# ToMCCA: A Toy Monte Carlo Coalescence Afterburner

Maximilian Mahlein1, Chiara Pinto1,2, Laura Fabbietti1

1Technical University of Munich, TUM School of Natural Sciences, Physics
Department,
James-Franck-Straße 1, 85748 Garching b. München, Germany
2 CERN

###### Abstract

The study of antinuclei in cosmic rays provides a unique opportunity to probe
physics beyond the Standard Model. Antinuclei in our Galaxy may stem either
from annihilation or decay of dark matter, or from collisions of cosmic rays
with the interstellar medium, which constitute the background of indirect dark
matter searches. Understanding the formation mechanism of (anti)nuclei is
crucial for setting limits on their production in space. Coalescence models,
which describe the formation of light nuclei from final-state interaction of
nucleons, have been widely employed in high-energy collisions. In this work,
we introduce ToMCCA (Toy Monte Carlo Coalescence Afterburner), which allows
for detailed studies of the nuclear formation processes without the overload
of general-purpose event generators. ToMCCA contains parameterizations of the
multiplicity dependence of the transverse momentum distributions of protons
and of the baryon-emitting source size, extracted from ALICE measurements in
pp collisions at $\sqrt{s}=5-13$ TeV, as well as of the event multiplicity
distributions, taken from the EPOS event generator. ToMCCA provides
predictions of the deuteron transverse momentum distributions, with an
agreement of $\sim 5\%$ with the experimental data. The results of ToMCCA show
that the coalescence mechanism in pp collisions depends only on the event
multiplicity rather than on the collision system or its energy. This allows
the model to be utilized for predictions at lower center-of-mass collision
energies, which are the most relevant for the production of antinuclei from
processes related to dark matter. This model can also be extended to heavier
nuclei as long as the target nucleus wave function and its Wigner function are
known. 111ToMCCA is available at https://github.com/HorstMa/ToMCCA-Public.

##  1 Introduction

The study and potential detection of antinuclei in cosmic rays has emerged to
be an exciting field of research, offering a unique opportunity to explore
possible signals from physics beyond the Standard Model. Antinuclei,
consisting of antiprotons and antineutrons, are exceedingly rare in cosmic
rays compared to their matter counterparts, and their presence in our Galaxy
could be due to annihilation or decays of dark matter (DM) candidates [1, 2].
Indeed, the antinuclear flux expected from Standard Model processes due to
collisions of high-energy cosmic rays (CRs) and the interstellar medium (ISM),
is expected to be two to four orders of magnitude lower than the dark matter
flux at low kinetic energies ($E_{\rm kin}<1$ GeV/$A$) [3, 4]. Both the CRs
and the ISM are mostly made of protons ($\sim$ 90$\%$) and 4He nuclei ($\sim$
9$\%$), and only in a small percentage of heavier nuclei. Hence, the main
contributing channels to the Standard Model background are collisions between
protons and 4He nuclei (p–p, p–4He, 4He–p, 4He–4He), with center-of-mass
energies of $\mathcal{O}(10)$ GeV [5]. Understanding the formation mechanism
of antinuclei is crucial for estimating the production cross-section from the
Standard Model and Dark Matter contributions and for interpreting any future
measurements of antinuclear flux from cosmic rays. Different experimental
efforts are ongoing to measure the flux of antinuclei in cosmic rays, e.g.
AMS-02 [6], GAPS [7] and BESS-Polar [8] experiments. However, antinuclei
heavier than antiprotons have not been measured yet.

Coalescence models have been widely used to study the formation of light
nuclei starting from the production of individual nucleons in high-energy
hadronic collisions [9, 10, 11, 12, 13, 14, 15]. In these models, light nuclei
are formed through the coalescence of nucleons that satisfy specific
conditions, such as small relative momentum, spatial proximity, and spin-
isospin alignment. In the simplest implementations of coalescence, only the
momentum correlations are considered and the bound states are formed if the
difference in momentum among the nucleons is below a given threshold, namely
the coalescence momentum $p_{0}$. In the state-of-the-art implementations of
the coalescence approach, the quantum-mechanical properties of the constituent
baryons and those of the final bound states are taken into account and the
coalescence probability is calculated from the overlap between the wave
functions of individual (point-like) baryons and the Wigner density of the
final-state cluster [15, 16, 17, 18, 19, 20]. Monte Carlo (MC) simulations
have proved to be powerful tools for investigating nuclear dynamics and
particle production in high-energy hadronic collisions, as they offer a
stochastic treatment of the complex dynamics in high-energy particle
collisions while maintaining the full information on the space-time position
and momenta of individual particles. By combining the principles of MC event
generators with coalescence models, recent descriptions of the momentum
distributions of nuclei have drastically improved [16, 20].

In light of these advancements, we propose a new approach to MC-based
coalescence, called ToMCCA (Toy Monte Carlo Coalescence Afterburner). ToMCCA
is a Toy Monte Carlo that allows for a detailed study of the nuclear formation
process without the overload of general-purpose event generators. In ToMCCA,
the production of bound-states is implemented through a coalescence
afterburner based on the Wigner function formalism. The event generation is
customizable and the kinematics of the produced nucleons are tuned using
parameterizations of (i) the multiplicity dependence of their transverse
momentum ($p_{\rm T}$) distributions, (ii) their angular momentum
correlations, (iii) the baryon-emitting source size, and (iv) the multiplicity
distributions. Using such parameterizations and computing the probability of
coalescence employing the Wigner function of the nucleus, ToMCCA provides
predictions for the transverse momentum distributions of deuterons, for
arbitrary multiplicity intervals covered by pp collisions. The
parameterizations included in ToMCCA, based on measurements carried out with
ALICE, make use of results from proton-proton (pp) collisions at a center-of-
mass energy of $\sqrt{s}=$ 5 TeV [21] and of those selected with the high
multiplicity (HM) trigger in pp collisions at $\sqrt{s}=$ 13 TeV [22]. The
performance of ToMCCA is tested against the deuteron $p_{\rm T}$ distributions
of pp collisions at energies not included in the parameterizations, namely
$\sqrt{s}=$ 13 TeV [23] (with a minimum-bias trigger) and $\sqrt{s}=$ 7 TeV
[24]. The ToMCCA model can be easily extended to heavier nuclei, as long as
the nucleus wavefunction is known and its Wigner function is computed, and to
lower center-of-mass collision energies, of the order of 10 GeV, that are the
most relevant for the production of antinuclei from processes related to dark
matter [5].

