# On non-commutativity in quantum theory (II):
toy models for non-commutative kinematics.

Curcuraci Luca  Department of Physics, University of Trieste, Strada Costiera
11 34151, Trieste, Italy
Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2 34127,
Trieste, Italy<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

In this article, we continue our investigation on the role of non-
commutativity in quantum theory. Using the method explained in _On non-
commutativity in quantum theory (I): from classical to quantum probability_ ,
we analyze two toy models which exhibit non-commutativity between the
corresponding position and velocity random variables. In particular, using
ordinary probability theory, we study the kinematics of a point-like particle
jumping at random over a discrete random space. We show that, after the
removal of the random space from the model, the position and velocity of the
particle do not commute, when represented as operators on the same Hilbert
space.

Quantum mechanics, Stochastic process, Foundation of quantum mechanics, Non-
commutativity, Stochastic geometry, Stochastic space, Particle jumps,.

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II Space and time in quantum mechanics
  3. III Model A: Discrete-time $1$-D kinematics on a random space
    1. III.1 The space process
    2. III.2 The particle process
    3. III.3 Remove the space from the model
    4. III.4 The entropic uncertainty relation for $X_{N}$ and $V_{N}$
    5. III.5 Construction of the Hilbert space structure for model A
    6. III.6 Final remarks and main limitations of model A
  4. IV Model B: Continuous-time 1-D kinematics on a random space
    1. IV.1 The space process
    2. IV.2 The particle process
    3. IV.3 The removal of the space process
    4. IV.4 Construction of the Hilbert space structure for model B
    5. IV.5 Final remarks and weak points of model B
  5. V Conclusion
  6. VI Acknowledgements
  7. VII Appendix

##  I Introduction

In LC  we proposed a method to construct a non-commutative probability theory
starting from a collection of ordinary probability spaces (i.e. a contextual
probability space khrennikov2009contextual ) using entropic uncertainty
relations. In this article, following this method, we try to shed some light
on the non-commutativity in quantum mechanics. In particular, we will focus on
the fundamental commutation relation

$[\hat{Q},\hat{P}]=i\hbar\hat{\mathbb{I}}$ (1)

between position and momentum operators in non-relativistic quantum mechanics.
The goal is to construct a Kolmogorov probabilistic model in which this
commutation relation and, more generally, non-relativistic quantum mechanics
can be recovered. To achieve this, we will present two toy models describing
the kinematics of a point-like particle jumping at random over a random
discrete space. As we will see, in these toy models a full derivation of the
commutation relation (1) is not possible (hence no direct comparison with non-
relativistic quantum mechanics is available) but from them, we may understand
how such a model should look like. Such model will be presented in LC3 .

The main idea is the following. In the framework of non-relativistic quantum
mechanics, the statistical description of a free particle in $\mathbb{R}^{3}$
is done by using an element of a separable infinite-dimensional Hilbert space,
$\psi\in L_{2}(\mathbb{R}^{3})$, and a set of non-commuting (in general) self-
adjoint operators on this Hilbert space. As we have seen, non-commutativity
has various consequences like Heisenberg uncertainty principle, CHSH
inequalities (when also the spin is considered) and, most important, the
impossibility to abandon the Hilbert space description***A notable exception
is the phase-space formulation of non-relativistic quantum mechanics
fairlie1964formulation ; baker1958formulation  which does not use Hilbert
spaces. However, it uses quasi-probability distributions where negative
probabilities are difficult to understand from the statistical point of view..
The basic assumption of the models presented here is that space (time is still
a parameter) plays an active role in the description of a particle. More
precisely, we will treat particles as point-like objects and the _physical
space_ as a random distribution of points. With the term “physical space”, we
mean the space on which particles actually move: for example the physical
space of classical mechanics is $\mathbb{R}^{3}$. The random distribution of
points used to describe the physical space is not static but evolves
stochastically in time according to some law. We also assume that a particle
moves by jumping at random from one point of the physical space to another.
The particle and physical space are described using random variables in the
framework of ordinary probability theory. When we want to describe _only_ the
particle, we have to remove (the exact meaning of this term will be clarified
later) the random variables describing the physical space: this will be the
origin of the non-commutativity between the position and the velocity
operators of the particle in these models.

The article is organized as follows. In section II we will give some physical
arguments supporting the basic assumptions of the model about the physical
space, then in section III, we will discuss a toy model where time is
discrete. This will be generalized in section IV to the continuous time case.
In both models, we will derive an entropic uncertainty relation for the
position and velocity random variables. This allows to conclude that they can
be represented on a common Hilbert space as two non-commuting operators (using
the results presented in LC ). For each toy model, we point out positive
aspects and limitations.

##  II Space and time in quantum mechanics

In the models proposed in the subsequent sections, space will be treated as a
stochastic process, while time will be a parameter. Here we try to argue our
choice of space and time using ordinary non-relativistic quantum mechanics.

Let us start with the _time_. In ordinary quantum mechanics, time is a
parameter, and we will treat it in the same way also in the proposed models.
It is known that, associate to time an operator $\hat{T}$ which is the
canonical conjugate of the Hamiltonian $\hat{H}$, i.e.
$[\hat{T},\hat{H}]=i\hat{\mathbb{I}}$, is problematic bunge1970so . Different
proposals are available wang2007introduce  however, none of them can be
considered as a satisfactory solution of the problem: 1) one may use an
operator $\hat{T}$ which is not self-adjoint and fulfil the commutation
relation, but then one has to deal with complex eigenvalues of such operator;
2) one may choose an Hamiltonian which is not bounded from below and fulfil
the commutation relation using a self-adjoint $\hat{T}$, but such Hamiltonian
does not describe stable physical systems. Giving up to fulfil the relation
$[\hat{T},\hat{H}]=i\hat{\mathbb{I}}$, another possible way to introduce a
time operator is the following. Suppose we have a quantum particle, described
at time $t$ by the vector $|\psi_{t}\rangle\in\mathcal{H}$, and whose time
evolution is given by the Schödingher equation, as usual. We want to define
the time operator $\hat{T}$ as the operator such that

$\hat{T}|\psi_{t}\rangle=t|\psi_{t}\rangle,$

for any $|\psi_{t}\rangle\in\mathcal{H}$. Since $t\in\mathbb{R}$ the spectrum
is real, hence $\hat{T}$ is self-adjoint. No commutation relation with the
Hamiltonian is assumed, hence we are free to assume the energy spectrum
bounded from below. In addition, we also assume that $\hat{T}$ commutes with
all the operators over $\mathcal{H}$. This is reasonable from the physical
point of view since we can always measure time together with any other
observable of the particle in a non-relativistic experiment. Indeed in non-
relativistic systems, the time in any clock of the laboratory is the time at
which the quantum particle is measured in the same experiment. Assuming that,
the spectral representation theorem LC ; moretti2013spectral  implies that

$\mathcal{H}=\int^{\bigoplus}_{\mathbb{R}}\mathcal{H}_{t}dt$

where $\int^{\oplus}\cdot$ means the continuous direct sum. The unitary time
evolution $\hat{U}_{t}$ induced by the Schödinger equation, can be seen as a
map between different Hilbert spaces in the direct sum above, namely
$\hat{U}_{s}:\mathcal{H}_{t}\rightarrow\mathcal{H}_{t+s}$. By the spectral
decomposition theorem LC ; moretti2013spectral , the operator $\hat{T}$ can be
written as

$\hat{T}=\int^{\bigoplus}_{\mathbb{R}}t\hat{\mathbb{I}}_{\mathcal{H}_{t}}dt$

where $\hat{\mathbb{I}}_{\mathcal{H}_{t}}$ is the identity on the Hilbert
space $\mathcal{H}_{t}$. In LC  we saw that non-commuting operators over a
Hilbert space are the non-commutative version of random variables and that,
their probability distributions are all encoded in a state defined on the
algebra that they form (typically represented using a vector of the Hilbert
space on which they are defined). In all attempts seen above to define a time
operator, we cannot consider time as a random variable with probability
distribution _induced by the quantum state used to describe the particle_.
Indeed, if $\hat{T}$ is not self-adjoint, it corresponds to a random variable
taking value on $\mathbb{C}$, which is hardly identifiable with physical time.
If $\hat{T}$ is self-adjoint but $\hat{H}$ is not bounded from below, the time
is a random variable but of an unphysical system. Finally, in the last
possibility, we can easily understand that no statistical information about
time is contained in $|\psi_{t}\rangle$. For these reasons in the proposed
models, we can safely treat time as a parameter “without neglecting possible
quantum effects”.

Now we turn our attention to _space_. In this case the situation is different.
Consider a quantum particle in $\mathbb{R}^{3}$, hence with Hilbert space
$L_{2}(\mathbb{R}^{3})$. Let
$\mathbf{x}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$, the position operator is
defined as

$\hat{Q}_{i}\psi(\mathbf{x})=x_{i}\psi(\mathbf{x})\mspace{30.0mu}i=1,2,3$

for all $\psi(\mathbf{x})\in\mathcal{S}(\mathbb{R}^{3})$, i.e. all the
Schwartz functions on $\mathbb{R}^{3}$. $\hat{Q}_{i}$ is self-adjoint and does
not commute with the momentum operator. The previous arguments does not apply
and it can be legitimately considered as a random variable whose statistical
properties are described by the wave function. However, $\hat{Q}_{i}$
represents on the Hilbert space the random variable describing the $i$-th
coordinate of the _particle position_ , and is not related to the underlying
physical space. The particle position is the random phenomena, not the
physical space where the particle lives. In appendix A, a simple model of
ruler described within the formalism of non-relativistic quantum mechanics is
given. In a nutshell, a quantum ruler can be considered as a collection of
quantum particles bounded together and localised in a given region of space.
Particles are assumed distinguishable, so they can be counted, and each
particle can be found in two different states, labeled by a spin variable.
Before any measurement, the spin variables of the ruler are in a known
configuration. The measurement is modelled with a contact interaction (between
the ruler and the particle we want to measure) which generates a spin-flip. A
(projective) measurement of the quantum ruler (as a photograph) right after
the interaction reveals which particle of the ruler “touches” the measured
quantum particle. We can then count the number of particles between the spin-
flipped particle and a chosen origin on the ruler (see the distance functions
in appendix B for some possible methods). Repeating this procedure many times,
we obtain that the probability to find the $i$-th particle of the ruler with
the spin flipped is well approximated by

$P[X_{A}=i]=\int\prod_{j=1}^{N}dy_{j}|\psi_{R}(y_{1},\cdots,y_{i},\cdots,y_{N})|^{2}|\phi_{A}(y_{i})|^{2},$

where $N$ is the number of particles of the quantum ruler,
$\psi_{R}(y_{1},\cdots,y_{i},\cdots,y_{N})$ is the wave function of the
quantum ruler and $\phi_{A}(x)$ is the wave function of the quantum particle
whose position is measured. In appendix A, it is argued that under reasonable
assumptions, the expression above reduces to $|\phi_{A}(x)|^{2}$ as expected.
Note that in the above expression, we have two contributions to the
probability: one due to the particle and one due to the ruler. Hence, if we
construct the physical space of a quantum system using a quantum mechanical
model of a ruler, we may legitimately think that the physical space of quantum
mechanics can be described by random variables.

We conclude by observing that the argument presented here about space _is not
loophole free_. One can always argue that the stochasticity we observe in the
physical space is an artifact of the ruler: the ruler is random, not the
space. This is clearly another legitimate possibility, but in this article, we
want to explore the consequences of _the choice of considering space as a
random phenomenon_.

##  III Model A: Discrete-time $1$-D kinematics on a random space

Here we will describe a discrete (and finite) random space and a particle
moving on it jumping at random from one point of space to another. Space,
position, and velocity of the particle at a given time will be treated in the
same way: using random variables. The whole model is 1-dimensional. We will
show that, once the space process is removed from the model, the position and
velocity of the particle can be jointly described in a non-commutative
probability space.

###  III.1 The space process

The process describing space in this model (Model A) we will be called _space
process_. The space process is assumed to be a discrete and finite set of
points distributed at random. More precisely, at each instant of time, space
is a random distribution of $M\in\mathbb{N}$ points over the real line. Such
points evolve in time as discrete-time random walks and, in this sense, space
is a _stochastic process_. This time evolution has a twofold interpretation. A
first possibility is to think it with respect to the real line: a point of the
space process is a random walk and it changes its position along the line as
time changes. A second possible way to see this time evolution is to look at
its effects on the “ordering among points”: the points change their distances
with respect to a chosen point (the origin) when this distance is “measured on
the points” (see the distances defined in appendix B). In some sense this
second point of view can be considered as an internal description: it
describes space as if the observer has no possibility to see the continuous
real line. On the other hand, the first possibility should be considered as an
external description†††An interesting analogy can be made between the two
possibilities explained here for the description of the space and the
description of a manifold. A manifold can be studied using a coordinate system
on it (internal description), or imagine that is embedded in a larger space
(external description), similarly to what happens here.. For simplicity, we
chose to describe the whole model from the first point of view. Nothing
forbids to adopt the second point of view for the description despite, at a
first look, it seems more complicated.

Let us recall some basic facts about the random walk rudnick2004elements .
Consider a lattice of points having spacing $l\in\mathbb{R}$, say
$\mathbb{Z}_{l}:=\\{x\in\mathbb{R}\mspace{5.0mu}|\mspace{5.0mu}x=ln,n\in\mathbb{Z}\\}$.
Then take a collection of independent, identically distributed Bernulli random
variables $\\{Y_{i}\\}_{i=1}^{\infty}$, characterised by the probabilities
$P[Y_{i}=-l]=p$ and $P[Y_{i}=+l]=q=1-p$ for all $i$. Using this collection, we
can define the random walk as the process

$S_{N}:=\sum_{i=0}^{N}Y_{i}$ (2)

where $Y_{0}$ is an arbitrary random variable with distribution $\pi(y_{0})$
taking value on $\mathbb{Z}_{l}$, representing the initial position of the
random walk. $N$ labels time (assumed discrete) and $S_{N}$ represents the
position of the random walk at time $N$. Let us now derive the probability
distribution of the random walk position at time $N$, i.e. $S_{N}$. Consider
the random walk at time $N$. Since at each time-step the random walk can move
by $+l$ or by $-l$ its position, if for $n<N$ times the random walk moves by
$-l$, its final position $d$ will be

$d=(N-n)l-nl=(N-2n)l$

Using this equation we can see that, if at time $N$ the random walk is found
in $d$, the number of times the random walk moves by $-l$ is

$n=\frac{1}{2}\left(\frac{d}{l}+N\right).$

Clearly, the number of times it moves by $+l$ will be $N-n$. Note that the
chronological order of the movements does not make any difference on the final
position. Assume, for the moment, that the initial position $Y_{0}$ is given,
and set it $Y_{0}=0$. Since for a given $d=(N-2n)l$ the random walk is just
the sum of Bernulli random variables, i.e. a binomial process, we can write

$\begin{split}P[S_{N}=d]&=\binom{N}{n}p^{n}(1-p)^{N-n}\\\
&=\binom{N}{\frac{1}{2}\left(\frac{d}{l}+N\right)}p^{\frac{1}{2}\left(\frac{d}{l}+N\right)}(1-p)^{N-\frac{1}{2}\left(\frac{d}{l}+N\right)}\\\
&=\binom{N}{\frac{d+N}{2l}}p^{\frac{d+Nl}{2l}}(1-p)^{\frac{Nl-d}{2l}}.\end{split}$

Nevertheless this formula holds only for $d\in[-lN,lN]$. If $d>lN$ or $d<-Nl$,
this probability must be zero because these regions of space cannot be reached
by the random walk in $N$ time-steps. Restoring $Y_{0}$ (hence we simply
translate the final position $d$ by $Y_{0}=y_{0}$), we can write that

$\begin{split}P[S_{N}-y_{0}=d-y_{0}]=\binom{N}{\frac{d+N}{2l}}p^{\frac{d+N}{2l}}(1-p)^{\frac{N-d}{2l}},\end{split}$
(3)

Note that (3) can be used as a probability only when the value of the random
variable $Y_{0}$ is given: hence it is a conditional probability with respect
to the value of $Y_{0}$, i.e. $P[S_{N}-y_{0}=d-y_{0}]=P[S_{N}=d|Y_{0}=y_{0}]$
. To complete the description of the random walk (2), using the Bayes theorem
we obtain

$P[S_{N}=d]=\sum_{y_{0}\in\mathbb{Z}_{l}}P[S_{N}=d|Y_{0}=y_{0}]\pi(y_{0}),$
(4)

which is the probability to find the random walk at time $N$ in the position
$d\in\mathbb{Z}_{l}$, given that at the initial time it started from the
position $Y_{0}$, random variable with distribution $\pi(y_{0})$. Without
loosing generality, we set $l=1$ for simplicity. We conclude our review on
basic facts about the random walk, formalising the description at measure-
theoretic level. As for any stochastic process, also for the random walk,
there exists a probability space $(\Omega,\mathcal{E},P)$. The sample space
$\Omega$ can be imagined as the set of all possible trajectories of the random
walk. It is a countable set (provided that the time of the random walk vary
over a finite interval), since the random walk is a discrete process.
$\mathcal{E}$ is a $\sigma$-algebra on $\Omega$, and can be thought as the
power set of $\Omega$, i.e. $\mathcal{E}=\mathcal{P}(\Omega)$‡‡‡ Given a set
$A$, with the symbol $\mathcal{P}(A)$ we label the power set of $A$., while
$P$ is the probability measure. The random walk on this probability space is
the identity random variable evaluated at a given time $N$, i.e. for
$s\in\Omega$ the position of the random walk at time $N$ is $S_{N}(s)=s(N)$.

