associated to the same physical quantity: atoms are
elementary propositions about this quantity, which are in one-to-one
correspondence with the closed subspace of $\mathcal{H}$. Orthogonality
between atoms translate in orthogonality between subspaces, thus we have an
infinte number of orthogonal subspace on which a single physical quantity
takes different values. This suggests that the dimension of $\mathcal{H}$
should be infinite for a reasonable physical theory. In this way, one can
motivate heuristically the application of the Soler theorem. Nevertheless we
still not select a particular Hilbert space. The Hilbert space over the field
of real numbers can be excluded from considerations about the Galilean
invariance, for a non-relativistic quantum theory. Hilbert spaces over the
field of quaternionic numbers, are still under studies, but they can be seen
as the direct sum of two complex Hilbert spaces. So it seem that the field of
complex number is in some sense special, and it is actually the arena where
standard quantum mechanics is formulated.
Despite we are not able to select uniquely the usual Hilbert space of quantum
mechanics, the quantum logic approach gives a strong argument supporting the
usual formulation of quantum mechanics in complex Hilbert space. One can see
that, as a consequence of a very basic physical consideration (the observation
O2), this structure cannot be avoided.

###  2.4 From Hilbert space to quantum mechanics

In this last paragraph, we briefly explain how it is possible to obtain all
the remaining postulate of quantum mechanics, as logical consequences of the
structure explained before plus some assumptions.

Let us recap the results obtained in the previous paragraph. Assuming that the
propositions about a quantum system fulfil O1 and O2, plus some other
reasonable hypothesis, we are lead to the following conclusion:

_To each quantum system we may associate a complex Hilbert space
$(\mathcal{H},\langle\cdot,\cdot\rangle)$. Propositions about a quantum system
are represented by closed subspaces of $\mathcal{H}$._

Over this Hilbert space, then we may introduce a probability-like measure
which tells us the truth value of each proposition. Such probability-like
measure, by the _Gleason theorem_ (assuming $\dim\mathcal{H}>2$), is uniquely
determined by a positive trace-class operator $\hat{\rho}$. Such operator can
be always normalised and it allows to say that the truth value of a
proposition $A\subset\mathcal{H}$ si given by

$P(A)=\mbox{Tr}\left[\hat{\rho}\hat{P}_{A}\right]$

where $\hat{P}_{A}$ is the projector associated to the closed subspace $A$.
This means that

_To each quantum system we may associate a positive, normalised, trace-class
operator $\hat{\rho}$, which allows us to compute the probability to find a
given proposition true as $\mbox{Tr}\left[\hat{\rho}\hat{P}_{A}\right]$_

Typically, such positive, normalised, trace-class operator is called _state_.
At this point we need to define the notion of observable. Among all the
propositions about a physical system, all the propositions regarding the same
physical quantity should have these features

  1. 1.

they are all simultaneously testable, since are all associated to the same
physical quantity. This implies that all the propositions commute and so in
this case the usual interpretation of logical connectors can be applied;

  2. 2.

call $m(A)$ is the outcome of a measurement of $A$, if $m(A)\in B$ is true and
$m(A)\in B^{\prime}$ is also true, then clearly $m(A)\in B\cap B^{\prime}$ is
true;

  3. 3.

the proposition $m(A)\in\mathbb{R}$ is always true, thus correspond to
$\mathbf{1}$;

  4. 4.

if $m(A)\in B$ and $B$ can be written as $B=\cup_{i\in I}C_{i}$, then the
proposition $m(A)\in B$ is equivalent to the proposition $\vee_{i\in
I}\\{m(A)\in C_{i}\\}$, where $\vee$ is interpreted as OR, since from the
point 1 we know that the usual interpretation of logical connectivities can be
applied.

Defining the set of propositions fulfilling all these physically reasonable
features, one can prove that to each physical quantity it is possible to
associate a projection-valued measure (PVM). Then by the _spectral theorem_ to
each PVM one can associate a self-adjoint operator to each physical quantity.
Moreover, one can always associate to PVM a probability measure which can be
used to compute the expectations (the so called _spectral measure_). More
precisely, one can prove the following

_Physical quantities are represented by self-adjoint operators over
$\mathcal{H}$. For a quantum system with state $\hat{\rho}$, the expectation
value of a physical quantity represented by a self-adjoint operator $\hat{A}$
is given by $\mbox{Tr}\left[\hat{\rho}\hat{{A}}\right]$_

At this point, one may be interesting in select among all the self-adjoint
operators, the ones that represent reasonable physical quantities. This can be
done using group theory. In particular, the symmetry group of a non-
relativistic physical theory is the Galilean group. Hence to obtain
interesting physical quantities, one have to represent this group over this
mathematical structure. In order to do that, one have to specify on what
Hilbert space we want to represent the group. The usual choice is the
following

_The Hilbert space describing a single quantum particle in $\mathbb{R}^{n}$
with $m$ internal degree of freedom is
$\mathcal{H}_{1p}=L_{2}(\mathbb{R}^{n},dx)\otimes\mathbb{C}^{m}$._

When one try to represent the Galilean group over the Hilbert space selected
above demanding that the transition probabilities are preserved (i.e. looking
for unitary representation), one is lead to consider the central extension of
this group. Then, by the _Stone-von Neumann-Mackey theorem_ one can prove that
the position and the momentum operator take its usual form, in addition from
the _Stone theorem_ one can obtain the usual (free) hamiltonian operator. All
these fact are contained in the following request

_The symmetry group of a non-relativistic quantum particle is the (central
extension) of the Galilean group._

Finally one need to explain how to deal with composite systems, namely system
where there are more particles. Requiring that the structure of the single
quantum particle systems is preserved, the measurement on one system does not
disturb the other and that the maximal information content is constant
irrespective to the way in which we gain the information, one is lead to the
notion of tensor product of Hilbert spaces.

_The Hilbert space for a quantum system composed by $N$ quantum particles is
given by $\mathcal{H}_{Np}=\bigotimes_{i=1}^{N}\mathcal{H}_{i,1p}$._

Once we have this method to treat composite system, then it is very reasonable
to think that the only propositions that make sense are the one that does not
depend on the ordering of the subsystems, namely the one that are invariant
under the permutation of the single subsystems. Thus we are lead to

_For a quantum system which consist in $n$ subsystems, the physically
admissible proposition are all the propositions that are invariant under the
action of the permutation group of $\\{1,\cdots,n\\}$._

Till now all the rules sketched to describe a quantum system, all consequence
of the assumption that we have to use an Hilbert space to describe a quantum
system and on that the invariance under a symmetry group is a reasonable
physical requirement. Nevertheless, to conclude this discussion one need to
introduce a last rule: the so called _measurement postulate_.

_If at the time $t$ we find that a particular proposition $A$ holds, the state
right after the measurement is given by_

$\hat{\rho}^{\prime}=\frac{\hat{P}_{A}\hat{\rho}\hat{P}_{A}}{\mbox{Tr}\left[\hat{P}_{A}\hat{\rho}\right]}$

_where $\hat{P}_{A}$ is the projector associated to the proposition $A$._

Different proposal have been done to derive this postulate from the others
like decoherence, dynamical collapses models and quantum bayesianism.

##  3 Modelling a lab: operational reconstruction of QM

In this section we will briefly review the algebraic formulation of quantum
mechanics, derived from operation considerations. The operational approach for
the construction of a physical theory can be summarised as the attempt to
formulate a physical theory defining each abstract mathematical operation as a
procedure that can be executed, at least in principle, in a laboratory. This
approach will lead to a formulation of quantum mechanics that is (almost)
equivalent to the one described in the previous section, but starting from
different assumptions. We will not speak anymore about propositions but we
will focus our attention on physical quantities we use to describe a system.
The main reference for this section are [5] for the operational arguments, [1]
for the mathematical theorems and [6] for the probabilistic notions.

###  3.1 A bit of math: some notions of algebra

In this paragraph we will concentrate mostly of the mathematical notions and
simple results we will use in what follows. The key concept is

######  Definition 10.

An _associative algebra_ $\mathcal{A}$ over a field $\mathbb{K}$, is a
$\mathbb{K}$-vector space equipped with a product
$\star:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ such that

  1. 1.

$a\star(b\star c)=(a\star b)\star c$, $\forall a,b,c\in\mathcal{A}$;

  2. 2.

$a\star(b+c)=a\star b+a\star c$, $\forall a,b,c\in\mathcal{A}$;

  3. 3.

$(a+b)\star c=a\star c+b\star c$, $\forall a,b,c\in\mathcal{A}$;

  4. 4.