##  2 Working principle of ToMCCA

###  2.1 Coalescence model

The coalescence afterburner model built in ToMCCA is based on the Wigner
function formulation of coalescence. For more details see Refs. [16, 20] and
references therein. For convenience, we summarize here the basics of the
coalescence model used in ToMCCA. The general idea is that the momentum
distribution of deuterons in the final state ($d^{3}N/dP^{3}$) can be
expressed by projecting the phase-space of the unbound proton-neutron (p-n)
system ($W_{np}$) onto the Wigner function of the deuteron
$D(\vec{q},\vec{r})$

$\frac{d^{3}N}{dP^{3}}=S\int d^{3}q\int d^{3}r_{d}\int
d^{3}rD(\vec{q},\vec{r})W_{np}(\vec{P}_{d}/2+\vec{q},\vec{P}_{d}/2-\vec{q},\vec{r}_{d},\vec{r}).$
(1)

The S is the spin-isospin statistics factor, which equals 3/8 for deuterons,
$\vec{P}_{d}$ is the total momentum, $\vec{q}$ is the relative momentum,
$\vec{r}_{d}$ is the center of mass position, and $\vec{r}$ the distance of
the p-n pair. The deuteron Wigner function depends only on the wavefunction of
the deuteron for which there are multiple options, e.g., Gaussian, Hulthen
[25], Double Gaussian, Argonne $v_{18}$ [26], etc. In this work, we use the
Argonne $v_{18}$ wavefunction, which has been shown to have the best agreement
with experimental data in Ref. [20]. For the deuteron Wigner function, we use
the parameterization reported in the appendix of Ref. [20] for the Argonne
$v_{18}$ wavefunction222The deuteron Wigner function for the Argonne $v_{18}$
wavefunction is available in the GitHub project at
https://github.com/HorstMa/ToMCCA-Public.. We make the assumption that one can
factorize the two-nucleon phase-space into its space and momentum terms. The
p-n phase-space assumes the form

$W_{np}(\vec{P}_{d}/2+\vec{q},\vec{P}_{d}/2-\vec{q},\vec{r}_{d},\vec{r})=H_{np}(\vec{r},\vec{r}_{d};r_{0})G_{np}(\vec{P}_{d}/2+\vec{q},\vec{P}_{d}/2-\vec{q}).$
(2)

Folding the deuteron Wigner function with the nucleon phase-space
distribution, it is possible to get the coalescence probability
$\mathcal{P}(r_{0},\vec{q})$ as a function of the source size $r_{0}$ and
$\vec{q}$

$\mathcal{P}(\vec{q},r_{0})=\int d^{3}q\int d^{3}r\int
d^{3}r_{d}D(\vec{q},\vec{r})H_{np}(\vec{r},\vec{r}_{d};r_{0}).$ (3)

The source size is described in Sec. 3. Such probability can then be applied
on any p-n pair, on a pair-by-pair basis, to predict deuteron transverse
momentum distributions.

###  2.2 The ToMCCA toy Monte Carlo

The use of a toy MC instead of traditional MC event generators, such as Pythia
[27, 28] or EPOS [29], allows for a much faster computation, as only processes
strictly necessary for coalescence are computed. Furthermore, such a
lightweight generator can be modified and customized easily by the end-user,
allowing for much more differential studies. In this work, we developed a Toy
Monte Carlo that reproduces the deuteron momentum distributions employing
parameterizations of the following parameters:

  * •

Charged-particle multiplicity of events,

  * •

Nucleon momentum distributions,

  * •

Angular correlations among nucleons,

  * •

Hadron-emitting source size,

  * •

Nuclear wavefunction.

The parameterizations included in ToMCCA are based on experimental
measurements from the ALICE collaboration, except for the parameterization of
the charged-particle multiplicity distributions, which is based on EPOS 3, as
described in detail in Sec. 2.2.4.

####  2.2.1 Event multiplicity

The starting point of ToMCCA is the determination of the charged-particle
multiplicity of the event that is generated. This number is usually given by
the number of charged particles per unit of pseudorapidity $\eta$, e.g.,
measured in a range of $|\eta|<0.5$ ($(dN_{ch}/d\eta)_{|\eta|<0.5}$). Particle
spectra are often measured in a different kinematic range, such as $|y|<0.5$,
where $y$ is the rapidity relative to the beam axis. To transform from
$|\eta|<0.5$ to $|y|<0.5$, a scaling constant $\alpha$ has to be determined
such that $\alpha\cdot(dN_{ch}/d\eta)_{|\eta|<0.5}=(dN_{ch}/dy)_{|y|<0.5}$
[30]. This can be computed using event generators. For the case of ToMCCA, a
value of $\alpha=1.192$ is determined using EPOS 3 [29].

There are multiple ways to reproduce each given $dN_{\mathrm{ch}}/d\eta$
value. Two simple approaches are included in ToMCCA: (i) fixed multiplicity,
i.e., generating exactly $\alpha dN_{\mathrm{ch}}/d\eta$ particles per event,
and (ii) Poissonian multiplicity, i.e., generating a Poissionian distribution
around the chosen multiplicity value. However, since coalescence is a non-
linear process, i.e., it depends on the number of pairs rather than single
particles, the overall shape of the distribution plays a significant role.
Thus, a more realistic approach is needed. An improvement can be reached by
using an event generator, such as EPOS 3. Using an unbiased triggering method,
such as the one employed by the ALICE experiment [22], the chosen mean
charged-particle multiplicities can be triggered outside of the nucleus
measurement region ($-3.7<\eta<-1.7$ and $2.8<\eta<5.1)$. The shapes of the
resulting midrapidity distributions can be well described using the Erlang
distribution [31], given by

$f(x;k,\lambda)=\frac{\lambda^{k}x^{k-1}e^{-\lambda x}}{(k-1)!}.$ (4)

While technically the Erlang distribution is only defined for
$k\in\mathbb{N}^{+}$, it can be extended to $k\in\mathbb{R}_{\geq 0}$ without
introducing any poles, as long as $x>0$. This further requires to generalize
the factorial $(k-1)!$ to the gamma function $\Gamma(k)$, valid for
$k\in\mathbb{R}_{\geq 0}$. With these modifications, the multiplicity
dependence of the parameters of the Erlang function $k$ and $\lambda$ can be
modeled with a power law function (given by Eq. 13). The parameters of the
power law function can be found in Tab. 1.

Parameter | k | $\lambda$
---|---|---
A | 0 | 0
B | 0.77544 | 0.72653
C | 0.74669 | -0.23343
Table 1: Parameterization of the Erlang distribution using power law
functions.