Let us now come back to the space process. As stated in the beginning, it
consists of a collection of $M$ random walks. At any time step $N$, the random
distribution of points of the random walks is the space process of model A at
time $N$. We may start with this preliminary definition.

######  Definition 1.

Let $\\{S^{(i)}_{N}\\}_{i\in I}$ be a collection of independent random walks,
where $|I|=M\in\mathbb{N}$, defined as in (2) and described with the
probability distributions (4). We call such a collection the _space process_
for the Model $A$.

A possible realisation of the space process is given in figure 1.

Figure 1: A realisation of an $M=5$ space process $\mathbb{S}^{A}$ is given.
The set of coloured point at any time step represents the random distribution
of points of the space process at a fixed time step, $\mathbb{S}^{A}_{N}$. The
dashed lines liking the points of the same color represent the random walk
evolution of each point of the space process. Such evolution changes the
random distribution of points of the space process as time changes. Note that
at a given time step, points may overlap.

We label the space process of model A with the symbol $\mathbb{S}^{A}$, while
$\mathbb{S}^{A}_{N}$ is the space process at the time-step $N$ (hence a random
variable describing the distribution of $M$ points in $\mathbb{R}$). The
outcome of the random variable $\mathbb{S}^{A}_{N}$, can be thought as a
$M$-tuple, i.e. $\mathbf{S}_{N}=(s_{1}(N),\cdots,s_{M}(N))$ where
$s_{i}(N)\in\mathbb{R}$ is the position of the $i$-th random walk at time $N$.
Call $\tilde{P}^{A}$ the probability measure for the space process
$\mathbb{S}^{A}$. Since the random walks are assumed to be independent, the
probability to obtain a specific configuration is given by

$\tilde{P}^{A}[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]=\prod_{i=1}^{M}P[S_{N}^{(i)}=s_{i}(N)].$
(5)

For the same reason, it may happen that for some realisation the points
overlap. In a similar manner, we can also construct the joint probabilities

$\begin{split}\tilde{P}^{A}[\mathbb{S}^{A}_{N}&=\mathbf{S}_{N},\mathbb{S}^{A}_{T}=\mathbf{S}^{\prime}_{T}]=\\\
&\prod_{i,j=1}^{M}P[S_{N}^{(i)}=s_{i}(N),S_{T}^{(j)}=s^{\prime}_{j}(T)].\end{split}$
(6)

Note that $P[S_{N}^{(i)}=s_{i}(N),S_{T}^{(j)}=s^{\prime}_{j}(T)]$ can be
constructed using the independence of random walks when $i\neq j$, while for
$i=j$ it is just the joint probability distribution of the $i$-th random walk.
Proceeding in this way, we may construct the whole family of finite-
dimensional distributions for the space process $\mathbb{S}^{A}$, which is
consistent since the probabilities of the single random walks belongs to
consistent families (in the sense of the Kolmogorov extension theorem, see Th.
2.1.5 in oksendal2013stochastic ). At this point we may replace the
preliminary definition of the space process with the following which is more
precise.

######  Definition 2.

Let $\\{S_{N}^{(i)}\\}_{i\in I}$ be a collection of $M=|I|\in\mathbb{N}$
independent random walks defined on probability spaces
$\\{(\Omega_{i},\mathcal{E}_{i},P_{i})\\}_{i\in I}$. Let us define

  1. i)

$\Omega_{\mathbb{S}^{A}}:=\Omega_{1}\times\cdots\times\Omega_{M}$;

  2. ii)

$\mathcal{E}_{\mathbb{S}^{A}}=\mathcal{P}(\Omega_{\mathbb{S}^{A}})$;

  3. iii)

$\tilde{P}^{A}:\mathcal{E}_{\mathbb{S}^{A}}\rightarrow[0,1]$ defined from the
$\\{P_{i}\\}_{i\in I}$, as in (5) or (6) and generalisation.

The space process is the stochastic process on
$(\Omega_{\mathbb{S}^{A}},\mathcal{E}_{\mathbb{S}^{A}},\tilde{P}^{A})$ defined
as the identity function, namely
$\mathbb{S}^{A}(s_{1},\cdots,s_{M})=(s_{1},\cdots,s_{M})$.

The set of all the possible configurations of points of the space process at a
given time $N$ will be labeled by $\mathcal{S}(N)$.

###  III.2 The particle process

In this model, a particle is considered as a point-like object. At any time
step $N$, it is completely described by its _position_ and its _velocity_ ,
which are assumed to be random variables.

The _position random variable_ , labeled by $X_{N}$, is interpreted as the
actual position of the particle at time $N$. Let
$(\Omega_{\mathbb{S}^{A}},\mathcal{E}_{\mathbb{S}^{A}},\tilde{P}^{A})$ be the
probability space for the space process. On a probability space
$(\Omega_{I},\mathcal{E}_{I},P_{I})$ define an integer value discrete-time
stochastic process $I_{N}:\Omega_{I}\rightarrow\\{1,\cdots,M\\}$, which we
call _selection process_. Assume that we place the origin of a reference frame
in $\mathbb{S}^{A}_{N}$ in the point $S^{(i_{O})}_{N}(\mathbf{S})$. Then we
define

$X_{N}(\omega):=\pi_{I_{N}(\omega_{I})}(\mathbb{S}^{A}_{N}(\mathbf{S}))-S^{(i_{O})}_{N}(\mathbf{S}),$
(7)

where $\pi_{i}$ is the projector of the $i$-th component of an $M$-tuple, and
$\omega=(\omega_{I},\mathbf{S})$ with $\omega_{I}\in\Omega_{I}$ and
$\mathbf{S}\in\Omega_{\mathbb{S}^{A}}$. Hence we can say that
$X_{N}=S^{(i_{N})}_{N}-S^{(i_{O})}_{N}$ for some $i_{N}\in\\{1,\cdots,M\\}$
and $S_{N}^{(i)},S^{(i_{O})}_{N}\in\mathbf{S}_{N}$ components of a given
realisation of the space process at time $N$ (the writing “$x\in\mathbf{y}$”,
where $x$ is a point and $\mathbf{y}$ is an $N$-tuple, should be interpreted
as “$x$ is a member of the $N$-tuple $\mathbf{y}$”). Thus we have the
following definition:

######  Definition 3.

Consider the probability space
$(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}},P^{A})$
and a measurable space $(\mathbb{Z},\mathcal{P}(\mathbb{Z}))$. The random
variable $X_{N}$ is the $\mathcal{P}(\mathbb{Z})$-measurable function

$X_{N}:\Omega_{I}\times\Omega_{\mathbb{S}^{A}}\rightarrow\mathbb{Z}$

defined as in (7). $X_{N}$ represents the position of the particle at time
$N$.

Note that on the probability space
$(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}},P^{A})$
we can describe also the space process $\mathbb{S}^{A}$ by simply demanding
that $P^{A}\circ[\mathbb{S}^{A}]^{-1}=\tilde{P}^{A}$. From now on in the whole
discussion of model A, instead of writing $P^{A}$ we simply write $P$ if no
confusion arises. By construction, $X_{N}$ is a function of the space process
$\mathbb{S}^{A}$. This implies that $X_{N}$ and $\mathbb{S}^{A}_{N}$ are not
two independent random variables. Indeed, assume

$\begin{split}\mathbb{S}^{A}_{N}(\mathbf{S})&=\mathbf{S}_{N}:=(x_{1},\cdots,x_{M}),\\\
\mathbb{S}^{A}_{N}(\mathbf{S}^{\prime})&=\mathbf{S}_{N}^{\prime}:=(x_{1}^{\prime},\cdots,x_{M}^{\prime}),\end{split}$

and that we can fix a common origin on them, say
$x_{o}\in\mathbf{S}_{N}\cap\mathbf{S}_{N}^{\prime}$. Choose $\mathbf{S}$ and
$\mathbf{S}^{\prime}$ such that there exists $z+x_{o}\in\mathbf{S}_{N}$ but
$z+x_{o}\notin\mathbf{S}_{N}^{\prime}$, i.e. $\mathbb{S}^{A}_{N}(\mathbf{S})$
and $\mathbb{S}^{A}_{N}(\mathbf{S}^{\prime})$ have at least one point which is
not in common. Then

$P[X_{N}=z|\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]\neq
P[X_{N}=z|\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}],$

since the second term vanishes by construction while the first can be non-zero
in general. Thus we cannot set
$P[X_{N}=z|\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]=P[X_{N}=z]$ in general, which
implies

$P[X_{N}=z,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]\neq
P[X_{N}=z]P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]$

Let us now describe the _velocity random variable_. In order to introduce this
process, we need to specify how the particle moves on a physical space
described with the space process introduced before. We assume that particle
moves by jumps: it jumps from one of the points of the space process at time
$N$ to another point of the space process at time $N+1$. These jumps are
described by the transition probabilities

$P[X_{N+1}=b|X_{N}=a]=\alpha(b,a),$ (8)

where $a,b\in\mathbb{Z}$. Once these transition probabilities are given, we
can define the velocity random variable $V_{N}$. We set

$V_{N}:=\frac{X_{N+1}-X_{N}}{N+1-N}=X_{N+1}-X_{N}.$ (9)

This is clearly the discrete-time version of the usual definition of velocity.
Note that this physical definition makes sense because, thanks to the
transition probabilities (8), we can describe $V_{N}$ from the probabilistic
point of view using only information available at time $N$. More formally, the
transition probabilities (8) allows to describe $V_{N}$ on the same
probability space of $X_{N}$, i.e.
$(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}},P^{A})$.

######  Definition 4.

Consider the probability space
$(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}},P^{A})$
and the measurable space $(\mathbb{Z},\mathcal{P}(\mathbb{Z}))$. The velocity
random variable $V_{N}$ is the $\mathcal{P}(\mathbb{Z})$-measurable function

$V_{N}:\Omega_{I}\times\Omega_{\mathbb{S}^{A}}\rightarrow\mathbb{Z}$

defined in (9). $V_{N}$ represents the velocity of the particle at time $N$.

Also the velocity random variable is a function of the space process and,
proceding as done for $X_{N}$, we may conclude that
$P[V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]\neq
P[V_{N}=c]P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]$. Let us now derive the
relation between the probabilities $P[V_{N}=c]$ and $P[X_{N}=a]$, in a way
that is consistent with the transition probabilities (8). It can be done
following this intuitive idea. Suppose that at time $N$ we know that the
particle is in the position $X_{N}=a$. Then the event $A:=\\{X_{N}=a\\}$ is
true, i.e. $P(A)=P[X_{N}=a]=1$, which means that
$P[X_{N}=a^{\prime}|A]=\delta_{a,a^{\prime}}$. Under the same conditions, one
should also write that $V_{N}=X_{N+1}-a$, and this suggests that the
probability to observe $V_{N}=c$ is equal to the probability to observe
$X_{N+1}=a+c$, when $A$ happens. Thus, using (8) we can write that

$P[V_{N}=c|A]=P[X_{N+1}=a+c|A]=\alpha(a+c,a).$

The equation above can be confirmed in a more rigorous way.

######  Proposition 1.

Let $X_{N}$ and $V_{N}$ be the position and the velocity random variables. If
$P[X_{N+1}=b|X_{N}=a]=\alpha(b,a)$, then $P[V_{N}=c|A]=\alpha(a+c,a)$ where
$A=\\{X_{N}=a\\}$.

###### Proof.

Since $V_{N}=X_{N+1}-X_{N}$, clearly $X_{N+1}$ and $X_{N}$ are conditionally
independent under the event $A=\\{X_{N}=a\\}$. Let
$\varphi_{V_{N}}(\lambda)|_{A}$, $\varphi_{X_{N+1}}(\lambda)|_{A}$ and
$\varphi_{X_{N}}(\lambda)|_{A}$ be the characteristic functions of the three
random variables considered here, computed with the conditional probabilities.
By conditional independence we can write that

$\varphi_{V_{N}}(\lambda)|_{A}=\varphi_{X_{N+1}}(\lambda)|_{A}\cdot\varphi_{-X_{N}}(\lambda)|_{A}.$

Since

$\begin{split}\varphi_{-X_{N}}(\lambda)|_{A}&=\sum_{a^{\prime}}e^{-i\lambda
a^{\prime}}\delta_{a,a^{\prime}}=e^{-i\lambda a}\\\
\varphi_{X_{N+1}}(\lambda)|_{A}&=\sum_{b}\alpha(b,a)e^{i\lambda b}\end{split}$

we have that

$\varphi_{V_{N}}(\lambda)|_{A}=\sum_{b}\alpha(b,a)e^{i\lambda(b-a)}.$

Because $b-a\in\mathbb{Z}$, clearly
$\varphi_{V_{N}}(\lambda)|_{A}=\varphi_{V_{N}}(\lambda+2\pi)|_{A}$ which means
that the random variable $V_{N}$ is a discrete random variable (as expected).
The inversion formula of the characteristic function, in this case is

$P[V_{N}=c|A]=\lim_{T\rightarrow+\infty}\frac{1}{2T}\int_{-T}^{+T}e^{-i\lambda
c}\varphi_{V_{N}}(\lambda)|_{A}d\lambda.$

Thus

$\begin{split}P[V_{N}=c|A]&=\lim_{T\rightarrow+\infty}\frac{1}{2T}\int_{-T}^{+T}e^{-i\lambda
c}\sum_{b}\alpha(b,a)e^{i\lambda(b-a)}d\lambda\\\
&=\sum_{b}\alpha(b,a)\lim_{T\rightarrow+\infty}\frac{1}{2T}\int_{-T}^{+T}e^{i\lambda(b-a-c)}d\lambda\\\
&=\sum_{b}\alpha(b,a)\lim_{T\rightarrow+\infty}\frac{e^{iT(b-a-c)}-e^{iT(b-a-c)}}{2Ti(b-a-c)}\\\
&=\sum_{b}\alpha(b,a)\lim_{T\rightarrow+\infty}\mbox{sinc}(T(b-a-c))\end{split}$

where $\mbox{sinc}(x)=\sin x/x$. Since
$\lim_{a\rightarrow\infty}\mbox{sinc}(ax)=\delta_{x,0}$ when $x\in\mathbb{Z}$,
we conclude that

$P[V_{N}=c|A]=\sum_{b}\alpha(b,a)\delta_{b-a-c,0}=\alpha(a+c,a).$

This concludes the proof. ∎

At this point, we may obtain $P[V_{N}=c]$ simply using the Bayes theorem,
namely

$P[V_{N}=c]=\sum_{a}\alpha(a+c,a)P[X_{N}=a]$ (10)

which is consistent with the transition probabilities given in the beginning.
The following assumption on the transition probabilities is done

$P[V_{N}=c|X_{N}=a]=P[X_{N}=a|V_{N}=c],$ (11)

i.e. the transition probabilities are _symmetric_ under the exchange of their
arguments. Having defined both the position and velocity random variables, we
may give a precise definition of what we call particle in Model A .

######  Definition 5.

A particle is a point like-object whose features at time $N$ are completely
specified by the position and velocity random variables. More formally, we can
say that a particle corresponds to the random vector
$\mathscr{P}_{N}:=(X_{N},V_{N})$. We will refer to $\mathscr{P}_{N}$ with the
name _particle process_ , when considered as a function of time.

An example of particle process is drawn in figure 2.

Figure 2: A possible realization of the particle process is drawn in red, over
the same realization of the space process considered in figure 1. The position
of the particle at a given time-step is given by the red point while its
velocity at the same time-step is represented by the outgoing arrow.

###  III.3 Remove the space from the model

In this section, we will explain what we mean with the expression “remove the
space from the model”. In LC  we observed that, given three random variables
$X$, $Y$ and $Z$ on the same probability space $(\Omega,\mathcal{E},P)$, one
can eliminate one of them, say $Z$, simply by conditioning with respect to the
outcomes of this random variable, i.e. conditioning on the events $\\{Z=z\\}$.
In the collection of probability spaces obtained after conditioning, the
description of the random variable $Z$ is not anymore possible unless one adds
the probabilities $P[Z=z]$. Such information cannot be obtained from the
collection of probability spaces one has after conditioning. The statistical
description of the remaining random variables can be done without influence
the random variable $Z$: in this sense $Z$ is not present anymore in the
probabilistic model used to describe $X$ and $Y$. However, when we deal with a
stochastic process the elimination of a random variable representing it at
given time, do not guarantee that we can manipulate all the remaining random
variables without influence the stochastic process (which means that we can
describe the remaining random variables without influence the removed
process). In this case we need to add additional conditions in order to be
sure that the stochastic process is not present anymore in the remaining
probabilistic model. Once that we apply a procedure that is capable to do so,
we say that the stochastic process is _removed from the model_.

Model A exhibits features that are interesting from the point of view of
quantum mechanics when we remove the space process from the model. Before
describing how to implement it mathematically, let us first explain the
physical principles that motivate this removal. Model A describes a particle
that jumps at random over a random distribution of points. Such a random
distribution of points is assumed to be the physical space in which the
particle moves: the physical space is not anymore a passive background against
which physical processes take place. Preparing the particle in a given state
means to perform an experimental procedure after which the statistical
properties of the particle’s observables are known. In other words, the state
preparation is an experimental procedure such that right after it terminates,
all random variables associated to the particle’s observables have a given
probability distribution. Hence, saying that a particle at time $N$ is in a
given state, means that at time $N$ all probability distributions of the
observables of the particle are fixed. However, assuming that the physical
space is random has big consequences. Any experimental procedure happens in
such a physical space. If the experimental procedure for the state preparation
ends at time $N$, the probability distributions of the observables are always
conditioned to the configuration of space at that time. This is because one
prepares the state of the particle at time $N$, in the configuration of space
that the space process assumes at that time. Hence the probability
distributions that describe the particle must be always conditioned to some
space configuration. If it is not so, to prepare the particle in a given state
we need to have control not only on it but also on the whole space. This means
that the probability distribution that describes the space process at a given
time does not depend on the probabilities describing the particle after a
preparation procedure. In other words, changing the probability distributions
describing the particle (i.e. changing the state) does not have to modify the
probability distributions describing the space process. When this happens we
say that the space process is removed from model A. In order to implement
that, we have to require the following:

  1. i)

The particle at time $N$ can be described only by using probabilities that are
conditioned with respect to some space configuration at that time. This means
that to describe the position and velocity random variables we have to use
only

$\begin{split}P_{\mathbf{S}_{N}}[X_{N}=a]&:=P[X_{N}=a|\mathbb{S}^{A}_{N}=\mathbf{S}_{N}],\\\
P_{\mathbf{S}_{N}}[V_{N}=c]&:=P[V_{N}=c|\mathbb{S}^{A}_{N}=\mathbf{S}_{N}],\end{split}$

where $\mathbf{S}_{N}$ is the configuration of the space process at time $N$.