$\alpha(a\star b)=(\alpha a)\star b=a\star(\alpha b)$,
$\forall\alpha\in\mathbb{K}$ and $a,b\in\mathcal{A}$.

Briefly, an algebra is a vector space equipped with a product operation which
is associative and distributive with respect the vector space operations.
Algebras can be classified according with the following definitions

######  Definition 11.

Given an associative algebra $\mathcal{A}$, then

  1. 1.

is said _normed algebra_ , if equipped with a norm $\|\cdot\|$ such that
$\|a\star b\|\leqslant\|a\|\|b\|$, $\forall a,b\in\mathcal{A}$;

  2. 2.

is said _banach algebra_ , if normed and at the same time $\mathcal{A}$ is a
Banach space under the norm;

  3. 3.

is said _∗ \- algebra_, if equipped with an involution ${}^{*}:a\mapsto
a^{*}$, $\forall a\in A$;

  4. 4.

is said _$C^{*}$ -algebra_, if banach and $\|a^{*}\star a\|=\|a\|^{2}$ (said
_$C^{*}$ -property_);

  5. 5.

is said _algebra with unit_ , if there exist an element
$\mathsf{I}\in\mathcal{A}$ such that $a=a\star\mathsf{I}=\mathsf{I}\star a$;

  6. 6.

is said _abelian (or commutative) algebra_ , if $[a,b]=a\star b-b\star a=0$,
$\forall a,b\in\mathcal{A}$.

The symbol $\star$ for the algebraic product will be omitted, if there are no
ambiguities with the usual product, and we also set $\mathbb{K}=\mathbb{C}$.
Another important algebra, which is not in general associative, is the _Jordan
algebra_

######  Definition 12.

A _jordan algebra_ $\mathcal{A}$ is a vector space, equipped with a bilinear
form $\circ:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ such that
$a\circ b=b\circ a$ and $a\circ(b\circ(a\circ a))=(a\circ b)\circ(a\circ a)$
(_jordan identity_).

In what follow we will consider an associative algebra, in particular a
∗-algebra. Its elements may be classified in similar manner of what is
typically done for operators: $a$ is said _normal_ if $aa^{*}=a^{*}a$, and
_self-adjoint_ if $a=a^{*}$. Again, following the similarity with operators
(which in may cases form an algebra) one can define also a notion of
_spectrum_ in algebraic contest.

######  Definition 13.

Given a Banach algebra with unit, $\mathcal{A}$, and consider
$a\in\mathcal{A}$. The _spectrum_ of $a\in\mathcal{A}$ is the set defined as

$\sigma(a):=\\{\xi\in\mathbb{C}|\nexists(a-\xi\mathsf{I})^{-1}\in\mathcal{A}\\}$

The following result about the spectrum of a $C^{*}$-algebra is interesting
for our discussion

######  Proposition 1.

Let $a\in\mathcal{A}$ be a self-adjoint element of a $C^{*}$-algebra with
unit, the $\sigma(a)\subset[-\|a\|,\|a\|]\subset\mathbb{R}$.

In addition, the notion of spectrum allows us to introduce a further
classification between the elements of a Banach algebra with unit:

######  Definition 14.

Given a Banach algebra with unit $\mathcal{A}$, an element $a\in\mathcal{A}$
is said _positive_ if self-adjoint and its spectrum is positive
$\sigma(a)\subset\mathbb{R}^{+}$. The set of all the positive elements will be
denoted by $\mathcal{A}^{+}$.

Positivity of the elements allows us to define the positivity of linear
functionals over the algebra. Such functionals play a special role in the
algebraic formulation of quantum mechanics since they are interpreted as
_states_.

######  Definition 15.

Let $\mathcal{A}$ be a Banach algebra with unit, a functional
$\omega:\mathcal{A}\rightarrow\mathbb{C}$ such that

  1. 1.

it is _positive_ , namely $\omega(a)\geqslant 0$, $\forall
a\in\mathcal{A}^{+}$;

  2. 2.

is _normalised_ , namely $\omega(\mathsf{I})=1$;

is called _state_.

It can be proved, that over a $C^{*}$-algebra, a linear functional is positive
if and only if it is bounded (in particular $\|\omega\|=\omega(\mathsf{I})$).
This implies that over a $C^{*}$-algebra, a state is a continuous (since
bounded) functional. Finally the following result about states is interesting
for our discussion

######  Proposition 2.

Let $\mathcal{A}$ be a $C^{*}$-algebra, take $a\in\mathcal{A}$ and consider
$\alpha\in\sigma(a)$. Then there exist a unique state
$\omega_{\alpha}:\mathcal{A}\rightarrow\mathbb{C}$, such that
$\omega_{\alpha}(a)=\alpha$.

###  3.2 Operational approach for a physical theory: description of a single
physical quantity

Any experimental science is based on the reproducibility of experiments. One
prepare the system in a certain _configuration_ and then perform a series of
measurements on some _physical quantity_ from which one can prove or disprove
a fact. Physics was the first science where this procedure was applied. The
theoretical models we use to describe a physical system should be based, as
much as possible, on the way one have access to the information that we learn
in a measurement. This is the heart of the operational approach for the
construction of a physical theory.

Let $\mathcal{C}$ be the set of all the possible configurations in which a
system can be prepared, and $\mathcal{A}$ be the set of all the physical
quantities of the system we can measure. Consider a physical quantity
$a\in\mathcal{A}$, the outcome of a measurement for a system prepared in the
configuration $\omega\in\mathcal{C}$, will be labeled by $m_{\omega}(a)$, and
can be operationally defined as the value that the pointer of the measuring
device assume when we measure the quantity $a$. The _result of a measurement_
of the physical quantity $a\in\mathcal{A}$ for a system prepared in the
configuration $\omega\in\mathcal{C}$, labeled $\omega(a)$, is defined to be
the average of all the measured value $m_{\omega}(a)$, repeating the
experiment (ideally) an infinite number of times, namely

$\omega(a):=\lim_{n\rightarrow+\infty}\frac{1}{n}\sum_{i=1}^{n}m_{\omega}^{(i)}(a)$

Two configurations $\omega_{1},\omega_{2}\in\mathcal{C}$ are _operationally
indistinguishable_ if the result of the two measurements is the same for all
the physical quantities. Mathematically speaking, on $\mathcal{C}$ we can say
that $\omega_{1}=\omega_{2}$ if and only if $\omega_{1}(a)=\omega_{2}(a)$,
$\forall a\in\mathcal{A}$. Similarly, two physical quantities are
operationally indistinguishable if preparing the system in all the possible
configurations, the results of the measurements are always identical. Hence on
$\mathcal{A}$ we can say that $a_{1}=a_{2}$ if and only if
$\omega(a_{1})=\omega(a_{2})$, $\forall\omega\in\mathcal{C}$. Using these last
equivalence relation, we can define operationally, the usual mathematical
operations over $\mathcal{C}$ and on $\mathcal{A}$. For example, consider
$a\in\mathcal{A}$ and $\lambda\in\mathbb{R}$, one can define the physical
quantity $\lambda a$ as the physical quantity measured by a measuring device
whose pointer scale is dilated by a factor $\lambda$ with respect to the
original scale. More formally, one can write $m_{\omega}(\lambda a):=\lambda
m_{\omega}(a)$ $\forall\omega\in\mathcal{C}$, which implies that
$\omega(\lambda a)=\lambda\omega(a)$. In similar way, given
$a,b\in\mathcal{A}$, one can operationally define the sum of two physical
quantities $a+b$ simply setting $m_{\omega}(a+b):=m_{\omega}(a)+m_{\omega}(b)$
$\forall\omega\in\mathcal{C}$, and so $\omega(a+b)=\omega(a)+\omega(b)$. We
can see that, by definition, $\omega$ is _linear_.

