#  An ontological description for relativistic, massive bosons.

Gerard ’t Hooft

(Faculty of Science, Department of Physics
Institute for Theoretical Physics
Princetonplein 5, 3584 CC Utrecht
The Netherlands
http://www.staff.science.uu.nl/~hooft101
)

Relativistic, scalar particles are considered, contained in a box with
periodic boundary conditions. Although interactions are not expected to be a
fundamental problem, we concentrate on free particles. By considering them to
be harmonic oscillators, it is found that their dynamical variables can be
replaced by a completely ontological set, which means that, here, quantum
mechanics does not deviate from a purely geometric, ontological particle
system. The effects of the mass terms are included. Locality holds for the
quantum theory, and seems to be fully obeyed also by the classical treatment,
although further discussion will be needed.
Quantised interactions are briefly speculated on, but mostly postponed to
later. We do discuss extensively the distinction between the quantum treatment
and the classical one, even though they produce exactly the same equations
mathematically. We briefly explain how this result can be squared with the
usual quantum no-go theorems. It is suggested to apply this theory for real
time quantum model simulations.

##  1 Introduction

Earlier publications [2]–[9] revealed how a totally classical system with a
completely ontological interpretation, can be represented by a Hilbert space
in such a way that it turns into a quantum theory without modifying the
physical phenomena described by these equations.. Here, we investigate the
reverse possibility. Do conventional quantum systems allow for a completely
classical interpretation, and under which conditions? The models proposed in
Ref. [10] , require some fast moving variables on the back ground, which have
to be assumed to fluctuate thermally. Simpler models were now searched for,
and found. Second quantised bosons appear to be fitting examples, since they
have the right energy spectra, which automatically include the fast variables
that were required in our previous papers.

A new problem arose however: locality.111The usual “proofs” that quantum
theories do not allow for a classical interpretation depend on ‘free will’
assumptions that cannot be employed at all in our theories. In Fourier space,
the match between the classical description and a quantum formalism appears to
be perfect, but going back to position space at first sight seems to be hard.
A miracle seemed to be asked for, because the classical form would have to
behave as a completely classical cellular automaton, and such a link is not
easy to realise. In this paper we explain how this can be done in a simple
case: the $D$+1-dimensional system of second-quantised scalar bosons, as yet
without interactions (interactions will be briefly discussed in Appendix C).
The issue encountered will be important: the classical equations of motion
that we find, at first sight deviate from locality in the sense that apparent
classical interactions spread over length scales of the order of the Compton
wave lengths. However, this appears to be merely the result of selecting out
states with positive energy, and indeed we shall find that the squared
equations are local again, as it also happens in the conventional quantum
field theories.

Our results suggest to search for promising models that may bring the Standard
Model of the particle interactions in the picture. Interactions range over
distance scales that are small enough to maintain the locality features of the
quantum theory.

This paper is set up as follows. Sections 2 and 3 are brief repetitions of
earlier work, and may be skipped by readers who are familiar with that.

But then, in section 4 we start our search for the ontological degrees of
freedom by regarding bosons in $D$+1 dimensions as harmonic coupled
oscillators (taking $D=3$ to be explicit). We explain how our search started
by first focusing on the annihilation and creation operators, $a$ and
$a^{\dagger}$, suspecting these to have ideal commutation relations. But we
spare the reader of the difficulties encountered. In an ontological theory,
all observables used to describe the evolution laws (except the hamiltonian
$H$, which strictly speaking is not observable) should commute with one
another, but $a$ and $a^{\dagger}$ do not commute. We do see good things in
$a$ and $a^{\dagger}$: the way they depend on the coordinates $q$ and momenta
$p$ does make us suspect that they do follow the circular motion of an
oscillator in $p,\,q$ space. This was the reason to replace $a$ and
$a^{\dagger}$ by very similar operators $b$ and $b^{\dagger}$ that do
completely commute with one another.

As long as we omit the interactions, a bosonic quantum field theory is nothing
but a gigantic harmonically coupled oscillator. In section 5, we search for
the ontological operators for that case. Diagonalising the oscillator amounts
to going to Fourier space. This involves the momenta $\vec{k}$. At each vector
$\vec{k}$ we have a circular observable $\tilde{b}$ (the tilde, $\tilde{\ }$,
reminds us that this field is defined in Fourier space). We Fourier transform
the field $\tilde{b}$ back to position space, and study its evolution.

There is actually a difficulty here that we partly postpone to the discussion,
section 6: the circular motion in Fourier space turns into more general motion
in 3-space, which could lead to different circles when we transform back. This
apparent flaw is cured.

In this section, we also ask for attention for the energy variable. In terms
of the ontological observables, the energy is vaguely defined, because it does
not commute with them. Therefore the question arises: how do we know when a
particle is surrounded by a vacuum? As it turns out, the vacuum state is a
(quantum) distribution where all probabilities regarding the ontological
variables are basically equal.

We briefly explain why our derivations are not contradicted by no-go theorems.
Then, in Appendix A, we show how, in general, the evolution laws for the $b$
fields can be transported to position space, and in Appendix B we carefully
analyse what we mean by locality, and how it has to be distinguished from
causality. It is important here to realise that our evolution equations are
compatible with the second-order classical field equation for the classical
version of the bosonic field theory.

It was not our intention to discuss at length our ideas about dealing with
interactions in terms of the philosophy advocated in this paper. There are
very important issues here. One might suspect that interactions will be beyond
reach, but this we do not believe. There seems to be a very natural and
physically appealing way to address interactions. and this idea is touched
upon in Appendix C.

Fermions have not yet been considered by this author. Some first steps in this
interesting field are described by Wetterich [8][9].

Our insights in this subject are still evolving, and further discussion is
kept for future publications.

##  2 Preliminaries

Let us begin with recapitulating what we have found before [2]–[4]. Any
elementary deterministic222In some of my work I characterised the concept of
determinism to be used, as: ‘it’s determinism all the way’. But in science
fiction literature and films, authors can go way beyond that point. They
assume highly conspirational modifications of ordinary logic that prevent
someone to go back to the past to kill his grandfather. ‘Determinism all the
way’ stops well before such stages are reached: we accept no modification of
cast-iron, down-to-earth logic., periodic process can be modeled in terms of a
sequence of $N$ classical states, to be indicated in terms of Dirac kets. The
evolution law can be characterised as

$\displaystyle|0\rangle\rightarrow|1\rangle\rightarrow\cdots|N-1\rangle\rightarrow|0\rangle\rightarrow.$
(2.1)

One may consider the continuum limit $N\rightarrow\infty$, which can also be
written as

$\displaystyle\frac{{\rm d}|x\rangle}{{\rm d}t}=|f(x)\rangle\ .$ (2.2)

In the language of quantum mechanics, this reads as:

$\displaystyle\frac{{\rm d}\hat{x}}{{\rm d}t}=-i[\hat{x},\,H]\ ,\qquad
H=\hat{p}\,f(\hat{x})\ ,$ (2.3)

where coordinates $x$ and momenta $p$ have been re-interpreted as being
operators, indicated by a caret ($\,\hat{}\,$) . The ordering of $\hat{p}$ and
$f(\hat{x})$ in Eq. (2.3) may be chosen freely, but symmetrisation of the
product would be preferred, as it would make the hamiltonian Hermitian. Eqs.
(2.2) and (2.3) may have non periodic solutions, while Eq. (2.1) is usually
periodic. In this case, we must take our classical equations as being less-
than-perfect approximations of the quantum mechanical ones. For the time
being, we consider only exactly periodic processes.