  2. ii)

The transition probabilities of any point of space (i.e.
$p_{i}=P[S^{(i)}_{N+1}=a+1|S^{(i)}_{N}=a]$ for all $i\in I$) cannot be changed
by the preparation procedure of the particle. This means that changing the
conditional probabilities of the particles, the transition probabilities of
the single point of space remains fixed.

These two conditions implement the idea that the space process cannot be
influenced by the preparation procedure of the particle. Note that the
requirement $i)$ is needed in order to avoid that the probability of the space
process at time $N$ is changed by the preparation procedure of the particle,
while the requirement $ii)$ avoids that such preparation procedure alters the
space process probabilities at times $N^{\prime}\neq N$ (i.e. in the past or
in the future). Since we are dealing with non-relativistic systems this last
requirement is reasonable from the physical point of view.

Let us now describe the effects of the removal of the space process in model A
from the mathematical point of view. We will focus first on the consequences
of the requirement $i)$. In order to do so we need to study better the effect
of conditioning on a probability space. For the interested reader, in appendix
C a short review on how conditioning is described in the measure-theoretic
formulation of probability theory is presented. However here we proceed
following a more intuitive approach. According to the removal procedure
explained above, we can describe the particle using only probabilities that
are conditioned to the event
$\\{\mathbf{S}_{N}\\}:=\\{\mathbf{S}\in\Omega_{\mathbb{S}^{A}}|\mathbb{S}^{A}_{N}(\mathbf{S})=\mathbf{S}_{N}\\}$.
At the level of the events, this means that for the random variable $X_{N}$
and $V_{N}$ we consider only events of this kind: $\\{X_{N}\in
A\\}\cap\\{\mathbb{S}^{A}_{N}=\mathbf{S}_{N}\\}$ and $\\{V_{N}\in
B\\}\cap\\{\mathbb{S}^{A}_{N}=\mathbf{S}_{N}\\}$. For the position random
variable, this means that the conditioning procedure effectively changes the
sample space and the $\sigma$-algebra of its starting probability space as

$(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}})\rightarrow(\Omega_{I}\times\\{\mathbf{S}_{N}\\},\mathcal{P}(\Omega_{I}\times\\{\mathbf{S}_{N}\\})).$

Let us call $\Omega_{X_{N}}:=\Omega_{I}\times\\{\mathbf{S}_{N}\\}$ and
$\mathcal{E}_{X_{N}}:=\mathcal{P}(\Omega_{I}\times\\{\mathbf{S}_{N}\\})$. It
is a known fact from probability theory that the measurable space
$(\Omega_{X_{N}},\mathcal{E}_{X_{N}})$ equipped with conditional probability
$P_{\mathbf{S}_{N}}[X_{N}=\cdot]$ defines a probability space. On this
probability space $(\Omega_{X_{N}},\mathcal{E}_{X_{N}},P_{\mathbf{S}_{N}})$
the random variable $X_{N}$ can be described after conditioning on the event
$\\{\mathbf{S}_{N}\\}$. Everythig we said till now, clearly also holds for the
velocity random variable: after conditioning it can be described in a
probability space $(\Omega_{V_{N}},\mathcal{E}_{V_{N}},P_{\mathbf{S}_{N}})$
defined in a similar manner.

Relevant for our goal is the study of the joint probabilities for $X_{N}$ and
$V_{N}$, and its link with the transition probabilities (8) after
conditioning. By definition $X_{N}$ and $V_{N}$ are two random variables
defined on the same probability space
$(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}},P)$.
This means that we can always find a joint probability distribution
$P[X_{N}=a,V_{N}=c]$, which can be used to derive the transition probabilities
(8) using the usual Bayes formula. Since the space process can be described on
the same probability space of $X_{N}$ and $V_{N}$, also the joint probability
distribution $P[X_{N}=a,V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]$ exists.
Applying the Bayes formula, we can derive the _conditional_ joint probability
for $X_{N}$ and $V_{N}$, namely

$P_{\mathbf{S}_{N}}[X_{N}=a,V_{N}=c]:=\frac{P[X_{N}=a,V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]}{P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]},$

from which one can derive _conditional_ transition probabilities

$\begin{split}\alpha_{\mathbf{S}_{N}}(c,a)&:=P_{\mathbf{S}_{N}}[V_{N}=c|X_{N}=a]\\\
&=\frac{P_{\mathbf{S}_{N}}[X_{N}=a,V_{N}=c]}{P_{\mathbf{S}_{N}}[X_{N}=a]}.\end{split}$

Note that $\alpha_{\mathbf{S}_{N}}(c,a)\neq\alpha(c,a)$. From the point of
view of the probability spaces, after conditioning we can always describe the
two random variables using a single probability space. Such probability space
is simply
$(\Omega_{X_{N}}\times\Omega_{V_{N}},\mathcal{E}_{X_{N}}\otimes\mathcal{E}_{V_{N}},P_{\mathbf{S}_{N}})$.
On it, we can define a joint probability distribution
$P_{\mathbf{S}_{N}}[X_{N}=\cdot,V_{N}=\cdot]$ such that
$P_{\mathbf{S}_{N}}[X_{N}=\cdot]$ and $P_{\mathbf{S}_{N}}[V_{N}=\cdot]$ are
the two marginals and $\alpha_{\mathbf{S}_{N}}(c,a)$ are the transition
probabilities between $V_{N}$ and $X_{N}$. Since the joint probability
distribution are symmetric under the exchange of the arguments, clearly

$\alpha_{\mathbf{S}_{N}}(c,a)P_{\mathbf{S}_{N}}[X_{N}=a]=\alpha_{\mathbf{S}_{N}}(a,c)P_{\mathbf{S}_{N}}[V_{N}=c],$

where $\alpha_{\mathbf{S}_{N}}(a,c)=P_{\mathbf{S}_{N}}[X_{N}=a|V_{N}=c]$.
According to khrennikov2009contextual , this is a signature that we are
working on a single measure-theoretic probability space. A more interesting
case happens when we use the the _unconditional_ transition probabilities
$\alpha(a+c,a)$ and $P[X_{N}=a|V_{N}=c]$. In this case we have that

$\alpha(a+c,a)P_{\mathbf{S}_{N}}[X_{N}=a]\neq
P[X_{N}=a|V_{N}=c]P_{\mathbf{S}_{N}}[V_{N}=c]$ (12)

in general, which means that we cannot describe $X_{N}$ and $V_{N}$ using a
single measure-theoretic probability space, _if we choose to use the
unconditional transition probabilities after conditioning with respect to the
space process at time $N$_. However, this does not mean that we cannot
describe $X_{N}$ and $V_{N}$ after conditioning using the transition
probabilities $\alpha(a+c,a)$ (we will come back on the physical reason for
the use of $\alpha(a+c,a)$ instead of $\alpha_{\mathbf{S}_{N}}(c,a)$ later).
We can do it using two different probability spaces: one for $X_{N}$ and one
for $V_{N}$. We have already seen that, after conditioning, we obtain a
probability spaces for each random variables, i.e.
$(\Omega_{X_{N}},\mathcal{E}_{X_{N}},P_{\mathbf{S}_{N}})$ for $X_{N}$ and
$(\Omega_{V_{N}},\mathcal{E}_{V_{N}},P_{\mathbf{S}_{N}})$ for $V_{N}$. However
we cannot construct a joint probability space where
$P_{\mathbf{S}_{N}}[X_{N}=\cdot]$ and $P_{\mathbf{S}_{N}}[V_{N}=\cdot]$ are
the two marginals of some joint probability distribution and $\alpha(a+c,a)$
are the transition probabilities that we obtain from the same joint
probability distribution. This is exactly the content of (12): the joint
probability we are looking for would not be symmetric in the exchange of the
arguments. This is something that it is not possible in an ordinary measure
space since the intersection of events in a sigma algebra is a symmetric
operation (i.e. commutative). As a consequence we may conclude that the Bayes
theorem cannot be used to relate the two marginals. However a relation between
$P_{\mathbf{S}_{N}}[X_{N}=\cdot]$ and $P_{\mathbf{S}_{N}}[V_{N}=\cdot]$ can
still be found khrennikov2005interference .

######  Theorem 1.

Let $\\{P_{\mathbf{S}_{N}}[X_{N}=a]\\}_{a\in\Omega_{X_{N}}}$ be the
probabilities describing the position of the particle at time $N$ under the
condition that the space process at time $N$ is $\mathbf{S}_{N}$. If
$P[X_{N+1}=b|X_{N}=a]=\alpha(b,a)$, then

$P_{\mathbf{S}_{N}}[V_{N}=c]=\sum_{a}\alpha(a+c,a)P_{\mathbf{S}_{N}}[X_{N}=a]+\delta(c|X_{N},\mathbf{S}_{N})$
(13)

where

$\begin{split}\delta(c|X_{N},\mathbf{S}_{N})&=\frac{1}{P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]}\sum_{\begin{subarray}{c}\mathbf{S}_{N}^{\prime}\\\
\mathbf{S}_{N}^{\prime}\neq\mathbf{S}_{N}\end{subarray}}\bigg{[}\sum_{a}\alpha(a+c,a)\cdot\\\
&\cdot
P[X_{N}=a,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]-P[V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]\bigg{]}\end{split}$
(14)

which is in general different from zero.

###### Proof.

Given $P[V_{N}=c]$, we can always write

$\begin{split}P[V_{N}=c]=\sum_{\mathbf{S}_{N}^{\prime}}P[V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]\end{split}$

and similarly

$P[X_{N}=a]=\sum_{\mathbf{S}_{N}^{\prime}}P[X_{N}=a,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}].$

Note that the sum over all possible configurations of the space process at
time $N$ is well defined, since the number of configurations is clearly
countable (it is a cartesian product of a discrete process taking value on the
integers). Substituting these expressions in (10) and dividing by
$P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]$, we get

$\begin{split}\sum_{\mathbf{S}_{N}^{\prime}}\frac{P[V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]}{P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]}&=\sum_{a}\alpha(a+c,c)\sum_{\mathbf{S}_{N}^{\prime}}\frac{P[X_{N}=a,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]}{P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]}\\\
P_{\mathbf{S}_{N}}[V_{N}=c]+\sum_{\begin{subarray}{c}\mathbf{S}_{N}^{\prime}\\\
\mathbf{S}_{N}^{\prime}\neq\mathbf{S}_{N}\end{subarray}}\frac{P[V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]}{P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]}&=\sum_{a}\alpha(a+c,c)P_{\mathbf{S}_{N}}[X_{N}=a]+\sum_{a}\alpha(a+c,c)\sum_{\begin{subarray}{c}\mathbf{S}_{N}^{\prime}\\\
\mathbf{S}_{N}^{\prime}\neq\mathbf{S}_{N}\end{subarray}}\frac{P[X_{N}=a,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]}{P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]}\end{split}$

Moving the second term of the LHS to the RHS, we obtain (13) and (14). Note
that in general (14) is non zero since

$\begin{split}\sum_{a}\alpha&(a+c,a)P[X_{N}=a,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}]\\\
&\neq P[V_{N}=c,\mathbb{S}^{A}_{N}=\mathbf{S}_{N}^{\prime}].\end{split}$

This concludes the proof. ∎

We can see that, after the conditioning on the space process, the Bayes
formula cannot be used anymore to compute $P_{\mathbf{S}_{N}}[V_{N}=c]$ from
the probabilities of the position random variable if we want to use the
transition probabilities $\alpha(a+c,a)$. We need to add a correction term
which contains statistical information about the space process. Note that this
correction term has the property

$\sum_{c}\delta(c|X_{N},\mathbf{S}_{N})=0,$ (15)

which is necessary in order to preserve the normalisation of probabilities,
i.e. $\sum_{c}P_{\mathbf{S}_{N}}[V_{N}=c]=1$. We also note that in general
$\delta(c|X_{N},\mathbf{S}_{N})\in[-1,1]$ and in particular it can be
negative. Summarising, given the transition probabilities $\alpha(a+c,a)$ we
cannot describe $X_{N}$ and $V_{N}$ on a single probability space after
conditioning on the space configuration at time $N$. However, the description
$X_{N}$ and $V_{N}$ in a single probability space after conditioning can be
always done: the price to pay is that we have to change the transition
probabilities from $\alpha(a+c,a)$ to $\alpha_{\mathbf{S}_{N}}(c,a)$.

At this point a legitimate question arises: can we motivate physically the
choice to use $\alpha(a+c,a)$ instead $\alpha_{\mathbf{S}_{n}}(c,a)$? Yes, if
we take into account the fact that we want to remove space from the model.
Indeed, in order to measure with an experimental procedure
$\alpha_{\mathbf{S}_{N}}(c,a)$, one would have control over space since one
has to be able to prepare the space process always in the configuration
$\mathbf{S}_{N}$, in order to measure $\alpha_{\mathbf{S}_{N}}(c,a)$. Since
the removal of space is done exactly to avoid such things, the use of
$\alpha(a+c,a)$ is more reasonable from the physical point of view. We want to
conclude our analysis on the consequence of the requirement $i)$ with a
comment on the particle process. Since it is a random vector parametrized by
time, one may be tempted to consider $\mathscr{P}_{N}$ as a stochastic
process. This is certainly possible considering also the space process, namely
before conditioning on $\mathbf{S}_{N}$. Nevertheless, after conditioning and
using the transition probabilities $\alpha(a+c,a)$, we just have a collection
of probability spaces and it is not trivial to assume that each of these
spaces can be seen as, part of a bigger probability space describing the
particle only (i.e. with no space process involved in the construction of such
probability space) as the Kolmogorov extension theorem oksendal2013stochastic
would imply. For this reason, considering the particle process as a stochastic
process, in this context, should be done with care.

Till now we explored the consequences of the requirement $i)$ for the removal
of space. Conditioning with respect to the space configuration
$\mathbf{S}_{N}$, we effectively eliminate the possibility to change the
$P[\mathbb{S}^{A}_{N}=\mathbf{S}_{N}]$ by varying the (conditional)
probability distributions in the collection of probability spaces that
describe the model. The requirement $ii)$ is added in order to avoid that by
varying the probabilities of the particle we can modify the probabilities
$P[\mathbb{S}^{A}_{N^{\prime}}=\mathbf{S}_{N^{\prime}}]$ when $N^{\prime}\neq
N$. The consequences of $ii)$ which are relevant for our analysis will be
analyzed in the next section.

###  III.4 The entropic uncertainty relation for $X_{N}$ and $V_{N}$

In this section we will analyze the basic consequence of the requirement $ii)$
for the removal of the space process in model A. From now on, we exclude that
the probabilities describing the space process and its constituents have
delta-like distributions. This implies that the space process of model A is
not a deterministic process. We note that the whole removal procedure, which
makes model A interesting to study, is meaningless in this case. The central
result of this section is the following.

######  Theorem 2.

Let $X_{N}$ and $V_{N}$ be the position and velocity random variables of model
A. Fixing the transition probabilities
$p_{i}=P[S_{N+1}^{(i)}=a+1|S_{N}^{(i)}=a]$ of the points of the space process
for all $i\in I$, then

$H_{\mathbf{S}_{N}}(X_{N})+H_{\mathbf{S}_{N}}(V_{N})\geqslant D,$ (16)

where $D$ is a positive constant which does not depend on
$\\{P_{\mathbf{S}_{N}}[X_{N}=a]\\}_{a\in X_{N}(\Omega_{X})}$ and
$\\{P_{\mathbf{S}_{N}}[V_{N}=c]\\}_{c\in V_{N}(\Omega_{V})}$.