One can operationally define also $a^{n}$, $n\in\mathbb{N}$ setting
$m_{\omega}(a^{n}):=[m_{\omega}(a)]^{n}$ $\forall\omega\in\mathcal{C}$.
Because of this definition, one can see that $m_{\omega}(a^{0})=1$ for any
$\omega$. Thus we can define an _identity_ $\mathsf{I}:=a^{0}$. We can also
see that it is quite reasonable to assume $a^{n+m}=a^{n}a^{m}$, since
$m_{\omega}(a^{n+m})=m_{\omega}(a)^{n}m_{\omega}(a)^{m}$. Finally, it is not
difficult to see that even for $\lambda\in\mathbb{C}$, $\lambda a$ is well
defined if we measure separately the real and imaginary part of it. Hence, all
the complex polynomials of $a\in\mathcal{A}$, like
$\alpha_{n}a^{n}+\alpha_{n-1}a^{n-1}+\cdots+\alpha_{0}\mathsf{I}$, are well
defined from the operational point of view. Let $\mathcal{A}_{a}$ denote the
set of all the possible complex polynomials of $a$. Clearly
$\mathsf{I}\in\mathcal{A}_{a}$ and on $\mathcal{A}_{a}$ one can naturally
define an _involution_ setting $(\lambda a)^{*}:=\overline{\lambda}a$, with
$\overline{\lambda}$ complex conjugate of $\lambda$, and
$(ab)^{*}=b^{*}a^{*}$. Thus we can conclude that _$\mathcal{A}_{a}$ is an
abelian ∗-algebra with unit_. We cannot say the same thing on $\mathcal{A}$,
because we may have troubles in the definition of $m_{\omega}(ab)$ and
$m_{\omega}(ba)$, because of the property O2: for this reason we concentrate
on $\mathcal{A}_{a}$ only. From the discussion done till now, we may conclude
that the following things holds for any $b,c\in\mathcal{A}_{a}$ (hence
polynomial of $a$)

  1. 1.

$m_{\omega}(\alpha b+c)=\alpha m_{\omega}(b)+m_{\omega}(c)$;

  2. 2.

$m_{\omega}(bc)=m_{\omega}(b)m_{\omega}(c)$;

  3. 3.

$m_{\omega}(\mathsf{I})=1$;

  4. 4.

$m_{\omega}(b^{*})=\overline{[m_{\omega}(b)]}$;

  5. 5.

$m_{\omega}(b^{*}b)\geqslant 0$.

We observe that the point 2, since we are dealing with polynomial of $a$ only,
is a consequence of 1 and the properties of powers for the elements of
$\mathcal{A}_{a}$ and does violate O2. These have consequences on the
configurations

  1. 1.

$\omega(\alpha a+b)=\alpha\omega(a)+\omega(b)$;

  2. 2.

$\omega(\mathsf{I})=1$;

  3. 3.

$\omega(a^{*})=\overline{[\omega(a)]}$;

  4. 4.

$\omega(a^{*}a)\geqslant 0$

This means that a configuration $\omega:\mathcal{A}_{a}\rightarrow\mathbb{C}$
is a positive, normalised linear functional over $\mathcal{A}_{a}$: for this
reason $\omega$ is a state on $\mathcal{A}_{a}$. At this point it is useful to
introduce the following map for any $a\in\mathcal{A}_{a}$

$\|\cdot\|:\mathcal{A}_{a}\rightarrow\mathbb{R}\qquad\qquad\|a\|:=\sup_{\omega\in\mathcal{C}}|\omega(a)|$

This number is the maximum value that a physical quantity may assume. Because
any real instrument have a finite scale, $\|a\|$ is a _finite_ , positive real
number. Without proving, we state the following proposition containing all the
key properties of $\|\cdot\|$.

######  Proposition 3.

Take $a\in\mathcal{A}$, $\lambda\in\mathbb{C}$ and $b,c\in\mathcal{A}_{a}$.
The map $\|\cdot\|$ on $\mathcal{A}_{a}$ defined as above fulfils

  1. 1.

$\|b+c\|\leqslant\|b\|+\|c\|$ and $\|\lambda b\|=|\lambda|\|b\|$;

  2. 2.

$\|b^{*}\|=\|b\|$;

  3. 3.

$\|bb^{*}\|\leqslant\|b\|^{2}$ with equality if and only if $b=b^{*}$;

  4. 4.

$\|bc\|\leqslant\|b\|\|c\|$ when $b=b^{*}$ and $c=c^{*}$.

Now, define following symmetric product on $\mathcal{A}_{a}$

$b\circ c:=\frac{bc+cb}{2}\qquad\qquad b,c\in\mathcal{A}_{a}$

It is not difficult to see that it is commutative, in addition also the Jordan
identity is fulfilled. Thus $\mathcal{A}_{a}$ equipped with this product is a
_Jordan algebra_. Now, because $|\omega(bc)|\leqslant|\omega(b)||\omega(c)|$
on $\mathcal{A}_{a}$, one can conclude that $\|b\circ c\|\leqslant\|b\|\|c\|$
(and so $\|\cdot\|$ defines a norm), and so $\mathcal{A}_{a}$ is also normed,
and after completion, $\mathcal{A}_{a}$ can be considered as a _abelian
Jordan-Banach algebra_ (with unit). From the definition given for the $\circ$
product, one can conclude that $\mathcal{A}_{a}$ is is a sub-algebra of a
larger abelian $C^{*}$-algebra with unit[5]. Thus, using operational
arguments, we arrived to the following conclusion: each physical quantity can
be described using an abelian $C^{*}$-algebra with unit. In the next
paragraphs, we will see how it is possible to describe algebraically the whole
physical system at the same time and not single physical quantities.

###  3.3 Another bit of math: represent an algebra

In the previous paragraph, we have seen that if when we want to describe a
physical quantity, it is reasonable to use a $C^{*}$-algebra. Nevertheless, to
explicitly perform any calculation, it should be useful to have more simple
(less abstract) mathematical objects. In this section, we will briefly review
how to represent an abstract $C^{*}$-algebra using concrete mathematical
objects, like functions or operators.

The first result in this sense, for the particular case of a commutative
$C^{*}$algebra, is the so called _commutative Gelfand-Naimark theorem_. To
formulate it we need the following definitions

######  Definition 16.

Let $\mathcal{A}$ be a commutative $C^{*}$-algebra with unit. A _character of
$\mathcal{A}$_ is a non-zero homomorphism between $A$ and $\mathbb{C}$, namely
a map $\phi:\mathcal{A}\rightarrow\mathbb{C}$ such that
$\phi(ab)=\phi(a)\phi(b)$, $a,b\in\mathcal{A}$. The set of all characters of
$\mathcal{A}$ is called _structure space_ (or _spectrum of the algebra_), and
is labeled by $\Delta(\mathcal{A})$.

Now, define a maps $\tilde{a}:\Delta(\mathcal{A})\rightarrow\mathbb{C}$ as
$\tilde{a}(\phi):=\phi(a)$, for $a\in\mathcal{A}$. This map is the so called
_Gelfand’s transform_ and can be proved that

$\sigma(a)=\\{\tilde{a}(\phi)\mspace{5.0mu}|\mspace{5.0mu}\phi\in\Delta(\mathcal{A})\\}$

which is the link between the spectrum of an element of the algebra
$\sigma(a)$ and the structure space $\Delta(\mathcal{A})$. At this point we
can state the aforementioned theorem

######  Theorem 3 (commutative Gelfand-Naimark theorem).

Let $\mathcal{A}$ be a commutative $C^{*}$-algebra with unit, and consider the
algebra of continuous functions of the structure space
$C(\Delta(\mathcal{A}))$ ($C^{*}$-algebra with respect to the norm
$\|\cdot\|_{\infty}$). Then the Gelfand’s transform is an isomorphism
preserving the involution and the norm (isometric ∗-isomorphism) between
$\mathcal{A}$ and $C(\Delta(\mathcal{A}))$

Thus this theorem guarantee that we can always ‟represent ˝a commutative
$C^{*}$-algebra with an algebra of continuous functions. As we will see this
is no longer true if we drop the commutativity of the algebra: in this case we
have to represent the algebra in a different manner.

Let us now consider the general case of a $C^{*}$-algebra, without assume
commutativity. It turns out that Hilbert spaces are the suitable concrete
mathematical object on which we can represent this algebra. This motivate the
following

######  Definition 17.

Consider a $C^{*}$-algebra with unit $\mathcal{A}$ and an Hilbert space
$\mathcal{H}$. An homomorphism, preserving the involution and unit,
$\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$, is called
_representation of $\mathcal{A}$ on $\mathcal{H}$_. The representation $\pi$
is said _faithful_ if it is one-to-one. Finally, a vector $\psi\in\mathcal{H}$
is said _cyclic_ for $\pi$, if
$\overline{\\{\pi(a)\psi\mspace{5.0mu}|\mspace{5.0mu}a\in\mathcal{A}\\}}=\mathcal{H}$.