If $f(x)$ is known, Eq. (2.3) may seem to be useful to build quantum
mechanical models out of deterministic building blocks, but then there is a
serious complication: quantum mechanical hamiltonians practically always have
a ground state, the state of lowest energy, which can often be interpreted as
the vacuum state. In contrast, the hamiltonian (2.3) often has no ground
state, because the spectrum of the momentum operator $\hat{p}$ usually is
unbounded both above and below. But already in sect. 3, we’ll see a counter
example.

The case of finite $N$ , Eq. (2.1), can also be written in hamiltonian form.
Here, we first define the evolution operator for one elementary time step
$t\rightarrow t+\delta t$, a unitary operator $U(\delta t)$ defined by

$\displaystyle|x\rangle_{t+\delta t}=U(\delta t)|x\rangle_{t}\ ,$ (2.4)

where $U(\delta t)$ is an $N\times N$ matrix having one entry 1 in each row
and in each column, while all remaining entries are 0.

So far, we considered only the case where there is exactly one cyclic orbit.
Now if we allow for more complicated evolution laws, solutions with different
periodicities may occur. These can be formally disentangled to yield simply
different, unmixed vector spaces, without the possibility to hop from one into
another; if hopping were possible, the entire set would consist of different,
exactly periodic cycles, and, at first sight, systems obtained this way will
not seem to be more complex to interpret than the elementary ones.

##  3 Setting up our procedures

Return to the simple, discrete, periodic, system, Eq. (2.1), with $N$
classical states. Let us call them

$\displaystyle|x\rangle^{{\mathrm{ont}}}\ ,\quad x=0,\,1,\,\cdots\,,N-1$ (3.1)

(the superscript ‘ont’ stands for ‘ontological’). When acting on a state
$|x\rangle^{\mathrm{ont}}$, the evolution operator $U(\delta t)$, defined in
Eq. (2.4) generates the ontological state it evolves into, after one time step
$\delta t$:

$\displaystyle U(\delta t)|x\rangle^{\mathrm{ont}}=|\,x\,+\,1\\!\mod
N\,\rangle^{\mathrm{ont}}\ ,\qquad U(n\,\delta t)=\big{(}U(\delta
t)\big{)}^{n}\ .$ (3.2)

We can always define an $N$ dimensional vector space with these $N$ states as
an orthonormal set as basis elements.333The case where different transitions
take different amounts of time $\delta t$ seems to be physically less
meaningful; in the present models, we do not consider such possibilities (One
might expect them to be relevant when gravity theories are considered). We
have

$\displaystyle U(N\,\delta t)=U(\delta t)^{N}=\mathbb{I}\ ,$ (3.3)

from which we deduce the eigen values of $U(\delta t)$. They are

$\displaystyle e^{\textstyle-iE_{n}}\ ,\qquad E_{n}=\frac{2\pi n}{N\delta t}\
,\qquad n=0,\,1,\,\cdots\,,\ N-1\,.$ (3.4)

As before, $N$ is the period of the motion, that is, the number of steps
needed to return to the point of departure.

The basis that diagonalises $U$ is a set of $N$ states $|n\rangle^{E}$, where
$E$ stands for ‘energy’. Eq. (3.3) allows us to compute a hamiltonian. It is
the operator with eigenvalues $E_{n}$ in the same basis:

$\displaystyle U(\delta t)\equiv e^{\textstyle-iH\,\delta t}\ ,\qquad
H|n\rangle^{E}=E_{n}|n\rangle^{E}\ .$ (3.5)

The basis states themselves are uniform superpositions of the ontological
states. From Eq. (3.3) we derive

$\displaystyle E_{n}=n\omega\ ,\quad\omega={\textstyle\frac{2\pi}{N\,\delta
t}}\ ;\qquad|x\rangle^{\mathrm{ont}}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{\sqrt{N}}}\sum_{n=0}^{N-1}e^{\textstyle-2\pi
inx/N\,}|n\rangle^{E}\ ;$ (3.6) $\displaystyle|n\rangle^{E}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{\sqrt{N}}}\sum_{x=0}^{N-1}e^{\textstyle\,2\pi
inx/N\,}|x\rangle^{\mathrm{ont}}\ .$ (3.7)

Notice that the ontological states used as a starting point in Eq. (2.1), were
_defined_ to be orthonormal. This is not at all trivial, since substituting
non-integer numbers for $x$ would yield conflicting results. Eventually, one
might wish to add intermediate states between the regular ones; this would
require reformulating our equations. But this is usually unnecessary.

Also note that the frequency $E_{0}$ attached to the lowest energy state may
be chosen arbitrarily; here we choose it to be zero. It must be observed
however, that the complex phases of all our states are to a large extent
artefacts of our procedure; the ontological states were defined to be written
as positive, real valued amplitudes, but, at best, the link to quantum
mechanics is as yet still artificial.

With Eq. (3.7), we find how $H$ acts on $|x\rangle^{\mathrm{ont}}$. We observe
that this is a spectrum of energy levels that is strictly linear in the rank
$n$. The universal energy gaps one finds are typical for the harmonic
oscillator. They are related to the linearity found in Eq. (2.3), and
discussed in section 1, but now we do have a ground state. As stated above,
one may shift the entire set of energy eigenvalues by adding the same amount
$\delta\omega$ to all energy levels, without the slightest modification of the
physical properties os the system.

One of our basic procedures will be that now we can turn these equations
around, to notice that:

> If we have a system where the hamiltonian $H$ is linear in the momenta $p$,
> as in Eq. (2.3), we may interpret this as a continuous, deterministic system
> of one or more variables $x$ obeying Eq. (2.2), and furthermore,
> if we have a system where the hamiltonian $H$ has a sequence of equidistant
> energy levels, with eigenstates $|n\rangle^{E}$, as in Eq. (3.4), then we
> take it to mean that the states $|x\rangle^{\mathrm{ont}}$ as defined in Eq
> (3.7) evolve ontologically, as in Eq. (3.1).

Thus, a hamiltonian that separates energy levels in steps of size $\omega$,
corresponds to a model where a variable $x$ takes steps of one unit in the
time span of $\delta t=2\pi/\,N\omega$.

The continuum limit is obtained as $N\rightarrow\infty,\ \delta t\rightarrow
0$. We keep the length of the period, $T=N\,\delta t$ constant. The distance
between the consecutive energy levels, given by Eq. (3.4), stays constant, but
the number of levels $N$ becomes infinite. The hamiltonian then stays non-
negative, and looks exactly as that of an ordinary, unperturbed harmonic
oscillator.444We ignored here the zero point oscillations, which move the
entire energy spectrum upwards by ${\textstyle\frac{1}{2}}\omega$, where
$\omega$ is the angular frequency of the oscillator. This is exactly one half
of the separation between consecutive energy levels.