####  2.2.2 The ToMCCA event

A schematic view of the event loop in ToMCCA is shown in Fig. 1. First, the
number of nucleons (protons and neutrons separately) in a given event is
calculated starting from the particle multiplicity of the event (Sec. 2.2.3).
In panel (1) of Fig. 1, an exemplary situation in which four protons (red) and
four neutrons (blue) are produced is depicted. Then, the 3-dimensional
momentum of the first proton and all neutrons is determined (Sec. 2.2.4), as
well as their relative distances (Sec. 3), as sketched in panel (2) of Fig. 1.
Using these momenta ($q$) and distances ($r_{0}$), the coalescence probability
$\mathcal{P(\it{q},r_{0})}$ is calculated for each pair using Eq. 3. A random
number $r\in[0,1]$ is drawn, and if $r<\mathcal{P(\it{q},r_{0})}$ the pair is
accepted as a deuteron. Otherwise, the proton gets checked against other
neutrons. If multiple neutrons satisfy the condition of coalescence with the
same proton, only the first pair forms the deuteron. However, this is a very
rare occurrence comparable to the 3He and 3H production rates ($\sim
1/10^{-7}$events [22]). After all p-n pairs have been combined, a new proton
and a completely new set of neutrons are created, and the same procedure is
repeated. If a deuteron is created in the first loop, the number of neutrons
available for the following protons in this event is reduced by one. In panel
(3) of Fig. 1, a sketch of a deuteron being formed is shown. The subsequent
event in Fig. 1 (4) shows the second proton with now only three, and
completely new, neutrons.

Figure 1: 1) Starting situation after the number of protons (red) and neutrons
(blue) have been determined. 2) For each proton, a new set of neutrons is
generated. 3) One proton-neutron pair that satisfies the coalescence condition
forms a deuteron. 4) Next proton of the event gets a new set of neutrons. The
amount of neutrons has been reduced by one since one neutron was bound to a
proton to form the deuteron.

####  2.2.3 Particle production

In ToMCCA, the exact composition of charged particles is not important. Only
the final number of protons and neutrons that are generated is of interest.
There are four methods implemented in ToMCCA to determine the number of
nucleons: uncorrelated emission, string fragmentation, quark recombination,
and tuned emission. The methods used for particle production are described in
detail below.

1\. Uncorrelated Emission
In this method, for every charged particle created, there is a probability for
it to be a proton given by

$p_{prot}=(dN/dy)/(dN_{ch}/dy),$ (5)

where d$N$/d$y$ is the production yield of protons, anchored to the measured
production spectra, as will be discussed in Sec. 2.2.4. A mirroring technique
is used to determine the number of neutrons, where $N_{ch}$ neutral particles
are created and assigned to be neutrons in the same manner as protons. Notably
this method creates no correlation between the event-by-event number of
protons and that of neutrons, and therefore between protons and deuterons. The
impact of this could be quantified by studying the deuteron number
fluctuations and correlations between nucleons and deuterons on an event-by-
event basis, as it has been done in Ref. [32] for Pb–Pb collisions.

2\. String Fragmentation
The second method implemented in ToMCCA is called String Fragmentation. The
method mimics much more complex hadronization methods employed by general-
purpose event generators, such as Pythia or EPOS. Here, one large string is
created, and either a u($\bar{\texttt{u}}$) or a d($\bar{\texttt{d}}$) quark
is placed at one end, and its antimatter counterpart at the other end. This
string is fragmented in a given number of locations, which equals the total
number of particles created $dN_{\mathrm{ch}}/dy$. On each fragmentation
location, a quark-antiquark ($\texttt{q}\bar{\texttt{q}}$) or diquark-
antidiquark ($\texttt{dq}\overline{\texttt{dq}}$) pair is created. The order
of the pairs is determined so that a color-neutral particle is created at all
times, i.e., diquarks are paired with quarks, and single quarks are paired
with single antiquarks. To avoid tetraquark configurations, no diquark can be
created directly after another diquark. After all the fragmentations, the
triplets created from diquarks are assigned to the corresponding particles
depending on the quark content. The combinations ddd and uuu are assigned to
neutrons and protons respectively, since these are $\Delta^{-}$ and
$\Delta^{++}$ resonances, which decay exclusively into the assigned nucleons.
Mesons that would be created with this method are ignored for simplicity. An
example of a string with an initial $\texttt{d}\bar{\texttt{d}}$ configuration
is shown in Fig. 2.

Figure 2: An example of the fragmentation of a $\texttt{d}\bar{\texttt{d}}$
string. Mesons are indicated for clarity but are ignored for coalescence in
ToMCCA.

This method determines the total proton yield starting from the probability of
creating a diquark. Due to edge effects in the fragmentation algorithm, baryon
production at low multiplicities gets suppressed. The diquark probability
needs to be modified and parameterized as a function of multiplicity, in order
to recover the production yield measured at low multiplicity. This probability
is parameterized such that the proton yields as a function of multiplicity
correspond to the one from the uncorrelated emission method through the
equation:

$p_{\mathrm{diquark}}=p_{\mathrm{prot,modified}}(dN/dy)=p_{\mathrm{prot}}(dN/dy)\cdot(A+B\
(dN/d\eta)^{C}),$ (6)

where $p_{\mathrm{prot}}$ is given by Eq. 5. The obtained parameters can be
found in Table 2.

3\. Quark recombination
The Quark Recombination method is based on the idea that a sea of u and d
quarks is created and recombined randomly to create hadrons. The total number
of each quark species is $N_{d}=N_{u}=dN_{\mathrm{ch}}/dy$, where
$dN_{\mathrm{ch}}/dy$ is the number of charged particles as described above.
First, a u and a d quark are combined randomly, and then, with probability
$p_{\mathrm{prot,modified}}$ given by Eq. 6, a third u(d) quark is added to
the ud pair, creating a proton (neutron). This is repeated until no ud pairs
are left. The fit parameters of Eq. 6, for the parameterization of the
probability of adding one quark to the ud pair, are also reported in Tab. 2.

Parameter | String Fragmentation | Quark recombination
---|---|---
A | 2.13957998 | 0.87627309
B | -3.8557517 | 1.01566945
C | -2.0928103 | -0.54537776
Table 2: Fit parameters of Eq. 6, for the parameterization of the diquark
probability (string fragmentation method) and of the probability of adding a
quark (quark recombination method).