At this point, take the $C^{*}$-algebra with unit $\mathcal{A}$, and a state
$\omega:\mathcal{A}\rightarrow\mathbb{C}$ on it. It is not difficult to see
that $\langle a,b\rangle_{\omega}:=\omega(a^{*}b)$ define pre-inner product
(in general it is degenerate). In order to have a well defined inner product,
one can define the set

$N_{\omega}:=\\{a\in\mathcal{A}\mspace{5.0mu}|\mspace{5.0mu}\omega(a^{*}b)=0,\forall
b\in\mathcal{A}\\}$

Then on $\mathcal{A}/N_{\omega}$, the product
$\langle\cdot,\cdot\rangle_{\omega}$ is a well defined inner product. Thus,
once can complete $\mathcal{A}/N_{\omega}$ to an Hilbert space, and we will
label such Hilbert space by $\mathcal{H}_{\omega}$. One may define for each
$a\in\mathcal{A}$

$[\pi_{\omega}(a)]([b]):=[a][b]\qquad\qquad\forall[b]\in\mathcal{H}_{\omega}$

This is by construction an homomorphism preserving the involution. Computing
the norm, one can prove that $\pi_{\omega}(a)$ is bounded, thus $\pi_{\omega}$
is a representation of $\mathcal{A}$ on $\mathcal{H}_{\omega}$. Finally, we
can understand the the unit of the algebra, defines a special vector in
$\mathcal{H}_{\omega}$, which is $\Psi_{\omega}=[\mathsf{I}]$. Such vector is
cyclic for $\pi_{\omega}$. Thus we can see that, given a $C^{*}$-algebra
$\mathcal{A}$ and a state $\omega$, we can always construct a triple
$(\mathcal{H}_{\omega},\pi_{\omega},\Psi_{\omega})$, called _GNS triple_. The
discussion done above, sketched a part of the proof of the so called _Gelfand-
Naimark-Segal theorem_.

######  Theorem 4 (GNS theorem).

Let $\mathcal{A}$ be a $C^{*}$-algebra with unit and
$\omega:\mathcal{A}\rightarrow\mathbb{C}$ a state. Then

  1. i)

there exist a triple $(\mathcal{H}_{\omega},\pi_{\omega},\Psi_{\omega})$ where
$\mathcal{H}_{\omega}$ is an Hilbert space,
$\pi_{\omega}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H}_{\omega})$ is a
∗-representation of $\mathcal{A}$ on the $C^{*}$-algebra of bounded operators
on $\mathcal{H}_{\omega}$, and $\Psi_{\omega}\in\mathcal{H}_{\omega}$ is a
vector, such that

    1. a)

$\Psi_{\omega}$ is unit vector, cyclic for $\pi_{\omega}$.

    2. b)

$\langle\Psi_{\omega}|\pi_{\omega}(a)\Psi_{\omega}\rangle=\omega(a)$ for any
$a\in\mathcal{A}$.

  2. ii)

If $(\mathcal{H},\pi,\Psi)$ is a triple such that

    1. a)

$\mathcal{H}$ is an Hilbert space,
$\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$ is a ∗-representation and
$\Psi\in\mathcal{H}$ is a unit vector cyclic for $\pi$;

    2. b)

$\omega(a)=\langle\Psi|\pi(a)\Psi\rangle$;

then there exit a unitary operator
$\hat{U}:\mathcal{H}\rightarrow\mathcal{H}_{\omega}$ such that
$\Psi=\hat{U}\Psi_{\omega}$ and $\pi_{\omega}(a)=\hat{U}\pi(a)\hat{U}^{-1}$
for any $a\in\mathcal{A}$.

Thus we have found a way to ‟represent˝a generic $C^{*}$-algebra over an
Hilbert space. Nevertheless, such representation depend on the state, hence
changing the state we change also the operators. In addition in general is not
one-to-one: we may find element in $\mathcal{B}(\mathcal{H}_{\omega})$ which
doesn’t represent any element of $\mathcal{A}$. The way out of this problem is
the _Gelfand-Naimark theorem_

######  Theorem 5.

Any $C^{*}$-algebra with unit $\mathcal{A}$ is isomorphic (there exist an
isomorphism preserving the involution) to a sub-algebra of
$\mathcal{B}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$.

In particular, one can prove that the Hilbert space considered in this theorem
is $\mathcal{H}=\bigoplus_{\omega\in\mathcal{C}}\mathcal{H}_{\omega}$ and the
isomorphism is ${\Pi(a):=\bigoplus_{\omega\in\mathcal{C}}\pi_{\omega}(a)}$.

Summarising we have that, any $C^{*}$-algebra with unit may be represented
over an Hilbert space using suitable bounded operators, if the algebra is also
abelian we can always find an algebra of function isomorphic to the original
algebra.

###  3.4 Operational approach for a physical theory: description of a physical
system

We have seen that, when we try to describe a single physical quantity, we are
naturally lead to introduce an abelian $C^{*}$-algebra with unit.
Commutativity, allows us to say that we can represent the elements of an
algebra using functions, while the _Riesz representation theorem_ allows us to
understand how to represent algebraic state over this space of function

######  Theorem 6 (Riesz representation theorem).

Let $\mathsf{X}$ be a locally compact Hausdorf space and
$\omega:C(\mathsf{X})\rightarrow\mathbb{C}$ a continuous functional. Then
there exist a unique Borel measure $\mu_{\omega}$ such that

$\omega(f)=\int_{\mathsf{X}}f(x)\mu_{\omega}(dx)$

This theorem allows us to conclude that, given the couple
$(\mathcal{A},\omega)$, where $\mathcal{A}$ is a commutative $C^{*}$-algebra
and $\omega$ is a state, then there exist a probability space
$(\mathsf{X},\mathscr{B}(\mathsf{X}),\mu_{\omega})$, and the elements of the
algebra are continuous function on $\mathsf{X}$ (which means that they are
random variables). Thus, using this different approach, one obtain that each
physical quantity is always described by a random variable over a probability
space. This is part of the content of the observation O1 about a quantum
system: we can see that it is a general feature of any physical system, since
no quantum assumption was done in its derivation.

From the discussion done above, one can see that $(\mathcal{A},\omega)$, and
the random variables over $(\mathsf{X},\mathscr{B}(\mathsf{X}),\mu_{\omega})$
describe the same thing: thus one can choose if deal with probability using
measure-theoretic notions or algebraic notions. More formally [6], the couple
$(\mathcal{A},\omega)$ where $\mathcal{A}$ is a ∗-algebra and $\omega$ is a
positive normalised functional on $\mathcal{A}$, is called _algebraic
probability space_. If the algebra is $C^{*}$, $(\mathcal{A},\omega)$ is said
_$C^{*}$ -probability space_. Finally, it is possible to prove that any
commutative algebraic probability space is equivalent, up to zero-measure
sets, to the usual measure-theoretic probability space [7].

In general a physical system is described by more than one physical quantity.
The mathematical object that we need to describe the whole systems should
contains all the abelian $C^{*}$-algebra of all the physical quantities. Thus,
in order to be conservative as much is possible, we may associate to each
physical system, a $C^{*}$-algebra $\mathcal{A}$ defined to be the smallest
$C^{*}$-algebra which contains all the abelian $C^{*}$-algebra with unit
associated to the single physical quantity $\mathcal{A}_{a}$ (clearly
$\mathcal{A}$ will have unit). In addition we may extend by continuity all the
states $\omega:\mathcal{A}_{a}\rightarrow\mathbb{C}$, on $\mathcal{A}$. Thus
we can summarise as follow: _to each physical system we may associate a
$C^{*}$-probability space $(\mathcal{A},\omega)$_. This in this way, one can
motivate operationally , the observation O1, seen in the introduction. As we
will see this $C^{*}$-probability space is not abelian in general. Given a
$C^{*}$-algebra $\mathcal{A}$, for each $a\in\mathcal{A}$ and
$\alpha\in\sigma(a)$, we may find a state $\omega_{\alpha}(a)=\alpha$. Thus we
can see that the spectrum of an element of the algebra can be interpreted as
the set of all the possible values that the physical quantity may assume
(which is not the set of all the possible outcome of an experiment). Because
the measurable quantities are always expressed using real numbers, we have to
conclude that the spectrum is a subset of $\mathbb{R}$. This happens if and
only if the physical quantities are represented by self-adjoint elements of
$\mathcal{A}$. Thus _physical quantities are the self-adjoint elements of the
algebra_.