The hamiltonian of a harmonic oscillator is

$\displaystyle
H={\textstyle\frac{1}{2}}\hat{p}^{2}+{\textstyle\frac{1}{2}}\omega^{2}\hat{q}^{2}=(n+{\textstyle\frac{1}{2}})\omega\
,\qquad n=0,\,1,\,2,\,\dots$ (3.8)

Here $\hat{p}$ is the momentum and and $\hat{q}$ the position operator of an
oscillating object, and $\omega$ is its angular frequency. We have
$[\hat{q},\hat{p}]=i$. The number $n$ counts its energy levels.555We shall
henceforth often ignore the carets $(\,\hat{}\,$) when $p$ and $q$ are
considered to be operators. One can now adopt Eqs. (3.6) and (3.7) to express
the ontological states $|x\rangle^{\mathrm{ont}}$ in terms of the energy eigen
states $|n\rangle^{E}$, and use the Hermite polynomials to relate the states
$|n\rangle^{E}$ to the eigen states of $\hat{p}$ and $\hat{q}$.

It is important to observe that both in the finite case, described by Eqs.
(2.1), and in the continuous case of the harmonic oscillator, where the limit
$N\rightarrow\infty$ is taken, the ontological variable $x$ runs in a circle,
with angular velocity $\omega$, the same one that describes the frequency of
the oscillator.

##  4 Creation and annihilation

Many of the elementary features of a harmonic oscillator, are reflected in the
annihilation operator $a$ and the creation operator $a^{\dagger}$. They are
directly related to the hamiltonian, $H=\omega a^{\dagger}a$, where $\omega$
is the angular frequency, and the original kinetic variables

$\displaystyle p=\sqrt{\frac{\omega}{2}}(a+a^{\dagger})\quad\hbox{and}\quad
q=\frac{1}{i\sqrt{2\omega}}(a^{\dagger}-a)\ .$ (4.1)

In the energy basis, where $E=\omega n\,,$ we have

$\displaystyle\langle n-1|\,a(t)\,|n\rangle=\sqrt{n}\,e^{\textstyle-i\omega
t}\ ,\qquad\langle n|\,a^{\dagger}(t)\,|n-1\rangle=\sqrt{n}\,e^{\textstyle
i\omega t}\ .$ (4.2)

All other matrix elements of $a$ and $a^{\dagger}$ in the energy basis are
zero. This allows us to read off the time dependence of $a$ and $a^{\dagger}$
directly from Eqs. (4.2). It is quite remarkable that these operators commute
with themselves at different times,

$\displaystyle{}\,[a(t),a(0)]=0\ ,\qquad[a^{\dagger}(t),\,a^{\dagger}(0)]=0\
,$ (4.3)

a property that they do not share with the momentum operator $p$ or the
position operator $q$ for the oscillator. It was our intention to use either
$a(t)$ as an ontological variable, or use $a^{\dagger}(t)$ , but _not_ both
$a$ and $a^{\dagger}$, because $[a,a^{\dagger}]\neq 0$. However, a more
troublesome feature of these operators is that their eigen values are
fundamentally complex.

At this point, we found more useful operators to describe ontological
variables. We shall call these the _truncated annihilation operator_ $b$ and
its conjugate $b^{\dagger}$. Their matrix elements in terms of the energy
eigen states $|n\rangle$ have properties more closely reflecting the original
equations (3.6) and (3.7) :

$\displaystyle\langle n-1\ \mathrm{mod}\,N\ \ |b(t)|n\rangle\ =\
e^{\textstyle-i\omega t}\ ,\qquad\langle n|b^{\dagger}(t)|n-1\
\mathrm{mod}\,N\ \rangle\ =\ e^{\textstyle i\omega t}\ .$ (4.4)

All other matrix elements in terms of the energy eigen states vanish.

These operators have better rotation symmetry,

$\displaystyle\langle n-m\ \mathrm{mod}\,N\ \ |b^{m}(t)|n\rangle\ =\
e^{\textstyle-i\omega\,mt}\ ,$ (4.5)

and, besides the equivalent of Eqs. (4.3), they obey

$\displaystyle{[b(t),\,b^{\dagger}(0)]}=0\ ,\qquad b^{-1}=b^{\dagger}\ .$
(4.6)

This is why both $b$ and $b^{\dagger}$ operators are more suitable for
describing ontological variables: their eigen values are all confined to the
unit circle.

An important question is how to define the operator $b(t)$ in any given
harmonic operator and how to return from the canonical variables $p(t)$ ,
$q(t)$ and the hamiltonian $H$ if we have some state in the $b$ basis. The
answers are derived from the first part of this section.

First, if we have some quantum state described in terms of the harmonically
oscillating variables $p$ and $q$, and we have the hamiltonian $H$ , (3.8)
(with the vacuum energy ${\textstyle\frac{1}{2}}\omega$ subtracted), we first
define the annihilation and creation operators $a$ and $a^{\dagger}$ as usual,
see Eq. (4.1):

$\displaystyle a^{\dagger},\ a\ =\frac{1}{\sqrt{2\omega}}p\pm
i\sqrt{{\scriptstyle\frac{1}{2}}\omega}\,q\ ,$ (4.7)

after which we write $H=\omega\,a^{\dagger}a$ .

Then, we define the operator $b$ by using Eq. (4.4), where the time dependence
is already given. We see that the energy states $|n=N\rangle$ and $n=0\rangle$
are identified. In the continuum limit, $N\rightarrow\infty$, this feature
will be insignificant, as in practice the hamiltonian is time-independent, and
the state $n=N$ has almost infinite energy. If this gives rise to any problem
we have to carefully take this discontinuity into account.

We can actually express $b$ directly in terms of $a$ and $a^{\dagger}$:

$\displaystyle b=(1+a^{\dagger}a)^{-{\scriptstyle\frac{1}{2}}}a\ ,\quad
b^{\dagger}=a^{\dagger}(1+a^{\dagger}a)^{-{\scriptstyle\frac{1}{2}}}\ .$ (4.8)

We could alter the order of the operators:

$\displaystyle b=a(a^{\dagger}a)^{-{\scriptstyle\frac{1}{2}}}\ ,\quad
b^{\dagger}=(a^{\dagger}a)^{-{\scriptstyle\frac{1}{2}}}a^{\dagger}\ ,$ (4.9)

but this would require an exceptional treatment for the $n=0$ state, which is
avoided in the Eqs. (4.8).

Conversely, we can start with he $b$ operator. Given $b$ and $b^{\dagger}$ we
can identify the operator $H$ as being the one that sends $b$ into circles
with frequency $\omega$:

$\displaystyle{\textstyle\frac{{\rm d}}{{\rm d}t}}b=[b,\,-iH]=-i\omega\,b\
,\quad[b,H]=b\ .$ (4.10)

We could write

$\displaystyle H=-i\omega\,b\frac{\partial}{\partial b}\ ,$ (4.11)

which shows a relation with our first-order deterministic approach as
described in section 2. Actually, one may regard the present work as a more
advanced justification for using quantum mechanics as a formalism for
deterministic first order equations.

Now to define the operators $a$ and $a^{\dagger}$ in terms of $b$ and
$b^{\dagger}$, we just invert Eqs. (4.8):

$\displaystyle a=(1+H)^{{\scriptstyle\frac{1}{2}}}b\ =\
bH^{{\scriptstyle\frac{1}{2}}}\ ;\qquad
a^{\dagger}=b^{\dagger}(1+H)^{{\scriptstyle\frac{1}{2}}}\ =\
H^{{\scriptstyle\frac{1}{2}}}b^{\dagger}\ .$ (4.12)

The operators $p$ and $q$ now follow from Eqs. (4.1).