###### Proof.

The entropy is a non-negative quantity by definition, hence varying with
respect to all $P_{\mathbf{S}_{N}}[X_{N}=a]$ clearly
$H_{\mathbf{S}_{N}}(X_{N})\geqslant 0$. Now consider the entropy for the
random variable $V_{N}$ and let us study what happens when we vary with
respect to all $P_{\mathbf{S}_{N}}[X_{N}=a]$. Given
$P_{\mathbf{S}_{N}}[X_{N}=a]$, the probability $P_{V_{N}}[V_{N}=c]$ can be
computed by means of the formula in theorem 1. On the other hand we are always
free to use the conditional transition probabilities
$\alpha_{\mathbf{S}_{N}}(c,a)$, i.e. to work on the joint probability space of
$X_{N}$ and $V_{N}$, to study how $H_{\mathbf{S}_{N}}(V_{N})$ change varying
with respect to $P_{\mathbf{S}_{N}}[X_{N}=a]$.This allow us to write

$H_{\mathbf{S}_{N}}(V_{N})\geqslant\sum_{a}P_{\mathbf{S}_{N}}[X_{N}=a]H_{\mathbf{S}_{N}}(V_{N}|X_{N}=a),$

with

$H_{\mathbf{S}_{N}}(V_{N}|X_{N}=a)=-\sum_{c}\alpha_{\mathbf{S}_{N}}(c,a)\log\alpha_{\mathbf{S}_{N}}(c,a).$

The conditional transition probabilities $\alpha_{\mathbf{S}_{N}}(c,a)$ can be
rewritten as follows. Consider the joint probability
$P_{\mathbf{S}_{N}}[X_{N+1}=b,X_{N}=a]$. In what follows, without loss of
generality we set $S_{N}^{i_{O}}=0$ at any time $N$. We can write the
following

$\begin{split}&\mspace{80.0mu}P_{\mathbf{S}_{N}}[X_{N+1}=b,X_{N}=a]=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=b,X_{N}=a]P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=b,X_{N}=a]\\\
&\mspace{50.0mu}=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=b,X_{N}=a]\bigg{(}\sum_{j=1}^{M}P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=b|X_{N}=a,I_{N}=j]P_{\mathbf{S}_{N}}[X_{N}=a,I_{N}=j]\bigg{)}\\\
&=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=b,X_{N}=a]\bigg{(}\sum_{j=1}^{M}P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=b|X_{N}=a,I_{N}=j]P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]P_{\mathbf{S}_{N}}[X_{N}=a]\bigg{)}\end{split}$

Since the event $\\{X_{N}=a\\}\cap\\{I_{N}=j\\}=\\{S_{N}^{(j)}=a\\}$ by
definition and because
$P_{\mathbf{S}_{N}}[X_{N+1}=a+c,X_{N}=a]=P_{\mathbf{S}_{N}}[V_{N}=c,X_{N}=a]$,
from this decomposition we can conclude that

$\alpha_{\mathbf{S}_{N}}(c,a)=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=b,X_{N}=a]\bigg{(}\sum_{j=1}^{M}P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=b|S^{(j)}_{N}=a]P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}.$

We also note that

$\begin{split}\sum_{i=1}^{M}P_{\mathbf{S}_{N}}&[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=b,X_{N}=a]=1,\\\
&\sum_{j=1}^{M}P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]=1.\end{split}$ (17)

In what follows, we set
$\gamma(i):=P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=b,X_{N}=a]$
and $\eta(i,j):=P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=b|S^{(j)}_{N}=a]$ in order to
keep the notation compact. From the above decomposition of
$\alpha_{\mathbf{S}_{N}}(c,a)$ we can write that

$\begin{split}H_{\mathbf{S}_{N}}&(V_{N}|X_{N}=a)=-\sum_{c}\left(\sum_{i=1}^{M}\gamma(i)\bigg{(}\sum_{j=1}^{M}\eta(i,j)P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}\right)\log\left(\sum_{i=1}^{M}\gamma(i)\bigg{(}\sum_{j=1}^{M}\eta(i,j)P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}\right).\end{split}$

Note that since only positive probabilities contribute to the entropy, all the
$\alpha_{\mathbf{S}_{N}}(c,a)$ are different from zero. This implies that all
the $\gamma(i)$, $\eta(i,j)$ and $P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]$ used to
compute the entropy are strictly positive. Since $f(x)=-x\log x$ is a concave
function, by the Jensen inequality and using (17), we have

$\begin{split}H_{\mathbf{S}_{N}}(V_{N}|X_{N}=a)&\geqslant\sum_{c}\sum_{i=1}^{M}\gamma(i)\left(-\bigg{(}\sum_{j=1}^{M}\eta(i,j)P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}\log\bigg{(}\sum_{j=1}^{M}\eta(i,j)P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}\right)\\\
&\geqslant\sum_{c}\left(\sum_{i=1}^{M}\gamma(i)\right)\min_{i}\left(-\bigg{(}\sum_{j=1}^{M}\eta(i,j)P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}\log\bigg{(}\sum_{j=1}^{M}\eta(i,j)P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}\right)\\\
&\geqslant\sum_{c}\min_{i}\left(\bigg{(}\sum_{j=1}^{M}P_{\mathbf{S}_{N}}[I_{N}=j|X_{N}=a]\bigg{)}\min_{j}\left(-\eta(i,j)\log\eta(i,j)\right)\right)\\\
&=\sum_{c}\min_{i,j}\left(-\eta(i,j)\log\eta(i,j)\right)\end{split}$

where $\min_{i,j}$ means the minimum over $i,j\in\\{1,\cdots,M\\}$ keeping $c$
constant. Summarising, we have that

$\begin{split}H(V_{N}|X_{N}=a)\geqslant\sum_{c}\min_{i,j}\left(-P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]\log
P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]\right).\end{split}$

Note that in the RHS there is still a dependence on $a$, which can be removed
by taking the minimum with respect to it. Thus we can write that

$H_{\mathbf{S}_{N}}(V_{N})\geqslant\sum_{a}P_{\mathbf{S}_{N}}[X_{N}=a]H(V_{N}|X_{N}=a)\geqslant
D_{1},$

where we set

$\begin{split}D_{1}:=\min_{a}\bigg{[}\sum_{c}\min_{ij}\bigg{(}-P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]\log
P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]\bigg{)}\bigg{]}.\end{split}$
(18)

$D_{1}$ is a positive number, since
$P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]\in(0,1)$ (we exclude the
case of _deterministic_ space process) and only positive probabilities
contribute to the entropy, as said above. To explicitly show that the $D_{1}$
does not depend on $P_{\mathbf{S}_{N}}[X_{N}=a]$ and
$P_{\mathbf{S}_{N}}[V_{N}=c]$, let us study in detail the terms
$P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]$. Recalling that the
random walks are independent and that
$P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]\neq 0$ only for the
$S^{(j)}_{N}\in\mathbf{S}_{N}$, we can write that

$P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=a]=\begin{cases}0\mbox{ if
$i=j$ and $c\neq\pm 1$;}\\\ p_{i}\mbox{ if $i=j$ and $c=1$;}\\\ 1-p_{i}\mbox{
if $i=j$ and $c=-1$;}\\\ P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c]\mbox{ if $i\neq
j$}\end{cases}$

where $p_{i}$ and $1-p_{i}$ are the transition probabilities of the $i$-th
random walk, which are fixed by hypothesis. Again, the first case is excluded
since only positive probabilities contribute to the entropy. What we need to
check is the last case, namely $P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c]$. Since
in the configuration $\mathbf{S}_{N}$ there is also the $i$-th random walk,
this term reduces to

$P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c]=P[S^{(i)}_{N+1}=a+c|S^{(i)}_{N}=e]$

for some $e\in\mathbb{Z}$. The only terms of this kind that contribute to the
entropy are those having $e=a+c\pm 1$, i.e. the transition probabilities of
the $i$-th random walks, which are fixed by hypothesis. Thus fixing $p_{i}$
for all $i\in I$ implies that $D_{1}$ is a positive constant. Summarising we
showed that

$H_{\mathbf{S}_{N}}(X_{N})+H_{\mathbf{S}_{N}}(V_{N})\geqslant D_{1},$

when we vary over any possible value of $P_{\mathbf{S}_{N}}[X_{N}=a]$ and when
the transition probabilities of the $M$ random walks are fixed.

To conclude the proof we need to study what happens when we vary over all
possible values of $P_{\mathbf{S}_{N}}[V_{N}=c]$. Similarly to the previous
case, $H_{\mathbf{S}_{N}}(V_{N})\geqslant 0$ while $H_{\mathbf{S}_{N}}(X_{N})$
changes according with the inequality

$H_{\mathbf{S}_{N}}(V_{N})\geqslant\sum_{c}P_{\mathbf{S}_{N}}[V_{N}=c]H_{\mathbf{S}_{N}}(X_{N}|V_{N}=c),$

where $H_{\mathbf{S}_{N}}(X_{N}|V_{N}=c)$ is the entropy computed using
$\alpha_{\mathbf{S}_{N}}(a,c)=P_{\mathbf{S}_{N}}[X_{N}=a|V_{N}=c]$. From the
definition of $X_{N}$ and $V_{N}$, one can conclude that

$\\{X_{N}=a\\}\cap\\{V_{N}=c\\}=\\{X_{N}=a\\}\cap\\{V_{N}=c\\}\cap\\{X_{N+1}=a+c\\}=\\{V_{N}=c\\}\cap\\{X_{N+1}=a+c\\}.$

This implies that
$P_{\mathbf{S}_{N}}[X_{N}=a,V_{N}=c]=P_{\mathbf{S}_{N}}[X_{N+1}=a+c,V_{N}=c]$,
i.e.

$\begin{split}\alpha_{\mathbf{S}_{N}}(a,c)=P_{\mathbf{S}_{N}}[X_{N}=a|V_{N}=c]=\frac{P_{\mathbf{S}_{N}}[X_{N}=a,V_{N}=c]}{P_{\mathbf{S}_{N}}[V_{N}=c]}=\frac{P_{\mathbf{S}_{N}}[X_{N+1}=a+c,V_{N}=c]}{P_{\mathbf{S}_{N}}[V_{N}=c]}.\end{split}$

As before, the whole analysis reduces to the study of this term. Given
$P_{\mathbf{S}_{N}}[X_{N+1}=a+c,V_{N}=c]$ we can write that

$\begin{split}&P_{\mathbf{S}_{N}}[X_{N+1}=a+c,V_{N}=c]=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N}^{(i)}=a+c,V_{N}=c]P[S_{N+1}^{(i)}=a+c,V_{N}=c]\\\
&=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=a+c,V_{N}=c]\bigg{(}\sum_{j,d}P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|V_{N}=c,I_{N}=j,X_{N+1}=d]\cdot\\\
&\mspace{580.0mu}\cdot
P_{\mathbf{S}_{N}}[V_{N}=c,I_{N}=j,X_{N+1}=d]\bigg{)}\\\
&=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=a+c,V_{N}=c]\bigg{(}\sum_{j,d}P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|V_{N}=c,I_{N}=j,X_{N+1}=d]\cdot\\\
&\mspace{500.0mu}P_{\mathbf{S}_{N}}[I_{N}=j,X_{N+1}=d|V_{N}=c]P_{\mathbf{S}_{N}}[V_{N}=c]\bigg{)}.\end{split}$

Observing that the event
$\\{V_{N}=c\\}\cap\\{I_{N}=j\\}\cap\\{X_{N+1}=d\\}=\\{S^{(j)}_{N}=d-c\\}$, we
conclude that

$\begin{split}\alpha_{\mathbf{S}_{N}}(a,c)=\sum_{i=1}^{M}P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=a+c,V_{N}=c]\bigg{(}\sum_{j,d}P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=d-c]\cdot\\\
\cdot P_{\mathbf{S}_{N}}[I_{N}=j,X_{N+1}=d|V_{N}=c]\bigg{)}.\end{split}$

Note that

$\begin{split}\sum_{i=1}^{M}P_{\mathbf{S}_{N}}&[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=a+c,V_{N}=c]=1\\\
&\sum_{j,d}P_{\mathbf{S}_{N}}[I_{N}=j,X_{N+1}=d|V_{N}=c]=1\end{split}$ (19)

Defining
$\tilde{\gamma}(i):=P_{\mathbf{S}_{N}}[X_{N+1}=S_{N+1}^{(i)}|S_{N+1}^{(i)}=a+c,V_{N}=c]$
and
$\tilde{\eta}(i,j):=P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=d-c]$,
the whole analysis done in the previous case can be repeated. One has simply
to replace $\gamma(i)$ with $\tilde{\gamma}(i)$, $\eta(i,j)$ with
$\tilde{\eta}(i,j)$ and use (19) instead of (17), obtaining

$H_{\mathbf{S}_{N}}(X_{N}|V_{N}=c)\geqslant\sum_{a}\min_{i,j,d}\left(-P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=d-c]\log
P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=d-c]\right).$

Setting

$D_{2}:=\min_{c}\sum_{a}\min_{i,j,d}\left(-P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=d-c]\log
P_{\mathbf{S}_{N}}[S_{N+1}^{(i)}=a+c|S^{(j)}_{N}=d-c]\right)$

which is a positive constant, we conclude that

$H_{\mathbf{S}_{N}}(X_{N})+H_{\mathbf{S}_{N}}(V_{N})\geqslant D_{2},$

when we vary over any possible value of $P_{\mathbf{S}_{N}}[V_{N}=c]$ and when
the transition probabilities of the $M$ random walks are fixed. Setting
$D:=\min\\{D_{1},D_{2}\\}$ the statement of the theorem follows. This
concludes the proof.

∎

We can better grasp the physical meaning of the inequality between entropies
proved above, considering a particular case of space process. Assume that all
the random walks of the space process are _identically distributed_. This
means that if $p_{i}=P[S_{N+1}^{(i)}=a+1|S_{N}^{(i)}=a]$ are the transition
probabilities and $\pi^{(i)}$ are the probability distributions of the initial
position of all random walks, we have

$\begin{split}p_{1}&=p_{2}=\cdots=p_{M}\\\
\pi^{(1)}&=\pi^{(2)}=\cdots=\pi^{(M)}.\end{split}$

This implies that $P[S^{(i)}_{N}=a]=P[S^{(j)}_{N}=a]$ for any
$a\in\mathbb{Z}$, for any $N\geqslant 0$, and any $i,j\in\\{1,\cdots,M\\}$.
Consider the value of the constant $D_{1}$. The $\min_{i}$ can be eliminated
since all the probabilities are equals. Hence

$D_{1}=D_{2}=-p\log p-(1-p)\log(1-p)$

Thus we can conclude that $D=-p\log p-(1-p)\log(1-p)$. This is the so called
binary entropy, which vanishes only if $p=0,1$ namely if that space is a
deterministic process, a case which is excluded. The physical meaning of the
inequality $H_{\mathbf{S}_{N}}(X_{N})+H_{\mathbf{S}_{N}}(V_{N})\geqslant D$,
in this case, is now clear: the uncertainty that we have on $X_{N}$ or $V_{N}$
must be at least equal to the uncertainty we have on a single point in the
future configurations of the space process (given that at time $N$ the
configuration is $\mathbf{S}_{N}$).

###  III.5 Construction of the Hilbert space structure for model A

Theorem 1 implies that after the removal of the space process, $X_{N}$ and
$V_{N}$ are described using two distinct probability spaces if we want to use
the unconditional transition probabilities $\alpha(a+c,a)$. Theorem 2 tell us
that under the same assumptions, the position and the velocity of the particle
in model A, fulfil an entropic uncertainty relation. At this point, we may
proceed algebraically and define the smallest $C^{*}$-algebra which is capable
to describe both $X_{N}$ and $V_{N}$ after conditioning, and the entropic
uncertainty relation (2) tells us that this algebra is non-commutative LC .
Then, we can represent these elements of the algebra as two non-commuting
operators over a Hilbert space via the GNS theorem. Despite this is a
legitimate way to proceed, in this section, using the results collected in LC
, we will use a more constructive approach. In particular, we show how to
construct the operators associated to these random variables and how to define
a suitable Hilbert space on which they are defined.

Consider the position random variable $X_{N}$. After conditioning on a
particular configuration of the space process $\mathbf{S}_{N}$, $X_{N}$ can be
seen as the as the following map between probability spaces

$\begin{CD}(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}},P)|_{\mathbf{S}_{N}}@>{X_{N}}>{}>(\Omega_{X_{N}},\mathcal{E}_{X_{N}},\mu_{X_{N}})\end{CD}$

where $\Omega_{X_{N}}=X_{N}(\Omega_{I}\times\\{\mathbf{S}_{N}\\})$,
$\mathcal{E}_{X_{N}}=\mathcal{P}(\Omega_{X_{N}})$ and
$\mu_{X_{N}}:=P_{\mathbf{S}_{N}}\circ X_{N}^{-1}$. As we have seen in LC ,
random variables over a probability space form a commutative von-Neumann
algebra which is isomorphic to an algebra of multiplicative operators over an
Hilbert space. In this particular case, the random variables over
$(\Omega_{X_{N}},\mathcal{E}_{X_{N}},\mu_{X_{N}})$ (on which $X_{N}$ is
represented by the identity map) form the abelian von-Neumann algebra
$\mathcal{V}_{c}(L_{2}(\Omega_{X_{N}},\mu_{X_{N}}))$. Seen as element of this
algebra, the random variables over
$(\Omega_{X_{N}},\mathcal{E}_{X_{N}},\mu_{X_{N}})$ are multiplicative
operators over $L_{2}(\Omega_{X_{N}},\mu_{X_{N}})$.

Similar considerations hold for the velocity of the particle. The main
difference is the definition of $\Omega_{V_{N}}$, i.e. the set of all the
elementary outcomes. It is not difficult to understand that, if we fix the
space process only, $\Omega_{V_{N}}$ seems to contain more outcomes of those
one should expect. The number of outcomes of the space process is $M^{2}$,
i.e. $\mbox{card }\Omega_{X_{N}}=M^{2}$. This because the origin and the point
of $\mathbb{S}^{A}_{N}$ selected by the selection process $I_{N}$, can take
$M$ different values. For the velocity process similar considerations lead to
$\mbox{card}\Omega_{V_{N}}=M^{4}$. However, we have to take into account that
we cannot detect the movement of the origin:
$S_{N+1}^{(i_{O})}-S_{N}^{(i_{O})}$ must be set equal to $0$, and all the
situations where this does not hold must be identified with it §§§More
precisely, we can define an equivalence relation between $X_{N+1}$ and
$X_{N}$: $X_{N+1}\sim X_{N}$ if
$X_{N+1}-X_{N}=S^{(i_{o})}_{N+1}-S^{(i_{o})}_{N}$. In this way we restrict our
attention to the intrinsic motion of the particle.. After that the velocity
can takes only $M^{2}$ different values (the $M$’s of $S_{N}^{(i_{N+1})}$
times the $M$’s of $S_{N}^{(i_{N})}$). Thus doing that we have $\mbox{card
}\Omega_{V_{N}}=M^{2}$. After this observation, we may see the velocity random
variable, after conditioning to $\mathbf{S}_{N}$, as the map

$\begin{CD}(\Omega_{I}\times\Omega_{\mathbb{S}^{A}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{A}},P)|_{\mathbf{S}_{N}}@>{V_{N}}>{}>(\Omega_{V_{N}},\mathcal{E}_{V_{N}},\mu_{V_{N}})\end{CD}$

where $\Omega_{V_{N}}=V_{N}(\Omega_{I}\times\\{\mathbf{S}_{N}\\})$,
$\mathcal{E}_{V_{N}}=\mathcal{P}(\Omega_{V_{N}})$ and
$\mu_{V_{N}}:=P_{\mathbf{S}_{N}}\circ V_{N}^{-1}$. Also in this case, the
random variables over $(\Omega_{V_{N}},\mathcal{E}_{V_{N}},\mu_{V_{N}})$, are
elements of a commutative von-Neumann algebra
$\mathcal{V}_{c}(L_{2}(\Omega_{V_{N}},\mu_{V_{N}}))$ (i.e. multiplicative
operators on $L_{2}(\Omega_{V_{N}},\mu_{V_{N}})$).