At this point we want to analyse more in detail the structure of the
$C^{*}$-algebra $\mathcal{A}$. In particular we want to discuss under which
condition it is abelian. The main tool we will use is the notion of _entropy
of a physical quantity_. Let $a\in\mathcal{A}$ be a physical quantity and
$\omega$ a state on it. By the GNS theorem, we may always find a triple
$(\mathcal{H}_{\omega},\pi_{\omega},\Psi_{\omega})$, and so the physical
quantity $a\in\mathcal{A}$, can be represented on $\mathcal{H}_{\omega}$ as
$\hat{\pi}_{\omega}(a)$. Using the spectral measure of this operator, one can
compute the probability to observe the value of $a$ in the set $A$ as

$P_{\omega}(a\in
A)=\int_{\sigma(a)}\chi_{A}(x)\mu^{(\hat{\pi}_{\omega}(a))}_{\omega}(dx)=\int_{[-\|a\|,\|a\|]}\chi_{A}(x)\mu^{(\hat{\pi}_{\omega}(a))}_{\omega}(dx)$

since $\sigma(a)\subset[-\|a\|,\|a\|]$ and the spectral measure has support on
$\sigma(a)$ only. Now, if we partition the spectrum using sets of diameter
$\epsilon$, namely $\cup_{i\in I}U_{i}^{(\epsilon)}=[-\|a\|,\|a\|]$, for a
given $\epsilon$ we obtain a set of probabilities
$\\{p_{\epsilon}(i;\omega)\\}_{i\in I}$, where
$p_{\epsilon}(i;\omega):=P_{\omega}(a\in U_{i}^{\epsilon})$, which describe
the probabilistic behaviour of the physical quantity in the state $\omega$.
The diameter $\epsilon$ of the above sets, can be thought as the width of the
bins of the measuring device of the given physical quantity. Then at this
point one can define consistently the _$\epsilon$ -entropy of $a$ in the state
$\omega$_ as

$H_{\omega}^{(\epsilon)}(a):=-\sum_{i\in I}p_{\epsilon}(i;\omega)\log
p_{\epsilon}(i;\omega)$

This is the entropy of the physical quantity $a\in\mathcal{A}$ for a system
prepared in the state (configuration) $\omega$, when is measured with an
measurement apparatus with resolution $\epsilon$. Given a second physical
quantity $b\in\mathcal{A}$, in the same way, one can define
$H_{\omega}^{(\delta)}(b)$. Then, excluding the trivial partitions, the
following theorem holds

######  Theorem 7.

Let $\mathcal{A}$ be a $C^{*}$-algebra with unit and consider
$a,b\in\mathcal{A}$. If for any state $\omega$, and any $\epsilon,\delta>0$

$H_{\omega}^{(\epsilon)}(a)+H_{\omega}^{(\delta)}(b)\geqslant
D(\epsilon,\delta)$

with $D(\epsilon,\delta)$ positive and fixed for any $\epsilon$ and $\delta$
positive, then $[a,b]\neq 0$.

Also the converse holds [8], thus the above relation is another way to demand
for non-commutativity between $a$ and $b$. Operationally, the meaning is
simple: even if we know $a$ with certainty ($H_{\omega}^{(\epsilon)}(a)=0$)
then we can’t know $b$ with arbitrary precision, and viceversa. This the
operational analogous of the observation O2. The above relations involve only
probability, and so it can be considered as an operational criterium for
establish if two physical quantities are represented by commuting element of
$\mathcal{A}$ or not. Hence, as claimed above, in general $\mathcal{A}$ may
not be commutative. A final observation: the partition of the spectrum is
necessary if we admit the possibility to have observable with $\sigma(a)$
which is not discrete (algebraically this cannot be established in advance,
because the usual classification of the spectrum depend on the state),
necessary to have a well defined notion of (Shannon) entropy. If one deal with
physical quantity having $\sigma(a)$ discrete (for instance by construction),
the theorem can be stated using only the usual Shannon’s entropy without
consider any partition of $\sigma(a)$. Relations like the one described above,
are called _entropic uncertainty relations_. Hence we can see that, _if we
find an entropic uncertainty relation, then the algebra cannot be abelian_. In
this case, such algebra can be represented only using a subset of bounded
operators over an Hilbert space $\mathcal{H}$, and not with functions.

###  3.5 Algebraic formulation of quantum mechanics

Now, we are able to reconstruct quantum mechanics from the operational point
of view. Below we will list the set of axioms needed to reconstruct quantum
mechanics, justified by the discussion done till now.

_Each physical system is described by an $C^{*}$-probability space
$(\mathcal{A},\omega)$. The physical quantities are the self-adjoint elements
of $\mathcal{A}$ and the possible way one can prepare the system are
represented by states $\omega$._

This is a general feature of a physical system and no quantum assumption are
done till this point. The probabilistic interpretation, allows us to conclude
that

_For a composite system, the algebraic probability space is constructed using
the $C^{*}$-tensor product of the algebras associated to each subsystem._

The $C^{*}$-tensor product, is the only possible tensor product between
algebras which preserves the $C^{*}$-property. This rules is just the
generalisation in $C^{*}$-algebraic contest of what is usually done in
measure-theoretic probability spaces: in fact, for commutative
$C^{*}$-algebra, it reduces exactly to the cartesian product of the sample
spaces. One need to specify if the algebra is abelian or not.

_For a quantum system, some entropic uncertainty relation holds, which means
that $\mathcal{A}$ is not abelian._

This requirement force us to remain over an Hilbert space, when we represent
the $C^{*}$-algebra. Positive, normalised linear functional are states and the
GNS theorem show that the expectation is computed as in the previous section.
Representing the algebra $\mathcal{A}$ over an Hilbert space, one is forced to
consider physical quantities only some class bounded operators. Thus, we have
no room for the position and momentum operators, because they are unbounded.
Nevertheless there is a way out: one can always consider succession of bounded
operators converging to the unbounded position and momentum operators. The
commutation relations between them can be obtained by Weyl relations defining
a _Weyl $C^{*}$-algebra_. Time evolution can be obtained using the additive
group $(\mathbb{R},+)$, which is the same subgroup of the Galilean group
generating the evolution in the previous section, to describe time
translations.

_Time evolution is described by the group of time translations
$(\mathbb{R},+)$._

On the GNS Hilbert space, such group is represented via Stone theorem,
obtaining the usual Schrodinger evolution. Finally one need to introduce the
measurement postulate. Using algebraic probabilistic considerations, the
measurement postulate can be obtained using conditional expectation (this is
the quantum bayesanisim solution of the measurement problem), nevertheless
this solution is not universally accepted. Thus one should also need

_If the observed value of $a\in\mathcal{A}$ in the state $\omega$ is in the
Borel set $F$, then the state after the measurement $\omega_{F}$ is_

$\omega_{F}:b\in\mathcal{A}\mapsto\frac{\langle\Psi_{\omega}|\hat{P}_{F}\pi_{\omega}(b)\hat{P}_{F}\Psi_{\omega}\rangle}{\langle\hat{P}_{F}\Psi_{\omega}|\hat{P}_{F}\Psi_{\omega}\rangle}$

##  4 What is the meaning of an Hilbert space? A toy-model proposal

In the two previous sections, we have seen in a rather informal way, how the
quantum mechanics can be formulated starting from basic principles. In both
cases, two are the basic assumptions: one about the intrinsic probabilistic
nature of a physical system (a very general feature), and one about the
limitations in this description due to the presence of quantities that cannot
be known at the same time. We have also seen that is this feature that render
unavoidable the Hilbert space formulation. In the quantum logic approach, is
this feature (O2) which for us to abandon the usual (boolean) logic, and so
the possibility to use the ordinary probability theory. In the operational-
algebraic approach, are the entropic uncertainty relations that make the
algebra not abelian, and so we cannot map it completely into an ordinary
probability space.

Quantum mechanics is usually formulated using the Hilbert space language, and
this give rise to all the famous problems in the interpretation (like state-
superposition or entanglement). We cannot abandon such formulation, because of
non-commutativity, so the proposal of this section, is to treat non-
commutativity as the reason because one is forced to use Hilbert space, and
doesn’t consider it as a consequence, as it is typically done. Once one accept
this view, one can recognise that the entropic uncertainty relations and
algebraic probability spaces are a very useful tools to derive the limitations
in the simultaneous description of physical quantities from the intrinsic
probabilistic nature of a physical system. In the following paragraph we will
describe a simple toy-model, where the origin of the uncertainty is clear,
which is able to reproduce the non-commutativity at algebraic level, of the
analogous position and momentum for a particle. This model is very simple, and
we do not claim that it reproduces completely the quantum analogous of it,
nevertheless we think that it suggests an interesting motivation on the reason
because nature at fundamental level, should be described using Hilbert spaces.