As the equations of motion for $b(t)$ and $b^{\dagger}(t)$,

$\displaystyle{\textstyle\frac{{\rm d}}{{\rm d}t}}b(t)=-i\omega\,b\
,\qquad{\textstyle\frac{{\rm d}}{{\rm
d}t}}b^{\dagger}=i\omega\,b^{\dagger}\,,$ (4.13)

do not involve any non-commuting operators anymore, we herewith established a
non quantum mechanical description, or interpretation, of the harmonic
oscillator.

Now how can we employ this observation in quantised field theories? Aren’t
they harmonic oscillators as well? This question is answered in section 5.

We end this section by calculating some matrix elements for the operator
$\hat{b}$. We put the caret $(\,\hat{\ }\,)$ on the letter $b$ to indicate the
_operator_ $b$, as opposed to its eigen value which we call $b$. The energy
eigen states are $|\,n\,\rangle$ with $H=n\omega=\omega a^{\dagger}a\,$. The
$\hat{b}$ eigen states with eigen value $b$ are indicated as $|b\rangle$.

We start with Eq. (4.5). From that, we derive,

$\displaystyle\langle n+1\ \mathrm{mod}\,N\,|b\rangle\ =\ \langle
n|\hat{b}|b\rangle\ =\ b\langle n|b\rangle\ \rightarrow\ \langle
n|b\rangle=C\,b^{n}\ ,$ (4.14)

where $C$ is a normalisation factor. Since $\langle N_{\,}|b\rangle=\langle
0|b\rangle$ , we have

$\displaystyle b^{N}=1\ \ \rightarrow\ \ b=e^{\textstyle-2\pi ik/N}\,,$ (4.15)

where $k$ is an integer between 0 and $N$. These are all $N$ eigen states of
$\hat{b}$. The relation between $k$ and $x$ in Eq. (3.6) is:

$\displaystyle k=n\,x\ \mathrm{mod}\,N\ .$ (4.16)

To normalise the eigen states to $\langle b|b\rangle=1$, we choose

$\displaystyle C=1/\sqrt{N}\ ;\qquad\langle
n|b\rangle={\textstyle\frac{1}{\sqrt{N}}}e^{\textstyle-2\pi ik/N}\ .$ (4.17)

The exponent gives the time dependence of these eigen states. To return to the
notation of section 3, we write the exponent as giving the time dependence of
these states, exhibiting the time steps $\delta t$ :

$\displaystyle e^{\textstyle-i\omega t}\ ;\qquad\omega\,\delta t=2\pi/_{\,}N\
.$ (4.18)

##  5 Relativistic free bosons as ontological objects

Elementary particles may be regarded as wave-like solutions of a quantised
field theory. They behave very much like harmonic oscillators, particularly
when we switch off the effects of interactions, which would turn the
oscillators into anharmonic ones.

The question we would like to ask is: can our energy spectra be compared with
those of the harmonically oscillating particle fields such as the ones
occurring in the Standard Model? Do they also contain sequences of equally-
spaced energy levels? If so, can we also attribute an ontological
interpretation to them? And is there a chance that we can continue along this
line? Would we arrive at ontologically trivial evolution equations, and could
such ideas be used to study the foundations of quantum mechanics?

In particular, we must identify ‘circular orbits’ along which an oscillator
moves, and find out how they are to be identified as ontological evolving
variables.

The same questions were posed by this author in 2001, see Ref. [2], section 3
there; but the proper answer was not discovered. In fact it seemed to be
complicated indeed. The oscillator represented by free bosons seems to be an
infinite set of strongly, harmonically coupled oscillators. How do we handle
that? Now, as is often he case in our field of science, more promising answers
are being found by going back to the basics even further.

In relativistic quantum field theory, at some moment in time $t$, one
introduces a field $\Phi(\vec{x})$ and its canonical associate,
$\Pi(\vec{x})$. In analogy with the canonical variables $q$ and $p$ of the
single oscillator, the commutation rules are:

$\displaystyle[\Phi(\vec{x}),\,\Pi(\vec{x}\,^{\prime})]=i\delta^{3}(\vec{x}-\vec{x}\,^{\prime})\
.$ (5.1)

One writes the hamiltonian $H$ for freely moving particles as

$\displaystyle H=\int_{V}{\rm
d}^{3}\vec{x}\big{(}{\textstyle\frac{1}{2}}(\Pi^{2}+(\vec{\nabla}\Phi)^{2}+M^{2}\Phi^{2})\big{)}\
,$ (5.2)

where $\vec{\nabla}=\partial/\partial\vec{x}$ , and $M$ will be the mass of
the particles. We consider the most elementary case of spinless, massive or
massless particles, as being our protoype. For our convenience we imagine our
bosonic field theory to be confined in a box with periodic boundary
conditions, defined by the periods $L_{x},\ L_{y},\ $ and $L_{z}$. The volume
of the box is $V=L_{x}L_{y}L_{z}$.

To stay in line with the philosophy handled in the previous sections, we first
have to disentangle the various oscillators, and this happens if we
diagonalise the hamiltonian (5.2).

This diagonalisation is realised by Fourier transforming the fields in
3-space, writing the Fourier transforms by putting a tilde on top:
$\tilde{\Phi}(\vec{k})$ and $\tilde{\Pi}(\vec{k})$,

$\displaystyle\Phi(\vec{x}\,)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{V}}\sum_{\vec{k}}\,e^{\textstyle
i\vec{k}\cdot\vec{x}}\,\tilde{\Phi}(\vec{k}\,)\ ,\quad\hbox{ so that}$ (5.3)
$\displaystyle\tilde{\Phi}(\vec{k}\,)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{V}}\int_{V}{\rm
d}^{3}\vec{x}\,e^{\textstyle-i\vec{k}\cdot\vec{x}}\,\Phi(\vec{x}\,)\ .$ (5.4)

where $\vec{k}$ only takes the values such that $k_{i}L_{i}/2\pi$ are
integers. The Fourier transforms $\tilde{\Pi}(\vec{k},t)$ of the fields
$\Pi(\vec{x},t)$ are defined in the same way.

In terms of these Fourier transforms the hamiltonian (5.2) reads

$\displaystyle
H=\sum_{k}{\textstyle\frac{1}{2}}\big{(}\,|\tilde{\Pi}(\vec{k})|^{2}+(\vec{k}^{2}+M^{2})|\tilde{\Phi}(\vec{k})|^{2}\big{)}.$
(5.5)

Now $\tilde{\Pi}(\vec{k})$ and $\tilde{\Phi}(\vec{k}\,^{\prime})$ all commute
except when $\vec{k}=\pm\vec{k}\,^{\prime}$, in which case $\tilde{\Phi}$ and
$\tilde{\Pi}$ obey exactly the commutation rules of positions $q$ and momenta
$p$. Clearly, we have one harmonic oscillator with
$\omega=\sqrt{\vec{k}^{2}+M^{2}}$, at every allowed value666A short-hand
notation is used: $\tilde{\Phi}(-\vec{k})=\tilde{\Phi}^{\dagger}(\vec{k})$ ,
in all three principal directions of $\vec{k}$. One can avoid this conceptual
complication by replacing the exponentials with sines and cosines of
$k_{i}\,x_{i}$, at the expense of more lengthy expressions.  of $\vec{k}$.

Figure 1 illustrates these discrete momentum values.