Thus both $X_{N}$ and $V_{N}$ can be represented by multiplicative operators
on suitable Hilbert spaces. Note that the two Hilbert spaces are different and
depend on the probability measure. In order to construct a common Hilbert
space on which both operators are defined, we should invoke the spectral
representation theorem, as explained in LC . We recall that the spectral
decomposition theorem tells that, given an operator $\hat{T}$, there exist a
surjective isometry $\hat{U}_{i}:\mathcal{H}_{i}\rightarrow
L_{2}(\sigma(\hat{T}),\mu_{i})$ such that
$\hat{U}_{i}^{*}\hat{T}|_{\mathcal{H}_{i}}\hat{U}_{i}$ is a multiplicative
operator on $L_{2}(\sigma(\hat{T}),\mu_{i})$, i.e. an element of
$\mathcal{V}_{c}(L_{2}(\sigma(\hat{T}),\mu_{i}))$. Consider the position
random variable $X_{N}$. We know that it is a multiplicative operator on
$L_{2}(\Omega_{X_{N}},\mu_{X_{N}})$, and let us now choose to parametrise the
probability measure of the position random variable with the outcome of
$X_{N}$. This can be achieved in the following way. Take $a\in\Omega_{X_{N}}$
and consider the probability measure $P_{\mathbf{S}_{N}}^{(a)}$, which is
defined such that $\mu_{X_{N}}(c)=P_{\mathbf{S}_{N}}^{(a)}\circ
X_{N}^{-1}(c)=\delta_{a,c}$. We can parametrise the probability measure of
$X_{N}$ with its outcomes defining
$\mu_{X_{N}|a}:=P_{\mathbf{S}_{N}}^{(a)}\circ X_{N}^{-1}$. Doing that we
obtain a collection of Hilbert spaces
$\\{L_{2}(\Omega_{X_{N}},\mu_{X_{N}|a})\\}_{a\in\Omega_{X_{N}}}$. Now, the
random variable $X_{N}$ can be represented with an operator $\hat{X}_{N}$,
having spectrum $\sigma(\hat{X}_{N})=\Omega_{X_{N}}$. The spectral
decomposition theorem tells that there exists a collection of Hilbert spaces
$\\{\mathcal{H}_{a}\\}_{a\in\Omega_{X_{N}}}$ and surjective isometries
$\hat{U}_{a}:\mathcal{H}_{a}\rightarrow L_{2}(\Omega_{X_{N}},\mu_{X_{N}|a})$,
which allows to define the Hilbert space

$\mathcal{H}(X_{N}):=\bigoplus_{a\in\Omega_{X_{N}}}\mathcal{H}_{a}$

on which $\hat{X}_{N}$ can be seen as a multiplicative operator. The spectral
representation theorem tells that if $\\{|x_{N}\rangle\\}$ is a basis of
$\mathcal{H}(X_{N})$ such that $|x_{N}\rangle\in\mathcal{H}_{x_{N}}$ for any
$x_{N}\in\Omega_{X_{N}}$, then $X_{N}$ can be represented by the operator

$\hat{X}_{N}=\sum_{x_{N}\in\Omega_{X_{N}}}x_{N}|x_{N}\rangle\langle x_{N}|.$

With similar considerations, for $V_{N}$ we obtain

$\mathcal{H}(V_{N}):=\bigoplus_{c\in\Omega_{V_{N}}}\mathcal{H}_{c}$

on which the operator $\hat{V}_{N}$ representing the velocity random variable,
is diagonal

$\hat{V}_{N}=\sum_{v_{N}\in\Omega_{V_{N}}}v_{N}|v_{N}\rangle\langle v_{N}|.$
(20)

At this point, we impose the condition

$\mathcal{H}(X_{N})=\mathcal{H}(V_{N})$

i.e. that the two Hilbert spaces are _unitary equivalent_. This is possible
since the dimension of both Hilbert spaces is $M^{2}$: both Hilbert spaces are
constructed from the spectrum of $\hat{X}_{N}$ or $\hat{V}_{N}$, and both have
the same number of elements. Since Hilbert spaces of equal dimension are
always isomorphic, there exists a unitary map between them, i.e. there exists

$\hat{U}:\mathcal{H}(V_{N})\rightarrow\mathcal{H}(X_{N}),$

such that $\hat{U}\hat{U}^{*}=\hat{\mathbb{I}}_{\mathcal{H}(X_{N})}$ and
$\hat{U}^{*}\hat{U}=\hat{\mathbb{I}}_{\mathcal{H}(V_{N})}$. This unitary
mapping allows to have, on the same Hilbert space, the operators representing
the position and the velocity random variables. More precisely, take the
velocity operator $\hat{V}_{N}$ on $\mathcal{H}(V_{N})$ defined in (20), then
the unitary map mentioned above allows us to write

$\begin{split}\hat{V}_{N}|_{\mathcal{H}(X_{N})}&=\hat{U}\bigg{(}\sum_{v\in\Omega_{V_{N}}}v_{N}|v_{N}\rangle\langle
v_{N}|\bigg{)}\hat{U}^{*}\\\
&=\sum_{v_{N}\in\Omega_{V_{N}}}v_{N}\hat{U}|v_{N}\rangle\langle
v_{N}|\hat{U}^{*},\end{split}$

which represents the velocity random variable on $\mathcal{H}(X_{N})$, the
Hilbert space constructed from the spectrum of the position operator (on which
$\hat{X}_{N}$ is diagonal). The entropic uncertainty relation, ensures that
$X_{N}$ and $V_{N}$ as operators on the same Hilbert space, do not commute. In
fact, it implies maassen1988generalized

$\max_{x_{N},v_{N}}|\langle x_{N}|v_{N}\rangle|\leqslant e^{-\frac{D}{2}}<1,$
(21)

as already observed in LC . Thus the two operators cannot be diagonalised on
the same basis, i.e. they do not commute.

We can also represent on $\mathcal{H}(X_{N})$ the velocity random variable
directly. Indeed on this Hilbert space, we may always consider a generic basis
$\\{|w_{N}\rangle\\}_{w_{N}\in\Omega_{V_{N}}}$ and impose that $\hat{V}_{N}$
is diagonal on this basis, i.e.

$\hat{V}_{N}=\sum_{w_{N}\in\Omega_{V_{N}}}w_{N}|w_{N}\rangle\langle w_{N}|.$

We can always parametrise the probability measure of $\mu_{V_{N}}$ using the
outcome of $X_{N}$ simply defining
$\mu_{V_{N}|a}:=P_{\mathbf{S}_{N}}^{(a)}\circ V_{N}^{-1}$. Then we obtain the
collection of Hilbert spaces
$\\{L_{2}(\Omega_{V_{N}},\mu_{V_{N}|a})\\}_{a\in\Omega_{X_{N}}}$. For a given
$a\in\Omega_{X_{N}}$, the entropic uncertainty relation (16) forbids to have
delta-like probability measure for both operators. Indeed, considering
$\hat{X}_{N}$, we have

$\begin{split}\mu_{X_{N}|a}:=\langle\psi|\hat{P}_{\mathcal{H}_{x_{N}}}\psi\rangle=\langle\psi|x_{N}\rangle\langle
x_{N}|\psi\rangle=\delta_{x_{N},a}\end{split}$

which is possible only if $|\psi\rangle=|a\rangle$. On the other hand for
$\hat{V}_{N}$ , if $\hat{P}_{\mathcal{H}_{w_{N}}}$ is the projector on the
subspace of $\mathcal{H}(X_{N})$ associated to the eigenvalue $w_{N}$ (i.e.
the outcome $w_{N}$ of the random variable $V_{N}$), we have

$\begin{split}\mu_{V_{N}|a}:&=\langle\psi|\hat{P}_{\mathcal{H}_{w_{N}}}\psi\rangle=\langle
a|w_{N}\rangle\langle w_{N}|a\rangle=|\langle a|w_{N}\rangle|^{2}.\end{split}$

Note that under the symmetry condition (11) on the unconditional transition
probabilities, i.e. $P[V_{N}=c|X_{N}=a]=P[X_{N}=a|V_{N}=c]$, the
$\mu_{V_{N}|a}$ probabilities are consistent with usual interpretation of
transition probabilities in quantum theory. Since the entropic uncertainty
relation hold, (21) forbids that $|w_{N}\rangle$ and $|x\rangle$ to be
orthogonal. Again, we conclude that $X_{N}$ and $V_{N}$ can be represented on
a common Hilbert space, $\mathcal{H}(X_{N})$, using two operators
$\hat{X}_{N}$ and $\hat{V}_{N}$ which cannot be diagonalised on the same
basis. Note that this $\hat{V}_{N}$ coincides exactly with
$\hat{V}_{N}|_{\mathcal{H}(X_{N})}$ thanks to the existence of the unitary map
$\hat{U}:\mathcal{H}(V_{N})\rightarrow\mathcal{H}(X_{N})$. Clearly, also
$X_{N}$ can be represented on $\mathcal{H}(V_{N})$ directly, following a
similar procedure. In this sense the whole description is consistent: starting
the construction of the Hilbert space from $X_{N}$ or $V_{N}$ does not change
anything, as it should be.

Finally, we conclude by observing that the probabilistic content is now
encoded in the vectors $|\psi\rangle$ of the constructed Hilbert spaces. In
fact, given $|\psi\rangle\in\mathcal{H}(X_{N})$ (or $\mathcal{H}(V_{N})$), we
can write that

$\begin{split}\mathbb{E}_{\mathbf{S}_{N}}[V_{N}]&=\mbox{Tr}\left[|\psi\rangle\langle\psi|\hat{V}_{N}\right]=\sum_{v}v\mbox{Tr}\left[|\psi\rangle\langle\psi|v\rangle\langle
v|\right]\end{split}$

where $\mbox{Tr}\left[|\psi\rangle\langle\psi|v\rangle\langle
v|\right]=P_{\mathbf{S}_{N}}[V_{N}=v]$ is the probability distribution for
$V_{N}$ after conditioning. The probability distribution for $V_{N}$ can be
related with the distribution of $X_{N}$ as follows

$\begin{split}P_{\mathbf{S}_{N}}[V_{N}=v]&=\sum_{x}\langle
x|\psi\rangle\langle\psi|v\rangle\langle v|x\rangle\\\
&=\sum_{x}\sum_{x^{\prime}}\langle
x|\psi\rangle\langle\psi|x^{\prime}\rangle\langle x^{\prime}|v\rangle\langle
v|x\rangle\\\ &=\sum_{x}\alpha(x+v,x)P_{\mathbf{S}_{N}}[X_{N}=x]\\\
&\mspace{20.0mu}+\sum_{x\neq x^{\prime}}\langle
x|\psi\rangle\langle\psi|x^{\prime}\rangle\langle x^{\prime}|v\rangle\langle
v|x\rangle\end{split}$

where we used $P_{\mathbf{S}_{N}}[X_{N}=x]=|\langle x|\psi\rangle|^{2}$ and
$\alpha(x+v,x)=|\langle x|v\rangle|^{2}$. Note that the second term in the
last sum (the interference term) corresponds to the correction term
$\delta(V_{N}|X_{N},\mathbf{S}_{N})$ in theorem 1. However, the method used
here does not provide a way to determine uniquely the objects on
$\mathcal{H}(X_{N})$ (or $\mathcal{H}(V_{N})$) associated to a given set of
probability distributions, as already noted. In fact, the method proposed does
not provide an explicit way to compute the phase of $\langle
x_{N}|v_{N}\rangle$ starting from the interference term. However QRLA may
indicate a possible way to do that khrennikov2005interference ;
khrennikov2009contextual

###  III.6 Final remarks and main limitations of model A

Let us conclude our presentation of model A, with some observations and a
discussion of some limitations of the model.

The model is surely interesting because it is capable to derive non-commuting
operators over an Hilbert space starting from a “classical” description (in
probabilistic sense). Such non-commutativity, at least mathematically, seems
to be related to the definition used for the two random variables of the
particle process. So despite they seem reasonable definitions, we should at
least argue why they seem to be related to the position and momentum operator
in ordinary (non-relativistic) quantum theory. In particular, once we choose
to describe a quantum particle in $L_{2}(\mathbb{R}^{n})$ and we decide that
the symmetry group of non-relativistic physics is the Galilean group, by the
Stone-von Neumann theorem, we can justify that the position and momentum
operator are defined as

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

for $\psi(\mathbf{x})\in\mathcal{S}(\mathbb{R}^{n})$. Consider the 1-D case
only. In quantum mechanics, the classical relation between position and
momentum for a point-like particle ($p=m\dot{x}$) does not seem a priori
valid. Nevertheless from the Ehrenfest theorem, we have that

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

Assume that for some reason time is discrete (for example because the limited
accuracy of the clock). The limit now can be replaced by an inferior and,
using the Wigner quasi-probability distribution $W(x,p)$, we can write that

$P_{t}=m\inf_{\delta t}\frac{X_{t+\delta t}-X_{t}}{\delta
t}\mspace{50.0mu}W(x,p)\mbox{-a.s.}.$

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 justifies, at
least at the qualitative level, the use of the two random variables described
in model A. An interesting feature of the model presented here is that the
square of the number of points of the space process (which is removed) is
equal to the dimension of the (minimal) Hilbert space on which we represent
the particle process, namely $X_{N}$ and $V_{N}$. Thus, the dimension of the
Hilbert space in model A seems to encode information on the removed process,
in this model. We do not expect this feature to be fundamental (the Hilbert
space structure seems more related to probabilistic rather than “geometrical”
considerations) but this observation will be useful in future.

A limitation of the model is that time is treated as a discrete parameter, a
choice that does not allow to compare the model directly with non-relativistic
quantum mechanics. In addition, also space appears discrete, in the sense that
it can take values only over a lattice and not on the whole $\mathbb{R}$.
Summarising we can neither derive the commutation relation (1), nor attempt a
comparison with non-relativistic quantum mechanics using Model A, because of
the following: $1)$ the Hilbert space of model A is finite dimensional, $2)$
$\hat{X}_{N}$ and $\hat{V}_{N}$ are bounded operators with discrete spectrum
and, $3)$ time is discrete. However, we was able to successfully represent
position and velocity (momentum) of the particle as non-commuting operators
over a common Hilbert space, a key feature of non-relativistic quantum
mechanics.

##  IV Model B: Continuous-time 1-D kinematics on a random space

We have shown in part III that model A exhibits very interesting features from
the point of view of quantum mechanics. Nevertheless, it also has some
limitations: time is a discrete parameter and the spectrum of the position
operator $\hat{X}_{N}$ is discrete. They do not allow for a direct comparison
with ordinary quantum mechanics. To allow for this comparison, one may try to
generalize Model A to continuous-time random variables. Here we will show how
to do it.

###  IV.1 The space process

In order to generalize model A to the continuous time case, we may start by
generalizing the space process. Instead of considering the space process as a
collection of random walks, we may consider their “continuous limits”, i.e.
Wiener processes. Let us recap the basic features of the Wiener process
klebaner2005introduction , as done for the random walk. A _Wiener process
$W_{t}$ starting at $y$_ is a Gaussian process with mean
$\mathbb{E}[W_{t}]=y$, and covariance $\mathbb{E}[W_{t}W_{s}]=\min(t,s)$. This
is one of the possible equivalent definitions of a Wiener process, and it
implies that (in the 1D case)

$P[W_{t}\in A]=\int_{A}\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^{2}}{2t}}dx.$

As consequence of its definition, the Wiener process $W_{t}$ is a continuous
function of the parameter $t$ for all $t\in\mathbb{R}^{+}$, in the sense that
there exists always a _continuous version_ of the Wiener process (with
“version of a process $X_{t}$ ” we mean that there exists another process
$Y_{t}$ such that $P[X_{t}=Y_{t}]=1$ for any $t\in\mathbb{R}^{+}$, i.e. the
two processes are statistically indistinguishable). For a Wiener process, the
trajectories (which can be thought as the function $\omega(t):=W_{t}(\omega)$)
have the following properties: $i)$ they are nowhere differentiable; $ii)$
they are never monotone; $iii)$ they have infinite variation in any interval;
$iv)$ they have quadratic variation equal to $t$ in the interval $[0,t]$. More
generally, let $C(\mathbb{R}^{+},\mathbb{R})$ be the space of all functions
$t\mapsto f_{t}$ taking value on $\mathbb{R}$ and continuous for any
$t\in\mathbb{R}^{+}$. $C(\mathbb{R}^{+},\mathbb{R})$ can be equipped with a
norm, which allows to define open sets (i.e. a topology). As usual these open
sets can be used to construct a Borel $\sigma$-algebra on
$C(\mathbb{R}^{+},\mathbb{R})$, say
$\mathscr{B}(C(\mathbb{R}^{+},\mathbb{R}))$. The Wiener process can be seen as
the identity function on
$(C(\mathbb{R}^{+},\mathbb{R}),\mathscr{B}(C(\mathbb{R}^{+},\mathbb{R}),\gamma)$
where $\gamma$ is the so called _Wiener measure_. The set of all continuous
functions $f_{t}\in C(\mathbb{R}^{+},\mathbb{R})$ which does not fulfil $i)$ –
$iv)$ have zero measure under $\gamma$. Such a probability space is called
_Wiener space_. Finally we conclude by observing that if also the starting
position $y$ is a random variable with distribution $\pi(dy)$ over
$\mathbb{R}$, then

$P[W_{t}\in A]=\int_{A}\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi
t}}e^{-\frac{(x-y)^{2}}{2t}}\pi(dy)dx.$

Let us now consider the space process for this model. As assumed for model A,
space is discrete and evolves with time. In particular, we have the following
preliminary definition which generalizes the one given for model A.