4\. Tuned Emission
Both the String Fragmentation and Quark Recombination methods by construction
have a fixed Pearson correlation coefficient $\rho_{pn}$ (Eq. 7), which is a
measure of the correlation between the number of protons and neutrons in a
given event. A negative coefficient means that the presence of a proton
suppresses the creation of new neutrons. To get around this feature, a fourth
method is developed, namely the tuned emission, which contains a free
parameter $a$ to adjust the level of nucleon suppression. It is based on the
uncorrelated emission, but the probability for a neutron $p_{neut}$ is
modified depending on the number of protons in the event via
$p_{neut,tuned}=(a(N_{p}-dN/dy)+1)p_{neut}$. Since the String Fragmentation
and Quark Recombination models show an unreasonably strong suppression of
nucleon pairs at low multiplicities, the value $a$ was set in such a way that
its Pearson coefficient lies roughly in the middle between String
Fragmentation and Uncorrelated Emission. The exact parameterization is
$a=1.508\,(dN_{ch}/dy)^{-1}$. The difference in the final deuteron yields
between all these methods (1 to 4) ranges from $<10\%$ for
$dN_{ch}/d\eta|_{|\eta|<0.5}>30$ to $300\%$ for
$dN_{ch}/d\eta|_{|\eta|<0.5}\sim 3$. This shows that the exact degree of
suppression is a valuable input to this model. The parameter $a$ can be tuned
to a measurement of the Pearson coefficient of proton and deuteron $\rho_{pd}$
as the correlation between proton and neutron is transferred into the
correlation between proton and deuteron. Such measurements have been performed
in the past on Pb–Pb data [33], but not yet in pp collisions.
The Pearson correlation coefficient can be calculated as

$\rho_{pn}=\langle(n_{p}-\langle n_{p}\rangle)(n_{n}-\langle
n_{n}\rangle)\rangle/\sqrt{\kappa_{2p}\kappa_{2n}},$ (7)

where $\kappa_{2a}$ is the second order cumulant of the particle multiplicity
distribution $\kappa_{2a}=\langle(n-\langle n\rangle)^{2}\rangle$. The
calculated proton-neutron Pearson coefficients $\rho_{pn}$ for each method can
be found in Tab. 3.
In the following, all results are obtained using the Tuned Emission method.

| Uncorrelated Emission | String Fragmentation | Quark Recombination | Tuned Emission
---|---|---|---|---
$\rho_{pn}$ | 0 | -0.052 | -0.058 | -0.024
Table 3: The Pearson correlation coefficient between protons and neutrons for
different hadronization methods.

####  2.2.4 Kinematics

Once the number of protons and neutrons is calculated, their kinematics have
to be determined. In the following, the procedure used to assign the three-
dimensional momentum to nucleons and the angular correlations to the nucleon
pairs is described.

1\. Nucleon momentum distributions
The $p_{\mathrm{T}}$ distributions of nucleons are described using a Levy-
Tsallis function (Eq. 16). However, the shape of the $p_{\mathrm{T}}$
distribution evolves with the multiplicity of the event [34]. Thus, ToMCCA
requires a parameterization of the shape of the transverse momentum
distributions of nucleons as a function of multiplicity. For this, the proton
spectra measured by ALICE in pp collisions at a center-of-mass energy of
$\sqrt{s}=$ 5 TeV [21] and those selected with the high multiplicity (HM)
trigger in pp collisions at $\sqrt{s}=$ 13 TeV [22] have been fitted using Eq.
16. The evolution of the Levy-Tsallis parameters (d$N$/d$y$, $n$, $T$) has
been modeled as a function of $dN/d\eta$ according to a power law function
(Eq. 13). The values of the parameters can be found in Tab. 4. Due to a high
correlation between the parameters T and n, the latter was constrained to Eq.
13, only letting the parameter A vary slightly ($7.3\pm 0.1$) and fixing
C=$-$1\. This is to ensure that the T parameter evolves smoothly as a function
of multiplicity. Furthermore, depending on the hadronization model used for
the particle production (Sec. 2.2.3), the parameter T needs to be modified
since the models can show edge effects that alter the yield, especially at low
multiplicities. For this, T gets modified as

$T_{\mathrm{mod}}(dN/d\eta)=T(dN/d\eta)\cdot(A+B(dN/d\eta)^{C}).$ (8)

The resulting parameterization can be found in Tab. 4. Since in the
modification term $A\approx 1$ and C $<$ 0, this correction only modifies low
multiplicity events, while the modification term tends to 1 for high
multiplicity events.

Parameter | dN/dy | n | T | $T_{\mathrm{mod}}^{\mathrm{QuarkReco}}$ | $T_{\mathrm{mod}}^{\mathrm{StringFrag}}$
---|---|---|---|---|---
A | – | 7.32306908 | – | 0.99534332 | 0.99472583
B | 0.06152 | 2.08554186 | 0.10805 | -0.73804899 | -4.34111446
C | 0.95621 | -1 | 0.36889 | -2.00693344 | -4.55584316
Table 4: Values of the parameters for the yield parameterization. The
parameters $T_{\mathrm{mod}}^{\mathrm{QuarkReco}}$ and
$T_{\mathrm{mod}}^{\mathrm{StringFrag}}$ refer to the modifications of the
temperature parameter $T$ of Eq. 8, for the quark recombination and string
fragmentation hadronization methods, respectively.

In addition to the transverse momentum, the rapidity $y$ is drawn from a flat
distribution $y\in[-0.5,0.5]$. This is well motivated for measurements at
midrapidity [35].

2\. Angular Correlations
To eventually fix the full 3-dimensional momentum information, the relative
angle between the proton and the neutron needs to be determined.
Experimentally, this is accessed using the angular correlation function [36]

$C(\Delta\varphi)=\frac{S(\Delta\varphi)}{B(\Delta\varphi)},$ (9)

where $S(\Delta\varphi)$ is the same event $\Delta\varphi$ distribution, i.e.,
the angular distribution of the azimuthal angle between two particles in the
same event. Conversely, $B(\Delta\varphi)$ is the $\Delta\varphi$ distribution
of pairs from mixed events, i.e., relative angles between particles from
different events. The latter removes trivial correlations from the same event
distribution and reduces them to the genuine correlation. However, for a model
like ToMCCA, the actual angle between particles in the same event is needed.
Hence only the same event angular distribution $S(\Delta\varphi)$ is required.
In order to isolate this distribution, the measured $C(\Delta\varphi)$
distribution is multiplied by a $B(\Delta\varphi)$ obtained from event
generators. This method is applicable since $B(\Delta\varphi)$ contains only
trivial correlations due to collision geometry and energy conservation that
are well reproduced in event generators. For ToMCCA, $B(\Delta\varphi)$ is
obtained again from EPOS 3 and assumed to be independent of the event
multiplicity. The $C(\Delta\varphi)$ for $pp+\overline{p}\overline{p}$ pairs
is measured by the ALICE collaboration [36], as a function of multiplicity.
However, since measurements in all multiplicity classes are compatible with
each other within their uncertainties, the data measured in the 20-40%
multiplicity class is used for all multiplicities. The final same event
distribution $S(\Delta\varphi)=C(\Delta\varphi)\times B(\Delta\varphi)$ is
expressed as

$S(\Delta\varphi)=\underbrace{(N_{0}\sin(\Delta\varphi-\pi/2)+1)}_{C(\Delta\varphi)}\times\underbrace{(N_{1}|\Delta\varphi|+B)(N_{2}e^{-\Delta\varphi^{2}/A^{2}})}_{B(\Delta\varphi)}.$
(10)

The parameters can be found in Tab. 5.