Figure 1:  The lattice of momentum space $\vec{k}$. The size of the pixels is
inversely proportional to the size of the box. This momentum space ranges to
infinity in all directions. Colouring indicates the relevant time scale:
$\omega=\sqrt{\vec{k}\,^{2}+M^{2}}$

Now in an earlier section, sec. 3, we already noted that the harmonic
oscillator with frequency $\omega$ may be regarded as a classical particle
rotating over a circle with angular velocity $\omega$. It is the same circular
motion as what we derived in section 4. Thus, at every value of $\vec{k}$, we
have such an object. But what kind of classical ‘particle’ are we looking at?
How do its properties depend on the sizes $L_{x},\,L_{y}$ and $L_{z}$ of the
box?

When $\vec{k}=0$ and $M>0$ , we know the answer: a ‘particle’ runs along a
circle. But this circle appears to be localised in some interior space. Its
angular velocity $\omega$ is $M$. It appeared to be difficult to understand
how it is localised in physical space-time, if really all we have is these
oscillators in Fourier space.

The position of the ‘particle’ on the circle is a physical parameter. We have
one such parameter for every allowed value of $\vec{k}$, but now comes a
difficulty: at $\vec{k}\neq 0$ , the angular frequency increases:

$\displaystyle\textstyle\omega=\sqrt{M^{2}+\vec{k}^{2}}\ .$ (5.6)

What kind of ontological orbit gives this frequency? During this
investigation, our first attempt was to find orbits of particles along some
closed paths, inside our periodic box, but this did not produce physically
sensible scenarios. A much more successful interpretation involves the
introduction of an abstract, but entirely ontological field
$\tilde{b}(\vec{k})$ (We put a tilde, $\tilde{\,}$, on top of $b$ or $\hat{b}$
to indicate that this operator is defined in Fourier space). Here, $\tilde{b}$
is exactly the truncated annihilation operator studied in the previous
section, 4. We found that this operator commutes with itself as well as its
conjugate, at other instants in time. Furthermore, of course it also commutes
with the same operator at other values of $\vec{k}$.

This is a novelty for a quantum theory, and we decided to study it from close.
First of all, we can Fourier transform $\tilde{b}(\vec{k})$ ‘back’ to the 3
dimensional physical space:

$\displaystyle b(\vec{x})=\frac{1}{\sqrt{V}}\sum_{\vec{k}}e^{\textstyle
i\vec{k}\cdot\vec{x}}\,\tilde{b}(\vec{k})\,,$ (5.7)

We also introduce the truncated creation operator
${\tilde{b}}^{\dagger}(\vec{k})$ , and Fourier transform that to position
space,

$\displaystyle
b^{\dagger}(\vec{x})=\frac{1}{\sqrt{V}}\sum_{\vec{k}}\,e^{\textstyle-i\vec{k}\cdot\vec{x}}\,{\tilde{b}}^{\dagger}(\vec{k})\,.$
(5.8)

Then one may verify that all these operators commute, for instance:

$\displaystyle[\tilde{b}(\vec{k}),\,{\tilde{b}}^{\dagger}({\vec{k}}^{\prime})]=0\
,\quad\hbox{and}\quad[b(\vec{x}),\,b^{\dagger}(\vec{x}\,^{\prime})]=0,$ (5.9)

Using Eqs. (4.4) and (4.6) we find777In Eqs. (4.4) and (4.6), we still have
the matrix elements; they all share the same time dependence. time dependence
of $\tilde{b}(\vec{k})$:

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\tilde{b}(\vec{k},t)=-i\omega(\vec{k})\,\tilde{b}(\vec{k},t)\ $ ,
$\displaystyle\qquad\tilde{b}(\vec{k},\,t)=e^{\textstyle-i\omega(\vec{k})t}\tilde{b}(\vec{k},\,0)\
;$ (5.10)
$\displaystyle{[\tilde{b}(\vec{k}),\,\tilde{b}(\vec{k}\,^{\prime})]}$
$\displaystyle=$ $\displaystyle
0\,\quad\forall\quad\vec{k},\,\vec{k}\,^{\prime}\ .$ (5.11)

This just confirms that the evolution law at any value of $\vec{k}$ does not
affect the laws at any other $\vec{k}\,^{\prime}$. However, for the $\vec{x}$
dependence the situation is a bit more complicated, since $\omega$ depends on
$\vec{k}$.

In the initial phases of this research the author thought that the original
operators $a$ and $a^{\dagger}$ had to be used at this point, but he fact that
$[a,\,a^{\dagger}]=1$ leads to expressions that are not classical, or
ontological. By switching to the truncated variables $b$ and $b^{\dagger}$
this problem could be overcome. $b$ and $b^{\dagger}$ commute.888A
disadvantage is that $b$ cannot be used to define the energy ground state,
while $a$ can: $a|0\rangle^{E}=0$.

But now what are the equations in $\vec{x}$ space? They can be read off from
the laws of Fourier transformations, starting with Eqs. (5.10):

$\displaystyle{\textstyle\frac{{\rm d}}{{\rm d}t}}b(\vec{x},t)$
$\displaystyle=$ $\displaystyle{-i}\int{\rm
d}^{3}\vec{y}\,F_{1}(\vec{x}-\vec{y})b(\vec{y},t)\ ;$ (5.12) $\displaystyle
b(\vec{x},\,t)$ $\displaystyle=$ $\displaystyle\int{\rm
d}^{3}\vec{y}\,F_{2}(\vec{x}-\vec{y},t)b(\vec{y},0)\,,\hbox{\qquad where}$
(5.13) $\displaystyle F_{1}(\vec{z})$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int{\rm
d}^{3}\vec{k}\,\omega(\vec{k})e^{\textstyle i\vec{k}\cdot\vec{z}}\
;\qquad\omega(\vec{k})\ =\ \sqrt{\vec{k}^{2}+M^{2}}\ ,$ (5.14) $\displaystyle
F_{2}(\vec{z},t)$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3}}\int{\rm
d}^{3}\vec{k}\ e^{\textstyle\,i\vec{k}\cdot\vec{z}-i\omega(\vec{k})t}\ .$
(5.15)

These equations are very similar to the quantum equations, but for one fact:
_all fields $b(\vec{x},\,t)$ in these equations commute. _They are classical
equations!

In principle, they are cellular automaton equations, but then one must note
that the equations are not strictly local: each cell ‘interacts’ with each
other cell, but the interaction strength $F_{1}(\vec{z})$ decreases fast when
the entry $\vec{z}\ =\ \vec{x}-\vec{y}$ increases.

In Appendix A we discuss the solution of Eqs. (5.12)–(5.15). In particular,
the direct interaction, (5.12), decreases exponentially as soon as
$|\vec{z}|>1/M$ , the Compton wavelength. Therefore, the system becomes more
typically classical, comparable to a cellular automaton, when we are dealing
with fundamental particles with the largest values for their intrinsic mass
$M$.

We do not have to consider only that limit, however. In spite of this apparent
non locality of the primary equation (5.12), one can prove that the evolution
law can also be cast in a local form, by squaring the evolution equation
(5.15). This results in

$\displaystyle\big{(}\vec{\partial}_{x}\\!^{2}-\partial_{t}^{2}\big{)}b=M^{2}b\
,$ (5.16)

which is of course local. A few more remarks on these subtle features are
added in Appendix B, but more discussions will be inevitable.

The most important result described in this section is that we have
represented the solutions of he quantum system of free bosons in a non-quantum
mechanical way, because all operators in Eqs. (5.12)–(5.15) commute with one
another.