###  4.1 The toy model: random jumps over a random space

In the model proposed here, there are two main actors: the space and the
particle. The space is assumed discrete and finite: it can be thought as a
random distribution of, say $N$, points over the real line (we will consider
the 1-dimensional case for simplicity). In addition the space is not assumed
static but stochastic: the initial distribution of space points is assumed to
bee know and it evolve in time as if each space point is a discrete-time
random walk. The particle is assumed to be a point-like object whose dynamics
can be thought as a succession of jumps from one space point to another. Now,
we formalise this ideas from the mathematical point of view,

Let us start briefly reviewing the mathematical formulation of the basic
stochastic process describing the space: _the random walk on a line_. Let
$\\{X_{i}\\}$ be a collection of random variables such that $P[X_{i}=+l]=p$
and $P[X_{i}=-l]=1-p=:q$, where $l$ is a fixed number. Such processes are
called _Bernulli random variables_ and typical example of it is the coin
tossing. We may describe the random walk in the following way. Suppose we have
a person in the initial position $x_{0}$ at the initial (discrete) time $0$.
At each instant of time this person tosses a coin: if he gets head he will
move on the left of $l$, otherwise he will move on the right of the same
amount. Thus the position of the person at the time $1$ will be
$S_{1}=x_{0}+l$ if he get head, $S_{1}=x_{0}-l$ otherwise. Repeating this
procedure for each instant of time we can say that the movement of this person
is a random walk. Hence we can say that the random walk starting at the point
$x_{0}$, $S_{N}$, is defined as

$S_{N}:=x_{0}+\sum_{i=0}^{N}X_{i}$

Central quantity we want to compute is the probability to have $S_{N}=d$, for
some $d$ integer multiple of $l$, at the time $N$ starting from the point
$x_{0}$. It can be proved that

$P[S_{N}-x_{0}=d]=\begin{cases}\frac{N!}{\left[\frac{1}{2}\left(\frac{d}{l}+N\right)\right]!\left[\frac{1}{2}\left(N-\frac{d}{l}\right)\right]!}p^{\left(\frac{d}{l}+N\right)}q^{\left(\frac{d}{l}+N\right)}&\mbox{
if }d\in[-Nl,+Nl]\\\ 0&\mbox{ otherwise}\end{cases}$

Suppose that the initial position $x_{0}$ is a random variable with
distribution $\pi(x_{0})$ over the real line (possibly discrete), then we can
write that

$P[S_{N}=d]=\int_{\mathbb{R}}dx_{0}P[S_{N}-x_{0}=d]\pi(x_{0})$

Consider now a collection of random walks on the line,
$\\{S_{N}^{(i)}\\}_{i\in I}$, starting in different points $x_{0}^{(i)}$,
where $I$ is a finite set. At each instant of time, this collection of random
walk will select at most $|I|$ points over the real line (two or more random
walks may overlaps because they are assumed independent: the presence of a
random walk in a give point doesn’t influence the probability of another
random walk to be found in the same point). Such collection of points,
randomly distributed over the real line, is our model of space.

The particle in this model can be described essentially by two quantities: the
first is the position of the particle, $X_{N}$, the second its velocity,
$V_{N}$, suitably defined over a space of this kind. Let us consider the
position first. Let $\Pi_{N}$ be the finite set of points selected by the
collection of random walks $\\{S_{N}^{(i)}\\}_{i\in I}$ at the time $N$.
Clearly $\Pi_{N}\subset\mathbb{R}$. The random variable describing the
position of a point-like particle at time $N$ will be a map
$X_{N}:\Omega_{X_{N}}\rightarrow\Pi_{N}$. At this level seems problematic
define the possible outcome of $X_{N}$ because the set $\Pi_{N}$ change at
each instant of time. To avoid this problem, we may think that the random
variable $X_{N}$ will select a single random walk in the collection
$\\{S_{N}^{(i)}\\}_{i\in I}$. Select means that at the time step $N$, we have
that $X_{N}=S_{N}^{(i)}$, in probability. Hence, in general we may relate the
probability to observe the particle in a given point of space, with the
probability to find a point of space where we observe the particle, as

$P[X_{N}=c]=\sum_{i\in I}P[X_{N}=S_{N}^{(i)}|S_{N}^{(i)}=c]P[S_{N}^{(i)}=c]$
(4)

Let us explain better the quantities involved in this equation. We have

  1. a)

$P[X_{N}=S_{N}^{(i)}|S_{N}^{(i)}=c]=:\gamma(N,i,c)$ represent the probability
that the particle select the $i$-th random walk in the collection, assumed
that this random walk at the time $N$ is in $x=c$. This is the probability
that can be changed if we act on the particle only. More precisely, we can
prepare the system (in our case the particle) in a configuration such that the
probability to observe it in a given point is higher respect to another
configuration, changing this term. For example, if we want to increase the
probability to observe the particle in $x=a$, we may select (hence choose the
index $i$) the random walks in the collection with an higher probability to be
found in $x=a$. Summarising, when we prepare the particle in a given
configuration we act on this object.

  2. b)

$P[S_{N}^{(i)}=c]$ is the probability to observe a point of space in $x=a$.
This quantity is given by the model and cannot be changed when we act on the
particle only.

The jumps of the particle between two different points of space will be
modelled with a simple discrete-time Markov chain with transition
probabilities

$P[X_{N+1}=b|X_{N}=c]=:\alpha(b,c)$ (5)

where $b$ and $c$ are integer multiple of $l$, the step of the random walk (in
what follow we will assume $l=1$, for simplicity, and so $b,c\in\mathbb{N}$).
These transition probabilities are assumed to fulfil some equation describing
the physical system’s dynamics. As we will see, the random space described
before will put some constraints on the possible values these transition
probabilities may assume. Let us consider now the second random variable we
are interested in: the velocity. The space described above is discrete and, in
addition even the time is assumed discrete, hence it seem reasonable to define
the velocity of a particle over this space as

$V_{N}:=\frac{X_{N+1}-X_{N}}{N+1-N}=X_{N+1}-X_{N}$

Clearly, if it is a random variable it should depends both on $X_{N}$ and
$X_{N+1}$. This fact render problematic the computation of the various
probability generating function (like the characteristic functions) as a
function of the random variable $X_{N}$ and $X_{N+1}$. Nevertheless, this
problem may be bypassed, in a certain sense, following this intuitive idea.
Suppose we know that the particle at the time $N$ is in the position
$X_{N}=c$, then we know that the event $C:=\\{X_{N}=c\\}$ is true, namely
$P[X_{N}=c]=1$ (which means $P[X_{N}=d]=\delta_{d,c}$). Then, in this case, we
can write that $V_{N}=X_{N+1}-c$ when $C$ is true. This suggests that the
probability to have $V_{N}=a$ assumed that $C$ is true is equal to the
probability that $X_{N+1}=a+c$ (where $a$ can vary and $c$ is fixed). Then,
using (5) we can conclude that $P[X_{N+1}=a+c]=\alpha(a+c,c)$ and so
$P[V_{N}=a|C]=\alpha(a+c,c)$. Finally we can write that $P[V_{N}=a]$ is given
by

$\begin{split}P[V_{N}=a]&=\sum_{c}P[V_{N}=a|X_{N}=c]P[X_{N}=c]\\\
&=\sum_{c}\alpha(a+c,c)P[X_{N}=c]\end{split}$

This complete the probabilistic description of the particle in this model.

#### Remark.

The computation of $P[V_{N}=a|C]$ can be done in a rigorous way using
characteristic function. Since $V_{N}=X_{N+1}-X_{N}$ and because when $C$ is
true ($P[C]=1$, hence $P[X_{N}=d|C]=\delta_{d,c}$), $X_{N}$ and $X_{N+1}$ are
independent, from the properties of the characteristic functions one can write
$\varphi_{V_{N}}(\lambda)|_{C}=[\varphi_{X_{N+1}}(\lambda)|_{C}][\varphi_{-X_{N}}(\lambda)|_{C}]$.
This means that

$\begin{split}\varphi_{V_{N}}(\lambda)|_{C}&=\left(\sum_{b}P[X_{N+1}=b|C]e^{i\lambda
b}\right)\left(\sum_{d}P[X_{N}=d|C]e^{i\lambda(-d)}\right)\\\
&=\sum_{b}P[X_{N+1}=b|C]e^{i\lambda(b-c)}\\\
&=\sum_{b}\alpha(b,c)e^{i\lambda(b-c)}\end{split}$

In the case of discrete random variable, one can recover the probability
measure from the characteristic function using the formula

$P[X=a]=\lim_{T\rightarrow+\infty}\frac{1}{2T}\int^{+T}_{-T}e^{ita}\varphi_{X}(t)dt$

Hence we have that

$\begin{split}P[V_{N}=a|C]&=\lim_{T\rightarrow+\infty}\frac{1}{2T}\int_{-T}^{+T}e^{-i\lambda
a}\varphi_{V_{N}}(\lambda)|_{C}d\lambda\\\
&=\lim_{T\rightarrow+\infty}\frac{1}{2T}\int_{-T}^{+T}e^{-i\lambda
a}\sum_{b}\alpha(b,d)e^{i\lambda(b-c)}d\lambda\\\
&=\sum_{b}\alpha(b,c)\lim_{T\rightarrow+\infty}\frac{1}{2T}\int_{-T}^{+T}e^{i\lambda(b-c-a)}d\lambda\\\
&=\sum_{b}\alpha(b,c)\delta_{0,b-c-a}\\\ &=\alpha(a+c,c)\end{split}$

Confirming the result obtained above.