Parameter | $B(\Delta\varphi)$
---|---
A | 0.85747601
B | 5.7197689
$N_{1}$ | -0.89304215
$N_{2}$ | 0.26041062
$N_{0}$ | 0.17701025
Table 5: Values for the parameterization of the angular correlation.

##  3 Source size

The last key ingredient for coalescence is the source size $r_{0}$, which is a
measure of the distance between the particles of the pair. In ToMCCA, the
source is assumed to be distributed according to a Gaussian function, and thus
the distances $r$ between particles follow the distribution [37]

$S(r;r_{0})=\frac{4\pi r^{2}}{(4\pi
r_{0}^{2})^{3/2}}e^{-\frac{r^{2}}{4r_{0}^{2}}},$ (11)

from which distances are drawn at random. The source size in small systems has
been measured as a function of the average transverse mass333The average
transverse mass is defined as $\langle
m_{\mathrm{T}}\rangle=\sqrt{\left(\frac{p_{\mathrm{T,1}}+p_{\mathrm{T,2}}}{2}\right)^{2}+\left(\frac{m_{1}+m_{2}}{2}\right)^{2}}$
of the pair [37]. In the case of deuteron coalescence, the masses of the
constituent nucleons are almost equal, $m_{1}=m_{2}=m_{N}\approx
0.938\leavevmode\nobreak\ $GeV$/c^{2}$, and the transverse momentum of the
deuteron is given by the sum of those of the constituents,
$p_{\mathrm{T,d}}=p_{\mathrm{T,1}}+p_{\mathrm{T,2}}$ in pp collisions at
$\sqrt{s}\leavevmode\nobreak\ =\leavevmode\nobreak\ 13\leavevmode\nobreak\
TeV$ with a high multiplicity trigger [37]. These collision events have a mean
charged-particle multiplicity of about $dN_{ch}/d\eta\sim 30$. On the other
hand, minimum bias pp collisions have a mean multiplicity of
$dN_{ch}/d\eta\sim 7$ and it is expected that the source size depends on
multiplicity [38]. Thus, a description of the source size for arbitrary
multiplicity is needed to accurately describe the deuteron production. For
this purpose, the coalescence model is reverted, leaving the source size as a
free parameter and including the deuteron momentum distributions in the
fitting procedure of ToMCCA. The deuteron spectra are taken from the ALICE
measurements in pp collisions at $\sqrt{s}=$ 5 TeV [21] and from the HM sample
of pp collisions at $\sqrt{s}=$ 13 TeV [22], spanning a range of
$dN_{\mathrm{ch}}/d\eta$ from 2.4 to 35.8 [22, 23].

The fitting procedure consists of several steps, as described in the
following. First, for a given multiplicity, the deuteron $p_{\mathrm{T}}$
distributions are generated for source sizes from $r_{0}$= 0 to 1.8 fm. Then,
for each value of $p_{\mathrm{T}}$, the
$\chi^{2}=\frac{Y_{\mathrm{ToMCCA}}-Y_{\mathrm{ALICE}}}{\Delta
Y_{\mathrm{ALICE}}}$ (being $Y$ the yield for a given $p_{\mathrm{T}}$, either
in ToMCCA or in data, and $\Delta Y$ the uncertainty of the data points) is
calculated for all tested source sizes. The resulting distribution of
$\chi^{2}$ as a function of the source size, $\chi^{2}(r_{0})$, can be fitted
with a polynomial function of second order, $y=a(x-c)^{2}$. The parameter $c$
gives the best fitting source size and $1/\sqrt{a}$ corresponds to the
$1\sigma$ deviation. This procedure is repeated for every value of
$p_{\mathrm{T}}$ covered by the measurement. The resulting source size scaling
as a function of $\langle m_{\mathrm{T}}\rangle$ is fitted using a power law
(Eq. 13, with parameter A=0). Finally, the procedure is repeated for all
multiplicities in the datasets of pp collisions at $\sqrt{s}=$ 5 TeV and HM at
$\sqrt{s}=$ 13 TeV. The resulting power law parameters $B$ and $C$ are fitted
as a function of $dN/d\eta$ with Eq. 14 and Eq. 15, respectively. The
functional form of Eq. 14 has been chosen to reproduce the observation of
$r_{0}\propto N_{ch}^{1/3}$ at large multiplicities [39] and its saturation to
a minimal source size $r_{0}\sim R_{p}=0.87$ fm [40] similar to the proton
radius ($R_{p}$). The parameters are reported in Tab. 6. With such
parameterization, it is possible to obtain the $\langle m_{T}\rangle$ scaling
of the source size for arbitrary multiplicity values, as shown as an example
in Fig. 3. The predictions of ToMCCA are compared to the source size measured
by ALICE [37] in the HM data sample of pp collisions, corresponding to an
average multiplicity of about 30 charged particles, showing very good
agreement with the results of the model. To date, there are no experimental
measurements of the $\langle m_{T}\rangle$-scaling of the source radii as a
function of the average charged-particle multiplicity in minimum bias pp
collisions at the LHC. Future measurements of this quantity with the LHC Run 3
data will be useful to make a quantitative comparison to the predictions of
ToMCCA.

The uncertainties associated with these predictions stem from the uncertainty
of the measurement of the proton and deuteron production yields. The model
uncertainties are estimated by repeating the fitting procedure using a
parameterization of the lower and upper bounds of the $dN/dy$ parameter in Eq.
16 and taking the uncertainty from the $\chi^{2}$ minimization described
above. The obtained global uncertainties are 3.5% from the protons and 4.4%
from the deuterons, which, added in quadrature, give $\delta
B(dN/d\eta)=5.7\%$. The C parameter is unchanged within these variations.
Propagating this uncertainty to the predictions of the deuteron spectra, by
varying $B(dN/d\eta)$ by 5.7%, changes the deuteron yield by $\pm 4.6\%$.