##  6 Interpretation and discussion of the result

We are making a claim here that, at first sight, seems to be irreconcilable
with well-known facts about quantum mechanics. First of all there is the well-
known Bell theorem, [11]–[13] the CHSH inequality[14], and other remarkable
‘quantum’ features[15], [16].

Theorems such as Bell’s theorem were challenged by many researchers, such as
L. Vervoort [17], who seems to have mechanisms in mind that resemble what will
be described here.

The fact that determinism or some version of ‘superdeterminism’ must be
considered more carefully was recently advocated by Hossenfelder and Palmer
[18][19].

Our point will be that Bell and his followers made the basic assumption that
an observer, at his or her free will, may choose which Hilbert space
components of a physical state to observe. One then can concoct a
configuration where the observed object is in an entangled state, a state that
cannot be realised in classical systems.

The calculations provided by conventional quantum mechanics are precise and
unambiguous, but their results are always expressed in terms of probability
distributions. The apparent contradictions come about when quantum physicists
do not seem to understand the classical model that they claim to pulverise.
The models there are quite different from what we propose. The ontological
models described in our recent papers are ones that will never produce
probabilities different from zero or one, _if the initial state is
ontologically precisely specified._ The entangled photons in Bell’s experiment
and others, are defined by the complex probability distributions they
generate.

If the final state has a non-trivial probability distribution, this must be
due to the fact that also the initial states are not in a single ontological
mode. To describe the correct initial state is impossible in practice, because
at all values of the Fourier parameter $\vec{k}$ we have to describe the
_ontological_ state of that oscillator, while, at large enough values for
$|\vec{k}|$, this is impossible in practice: the oscillations there are too
fast.

Instead, we usually state how many photons there are at any momentum state
$\vec{k}$. Usually the answer to that is that we have zero photons at the
higher energy states.

The state with zero photons is not at all ontological; to the contrary, it is
an even distribution of _all possible_ ontological configurations. This is
because the zero photon state is an energy eigen state, which as such smears
the situation over all different times.

In classical physics, it is obvious that one can generate ‘uncertainties’ by
not giving the proper, complete, description of an initial state. This is a
general feature of classical models that quantum physicists often do not
consider.

In our paper it is shown that, by taking such complications into account, one
can explain ‘quantum behaviour’ exactly. We found a basis of Hilbert space by
searching for a set of mutually commuting operators. This is not as difficult
as it sounds. It is done here. In our models, one can, in principle, choose a
completely certain initial state. Then the final state will be a single,
certain outcome as a result.

Why can experimenters not choose such an initial state? Simply because there
are far too many ontological modes to consider. If we choose a beam of
particles, it is surrounded by an ocean of ontological data that we cannot all
take into account. In practice, we put all these unseen objects in their zero
energy state. That is the vacuum, a highly mixed state with probability
contours that are totally flat, far from what would be needed to generate
final states without uncertainties.

We borrowed from quantum theory the concept of the vacuum state. Since the
ingredients in this quantum state interact with the particles that we study,
one finds that the results of experiments now require auxiliary states that
are non commuting. In other words, the fact that we end up using non-commuting
operators is caused by our inability to control everything; if we could
control every single degree of freedom, we would not need quantum mechanics.

In appendix A we also derive the speed by which this automaton emits signals
(over distances much larger than the Compton wavelength, finding the speed to
be exactly the group velocity of the quantum wave equation, which is slower
than light.

_The vacuum state is the completely uniform probabilistic distribution of
_all_ possible realistic states._

Now this is the way we justify our limit $\Lambda$ for the momenta $\vec{k}$
in Fourier space: at all momenta $\vec{k}$ with $|\vec{k}\,|>\Lambda$, it is
assumed that the state there is not known. All possible real states have equal
(low) probabilities. When we do experiments, it will always be useful to
consider these unobserved states as being in the $n=0$ state, where the
quantum wave function ${}^{E}\langle 0|x\rangle^{\mathrm{ont}}$ takes the same
value at all values of $x$.

Thus we see that, before trying to interpret the solutions that we
constructed, only the particles actually observed, stick out as being much
more probable than all other states. It is their wave functions that have to
be different from the universally flat wave functions. Therefore, quantum
mechanics can now be seen as describing the salient probability distributions
needed for describing an experiment.

This brings us to our previous publications [7][10] where we came close to
saying the same things. There, what we needed was some variable that carries
very fast fluctuating realistic variables. By having these interact with the
particles that we wish to study at much lower energies – read: longer time
scales – one introduces the effects of genuine quantum states to the dynamical
phenomena of slow particles. The slow particles will cease to behave as
classical objects because they are coupled to quantum objects. The quantum
objects are just the high energy quantum states of the $|\vec{k}\,|>\Lambda$
variables of a particle type that only needs to be a virtual particle,
something comparable to a super high mass unifying gauge particle. We do not
see such particles directly, but we see how their interactions generate
quantum features in all other, slow, particles in our world.

Returning to our cellular automaton, we see that we did not copy the discrete
nature of the usual automata speculated on by other researchers [20][21][22].
Our present evolution law is as classical as a cellular automaton, but it is
not discrete. Making the step to discrete physics should not be too difficult
conceptually. We may suspect that, at the Planck scale, all physics might be
discrete. Then, our construction will automatically lead to a ‘real’ cellular
automaton.

The thing to do at the point we are now, is to study other particles, notably
fermions [9], and to study interactions. Of course, models adhered to
presently , such as the Standard Model of the fundamental particles, have
interaction terms. It is well known how to describe these, and how to measure
them. Conversely, we may study classical interaction phenomena that will turn
our system of bosons into interacting bosons. The result of these
considerations will not automatically lead to a complete one-to-one comparison
between classical and quantum field equations, but it seems fair to suggest to
make attempts. This author is optimistic about the future of these attempts.
We continue on the topic of interactions in appendix C.

##  Appendix A The classical interaction $F_{i}(\vec{z}\,)$

In order to understand the physical features of our calculation, we must
calculate the functions $F_{1}$ and $F_{2}$ in Eqs. (5.14) and (5.15). The
equations are both of the form

$\displaystyle F(\vec{x})$ $\displaystyle=$
$\displaystyle\int_{|k|<\Lambda}{\rm d}^{3}\vec{k}\ f(\vec{k}\,^{2})\
e^{\textstyle i\vec{k}\cdot\vec{x}}\ .$ (A.1)

In view of rotation symmetry, we may assume $\vec{x}$ to point in the
3-direction, and the component of $\vec{k}$ parallel to $\vec{x}$ is called
$k$. Orthogonally to $\vec{x}$ is a 2-vector $\tilde{\ell}$ :

$\displaystyle\vec{x}=\begin{pmatrix}\tilde{0}\\\ z\end{pmatrix}\
,\qquad\vec{k}=\begin{pmatrix}\tilde{\ell}\,\\\ k\end{pmatrix}\ .$ (A.2)

Introducing a temporary cut-off $\Lambda$, and writing
$|\tilde{\ell}|\,=\ell\,$, we have

$\displaystyle F(\vec{x})$ $\displaystyle=$
$\displaystyle\int_{|k|<\Lambda}{\rm d}k\int_{0}^{\Lambda^{2}-k^{2}}{\rm
d}\ell\ 2\pi\ell\,f(k^{2}+\ell^{2})\,e^{\textstyle ikz}\ .$ (A.3)
$\displaystyle F(\vec{x})$ $\displaystyle=$
$\displaystyle\pi\int_{|k|<\Lambda}{\rm d}k\int_{0}^{\Lambda^{2}-k^{2}}{\rm
d}y\,f(k^{2}+y)\,e^{\textstyle ikz}\ =\ \pi\int_{|k|<\Lambda}{\rm d}k\
g(y)\Big{|}_{y=k^{2}}^{\Lambda^{2}}\,e^{\textstyle ikz}\,.$ (A.4)

where$\quad\ell\,^{2}=y\,;\qquad f(y)=\displaystyle\frac{{\rm d}g(y)}{{\rm
d}y}$ .