###  4.2 Entropy for $V_{N}$ and entropy for $X_{N}$

In this paragraph, we will compute the entropy of the two random variables
described before. Consider the following situation: suppose that we know that
at time $N$ the particle is in the position $x=c$. Thus we can conclude that
$P[X_{N}=d|C]=\delta_{d,c}$. In addition, using (4.1), we also know that
$P[V_{N}=a|C]=\alpha(a+c,c)$. Now, if $\alpha(a+c,c)$ can be changed
continuously to a delta, then we should obtain $P[V_{N}=a|C]=\delta_{a+c,c}$.
The computation of the two entropies with these probabilities will give us
$H(X_{N}|C)=0$ and $H(V_{N}|C)=0$. Nevertheless we should observe the
following fact. Recalling (4), we can write

$\alpha(a+c,c)=\sum_{j\in
I}P[X_{N+1}=S^{(j)}_{N+1}|S^{(j)}_{N+1}=a+c,C]P[S^{(j)}_{N+1}=a+c]$ (6)

The quantity
$P[X_{N+1}=S^{(j)}_{N+1}|S^{(j)}_{N+1}=a+c,C]=:\gamma(N+1,j,a+c|C)$ have the
following property

$\sum_{j\in I}\gamma(N+1,j,a+c|C)=1$ (7)

Recalling that $\gamma(N+1,j,a+c|C)\in[0,1]$ because is a transition
probability, the equation (6) can be considered as the average, with respect
the probability distribution $\\{\gamma(N+1,j,a+c|C)\\}_{j\in I}$, of the
probability to find a point of space in $x=a+c$. This observation is very
important: it allows us to find a bound for the transition probabilities. In
fact, by the property of the average, for all the $\alpha(a+c,c)\neq 0$ (which
gives a non zero contribution to the entropy) we can write that

$\alpha(a+c,c)\leqslant\max_{j\in I}P[S^{(j)}_{N+1}=a+c]$ (8)

If we assume that the random space is a _purely random process_ , namely that
$P[S^{(j)}_{N}=b]<1$ for any $N$, $b$ and $j\in I$, then we can see that the
$\alpha(a+c,c)$ cannot be $1$ for any value of $a$ (from now on it will be
always assumed so). This implies that we cannot have a delta-like probability
distribution for both the observables $V_{N}$ and $X_{N}$ if the space is a
purely random process. The bound found above cannot be changed if we act on
the particle only: it is linked to the process describing the random space.
This observation suggests that we can find a bound for the information that we
can have on these two random variables at the same time.

Let us find a bound for the sum of the two entropies. It is a known fact that
the entropy is a concave function of the probability distribution. Thus, the
Jensen inequality holds, namely if $g$ is a concave function

$g\left(\frac{\sum_{i=1}^{n}a_{i}x_{i}}{\sum_{i=1}^{n}a_{i}}\right)\geqslant\frac{\sum_{i=1}^{n}a_{i}g(x_{i})}{\sum_{i+1}^{n}a_{i}}$

where $a_{i}\in\mathbb{R}$ are arbitrary numbers. Then, we can write

$\begin{split}H(V_{N}|C)&=-\sum_{a}\alpha(a+c,c)\log\alpha(a+c,c)\\\
&=-\sum_{a}\left(\sum_{j\in
J}\gamma(N+1,j,a+c|C)P[S^{(j)}_{N+1}=a+c]\right)\log\left(\sum_{j\in
J}\gamma(N+1,j,a+c|C)P[S^{(j)}=a+c]\right)\\\ &\geqslant-\sum_{a}\sum_{j\in
J}\gamma(N+1,j,a+c|C)\left(P[S^{(j)}_{N+1}=a+c]\log P[S^{(j)}=a+c]\right)\\\
&\geqslant-\sum_{a}\min_{j\in J}|_{a}\left(P[S^{(j)}_{N+1}=a+c]\log
P[S^{(j)}=a+c]\right)\end{split}$

Where we used the Jensen inequality and (7), while $\min_{j\in J}|_{a}$ is the
minimum for $a$ fixed. This is already a bound on $H(V_{N}|C)$ which doesn’t
involve processes related to the particle, nevertheless we can further
simplify this result if we add some reasonable assumption on the random walks
describing the space process. We may require that

  1. a)

The initial positions $\\{x^{(i)}_{0}\\}_{i\in J}$ are $i.i.d.$ random
variables, namely $x^{(i)}_{0}\sim\pi^{(i)}=\pi$ for any $i\in J$,

  2. b)

The left and right probabilities (i.e. $p^{(i)}$ and $q^{(i)}$) of the random
walks are all equal, namely $p^{(i)}=p$ for any $i\in J$.

Under these two assumption, we can say that all the random walks are
_statistically equivalent_ , in the sense that
$P[S_{N}^{(i)}=a]=P[S_{N}^{(j)}=a]$ for any $i,j\in J$ ("observing" the space
process only at time-step $N$). This implies that

$-\sum_{a}\min_{j\in J}|_{a}\left(P[S^{(j)}_{N+1}=a+c]\log
P[S^{(j)}=a+c]\right)=-\sum_{a}\left(P[S^{(j)}_{N+1}=a+c]\log
P[S^{(j)}=a+c]\right)$

namely

$H(V_{N}|C)\geqslant H(S_{N+1})$

Finally, it is not difficult to see that $H(X_{N}|C)=0$ since
$P[X_{N}=d|C]=\delta_{d,c}$. Hence we can write that

$H(X_{N}|C)+H(V_{N}|C)\geqslant H(S_{N+1})$

and because $H(X|Y)=\sum_{i}P[Y=i]H(X|\\{Y=i\\})$ and $H(X|Y)\leqslant H(X)$
(conditioning reduces entropy), the above inequality implies that

$\begin{split}H(X_{N})+H(V_{N})&\geqslant\sum_{c}P(X_{N}=c)\left[H(X_{N}|C)+H(V_{N}|C)\right]\\\
&\geqslant\sum_{c}P[X_{N}=c]H(S_{N+1})=H(S_{N+1})\end{split}$ (9)

The RHS has the following features

  1. a)

$H(S_{N+1})$ is a positive quantity which cannot be changed if we act on the
particle only: $H(S_{N+1})$ is fixed once that the model of the space is
given;

  2. b)

$H(S_{N+1})$ is zero only for a deterministic space. This case is excluded if
we assume that the random space is a purely random process (the space is not
deterministic). As expected from the initial discussion on the transition
probabilities, we can see that the bound in the entropies is related to the
random nature of the space;

  3. c)

It is not guarantee that this bound is optimal.

The above bound can be explained in the following manner: we may change the
system configuration (namely the $\\{\gamma(N,i,c)\\}$) in order to know
completely the position of the particle, nevertheless the velocity of the
particle remains uncertain at least as the future position of a space point.
Such uncertainty cannot be reduced acting on the particle only. The discussion
done till now, should prove that position and velocity of a point-like
particle which jumps at random in different points over a random space satisfy
an entropic uncertainty relation.

###  4.3 Algebraic description of a point-like particle

The final result of the previous paragraph, suggests that position and
velocity doesn’t commute. In order to point out explicitly this non-
commutativity, we will describe the particle (not the space) using algebras.
The position is a random variable, namely a measurable map between a
probability space $(\Omega_{X},\mathcal{E}(\Omega_{X}),P_{X})$ and a measure
space, say $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Thus we can write
$X_{N}:\Omega_{X}\rightarrow\mathbb{R}$. Same things can be say for the
velocity: it is defined over a probability space
$(\Omega_{V},\mathcal{E}(\Omega_{V}),P_{V})$, take value over the measure
space $(\mathbb{R},\mathscr{B}(\mathbb{R}))$ and is a measurable map, thus
$V_{N}:\Omega_{V}\rightarrow\mathbb{R}$. This description is equivalent, up to
null sets, to the algebraic probability spaces $(\mathcal{A}_{X},\varrho)$ and
$(\mathcal{A}_{V},\varsigma)$, for $X_{N}$ and $V_{N}$ respectively. In this
description, $X_{N}$ and $V_{N}$ are algebraic random variable, namely
(involution preserving) homomorphism between two algebras: $X_{N}$ is defined
as the homomorphism $X_{N}:\mathcal{A}_{X}\rightarrow
L_{\infty}(\mathbb{R},\mathscr{B}(\mathbb{R}))_{X}$ and similarly for $V_{N}$
($L_{\infty}(\mathbb{R},\mathscr{B}(\mathbb{R}))$ is the canonical
$C^{*}$-algebra associated to the measurable space
$(\mathbb{R},\mathscr{B}(\mathbb{R}))$).