Figure 3: Scaling of the baryon-emitting source size as a function of $\langle m_{T}\rangle$, for arbitrary multiplicity values. The predictions of ToMCCA are compared to the measured source size corresponding to an event multiplicity of about 30 [37].  Parameter | B($dN/d\eta$) | C($dN/d\eta$)
---|---|---
A | 0.59781 | -0.85246
B | 0.58983 | 0.83824
C | 1.26481 | 4.25301
D | -0.48262 | 0.182242
N | – | 0.99022
M | 10.8889 | –
Table 6: Parameterization as a function of multiplicity of the power law
parameters describing the source size $r_{0}$.

##  4 Results

###  4.1 Transverse momentum distributions and coalescence parameter

Using all the parameterizations as a function of multiplicity described in the
previous Sections and the coalescence afterburner based on the Wigner
function, it is possible to obtain the proton and deuteron momentum
distributions for any value of multiplicity. In Fig. 4, the resulting proton
and deuteron momentum distributions obtained with ToMCCA are shown in the top
panels and compared to the $p_{\mathrm{T}}$ distributions measured by ALICE
[23, 41, 34, 24, 21]. The different colors refer to different multiplicity
classes, as reported in the legend, taken from different pp collision
energies. All results show excellent agreement with the data points, being
able to reproduce data within $1\sigma$ for all the multiplicity classes
covered by pp collisions (up to $\sim 36$). These results demonstrate that the
coalescence model implemented in ToMCCA works for any collision energy and
depends only on the chosen multiplicity class. This can be clearly seen in
Fig. 5, where ToMCCA predictions are compared to results from three different
energies of pp collisions, corresponding to similar multiplicity values
($dN_{\mathrm{ch}}/d\eta\sim 8$). Consequently, the model presented in this
article can be used for any collision energy of pp collisions, finding
applicability for the astrophysics searches of indirect dark matter. Indeed,
to correctly interpret future measurements of the flux of antinuclei from
cosmic rays, it is crucial to model the flux of antinuclei produced in
collisions between the ISM and the primary CRs, which constitute the
background for the DM searches. It has been shown in [5] that antideuterons in
cosmic rays most likely originate from collisions of center-of-mass energy of
$\sqrt{s}=20-30$ GeV. The multiplicities corresponding to these collision
energies have been measured by the ISR experiment [42] and correspond to
values between $dN/d\eta=(1.50\pm 0.01)$ and $dN/d\eta=(1.66\pm 0.02)$, for
energies between $\sim 23.6\,\mathrm{GeV}$ and $\sim 30.8\,\mathrm{GeV}$.
Hence, ToMCCA can be used to model the production of antideuterons at such
multiplicities and to predict the expected flux of antinuclei due to the
background.

Using the resulting proton and deuteron momentum distributions, the
coalescence parameter $B_{A}$ can be computed. It is obtained from the ratio
of the invariant yield of the nucleus with mass number $A$ and that of the
protons squared, assuming protons and neutrons have the same momentum
distributions as they are isospin partners, and $p_{\mathrm{T}}^{\mathrm{p}}$
= $p_{\mathrm{T}}^{A}$/$A$

$B_{A}={\biggl{(}\dfrac{1}{2\pi
p^{A}_{\mathrm{T}}}\biggl{(}\dfrac{d^{2}N}{dydp_{\mathrm{T}}}\biggr{)}_{A}\biggr{)}}\bigg{/}{\biggl{(}\dfrac{1}{2\pi
p^{p}_{\mathrm{T}}}\biggl{(}\dfrac{d^{2}N}{dydp_{\mathrm{T}}}\biggr{)}_{p}\biggr{)}^{A}}.$
(12)

The resulting coalescence parameters are shown in the bottom panels of Figs. 4
and 5, in comparison to ALICE measurements. Also in this case, the predictions
of ToMCCA are in excellent agreement with the experimental results
($<1\sigma$).

Figure 4: Results of ToMCCA in different multiplicity classes, for the
transverse momentum spectra of protons (top left) and deuterons (top right),
and the coalescence parameter $B_{2}$ (bottom), compared to data measured by
ALICE. The values of the average charged-particle multiplicity corresponding
to each multiplicity class in the three different collision energies are
indicated in the legend. The values of $B_{2}$ are scaled for better
visibility, by the following factors, going from the top to the bottom: 42,
16, 8, 4, 2, 1.  Figure 5: Results of ToMCCA in the multiplicity class VI, VII
and VI+VII respectively for pp $\sqrt{s}=$ 5, 13 and 7 TeV, for the transverse
momentum spectra of protons (top left) and deuterons (top right), and the
coalescence parameter $B_{2}$ (bottom), compared to data measured by ALICE.
The values of the average charged-particle multiplicity corresponding to each
multiplicity class (VI, VII, and VI+VII) in the three different collision
energies are indicated in the legend.

###  4.2 Comparison with data as a function of multiplicity

The results shown in the previous Sect. 4.1 can be obtained for arbitrary
multiplicity. Hence, it is possible to scan the full range of average charged-
particle multiplicity per event covered by pp collisions from $\sim 1$ up to
$\sim 70$. Using the deuteron and proton $p_{\rm T}$ distributions obtained
with ToMCCA, the coalescence parameter $B_{2}$ is computed, as well as the
deuteron-to-proton ratio of the integrated yields (d/p). Selecting one value
of $p_{\rm T}/A$, the predictions for the $B_{2}$ of ToMCCA are shown as a
function of multiplicity in Fig. 6, and compared to the data measured by
ALICE. Similarly, the d/p integrated yield ratios are shown as a function of
multiplicity in Fig. 7. The predictions from ToMCCA show an excellent
agreement with data, within 1$\sigma$, for both the $B_{2}$ and the d/p ratio.

Figure 6: Coalescence parameter $B_{2}$ as a function of multiplicity
predicted by ToMCCA (red line), compared with the $B_{2}$ measurements from
ALICE. The blue markers show the data included in the parametrizations, while
the green markers show the ones not included. The dotted line is the model
prediction of Ref. [38], where the coalescence parameter for deuterons with a
size of 3.2 fm is computed.  Figure 7: Deuteron-to-proton ratio of integrated
yields as a function of multiplicity obtained with ToMCCA (red solid line),
compared to measurements of the ALICE Collaboration. The width of the model
predictions reflects the statistical uncertainties. The dotted line is the
model prediction of Ref. [14].