The computation of the case $F(\vec{x})=F_{1}$ is arrived at by choosing

$\displaystyle f(k)=\omega(k)=(k^{2}+M^{2})^{1/2}\,,\quad
g(y)={\textstyle\frac{2}{3}}(k^{2}+M^{2})^{3/2}\,,$ (A.5)

so that

$\displaystyle
F(\vec{x})={\textstyle\frac{2\pi}{3}}\int_{k=-\Lambda}^{\Lambda}{\rm
d}k(y+M^{2})^{3/2}\Big{|}_{y=k^{2}}^{y=\Lambda^{2}}e^{\textstyle ikz}\ .$
(A.6)

Assuming $z>0$, we can deform the integration contour to run from
$k=i\Lambda$, around the pole at $k=iM$, back to $i\Lambda$ to get two
integrals from $k=iM$ to $i\Lambda$ . Writing $k=ip$, we then get (the
$\Lambda$\- dependence quickly disappears):

$\displaystyle(2\pi)^{3}F_{1}(\vec{x})$ $\displaystyle=$
$\displaystyle\frac{2\pi}{3}\int_{M}^{\infty}i{\rm d}p\,e^{\textstyle-
pz}\,\big{\\{}(\Lambda^{2}+M^{2})^{3/2}-(M^{2}-p^{2})^{3/2}\big{\\}}\ =$ (A.7)
$\displaystyle=$ $\displaystyle-\frac{2\pi}{3}\int_{M}^{\infty}i{\rm
d}p\,e^{\textstyle-pz}\,(p^{2}-M^{2})^{3/2}\
\big{(}e^{\textstyle{\textstyle\frac{3}{2}}i\pi}-e^{\textstyle-{\textstyle\frac{3}{2}}i\pi}\big{)}\
=$ (A.8) $\displaystyle=$ $\displaystyle-\frac{4\pi}{3}\,\int_{M}^{\infty}{\rm
d}p\,e^{\textstyle-pz}\,(p^{2}-M^{2})^{3/2}\qquad\hbox{where }\quad
z=|\vec{x}\,|\ .$ (A.9)

The function $F_{2}(\vec{x}\,)$ is obtained in a similar way:

Here, the function $f(\vec{k})$ is

$\displaystyle e^{\textstyle-i\omega(k)t}$ $\displaystyle=$ $\displaystyle
e^{\textstyle-it\sqrt{k^{2}+M^{2}}}\ ;\qquad y=k^{2}\ ,$ (A.10) $\displaystyle
g(y)$ $\displaystyle=$ $\displaystyle-\frac{2}{t^{2}}\,e^{\textstyle-
it\sqrt{y+M^{2}}}(-it\sqrt{y+M^{2}}-1)\ ;$ (A.11)

The integral (A.4) shrinks rapidly except when the fluctuations in the $z$
variable cancel the fluctuations in the $t$ variable. In Eq. (5.15) and
(A.11), consider a small region $\delta\vec{k}$ in $\vec{k}$ space
contributing to the integral (5.15). We look at one point $z$ in 3-space and
one moment $t$ in the elapsed time. The exponent then approaches the integral
of

$\displaystyle e^{\textstyle i(\vec{z}\cdot\delta\vec{k}-t\,\delta\omega)}$
(A.12)

This quantity balances to a stationary point if

$\displaystyle\frac{{\rm d}\omega}{{\rm d}\vec{k}}=\frac{\vec{z}}{t}\ ,$
(A.13)

This means that points where the integral becomes substantially enhanced occur
where ${\vec{z}}/{t}={{\rm d}\omega}/{{\rm d}\vec{k}}\,,$ or:
 _The signal expands with velocity equal to what is well-known as being the
group velocity;_ for relativistic bosons, this velocity is

$\displaystyle\vec{v}\,^{\mathrm{group}}=\frac{\vec{k}}{\sqrt{k^{2}+M^{2}}}\,,$
(A.14)

which is always slower than light.

##  Appendix B Locality

Locality is a statement not about the data, but about the evolution law
believed to explain the data. If, to compute the predicted data in some
region, the only knowledge required is the data in its immediate past, the
evolution law may be declared to be a local and causal one. In Figure 2, it is
surmised that all data within the past light cone shown, populating region
$A$, suffice to predict what happens in regions $C$ and $B$, up to the point
$Q$. One then does not need to know the particles that enter the regions from
outside, because their effects have already been accounted for. If the
boundaries are local light cones, this law may well be relativistically
invariant. In many cases, the velocity by which data can be transferred is
limited by the velocity of the messenger particles, which must be going slower
than light. But this velocity, called group velocity (as it can be computed by
considering pakets of de-Broglie waves, which tend to form groups, with their
own computable velocities), often depends on the particles concerned, and as
long as it does not exceed the velocity of light, those details have little
effect on the locality property.

According to this definition of locality, not only the quantum theory of free
bosons obeys locality, but also the ontological variables studied in this
paper, since they are directly related. These data are also ontological. They
can all be computed, first by identifying all values of $\vec{k}$ being
involved, all of which are represented if we consider regions larger than the
inverse of the mass $M$.

Figure 2: Locality.

See text.

At first sight, there might be a restriction. Our evolution equations, in
particular Eq. (5.12), seems to transport information instantaneously, since
the function $F_{1}(\vec{x}-\vec{y}\,)$ connects space-like separated points,
even if these points cannot be further apart than a few Compton wavelengths.
However, the resulting law over intervals in time-like directions, Eq. (5.15)
does not require knowledge over any space-like separations, as is well-known
in ordinary field theory. The proof of this is as follows. Consider the vector
$\vec{k}$ to be in the 3-direction. Let us shift the contour of $k_{3}$ to
become

$\displaystyle k_{3}=p+ia\ ,\qquad p,\ a\ \hbox{ real}\ ,$ (B.1)
$\displaystyle z>t\ ,\qquad\omega=\sqrt{(p+ia)^{2}+M^{2}}\ =q+ib\ ,\quad M,\
q,\ b\ \ \hbox{ real };$ (B.2)

then it is easy to prove that

$\displaystyle b<a\ .$ (B.3)

Then when we send the contour to the top of the complex plane,
$a\rightarrow\infty$ , the integration over $k_{3}$ converges to zero. This is
because, as long as $z>t$, the imaginary part of the exponent
$ik_{3}\,z-i\omega(k)t$ keeps the same sign and tends to infinity.

Indeed, being continuous rather than discrete, gives us this important
advantage: automatic Lorentz invariance of the entire evolution equation over
finite distances in space-time.