######  Definition 6.

Let $\\{W_{t}^{(i)}\\}_{i\in I}$ be a collection of independent Wiener
processes, where $|I|=M\in\mathbb{N}$. Such collection will be called _space
process_ for model B.

We will label this process by $\mathbb{S}^{B}$. At any given time
$t\in\mathbb{R}^{+}$, the space process is a collection of $M$ points on
$\mathbb{R}$, which are the positions of the $M$ Wiener processes: in this
sense the space is discrete and evolves, in a continuous way, in time. Also in
this case we may have two possible descriptions of the space: one is to
consider the points with respect to the real line (we will choose this point
of view, as done for model A), while the second is to describe the effects of
the time evolution from the point of view of the “ordering among the points”
(as explained in appendix B). Because of independence, equation (5) holds true
if we simply substitute $P[S_{N}^{(i)}=s_{i}]$ with $P[W_{t}^{(i)}\in A_{i}]$,
where $A_{i}\subset\mathbb{R}$ for any $i\in\\{1,\cdots,M\\}$, and similarly
for (6) and generalisation. Because $P[W_{t}^{(i)}\in A_{i}]$ can be written
as the integral over $A_{i}$ with respect to a probability density
$\rho_{W_{t}^{(i)}}(x_{i})$, equation (5) is replaced by the following

$\rho_{\mathbb{S}^{B}_{t}}(\mathbf{S}_{t})=\prod_{i=1}^{M}\rho_{W_{t}^{(i)}}(x_{i}),$
(22)

where $\rho_{\mathbb{S}^{B}_{t}}(\mathbf{S}_{t})$ is the probability density
of the probability measure $\tilde{P}^{B}[\mathbb{S}^{B}_{t}\in A]$. In a
similar way one can generalise (6) and any other density for the space
process. At this point, as done for model A, we may give the following
definition for the space process.

######  Definition 7.

Let $\\{W_{t}^{(i)}\\}_{i\in I}$ be a collection of $M=|I|\in\mathbb{N}$
Wiener processes defined on the Wiener spaces
$\\{(\Omega_{i},\mathcal{E}_{i},P_{i})\\}_{i\in I}$. Let us define

  1. i)

$\Omega_{\mathbb{S}^{B}}:=\Omega_{1}\times\cdots\times\Omega_{M}$;

  2. ii)

$\mathcal{E}_{\mathbb{S}^{B}}$ is the Borel $\sigma$-algebra generated by the
open sets of $\Omega_{\mathbb{S}^{B}}$¶¶¶To define an open set on
$\Omega_{\mathbb{S}^{B}}$ we may use the topology induced by the norm
$\|\mathbf{S}\|:=\sup_{t\in[0,T]}|\mathbf{S}_{t}|_{M}$, where $|\cdot|_{M}$ is
the $M$-dimensional euclidean norm. This is what is typically done on Wiener
spaces.;

  3. iii)

$\tilde{P}^{B}:\mathcal{E}_{\mathbb{S}^{B}}\rightarrow[0,1]$ defined from the
$\\{P_{i}\\}_{i\in I}$, via the densities as in (22) and generalisations.

The space process is the stochastic process on
$(\Omega_{\mathbb{S}^{B}},\mathcal{E}_{\mathbb{S}^{B}},\tilde{P}^{B})$ defined
as the identity function, namely
$\mathbb{S}^{B}(\omega_{1},\cdots,\omega_{M})=(\omega_{1},\cdots,\omega_{M})$.

The set of all possible configurations of the space process at time $t$ will
be labeled by the symbol $\mathcal{S}(t)$. This completes our description for
the space process in model B.

###  IV.2 The particle process

Again, a particle is considered as a point-like object. It jumps from one
point of space to another and it is completely characterized by the _position_
and _velocity_ random variables.

The _position random variable_ , labeled by $X_{t}$, is interpreted as the
actual position of the particle at time $t$ with respect to a chosen origin.
Hence, if
$(\Omega_{\mathbb{S}^{B}},\mathcal{E}_{\mathbb{S}^{B}},\tilde{P}^{B})$ is the
probability space of the space process, $(\Omega_{I},\mathcal{E}_{I},P_{I})$
is a probability space on which an integer value stochastic process
$I_{t}:\Omega_{I}\rightarrow\\{1,\cdots,M\\}$ is defined (called _section
process_), and $W_{t}^{(i_{0})}$ is a chosen origin of a reference frame on
$\mathbb{S}^{B}_{t}$, then

$X_{t}(\omega_{X}):=\pi_{I_{t}(\omega_{I})}(\mathbb{S}^{B}_{t}(\mathbf{S}))-W_{t}^{(i_{0})},$
(23)

where $\pi_{i}$ is the projector of the $i$-th component of an $M$-tuple, and
$\omega_{X}=(\omega_{I},\mathbf{S})$ with $\omega_{I}\in\Omega_{I}$ and
$\mathbf{S}\in\Omega_{\mathbb{S}^{B}}$. Thus we have the following definition.

######  Definition 8.

Consider the probability space
$(\Omega_{I}\times\Omega_{\mathbb{S}^{B}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{B}},P^{B})$
and a measurable space $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. The random
variable $X_{t}$ is the $\mathscr{B}(\mathbb{R})$-measurable function

$X_{t}:\Omega_{I}\times\Omega_{\mathbb{S}^{B}}\rightarrow\mathbb{R}$

defined as in (23). $X_{t}$ represents the position of the particle at time
$t$.

Clearly, as any random variable $X_{t}$ induces a probability distribution
$\mu_{X_{t}}=P^{B}\circ X_{t}^{-1}$ and, on the probability space
$(\mathbb{R},\mathscr{B}(\mathbb{R}),\mu_{X_{t}})$ it can be considered as the
identity function. Also in this case the space process can be described on
$(\Omega_{I}\times\Omega_{\mathbb{S}^{B}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{B}},P^{B})$,
by simply demanding that $P^{B}\circ[\mathbb{S}^{B}]^{-1}=\tilde{P}^{B}$.
Again if no confusion arises, we omit the suffix B in the probability measure
$P^{B}$.

In model B, the particle moves by jumps from one point to another. This time
the frequency of the jumps is assumed to be infinite, which means that the
particle jumps from one point to another at each instant of time. In this way,
we can say that it is the continuous time generalization of the kinematics
described in model A. We do not generalize the definition of the velocity
process given before directly. This time we use the following definition:

$V_{t}(t^{\prime}):=\frac{X_{t^{\prime}}-X_{t}}{t^{\prime}-t}$ (24)

where we always assume $t^{\prime}>t$. More formally we adopt the following
definition for $V_{t}(t^{\prime})$.

######  Definition 9.

Consider the probability space
$(\Omega_{I}\times\Omega_{\mathbb{S}^{B}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{B}},P^{B})$
and a measurable space $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Let
$t,t^{\prime}\in\mathbb{R}$ such that $t^{\prime}>t$, the random variable
$V_{t}(t^{\prime})$ is the $\mathscr{B}(\mathbb{R})$-measurable function

$V_{t}(t^{\prime}):\Omega_{I}\times\Omega_{\mathbb{S}^{B}}\rightarrow\mathbb{R}$

defined in (24). $V_{t}(t^{\prime})$ represents the mean velocity of the
particle in the interval $[t,t^{\prime}]$.

Also in this case $V_{t}(t^{\prime})$ can be seen as the identity random
variable on the probability space
$(\mathbb{R},\mathscr{B}(\mathbb{R}),\mu_{V_{t}(t^{\prime})})$, where
$\mu_{V_{t}(t^{\prime})}=P^{B}\circ V_{t}(t^{\prime})^{-1}$. As in model A,
for the description of the particle we need to introduce the transition
probabilities. These allow to write that

$\mu_{V_{t}(t^{\prime})}(v)=\int_{\mathbb{R}}\alpha(v,x;t^{\prime})\mu_{X_{t}}(x)dx$
(25)

where $\alpha(v,x;t^{\prime})$ are the probability densities of
$V_{t}(t^{\prime})$ _given the event_ $\\{X_{t}=x\\}$. Note that they depend
also on the value of $t^{\prime}$ used to define $V_{t}(t^{\prime})$. Also in
this case we assume that they are _symmetric_ under the exchange of their
arguments, namely $\alpha(v,x;t^{\prime})=\alpha(x,v;t^{\prime})$, where
$\alpha(x,v;t^{\prime})$ is the probability density of $X_{t}$ _given th
event_ $\\{V_{t}(t^{\prime})=v\\}$. Note that this expression is nothing but
the Bayes theorem for continuous random variables (see appendix C, Prop. 4).
In what follows we will omit $t^{\prime}$ in $\alpha(v,x;t^{\prime})$ and
$\alpha(x,v;t^{\prime})$ if no confusion arises.

We conclude this section defining the particle process for this model. With
the definitions 8 and 9 we gave a meaning to the position and velocity _random
variables_. However, they are parametrised by $t$, the physical time which we
choose to treat as an external parameter. Since we have a collection of random
variables parametrised by $t$, we can speak of _position and velocity
stochastic process_ , but with the same caution explained in section III.2.

######  Definition 10.

Let $X_{t}$ and $V_{t}(t^{\prime})$ be the position and velocity process. The
couple $\mathscr{P}_{t}(t^{\prime})=(X_{t},V_{t}(t^{\prime}))$ is called
_particle process_ of model B.

An example of particle process over a space process is drawn in figure 3.

Figure 3: The particle process in model B is drawn in red. The position of the
particle at a given time is given by the red point while its velocity at the
same time is represented by the outgoing arrow. On the back, a possible
realization of an $M=6$ space process.

###  IV.3 The removal of the space process

The removal of the space process in model B is done exactly as before:

  1. i)

We consider only the conditional probability densities
$\mu_{X_{t}|\mathbf{S}_{t}}(x)$ and
$\mu_{V_{t}(t^{\prime})|\mathbf{S}_{t}}(v)$ for the random variables $X_{t}$
and $V_{t}(t^{\prime})$;

  2. ii)

We fix the transition probabilities of the single point of space, i.e. we fix
the transition probabilities
$p^{(i)}(x,t^{\prime};y,t):=\rho_{W_{t^{\prime}}^{(i)}|W_{t}^{(i)}=y}(x)$ of
all the $M$ Wiener processes.

As in Model A, requirement $i)$ implies that we will always work with the
densities $\mu_{X_{t}|\mathbf{S}_{t}}(x)$ and
$\mu_{V_{t}(t^{\prime})|\mathbf{S}_{t}}(v)$, namely the probability
distributions for $X_{t}$ and $V_{t}(t^{\prime})$ _given the event_
$\\{\mathbb{S}^{B}_{t}=\mathbf{S}_{t}\\}$. Clearly, we can define a joint
probability space for $X_{t}$ and $V_{t}(t^{\prime})$ after conditioning on
$\\{\mathbb{S}^{B}_{t}=\mathbf{S}_{t}\\}$ and, on this joint probability
space, some _conditional_ transition probabilities
$\alpha_{\mathbf{S}_{t}}(v,x)$ can be defined. However, if we insist in using
the _unconditional probability density_ $\alpha(v,x)$, no joint probability
space can be defined. Indeed, we can prove the analogue of theorem 1. Below we
will prove the theorem in a slightly more general setting of what we need
later: the simple case of absolute continuous measure with respect to Lebesgue
will be discussed as an example later.

######  Theorem 3.

Let
$(\Omega_{I}\times\Omega_{\mathbb{S}^{B}},\mathcal{E}_{I}\otimes\mathcal{E}_{\mathbb{S}^{B}},P)$
be the probability space on which $\mathbb{S}^{B}_{t}$, $X_{t}$ and
$V_{t}(t^{\prime})$ are defined. Assume that the probability spaces for each
of these random variables has the regular conditional probability property.
Then

$\begin{split}P_{\mathbf{S}_{t}}[V_{t}(t^{\prime})\in
B]=&\int_{B}\int_{\Omega_{X_{t}}}\alpha(v,x)P_{\mathbf{S}_{t}}[X_{t}\in dx]\\\
&+\delta(B|X_{t},\mathbf{S}_{t})\end{split}$ (26)

where $\delta(B|X_{t},\mathbf{S}_{t})$ is the Radon-Nikodym derivative with
respect to $P\circ[\mathbb{S}^{B}_{t}]^{-1}=\mu_{\mathbb{S}^{B}_{t}}$, of the
measure $\Gamma\mapsto\Delta(B|X_{t},\Gamma)$ defined as

$\begin{split}\Delta(B|X_{t},\Gamma):=&\int_{\mathcal{S}(t)/\Gamma}\bigg{[}\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\cdot\\\
&\cdot\mu_{X_{t}|\mathbf{S}_{t}^{\prime}}(dx)-P_{\mathbf{S}_{t}^{\prime}}[V_{t}(t^{\prime})\in
B]\bigg{]}\mu_{\mathbb{S}^{B}_{t}}(d\mathbf{S}_{t}^{\prime}).\end{split}$ (27)

when $\mathbf{S}_{t}\in\Gamma\subset\mathcal{S}_{p}^{\epsilon}(t)$.

###### Proof.

Since $\\{V_{t}(t^{\prime})\in B\\}=\\{V_{t}(t^{\prime})\in
B\\}\cup\\{\mathbb{S}^{B}_{t}\in\mathcal{S}(t)\\}$, using the regular
conditional probability property of probability space, we have

$\begin{split}P[&\\{V_{t}(t^{\prime})\in
B\\}\cup\\{\mathbb{S}^{B}_{t}\in\mathcal{S}(t)\\}]=P[V_{t}(t^{\prime})\in
B]\\\ &=\mathbb{E}[\chi_{\\{V_{t}(t^{\prime})\in B\\}}(\omega)]\\\
&=\int_{\mathcal{S}(t)}P_{\mathbf{S}_{t}^{\prime}}[V_{t}(t^{\prime})\in
B](P\circ[\mathbb{S}^{B}_{t}]^{-1})(d\mathbf{S}_{t}^{\prime})\end{split}$

where $P_{\mathbf{S}_{t}^{\prime}}[V_{t}(t^{\prime})\in B]$ is the regular
conditional probability $P[V_{t}(t^{\prime})\in
B|\mathbb{S}^{B}_{t}=\mathbf{S}_{t}^{\prime}]$. Take
$\Gamma\subset\mathcal{S}(t)$ and set
$\mu_{\mathbb{S}^{B}}(d\mathbf{S}_{t}^{\prime})=(P\circ[\mathbb{S}^{B}_{t}]^{-1})(d\mathbf{S}_{t}^{\prime})$,
then we can write

$\begin{split}P[V_{t}(t^{\prime})\in
B]&=\int_{\Gamma}P_{\mathbf{S}_{t}^{\prime}}[V_{t}(t^{\prime})\in
B]\mu_{\mathbb{S}^{B}}(d\mathbf{S}_{t}^{\prime})\\\
&+\int_{\mathcal{S}(t)/\Gamma}P_{\mathbf{S}_{t}^{\prime}}[V_{t}(t^{\prime})\in
B]\mu_{\mathbb{S}^{B}}(d\mathbf{S}_{t}^{\prime}).\end{split}$

Consider now the random variable $X_{t}$ and the $\sigma$-algebra
$\mathcal{E}_{X_{t}}$. Then, by the law of conditional expectation, we can
write that

$\begin{split}P[&\\{V_{t}(t^{\prime})\in
B\\}\cup\\{\mathbb{S}^{B}_{t}\in\mathcal{S}(t)\\}]=P[V_{t}(t^{\prime})\in
B]\\\ &=\mathbb{E}[\chi_{\\{V_{t}(t^{\prime})\in B\\}}(\omega)]\\\
&=\mathbb{E}[\mathbb{E}[\chi_{\\{V_{t}(t^{\prime})\in
B\\}}(\omega)|\mathcal{E}_{X_{t}}]]\\\
&=\int_{\mathcal{S}(t)}\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\mu_{X_{t}|\mathbf{S}_{t}^{\prime}}(dx)\mu_{\mathbb{S}^{B}_{t}}(d\mathbf{S}_{t}^{\prime})\end{split}$

where we used the fact that the conditional expectation
$\mathbb{E}[\chi_{\\{V_{t}(t^{\prime})\in B\\}}(\omega)|\mathcal{E}_{X_{t}}]$
is a random variable and the disintegration theorem (see appendix C, Th. 6).
Then, as done before, for a given $\Gamma\subset\mathcal{S}(t)$ we can write
that:

$\begin{split}P[V_{t}(t^{\prime})\in
B]=\int_{\Gamma}\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\mu_{X_{t}|\mathbf{S}_{t}^{\prime}}(dx)\mu_{\mathbb{S}^{B}_{t}}(d\mathbf{S}_{t}^{\prime})+\int_{\mathcal{S}(t)/\Gamma}\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\mu_{X_{t}|\mathbf{S}_{t}^{\prime}}(dx)\mu_{\mathbb{S}^{B}_{t}}(d\mathbf{S}_{t}^{\prime}).\end{split}$

Comparing the two expressions found for $P[V_{t}(t^{\prime})\in B]$ we obtain

$\int_{\Gamma}P_{\mathbf{S}_{t}^{\prime}}[V_{t}(t^{\prime})\in
B]\mu_{\mathbb{S}^{B}}(d\mathbf{S}_{t}^{\prime})=\int_{\Gamma}\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\mu_{X_{t}|\mathbf{S}_{t}^{\prime}}(dx)\mu_{\mathbb{S}^{B}_{t}}(d\mathbf{S}_{t}^{\prime})+\Delta(B|X_{t},\Gamma)$
(28)

with

$\begin{split}\Delta(B|X_{t},\Gamma)=\int_{\mathcal{S}(t)/\Gamma}\bigg{[}\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\mu_{X_{t}|\mathbf{S}_{t}^{\prime}}(dx)-P_{\mathbf{S}_{t}^{\prime}}[V_{t}(t^{\prime})\in
B]\bigg{]}\mu_{\mathbb{S}^{B}_{t}}(d\mathbf{S}_{t}^{\prime}).\end{split}$

Note that the map $\Gamma\mapsto\Delta(B|X_{t},\Gamma)$ is a (signed) measure.
Now we take the Radon-Nikodym derivative of (28) with respect to the measure
$\mu_{\mathbb{S}^{B}_{t}}$, getting

$\begin{split}P_{\mathbf{S}_{t}}[V_{t}(t^{\prime})\in
B]&=\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\mu_{X_{t}|\mathbf{S}_{t}}(dx)\\\
&+\delta(B|X_{t},\mathbf{S}_{t})\end{split}$

for some fixed $\mathbf{S}_{t}\in\Gamma$ and where

$\delta(B|X_{t},\mathbf{S}_{t})=\frac{d}{d\mu_{\mathbb{S}^{B}_{t}}}\Delta(B|X_{t},\Gamma).$

Using (25) in the last expression of $P_{\mathbf{S}_{t}}[V_{t}(t^{\prime})\in
B]$, we obtain the claimed result. This concludes the proof. ∎

On the contrary to what we found for model A, where an explicit expression of
$\delta(c|X_{N},\mathbf{S}_{N})$ was given, the expression we find in general
for model B is purely formal. A simple case where we may compute the Radon-
Nikodym derivative $\delta(B|X_{t},\mathbf{S}_{t})$ is described here. Note
that trivially

$\begin{split}\Delta(B|X_{t},\mathbf{S}_{t})=-\bigg{[}&\int_{\Omega_{X_{t}}}P[V_{t}(t^{\prime})\in
B|X_{t}=x]\mu_{X_{t}|\mathbf{S}_{t}}(dx)\\\
&-P_{\mathbf{S}_{t}}[V_{t}(t^{\prime})\in B]\bigg{]}.\end{split}$

Assuming that all measures in the above expression admit a density with
respect to the Lebesgue measure and also $\Delta(B|X_{t},\Gamma)$ admits this
density, i.e.

$\Delta(B|X_{t},\Gamma)=\int_{B}\int_{\Gamma}\delta(v|X_{t},\mathbf{S}_{t})dvd\mathbf{S}_{t},$

one can immediately derive the following relation:

$\begin{split}\delta(v|X_{t},\mathbf{S}_{t})&=-\bigg{[}\int_{\Omega_{X_{t}}}\alpha(v,x)\mu_{X_{t}|\mathbf{S}_{t}}(x)dx-\mu_{V_{t}(t^{\prime})|\mathbf{S}_{t}}(v)\bigg{]}\\\
&=-\frac{1}{\mu_{\mathbb{S}^{B}_{t}}(\mathbf{S}_{t})}\bigg{[}\int_{\Omega_{X_{t}}}\alpha(v,x)\mu_{X_{t},\mathbb{S}^{B}_{t}}(x,\mathbf{S}_{t})dx-\mu_{V_{t}(t^{\prime}),\mathbb{S}^{B}_{t}}(v,\mathbf{S}_{t})\bigg{]}.\end{split}$
(29)

This expression can be considered as the analogous of equation (14), found for
model A. In fact, as for $\delta(c|X_{N},\mathbf{S}_{N})$, it can be computed
using the joint probability densities, information that is lost after the
conditioning with respect to $\mathbb{S}^{B}_{t}$. Thus also here,
$\delta(v|X_{t},\mathbf{S}_{t})$ contains information about the space process
(as in model A).

Let us now analyze the consequences of $ii)$, the following observation is
useful. Consider the velocity random variable of model B. Setting $\delta
t:=t^{\prime}-t$ we can write

$V_{t}(t^{\prime})=V_{t}(t+\delta t)=\frac{X_{t+\delta t}-X_{t}}{\delta t}.$

Since $t$ is a parameter, we can always rescale it in order to have $\delta
t=1$. In this case, $V_{t}(t+1)$ resemble the velocity random variable $V_{N}$
of model A. This can be done for any value of $t^{\prime}>t$. To make this
correspondence more concrete, we may also discretize the space process. More
precisely, since the points of the space process are Wiener processes taking
values on $\mathbb{R}$, we can partition $\mathbb{R}$ in intervals
$\\{\Delta_{k}\\}_{k\in K}$ (i.e. $\cup_{k\in K}\Delta_{k}=\mathbb{R}$) where
$K\subset\mathbb{N}$. At this point one can consider the discretized random
variable for the space process and the position random variable. If
$W_{t}^{(i)}$ is a point of the space process $\mathbb{S}^{B}_{t}$, one can
define a new random variable $S_{t}^{(i)}:\Omega_{\mathbb{S}^{B}}\rightarrow
K$ as

$S_{t}^{(i)}(\omega):=k\mspace{30.0mu}\mbox{if}\mspace{10.0mu}W_{t}^{(i)}(\omega)\in\Delta_{k},$

which simply reveals in which $\Delta_{k}$ the Wiener process is. Clearly,
$P[S_{t}^{(i)}=k]=\int_{\Delta_{k}}\rho_{W_{t}^{(i)}}(x)dx$ and given the
transition probability densities for the Wiener process, say
$p^{(i)}(x,t+1;y,t)$, the transition probabilities for $S_{t}^{(i)}$ are
given, i.e.

$\begin{split}P[S_{t+1}^{(i)}&=k|S_{t}^{(i)}=j]\\\
&=\frac{\int_{\Delta_{k}}dx\int_{\Delta_{j}}dyp^{(i)}(x,t+1;y,t)\rho_{W_{t}^{(i)}}(y)}{\int_{\Delta_{j}}\rho_{W_{t}^{(i)}}(y)dy}.\end{split}$

At the end of this procedure one ends up with a discretized version of the
space process of model B, which is equivalent to the one used in model A. The
same discretization procedure can be done for the position and velocity random
variables $X_{t}$ and $V_{t}(t+1)$. It is not difficult to realize that
theorem 2 can be applied and its application does not depend on the size of
the sets $\\{\Delta_{k}\\}_{k\in K}$. Thus, as in the previous model, the
requirement $ii)$ implies the entropic uncertainty relation between the
position and the velocity random variables. Then as in model A, this relation
can be used to prove that $X_{t}$ and $V_{t}(t^{\prime})$, after conditioning
on $\mathbf{S}_{t}$, are representable as two non-commuting operators on the
same Hilbert space. This will be discussed in the next section. Let us now
describe a bit further how to obtain the entropic uncertainty relation from
the discretization of model B. First of all, if we want to apply the results
listed in LC , we need to be sure that the two random variables are bounded,
i.e. the set of all values they can assume is a bounded set. In fact, only in
this case, they can be associated to two bounded self-adjoint operators, which
are elements of a $C^{*}$-algebra, and the relation between non-commutativity
and the entropic uncertainty relation holds true. In order to do that, we
consider the restriction of the two random variables to a given subset. More
precisely, given $\Lambda\subset\mathbb{R}=\Omega_{X_{t}}$, the _bounded
version_ of $X_{t}$ will be the random variable

$X_{t}|_{\Lambda}(\omega):=X_{t}(\omega)\chi_{\Lambda}(X_{t}(\omega))$

where $\chi_{\Lambda}(x)$ is the indicator function of the set $\Lambda$.
Similarly, we can define the bounded version of $V_{t}(t^{\prime})|_{\Gamma}$.
At this point we consider the discrete version of these random variables,
similarly to what we did for the space process. Given $X_{t}|_{\Lambda}$, we
can discretise it simply by dividing the set $\Lambda$ in $N$ parts of equals
size, obtaining a partition $\\{\Delta_{N,k}^{X}\\}_{k\in K}$,
$K\subset\mathbb{N}$, such that
$|\Delta_{N,k}^{X}|=|\Delta_{N,k^{\prime}}^{X}|$ for any possible $k$. We can
see that the number of subsets of the partition (i,e. $N=|K|$) determines the
width of the sets $\Delta_{N,k}^{X}$. The bounded and _discrete version_ of
$X_{t}$ is then defined as

$X_{t}^{\Delta}|_{\Lambda}(\omega):=k\mspace{30.0mu}\mbox{if}\mspace{10.0mu}X_{t}(\omega)|_{\Lambda}\in\Delta_{N,k}^{X}.$

The same construction can be done for the bounded version of
$V_{t}(t^{\prime})$, using in general a different partition
$\\{\Theta_{N^{\prime},j}^{V}\\}_{j\in J}$, obtaining
$V_{t}(t^{\prime})^{\Theta}|_{\Gamma}$. It is useful to choose the partitions
for $X_{t}$ and $V_{t}(t^{\prime})$ compatible with the partition used for the
space process. To do that it is enough to set the partition for $X_{t}$ and
$\mathbb{S}^{B}_{t}$ equal and choose the partition for $V_{t}(t^{\prime})$
consequently. Finally we also chose to set $|\Gamma|=|\Lambda|$, i.e. the size
of the two set used to bound the position and velocity random variable
coincides. At this point, by discretising time as explained above
$X_{t}^{\Delta}|_{\Lambda}$ and $V_{t}(t+1)^{\Theta}|_{\Gamma}$ (for
simplicity we simply write $V_{t}^{\Theta}|_{\Gamma}$) become discrete random
variables similar to those used for model A. Then, applying theorem 2, we know
that

$H_{\mathbf{S}_{t}}(X_{t}^{\Delta}|_{\Lambda})+H_{\mathbf{S}_{t}}(V_{t}^{\Theta}|_{\Gamma})\geqslant
D,$ (30)

where $D$ is a positive constant that in general can depend on the partition
chosen but not on the probability distribution of $X_{t}^{\Delta}|_{\Lambda}$
and $V_{t}^{\Theta}|_{\Gamma}$, hence thet do not depend on
$\mu_{X_{t}|\mathbf{S}_{t}}$ and $\mu_{V_{t}|\mathbf{S}_{t}}$. The whole
construction does not depend on the partitions chosen, once they are chosen in
the consistent way explained above. In particular, the above inequality holds
for arbitrary partitions having small but finite size.

###  IV.4 Construction of the Hilbert space structure for model B

The construction of the Hilbert space structure for model B goes more or less
as in Model A. However, in this case, we have some additional technicalities
due to the use of the partitions for the description of the two random
variables involved. The entropic uncertainty relation (30), ensures that
$X_{t}^{\Delta}|_{\Lambda}$ and $V_{t}^{\Theta}|_{\Gamma}$, after conditioning
on $\mathbf{S}_{t}$, can be jointly described only on a non-commutative
probability space, i.e. with non-commuting operators. Let us fix for the
moment the partitions used. As in model A, the bounded and discrete version of
the position random variable can be represented on the Hilbert space

$\mathcal{H}(X_{t}|N,\Lambda)=\bigoplus_{k=1}^{N}\mathcal{H}_{k},$

as the diagonal operator

$\hat{X}_{t}(N,\Lambda)=\sum_{k=1}^{N}k|k\rangle\langle k|.$

Here $|k\rangle\in\mathcal{H}_{k}$ and
$\hat{P}^{(\hat{X}_{t}(N,\Lambda))}_{k}:=|k\rangle\langle k|$ is the PVM such
that

$P[X_{t}^{\Delta}|_{\Lambda}=k]=P[X_{t}|_{\Lambda}\in\Delta^{X}_{N,k}]=\langle\psi|\hat{P}^{(\hat{X}_{t}(N,\Lambda))}_{k}|\psi\rangle$
(31)

for some $\psi\in\mathcal{H}(X_{t}|N,\Lambda)$. Similarly, the bounded and
discrete version of the velocity random variable can be represented on the
Hilbert space

$\mathcal{H}(V_{t}|N,\Gamma)=\bigoplus_{j=1}^{N}\mathcal{H}_{j}$

(note that particular partitions considered implies that $N=N^{\prime}$) as
the diagonal operator

$\hat{V}_{t}(N,\Gamma)=\sum_{j=1}^{N}j|j\rangle\langle j|.$

The two Hilbert spaces $\mathcal{H}(X_{t}|N,\Lambda)$ and
$\mathcal{H}(V_{t}|N,\Gamma)$ have the same dimension and so they are unitary
equivalent, i.e. there exists a unitary map
$\hat{U}:\mathcal{H}(V_{t}|\Gamma)\rightarrow\mathcal{H}(X_{t}|\Lambda)$.
Hence we can represent $\hat{V}_{t}(N,\Gamma)$ on
$\mathcal{H}(X_{t}|N,\Lambda)$ and viceversa. The entropic uncertainty
relation (30) ensures that

$[\hat{X}_{t}(N,\Lambda),\hat{V}_{t}(N,\Gamma)]\neq 0$ (32)

Let us now analyse what happens when we change the size of the partition.
First, we consider the limit $N\rightarrow\infty$ which means that the size of
the partitions goes to zero. Because the sets $\Delta^{X}_{N,k}$ shrink to a
point, say $\\{x\\}$, we have

$\lim_{N\rightarrow\infty}\langle\psi|\hat{P}^{(\hat{X}_{t}(N,\Lambda))}_{k}|\psi\rangle=\lim_{N\rightarrow\infty}[X_{t}|_{\Lambda}\in\Delta^{X}_{N,k}]=0$
(33)

for any $\psi$, i.e. any $P$. This means, by prop 9.14 of moretti2013spectral
, $x\in\sigma_{c}(\hat{X}_{t}(\Lambda))$ (here
$\hat{X}_{t}(\Lambda):=\hat{X}_{t}(\infty,\Lambda)$ ). By the arbitrariness of
$x$ we conclude, as expected, that $\hat{X}_{t}(\Lambda)$ is a bounded
operator with purely continuous spectrum. Note that the Hilbert space on which
we can define $\hat{X}_{t}(\Lambda)$ is

$\mathcal{H}(X_{t}|\Lambda):=\int^{\oplus}_{\Lambda}\mathcal{H}_{x}dx$

which is not separable in general. Here, $\hat{X}_{t}(\Lambda)$ can be written
as

$\hat{X}_{t}(\Lambda)=\int_{\Lambda}xP^{(\hat{X}_{t}(\Lambda))}(dx).$

Similar conclusions hold for the operator representing the bounded and
discrete velocity random variable:
$\hat{V}_{t}(\Gamma):=\hat{V}_{t}(\infty,\Gamma)$ is a bounded operator with
continuous spectrum. Since for any value of $N$, $\hat{X}_{t}(N,\Lambda)$ is
the operator representing the random variable obtained by discretizing the
_same_ random variable $X_{t}|_{\Lambda}$, also the operators
$\hat{X}_{t}(N,\Lambda)$ can be obtained by discretising the same operator
$\hat{X}_{t}(\Lambda)$. The same holds for $\hat{V}_{t}(N,\Gamma)$. At this
point because (32) is valid for any possible partition chosen in the
consistent way explained in the previous section (i.e. for any $N$), we can
conclude that

$[\hat{X}_{t}(\Lambda),\hat{V}_{t}(\Gamma)]\neq 0.$

Since $\Gamma$ and $\Lambda$ are arbitrary, with similar considerations we may
conclude that

$[\hat{X}_{t},\hat{V}_{t}]\neq 0$ (34)

where $\hat{X}_{t}$ is the unbounded operator on a Hilbert space
$\mathcal{H}(X_{t}):=\mathcal{H}(X_{t}|\mathbb{R})$ such that
$\hat{X}_{t}(\Lambda)=\hat{P}_{\Lambda}\hat{X}_{t}\hat{P}_{\Lambda}$ (here
$\hat{P}_{\Lambda}$ is the projector from $\mathcal{H}(X_{t})$ to the Hilbert
space $\mathcal{H}(X_{t}|\Lambda)$) and $\hat{V}_{t}$ is defined in a similar
manner.

We conclude by observing that $\mathcal{H}(X_{t}|\Lambda)$ and
$\mathcal{H}(V_{t}|\Gamma)$ may be not separable (and so also
$\mathcal{H}(X_{t})$ and $\mathcal{H}(V_{t})$). In general, non-separable
infinite-dimensional Hilbert spaces are not mutually isomorphic. Thus in this
case we cannot define a unitary map
$\hat{U}:\mathcal{H}(V_{t})\rightarrow\mathcal{H}(X_{t})$ which maps the
operator representations of $X_{t}$ and $V_{t}$ on $\mathcal{H}(V_{t})$ into
the corresponding operators in $\mathcal{H}(X_{t})$. This is an effect of the
possible lack of separability of the Hilbert spaces $\mathcal{H}(X_{t})$ and
$\mathcal{H}(V_{t})$. However this does not mean that we cannot represent the
velocity random variable on $\mathcal{H}(X_{t}|\Lambda)$ and vice-versa: one
simply represents the velocity random variable on
$\mathcal{H}(X_{t}|N,\Lambda)$ and then takes the limit. However, to have a
consistent description the velocity operator obtained in this limit must be
isomorphic to the operator $\hat{V}_{t}$ diagonal on
$\mathcal{H}(V_{t}|\Gamma)$. We will refer to this problem with the name
“separability problem” and we will comment on it in the next section. We
conclude by observing that the result obtained here, as explained in the
previous section, holds for any value of $t^{\prime}>t$.