##  5 Conclusions

In this paper, we have introduced ToMCCA (Toy Monte Carlo Coalescence
Afterburner), a toy Monte Carlo model for coalescence based on the Wigner
function formalism. Our results show that ToMCCA is able to predict deuteron
yields with a precision better than 5% if all the required parameters, namely
the charged particle multiplicity, nucleon momentum distributions, angular
correlations, and the two-nucleon source size, are provided from
parameterizations of experimental measurements. The results of this model
demonstrate that ToMCCA can be used for arbitrary center-of-mass energy of pp
collisions, as the coalescence predictions depend only on the average
multiplicity of the event rather than on the collision energy. This would be
useful for astrophysical studies, enabling the prediction of antinuclear
fluxes in cosmic rays at the multiplicities corresponding to the energies at
which antinuclei are produced in the collisions of ISM and CRs. This is key
for disentangling the contribution of Dark Matter annihilations from the
Standard Model background, for correctly interpreting future measurements of
antinuclear fluxes in cosmic rays.

Due to its lightweight nature and minimum complexity, ToMCCA can also be
easily extended to study more complex phenomena related to light nuclei
formation, either towards other collision systems, lower collision energies
and multiplicities, or towards heavier nuclei, as long as their wavefunction
is known.

##  Appendix A Mathematical functions

Power Law:

$F(x)=A+B(x)^{C}.$ (13)

Root-3-Sigmoid function, representing a smooth transition from cube-root
behavior to zero-degree polynomial:

$F(x)=\frac{A}{1+e^{-B(x-M)}}x^{1/3}+(1-\frac{1}{1+e^{-B(x-M)}})C+D.$ (14)

Sigmoid Function:

$F(x)=\frac{A}{(B+e^{D\cdot x-C})}+N.$ (15)

Levy-Tsallis, describing the transverse momentum distributions of particles
with mass $m$, described by a temperature parameter $T$, a shape parameter
$n$, and with total yield $dN/dy$:

$\dfrac{d^{2}N}{dp_{\mathrm{T}}dy}=\dfrac{dN}{dy}\dfrac{p_{\mathrm{T}}}{nT}\dfrac{(n-1)(n-2)}{(nT+m(n-2))}\left(1+\dfrac{m_{T}-m}{nT}\right)^{-n}.$
(16)

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11affiliationtext: Department of Science and Mathematics, Texas A&M
University-Central Texas
<EMAIL_ADDRESS>
# Sliced, not Splitted: a Better Alternative to Many-Worlds?

Christopher Thron and Braeden Welsch

Abstract: The many-worlds interpretation (MWI) of quantum mechanics is
currently experiencing a popular resurgence, propelled by such prominent and
articulate physicists as Sean Carroll, David Deutsch, Max Tegmark, and Lev
Vaidman. The consequences of MWI are mind-boggling: the spacetime universe of
our experience is only one branch of an unimaginably fast-multiplying plethora
of alternative universes held incommunicado. In this paper, we propose that
the mass of Medusa’s hair served up by MWI is due to a failure to embed the
spacetime universe in the right space: spacetime is a slice of bread, not a
splitting strand of pasta. By way of motivation, we first give a very simple
presentation of Bell’s inequality by comparing it to a “quantum game show”,
followed by a simple description of Aspect’s 1985 experiment involving
entangled photons which confirms the inequality. We interpret the paradoxical
correlation between measurements as resulting from a process outside of
spacetime that produces both the original entanglement and the measurements.
This is followed by a brief presentation of MWI, and then by a pictorial
comparison of the proposed process model and MWI. The final section lists a
number of potential consequences of the model related to causality,
determinism, free will, and consciousness, and points to further references
that give a more in-depth and rigorous presentation of the process dimension
model.

The entire article is non-technical and requires no mathematical background
other than high school mathematics and an understanding of basic concepts in
probability. The physics involved in Aspect’s experiment is also explained.

Keywords: quantum mechanics; interpretation; Bell’s inequality; Aspect
experiment; entanglement; correlation; causality; process; spacetime;
universe; many-worlds; free will.

AMS 2020 Subject Classification: 81-01, 81P05, 81P40, 81Q65

##  1 A quantum game show

The following analogy will elucidate a fundamental difference between
classical and quantum mechanics. The example involves a game show in which a
single participant chooses two of three doors, each concealing one of two
prizes (cash or car). Let us label our three doors as green (G), red (R), and
blue (B). Figure 1 shows the setup.

Figure 1: The doors, in order from left to right, are green (G), red (R), and
blue (B). The arrows point to the two possible prizes (cash or car), one of
which is behind each door.

The goal of this “Quantum Game Show” is to choose two doors with the _same_
prize (similar to Wheel of Fortune’s “half-car” prize, where you only win the
car if you get both halves). Dedicated fans of the show who’ve watched
thousands of episodes have calculated the probability of winning by choosing
adjacent doors is $85\%$, For example, since the green door G is next to the
red door R, it follows that that the probability that the green and red prizes
are the same is $0.85$. For short, we’ll write this as
$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.7,0,0}R}})=0.85$
Similarly, we also have
$P(\textbf{{\color[rgb]{0.7,0,0}R}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})=0.85$.

Now comes the big question. Knowing this information (and only this
information), can we estimate the probability of winning by choosing the green
and blue doors (i.e.
$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})$)?

Let’s consider the possibilities. A little thought reveals that it must be
true that at least two doors have the same prize at all times, and possibly
all three doors have the same prize. To represent the different possibilities
(and to brighten up the text), let’s introduce some colorful notation. We’ll
indicate which door has a different prize from the other two by greying out
its color and using lowercase. For example, GRb indicates the case where the
green and red doors have the same prize, and the blue door has a different
prize. Alternatively, GRB is the case where all three doors have the same
prize. Table 1 summarizes all possible arrangements.

|  G & R outcomes
---|---
|  | $~{}~{}\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.7,0,0}R}}~{}~{}$ | $~{}~{}\textbf{{\color[rgb]{0.2,0.85,0}G}}\neq\textbf{{\color[rgb]{0.7,0,0}R}}~{}~{}$
R&B outcomes | $\textbf{{\color[rgb]{0.7,0,0}R}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}}$ | GRB | gRB
$\textbf{{\color[rgb]{0.7,0,0}R}}\neq\textbf{{\color[rgb]{0.03,0.63,0.73}B}}$ | GRb | GrB

Table 1: Possible arrangements of matching prizes behind the three doors. For
example, gRB represents the case when the red and blue doors have the same
prize while the green door is different.