At this point, using the algebraic probability language, we can define the
$C^{*}$-probability space describing the point-like particle of the model as
$C^{*}$-probability space $(\mathcal{A},\omega)$, where $\mathcal{A}$ is the
smallest $C^{*}$-algebra which contain both
$L_{\infty}(\mathbb{R},\mathscr{B}(\mathbb{R}))_{X}$ and
$L_{\infty}(\mathbb{R},\mathscr{B}(\mathbb{R}))_{V}$, and $\omega$ is a state.
Thus $X_{N}$ and $V_{N}$ are elements of $\mathcal{A}$, and the discussion
done in the previous paragraph, tell us that they fulfil an entropic
uncertainty relation and, by the theorem 7 we have that $[X_{N},V_{N}]\neq 0$.
Thus, recalling the previous discussion about non-abelian algebras, we can see
that the description of our point-like particle must be done over an Hilbert
space, and cannot be mapped to an ordinary probability space.

It is worth to remark that this apparently strange result, is possible because
we eliminated the space from the description. In fact, we start using an
ordinary probability space, and we found a relations between the entropies of
the three main objects of the model (the space, the position and the
velocity). Then eliminating the space from the model, we found an entropic
uncertainty relation. It is interesting to observe that in ordinary quantum
mechanics the space doesn’t play any role, exactly as in the final model of
the particle.

###  4.4 Conclusions, weak points of the model and possible further
development

The model presented in the previous sections have the following interesting
feature: the space as well as the particle are treated in the same way.
Thinking that any observation we can do in a laboratory give us only a
probabilistic outcome (it is a nonsense to say "the quantity $A$ has the value
$b$") this feature can be justified using an operational approach for the
construction of a physical theory. In the model presented, the evolution of
the particle is the only possible, in the sense that if the particle change
position, it must happens with a jump. Thus in this sense we derived O1 form
O2, in the particular model considered. Of course, from the discussion done
till now we cannot conclude that the quantum mechanics is equivalent to
something similar to the model presented here. In particular, after a bit of
though, the following aspects may look strange

  1. a)

_Time is discrete_. The discreteness of time allows us to define $V_{N}$ as is
done and non-commutativity seem to be a consequence of it. Nevertheless, in
the proof of the entropic uncertainty relation between $X_{N}$ and $V_{N}$
time doesn’t play any role. So the proof seem to be quite robust to possible
change in the definition of $V_{N}$, due to change in the time assumptions.

  2. b)

_Space is discrete_. The structure of space is the core of the proof.
Nevertheless, discreteness limit only the place where a point of space can be
found. This is problem can be solved switching the description of space from
random walks to wiener(-like) processes, something that at the moment is not
available. Thus this weak point still remain but, we will discuss later an
interesting mathematical object that can be used to treat this problem.

  3. c)

_The constant in the entropic uncertainty relation change with time_. Looking
back to the entropic uncertainty relation (9) one can easily see that the
constant depend on time. This seem rather strange despite it doesn’t influence
the result: the constant remains positive for each time at it is (particle)
state independent. Nevertheless, one should observe this fact: the constant is
related to the stochastic process describing the space. In particular, in (9)
only the RHS depend on the space, so if we change the model of space, only
this side change. For example one can substitute the usual random walk (where
the possible position of the walker can be any integer number, this cause the
increasing of the value of the constant) with a reflexing boundary random
walk. In this case the number of possible position of the point of space are a
finite number, and so the constant can be fixed. This suggests that this
problem is not so fundamental. In addition the bound is not assumed to be
optimal, as already observed.

From the above arguments, one can see that a better model for the space should
be desirable. Keeping in mind this we may list a series of fact, suggesting
that further studies in this direction can be interesting

  1. a)

_Determinantal random point field._ A determinantal random point field is a
stochastic process which describe the random distribution of $N$ (possibly
infinite) points over $\mathbb{R}^{n}$ (or more generally over a Polish
space). This process is said deteriminantal because any point correlation
function can be expressed as Fredoholm’s determinant of a locally trace class
operator $\hat{\sigma}$ over an Hilbert space (in particular
$\hat{\sigma}:L_{2}(\mathbb{R}^{n})\rightarrow\mathcal{K}$, with
$\mathcal{K}\subset L_{2}(\mathbb{R}^{n})$ and $\dim\mathcal{K}=N$) [9]. Such
property is called _determinantal property_. Locally trace class operator are
operators whose trace may be infinite but, if restricted to suitable subspace,
it is finite. Clearly any trace class operator can be considered as a locally
trace class operator. Thus given a quantum system described by an Hilbert
space $\mathcal{H}$ and a trace class operator $\hat{\rho}$ we may associate
to it a determinantal random point field. Other similarity of these
mathematical objects with quantum mechanics is that also that can be described
using the second quantisation language. Finally the unitary evolution of
quantum mechanics preserves the determinantal property.

  2. b)

_$L_{2}(\mathbb{R}^{3})$ for a particle._ If we assume that the space where
the particles live is a random distribution of points over $\mathbb{R}^{3}$
described by a determinantal random point field, then we may justify the
postulate of quantum mechanics which tells us that $L_{2}(\mathbb{R}^{3})$ is
the Hilbert space for a single particle.

  3. c)

_Position and momentum operator and Galilean group._ From the Stone-von
Neumann theorem we may justify that the position and momentum operators in
quantum mechanics are defined as

$\hat{X}_{i}\psi(x)=x_{i}\psi(x)\mspace{50.0mu}\hat{P}_{i}\psi(x)=-i\frac{\partial}{\partial
x_{i}}\psi(x)$

for $\psi(x)\in\mathcal{S}(\mathbb{R}^{n})$. In quantum mechanics doesn’t seem
a priori valid the classical relation between position and momentum for a
point-like particle ($p=m\dot{x}$). Nevertheless from the Galilean group one
can prove that[1]

$\langle P_{t}\rangle=m\frac{d}{dt}\langle X_{t}\rangle$

Assuming, because a limited accuracy for any clock, that the time is discrete
(hence the limit is replaced by an inferior) and using the Wigner quasi
probability distribution, we can write that

$P_{t}=m\inf_{\delta t}\frac{X_{t+\delta t}-X_{t}}{\delta
t}\mspace{50.0mu}dW\mbox{-a.s.}$

Setting $\delta t=1$ ($\delta t$ is our unit of time) and $t=N\delta t$, we
can write $P_{N}=m(X_{N+1}-X_{N})=mV_{N}$. This consideration justify at
qualitative level the interest in the two random variable analysed in these
notes.

We conclude this writing, with a quotation whose author consider it contains
the spirit of what was written in this last section

‟ _Probability is the most important concept in modern science, especially as
nobody has the slightest notion what it means_ ˝

(Bertrand Russel in a lecture, 1929)

## References

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  * [3] M. P. Solèr - _Characterisation of Hilbert spaces with orthomodular spaces_ ; Comm. Algebra 23 (1995)
  * [4] G. Birkhoff, J. von Neumann -_The logic of quantum mechanics_ ; Ann. Math. 37, No. 4 (1936)
  * [5] F. Strocchi - _An introduction to the mathematical structure of quantum mechanics_ ; Advanced series in mathematical physics, Vol. 28, World Scientific (2008)
  * [6] L. Accardi - _Probabilita’ quantistica_ ; Storia della Matematica, Vol. 4, Einaudi (2010)
  * [7] H. Maassen - _Quantum probability applied to the dumped harmonic oscillator_ ; arXiv:quant-ph/0411024, (2004)
  * [8] H. Maassen, J. Uffink -_Generalized entropic uncertainty relations_ ; Phys. Rev. Lett., Vol. 60 (1988)
  * [9] A. Soshnikov - _Determinantal random point field_ ; Russian Math. Surveys, 55:5, 923-975 (2000)