###  IV.5 Final remarks and weak points of model B

We completed the description of model B, which can be considered as the
continuous time generalization of model A. The discreteness of time was
recognized as a limitation of model A for a direct comparison to ordinary
quantum mechanics. Here time is a continuous parameter as in ordinary quantum
mechanics but a direct comparison is still not possible. The construction
presented here leads to an infinite dimensional Hilbert space which may be not
separable while in ordinary quantum mechanics the Hilbert space always is.
Comparing this model with model A, we can understand that this time the number
of points in the space process, $M$, does not determine the dimension of the
Hilbert space. After a bit of thought, one can realize that this is a
consequence of the fact that we are using probability measures which admit a
density with respect to the Lebesgue measure. Another consequence of this fact
is the continuous spectrum of the operators representing the particle process.
However, one can always imagine that, if we let the support of the probability
measure shrink to a single point (hence obtainining a Dirac measure, which is
not absolutely continuous with respect to the Lebesgue measure), the operators
have a pure point spectrum. This suggests that the “real” continuity of the
spectrum is obtained only in the limit $M=\infty$ and the absolute continuity
is possible only in this case. One may observe the following. When
$M\rightarrow\infty$ we can have two cases:

  1. a)

the points increase in a _non dense_ way: their number is infinite but in any
subset of $\mathbb{R}$ these is just a finite number of them (they behaves as
numbers in $\mathbb{N}$ or $\mathbb{Z}$);

  2. b)

the points increase in a _dense_ way: their number is infinite and in any
subset of $\mathbb{R}$ there is an infinite number of them (like numbers in
$\mathbb{Q}$). We will refer to this case with the name _dense-point limit_.

Note that in both cases they are assumed to be countable. In the first case,
$\hat{X}_{t}$ can be seen as the limit of a sequence of compact operators: the
spectrum is purely point-like. However, this possibility does not seem to be
comparable with the usual position operator in quantum mechanics, which is
just bounded (and not compact) when we restrict it to a subset of
$\mathbb{R}$. On the other hand, the second case is more interesting. Indeed,
it may give rise to bounded operators which are not compact. This suggests
that to completely recover quantum mechanics, the dense-point limit must be
taken.

Despite the observations done above, we still want to try a comparison with
non-relativistic quantum mechanics. This time we are really closer to deriving
the canonical commutation relation between position and momentum from the
quantities of the model, as we will see. Assume the following:

  1. i)

The Hilbert space on which we can represent $\hat{X}_{t}$ is separable and
infinite-dimensional, i.e. $L_{2}(\mathbb{R})$;

  2. ii)

There exists a self-adjoint operator

$\hat{H}=\frac{1}{2m}\nabla_{x}^{2}+V(x)$

which, together with $\hat{X}_{t}$, fulfils all the mathematical requirements
needed to apply the Ehrenfest theorem (see friesecke2009ehrenfest ).

Clearly $\hat{H}$ is nothing but the ordinary hamiltonian operator in quantum
mechanics. At this point, by the Ehrenfest theorem, we have the equation

$\frac{d}{dt}\langle\hat{X}_{t}\rangle_{\psi}=\frac{1}{m}\langle\hat{P}_{t}\rangle_{\psi},$

where $m$ is the mass of the quantum particle and $\psi\in L_{2}(\mathbb{R})$.
Consider now the velocity random variable of the model B

$V_{t}(t^{\prime})=\frac{X_{t^{\prime}}-X_{t}}{t^{\prime}-t}.$

Note that, after the removal of the space process, the three random variables
lies in three different probability spaces and there does not exist a joint
probability space where we can describe all of them (we recall that when we
remove the space we use the _unconditional_ transition probabilities). Thus
this expression is purely formal and, in particular, it is not expected to
hold at the level of the outcomes of these random variables. However, the
following expression makes sense

$\mathbb{E}[V_{t}(t^{\prime})|\mathbf{S}_{t}]=\frac{\mathbb{E}[X_{t^{\prime}}|\mathbf{S}_{t}]-\mathbb{E}[X_{t}|\mathbf{S}_{t}]}{t^{\prime}-t}$

since the probability measures of each expectation are defined on different
probability spaces. Using the procedure explained in the previous section and
under the assumption $i)$, we can jointly describe these three random variable
using a non-commutative probability space. In particular, we compute the
expectation using the Hilbert space structure, writing

$\mathbb{E}[V_{t}(t^{\prime})|\mathbf{S}_{t}]=\langle\hat{V}_{t}(t^{\prime})\rangle_{\psi}=\frac{\langle\hat{X}_{t^{\prime}}\rangle_{\psi}-\langle\hat{X}_{t}\rangle_{\psi}}{t^{\prime}-t},$

where $\psi\in\mathcal{H}$ with $\mathcal{H}$ Hilbert space constructed as in
section IV.4. This time an explicit procedure to construct $\psi$ is not
known. From this equation we can write that

$\lim_{t^{\prime}\rightarrow
t}\langle\hat{V}_{t}(t^{\prime})\rangle_{\psi}=\frac{d}{dt}\langle\hat{X}_{t}\rangle_{\psi}=\frac{1}{m}\langle\hat{P}_{t}\rangle_{\psi},$

which means that the _weak_ -limit $t^{\prime}\rightarrow t$ of velocity
operator in model B, under assumptions $i)$ and $ii)$, coincides with the
momentum operator of non-relativistic quantum mechanics. Note that the
assumption $i)$ on the separability of the Hilbert space is crucial for this
consideration. Finally, we also note that separability also solves the problem
of the non-unitary equivalence of $\mathcal{H}(X_{t})$ and
$\mathcal{H}(V_{t})$ mentioned at the end of section IV.4. Summarising,
despite Model B is capable to reproduce the commutation relation between the
position and velocity operators of the particle, which resembles the quantum
mechanical commutation relation, it did not succeed in the derivation of (1).
However, if in some other model (similar to model B) we can justify $i)$ and
$ii)$ in some way, we can have a correspondence of the model with non-
relativistic quantum mechanics.

##  V Conclusion

In this paper, we show how a jump-type kinematics of a point like particle
together with an intrinsic stochasticity of the physical space (on which the
particle moves) can give rise to a non-commutative description of the two
basic observables which define the particle: its position and its velocity. In
model A time is treated as a discrete parameter and this makes it impossible
to have a precise comparison with the ordinary quantum mechanics. Generalizing
to the continuous time case, we obtained model B. However, as pointed out in
section IV.5, even in this case we can not compare the two theories. Indeed,
the Hilbert space on which we represent model B is non-separable in general.
In both models we are able to obtain the same non-commutativity of ordinary
quantum mechanics (at the algebraic level) but we have to conclude that this
is not sufficient. It is worth to recall how this non-commutativity was
obtained: _by removing the space process at a given time $t$_. Physically this
requirement is very natural: any experiment which can be done to measure the
probability (via frequency), can be done _in a given configuration of the
space_. This means that if we assume that space really is the stochastic
process described in this paper, any probability that we can measure in a
laboratory is somehow conditioned to the configuration of space that we have
at the time of this measurement. The fact that in our models, space is not
described but removed (essentially via conditioning and not by averaging),
expresses exactly this fact and is the origin of non-commutativity. We also
note that in order to obtain such non-commutativity, the space process must be
_random_ , as the entropic uncertainty relation obtained show. Indeed, if
space is a deterministic phenomenon we obtain a trivial bound. The space
process seems to be central, despite it must be removed to obtain a non-
commutative probability space for the particle: in this sense it plays an
active role in the description of the particle despite the non-commutative
probability theory obtained after its removal is not capable to describe it.
In addition, in model A, the space process determines the dimension of the
Hilbert space, a feature which is lost in model B. This suggests that a better
understanding of the space process may show a possible solution to the non-
separability problem. In particular, it can be that a careful selection of a
particular class of space processes, may “force” the Hilbert space to be
separable. This possibility will be discussed in LC3 .

##  VI Acknowledgements

The author would like to thank S. Bacchi, for many useful discussions and
comments on the manuscript, G. Gasbarri, for its patience during the
corrections and S. Marcantoni for many advices and his fundamental support
during the final stage of this work. Other persons which indirectly contribute
to this works are L. Bersani, N. D’Andrea, A. Motta, R. Truglia e M. Viganó.

##  VII Appendix

### A - Quantum ruler

Here we will describe an attempt to give a quantum mechanical description of a
ruler, a model that will be called _quantum ruler_. Let us start with the
formal definition. A quantum ruler of length $N\in\mathbb{N}$ is an
$N$-particle quantum system with the following features:

  1. i)

the Hilbert space of a quantum ruler is
$\mathcal{H}_{R}:=\bigotimes_{i=1}^{N}\left(L_{2}(\mathbb{R}^{3})\otimes\mathbb{C}^{2}\right)_{i}$
and a generic state take this form

$|\psi\rangle=\psi(y_{1},\cdots,y_{N})\otimes|-,\cdots,-\rangle;$

  2. ii)

there exists a region of space $L\subset\mathbb{R}^{3}$ such that
$\langle\psi|\hat{P}^{(\hat{\mathbf{X}})}(L)|\psi\rangle=1$, where

$\hat{P}^{(\hat{\mathbf{X}})}(L)=\hat{P}^{(\hat{X}_{1})}(L)\otimes\cdots\otimes\hat{P}^{(\hat{X}_{N})}(L)$

with $\hat{P}^{(\hat{X}_{i})}(\cdot)$ is the PVM associated to the position
operator of the $i$-th particle;

  3. iii)

the time evolution of a quantum ruler is determined by the hamiltonian
$\hat{H}_{R}:=\sum_{i=1}^{N}\hat{T}_{i}+\hat{V}$, where $\hat{T}_{i}$ are the
$i$-th particle kinetic terms and $\hat{V}\neq\sum_{i=0}^{N}\hat{V}_{i}$ is
some potential (chosen in order to have $\hat{H}$ bounded from below);

  4. iv)

before any measurement the quantum ruler is described by a bounded state of
the hamiltonian operator, namely $\hat{H}_{R}|\psi\rangle=E_{0}|\psi\rangle$;

  5. v)

the measurement process of the position of a particle (call it $A$) with wave
function $\phi_{A}(x)\in\mathcal{H}_{A}$ (which is not a particle of the
quantum ruler) occurs with the following interaction hamiltonian (defined on
the tensor product Hilbert space $\mathcal{H}_{R}\otimes\mathcal{H}_{Part}$)

$\hat{H}_{I}\cdot:=\sum_{i=1}^{N}g\delta(y_{i}-x)|+_{i}\rangle\langle-_{i}|$

where $g$ is a real constant.

Let us explain the physical meaning of these requirements. A quantum ruler is
a quantum system composed of $N$ distinguishable particles with spin. The spin
degree of freedom should not be considered as the real spin of the particles,
rather as labels that model the possibility to find the $i$-th particle in two
distinguishable states. This is the content of the requirement $i)$. The
condition $ii)$ simply means that, when considered as a single object, a
quantum ruler is localised in a specific region $L$ of space. This assumption
plays a marginal role in the rest of the analysis, nevertheless it expresses
the basic fact that we cannot measure arbitrarily long distances with a given
quantum ruler. Requirements $iii)$ and $iv)$ simply express that these
particles are bounded together. Finally $v)$ describes how the quantum ruler
measures a distance: by a spin flip. Given the particle $A$, the measurement
of its position by the quantum ruler happens when “$A$ touches one of the
particles of the quantum ruler” causing a spin flip. After this interaction
the quantum ruler undergoes a projective measurement. Since the particles are
distinguishable we can label them by establishing an order, and chosing an
origin (the “zero” of the ruler). The position of $A$ is measured by counting
the number of particles between the origin and the particle of the ruler with
the spin flipped, i.e. the particle of the ruler which interacted with the
particle we want to measure. If this model is correct, at least in some limit,
we should be able to recover the statistics of the position of particle $A$,
namely $|\phi_{A}(x)|^{2}$. Here we will show how this is possible.

Let $A\subset L$ be a set. The probability to find the $i$-th particle of the
ruler in this set is given by

$P[x_{i}\in
A]=\langle\psi|\hat{\mathbb{I}}\otimes\cdots\otimes\hat{\mathbb{I}}\otimes\hat{P}^{(\hat{X}_{i})}(A)\otimes\hat{\mathbb{I}}\otimes\cdots\otimes\hat{\mathbb{I}}|\psi\rangle$

where $|\psi\rangle$ is the state of the quantum ruler. From ordinary quantum
mechanics, we know that

$P[x_{i}\in A]=\int_{A}\rho_{i}(x)dx$

where $\rho_{i}(x)$ is a density with respect to the Lebesgue measure. An
example of possible probability densities of a quantum ruler is drawn in
figure 4-$a)$.

Figure 4: In figure $a)$ the $\rho_{i}(x)$ of a $N=7$ quantum ruler are drawn:
the $\rho_{i}$ of all particles of the ruler is drawn except the origin. The
origin is assumed to be a particle in the point $O$, i.e.
$\rho_{O}(x)=\delta(x)$ (and it is not drawn), for simplicity. In figure $b)$
the probability distribution for the position of the point $A$ (obtained by
the same quantum ruler. The probability $P[d_{AO}=i]$ drawn should be
considered as the probability distribution of the point $A$ (naïvely, a
particle with a delta-like wave function centred in $x_{A}$) measured with the
quantum ruler with origin chosen to be the green particle.

From the figure, it is easy to understand that the quantum ruler can give us
only statistical information about a possible measurement of the point $A$.
Such probability distribution is showed in figure 4-$b)$. Note that we did not
use the requirement $v)$ to arrive at this conclusion, hence we are not really
describing the measurement of the position of the particle. In order to do
that precisely, let us assume that we have a particle $A$, described by a wave
function $\phi_{A}(x)\in\mathcal{H}_{A}$. The whole system (quantum ruler plus
particle) will be described by the state

$|\psi_{t}\rangle=\psi_{R}(y_{1},\cdots,y_{N})\otimes|-,\cdots,-\rangle\otimes\phi_{A}(x)$

which is an element of $\mathcal{H}_{R}\otimes\mathcal{H}_{A}$. The
measurement process is modelled by the interaction hamiltonian as in point
$v)$. After the interaction we perform a projective measurement to learn which
spin is flipped: the number of particles between the origin (which is an
arbitrarily chosen particle of the quantum ruler) and the particle with the
spin flipped is our distance. Hence, if $d(O,A)$ is the distance between the
origin and the particle with the spin flipped, we can write that

$P[d(O,A)=i]=P[S_{i}=+_{i}]=\langle\psi_{s}|(|+_{i}\rangle\langle+_{i}|)|\psi_{s}\rangle$

where $|\psi_{s}\rangle$ is the total quantum state right after the
interaction. We will compute $|\psi_{s}\rangle$ in interaction picture for
$s-t=\delta\tau$ such that $g\delta\tau\ll 1$. In this regime we have that

$e^{-i\delta\tau\hat{H}_{I}}\approx\hat{\mathbb{I}}-ig\delta\tau\sum_{i=1}^{N}\delta(y_{i}-x)|+_{i}\rangle\langle-_{i}|+O(g^{2}\delta\tau^{2})$

Thus, setting $t=0$

$\begin{split}|\psi_{\delta\tau}\rangle\approx|\psi_{0}\rangle-
ig\delta\tau&\sum_{i=1}^{N}\delta(y_{i}-x)\psi_{R}(y_{1},\cdots,y_{i},\cdots,y_{N})\otimes\\\
&\otimes|-,\cdots,+_{i},\cdots,-\rangle\otimes\phi_{A}(x)+O(g^{2}\delta\tau^{2})\end{split}$

Using this result, we obtain

$\begin{split}P[&d(O,A)=i]=\langle\psi_{\delta\tau}|(|+_{i}\rangle\langle+_{i}|)|\psi_{\delta\tau}\rangle\\\
&\approx
g^{2}\delta\tau^{2}\int\prod_{j=1}^{N}dy_{j}dx\delta^{2}(y_{i}-x)|\psi_{R}(y_{1},\cdots,y_{N})|^{2}|\phi_{A}(x)|^{2}\\\
&=g^{2}\delta\tau^{2}\delta(0)\int\prod_{j=1}^{N}dy_{j}|\psi_{R}(y_{1},\cdots,y_{i},\cdots,y_{N})|^{2}|\phi_{A}(y_{i})|^{2}.\end{split}$

The $\delta(0)$ is due to the presence of the square of the Dirac-delta in the
probability density computed using the wave function evolved in the
interaction picture. In order to deal with this divergent term, we require
that $g^{2}\delta\tau^{2}\delta(0)\approx 1$. This requirement is in agreement
with the fact that the terms $g^{2}\delta\tau^{2}$ are infinitesimal and they
can be neglected in the expression of $|\psi_{\delta\tau}\rangle$. Thus we
conclude that

$\begin{split}P[&d(O,A)=i]=\\\
&=\int\prod_{j=1}^{N}dy_{j}|\psi_{R}(y_{1},\cdots,y_{i},\cdots,y_{N})|^{2}|\phi_{A}(y_{i})|^{2}\end{split}$
(35)