##  Appendix C Interactions

The field $b(\vec{x},t)$ indeed obeys a local equation, (5.16). This equation
happens to coincide with the classical equation that was at the source of the
definition of our quantum field theory. And indeed, by replacing this equation
with the non-local, first order equation, Eqs. (5.12) and (5.14), one can
select a special class of solutions that obey positivity. The time derivative
now has a fixed sign, so that the emerging particles are guaranteed to have
positive energies. This is an encouraging sign that these equations combined
generate the correct ontological solutions. Since all operators $b(\vec{x},t)$
commute with one another, we have a _classical_ theory here.

At first sight, to introduce interactions at this level might seem to be too
audacious, but there is hope. Our general philosophy is simple:

> If the classical free field theory generates quantum effects simply by
> choosing the ‘chaotic’ boundary conditions at high $\vec{k}$ values, then
> why should the same not also be true if we add the usual interaction term to
> Eq. (5.16)?

$\displaystyle(\vec{\partial}_{x}^{2}-\partial_{t}^{2})b(x)=M^{2}b(x)+{\textstyle\frac{1}{6}}\lambda\,b^{3}(x)\
,\qquad x\equiv(\vec{x},t)\ .$ (C.1)

This is undoubtedly a totally classical equation, but one may suspect that we
can transform it back to the quantum model, first only in the unperturbed
sector by writing

$\displaystyle b(\vec{k})=G\,a\ ,\qquad b^{\dagger}(\vec{k})=a^{\dagger}\,G\
,\qquad
G=\big{(}1+{\mathcal{H}}^{\mathrm{free}}(\vec{k})\big{)}^{-{\scriptstyle\frac{1}{2}}}\
.$ (C.2)

The unperturbed theory has

$\displaystyle{\textstyle\frac{\partial}{\partial
t}}b(\vec{k})=-i\omega(\vec{k})b(\vec{k})$ (C.3)

To compare the effects of an interaction with the corresponding quantum
theory, we must use the same strategy in both. Here this means: perform the
perturbation expansion with respect to the small parameter $\lambda$ in Eq.
(C.1). The equation is of course classical, but this is what we are looking
for. If the effects of $\lambda$ shrink when we go to high energies, we may
use the same boundary conditions at high energies as in quantum field theory.
Therefore we expect our mathematical method to work particularly well in
asymptotically free theories such as Quantum Chromodynamics, QCD.

It is conceivable that our ideas may lead to new alleys towards efficient
lattice simulation methods for such theories. There would be one great
advantage: in quantum theories our refuge into perturbation expansions is
almost inevitable. Perturbation expansions, however, are fundamentally
divergent when applied to quantum field theories. In the ontic theories we can
also do the perturbation expansion, but here, we do not have to, and we can
set up simulations in lattice configurations, using the classical equation
(C.1) non-perturbatively. There is no danger for divergences in the underlying
classical models.

More speculative is the notion that the necessary discretisation of classical
equations such as Eq. (C.1) may also lead to discretisation of perturbation
parameters such as $\lambda$, and other expansion parameters in the Standard
Model.

## References

  * [1]
  * [2] G. ’t Hooft, _Determinism in Free Bosons_ , Int.J.Theor.Phys. 42 (2003) 355-361, arXiv:hep-h/0104080 (hep-th)
  * [3] G. ’t Hooft, _Duality between a deterministic cellular automaton and a bosonic quantum field theory in 1+1 dimensions_ quant-ph arXiv:1205.4107.
  * [4] G. ’t Hooft, _The Cellular Automaton Interpretation of Quantum Mechanics_ , Fundamental Theories of Physics, Vol. 185, Springer International Publishing, 2016. eBook ISBN 978-3-319-41285-6, DOI 10.1007/978-3-319-41285-6, http://arxiv.org/abs/1405.1548; see Springer home page.
  * [5] G. ’t Hooft, _Ontology in Quantum Mechanics, in: Topics on Quantum Information Science,_ Edited by Sergio Curilef and Angel Ricardo Plastino, IntechOpen, DOL: 105772/intechopen.99852. Available from: https://www.intechopen.com/chapters/78365; http://arxiv.org/abs/2107.14191.
  * [6] G. ’t Hooft, The Ontology Conservation Law as an Alternative to the Many World Interpretation of Quantum Mechanics, Presented at the Conference PAFT2019, on ‘Current Problems in Theoretical Physics’, session on ’Foundations of Quantum Theory’, Vietri sul Mare, Salerno, Italy, April 13-17, 2019, arxiv:1904.12364.
  * [7] G. ’t Hooft, _Fast Vacuum Fluctuations and the Emergence of Quantum Mechanics,_ Found Phys 51, 63 (2021). https://doi.org/10.1007/s10701-021-00464-7; arxiv:2010.02019.
  * [8] C. Wetterich, _Probabilistic cellular automata for interacting fermionic quantum field theories,_ Nuclear Physics B 963115296 (2021), arXiv:2007.06366 [quant-ph];
  * [9] C. Wetterich, _Fermionic quantum field theories as probabilistic cellular automata_ , arXiv:2111.06728v1 [hep-lat] (2021).
  * [10] G. ’t Hooft, _Explicit construction of Local Hidden Variables for any quantum theory up to any desired accuracy_ , arxiv:2103.04335[quant-ph], to be pub.
  * [11] J. S. Bell, _On the Einstein–Podolsky–Rosen Paradox_ , Physics 1 (1964) 195
  * [12] J. S. Bell, _On the impossible pilot wave_ , Foundations of Physics 12 (1982) 989
  * [13] J. S. Bell, _Speakable and unspeakable in quantum mechanics_ (Cambridge Univ. Press, Cambridge, 1987).
  * [14] J.F. Clauser; M.A. Horne; A. Shimony; and R.A. Holt, Proposed experiment to test local hidden-variable theories, DOI:https://doi.org/10.1103/PhysRevLett.23.880 (1969)
  * [15] D.M. Greenberger, M.A. Horne, and A. Zeilinger (2007), Going beyond Bell’s Theorem, arXiv:0712.0921, Bibcode:2007arXiv0712.0921G.
  * [16] J. Conway and S. Kochen, The Strong Free Will Theorem, Notices of the AMS Volume 56, Number 2, 2009, p. 226.arXiv:0807.3286 [quant-ph].
  * [17] L. Vervoort, Bell’s Theorem: Two Neglected Solutions, Found. Phys. DOI 10.10701-013-9715-7 (2013), ArXiv 1203.6587v2.
  * [18] S. Hossenfelder and T.N. Palmer _Rethinking Superdeterminism_ , Front.in Phys. 8 (2020) 139, e-Print: 1912.06462 [quant-ph];
  * [19] S. Hossenfelder, _Superdeterminism: A Guide for the Perplexed_ , arXiv: 2010.01324v2 [quant-ph] 6 Oct 2020.
  * [20] K. Zuse, _Rechnender Raum._ Braunschweig: Friedrich Vieweg & Sohn, transl: _Calculating Space_ , MIT Technical Translation AZT-70-164-GEMIT, Massachusetts Institute of Technology (Project MAC), Cambridge, Mass. 02139 (1969). Adrian German and Hector Zenil (eds).
  * [21] S. Wolfram, _A New Kind of Science_ , Champaign, IL: Wolfram Media, Inc. (2002). ISBN 1-57955-008-8. OCLC 47831356; http: www.wolframscience.com
  * [22] E. Fredkin, _An Introduction to Digital Philosophy_ , Int. J. Theor. Phys. 42, No. 2, February 2003.