The four entries in the table represent all possible prize arrangements, and
exactly one of these arrangements must be realized each time the game is
played. It follows that the sum of the probabilities of the four outcomes must
be 1:

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})+P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.5,0.5,0.5}b}})+P(\textbf{{\color[rgb]{0.5,0.5,0.5}g}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})+P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.5,0.5,0.5}r}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})=1$
(1)

The first column of Table 1 lists the two possible ways that the green prize
can be the same as the red prize. As mentioned above, game fans have
determined the probability of this happening is 0.85. We may then add the
probabilities and obtain:

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})+P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.5,0.5,0.5}b}})=P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.7,0,0}R}})=0.85$
(2)

Also, the first row of Table 1 lists the two possible ways that the red prize
can be the same as the blue prize. We know already this happens with
probability 0.85. So by similar reasoning we have

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})+P(\textbf{{\color[rgb]{0.5,0.5,0.5}g}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})=P(\textbf{{\color[rgb]{0.7,0,0}R}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})=0.85.$
(3)

Now for some algebra. Taking (2) + (3) $-$ (1) and cancelling terms gives us:

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})-P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.5,0.5,0.5}r}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})=0.7,$

and adding
$2P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.5,0.5,0.5}r}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})$
to both sides of the equation gives:

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.7,0,0}R}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})+P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.5,0.5,0.5}r}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}})=0.7+2P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.5,0.5,0.5}r}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}}).$
(4)

The left-hand side of (4) represents all possible arrangements where G is
equal to B (i.e.
$\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}}$);
while the right hand is 0.7 plus a non-negative number, which must be greater
than or equal to 0.7. This leads us to our final result:

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})\geq
0.7.$ (5)

In conclusion, the probability of having the same prize behind the green and
blue doors must be at least $70\%$. So although we can’t find the exact
probability, at least we can get a lower estimate.

If we’d worked out the formula without putting in specific numbers, we’d have
found that in general:

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})=P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.7,0,0}R}})+P(\textbf{{\color[rgb]{0.7,0,0}R}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})-1+2P(\textbf{{\color[rgb]{0.2,0.85,0}G}}\textbf{{\color[rgb]{0.5,0.5,0.5}r}}\textbf{{\color[rgb]{0.03,0.63,0.73}B}}),$
(6)

or

$P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})\geq
P(\textbf{{\color[rgb]{0.2,0.85,0}G}}=\textbf{{\color[rgb]{0.7,0,0}R}})+P(\textbf{{\color[rgb]{0.7,0,0}R}}=\textbf{{\color[rgb]{0.03,0.63,0.73}B}})-1.$
(7)

Equation (7) is one way of stating _Bell’s inequality_.

There are no tricks in what we’ve done so far–it’s all up-and-up mathematics,
straight out of any probability textbook. But what’s actually measured in the
real-life quantum experiments doesn’t agree, as we shall see shortly.

For an alternative presentation of Bell’s inequality (without the game show,
and with more physics) see Lorenzo Maccone’s article [12].

##  2 Aspect’s Experiment

A real-life “Quantum Game Show” experiment was conducted by French physicist
Alain Aspect during the early 1980s [1]. In order to understand the
experiment, we first need some background information about physics. The
following sections introduce two key physical phenomena that play important
roles in Aspect’s experiment: polarization and photons.

### Polarization of light waves

In physics, there are two basic wave types: longitudinal and transverse. Both
types of waves have a direction of travel and direction of oscillation (the
“back-and-forth” motion of the wave). The relation between these directions
determines which type of wave it is. Longitudinal waves travel _parallel_ to
the direction of oscillation, whereas transverse waves travel _perpendicular_
to the direction of oscillation. Traffic jams and sound are common examples of
longitudinal waves, while light and ripples on the surface of water are
examples of transverse waves.

Polarization is a property of transverse waves in which the direction of
oscillation is consistent. For example, water waves are polarized because the
oscillation is always in the up-and-down direction, and never from side to
side. This is not true of light: for example the Sun (and other common light
sources) emits _unpolarized_ light waves, or _electromagnetic waves_ , which
we see in Figure 2. In this case, the direction of oscillation can be random,
as long as it’s always perpendicular to the direction of travel. Unpolarized
light waves can become polarized by passing through a _polarizer_ which fixes
the oscillation in a consistent direction.

Figure 2: Light source producing unpolarized light which passes through a
polarizer to produce a polarized light wave. Web Reference:
https://www.researchgate.net/publication/328100369_Polarised_infrared_light_enables_enhancement_of_histo-
morphological_diagnosis_of_prostate_cancer

There is more than one way that a transverse wave can be polarized. Linear
polarization is the simplest type of polarization. Suppose you are holding one
end of a long rope and begin to move your hand directly up and down. The
rope’s consequential oscillating motion corresponds to a linearly polarized
transverse wave. A mathematician would recognize the produced wave as a
sinusoidal wave. If you were to move your hand continuously around in a
circle, the rope’s oscillation would then represent a circularly polarized
transverse wave. Figure 3 demonstrates these two distinct types of polarized
transverse waves (it’s also possible to have elliptical polarization, but
that’s more complicated than we need to consider right now).

Figure 3: Linear and circular polarization of electromagnetic waves. The red
arrow in each diagram shows the direction of the electric field in the
electromagnetic wave (light wave). Taken from
https://www.emedicalprep.com/study-material/physics/wave-optics/polarization/

Light beams should actually not be considered as a single wave, but as a
collection of many individual waves travelling in the same direction. Each
individual wave carries a small amount of energy: the amount of energy is
determined by frequency of the wave (the frequency of the wave also determines
its color: for example, red light has a lower frequency (and less energy) than
blue light). Each one of these individual waves is called a _photon_. Photons
are often thought of as “particles” of light, and in some respects they behave
like discrete objects. However, this conception is inaccurate and misleading,
so we will avoid referring to photons in this way.

Each individual photon has a definite polarization, but a light beam may
consist of many photons with different polarizations that happen to be
traveling in the same direction. If you send a single photon through a
polarizer, it either makes it through or it doesn’t. If the photon makes it
through, then the outgoing photon has a polarization that is parallel to the
polarizer’s orientation. If the photon doesn’t make it through, then the
photon is reflected by the polarizer, and its outgoing polarization is
perpendicular to the polarizer’s orientation.

If a light beam consists of circularly polarized photons, then regardless of
polarizer’s orientation half of the photons will be measured as parallel and
half will be measured as perpendicular. A key point here is that the
measurement of a circularly polarized photon necessarily changes the
polarization of the photon. Unlike ordinary ‘classical’ physics, in quantum
physics it is impossible to make a measurement without changing the thing
you’re measuring: this is one of the biggest differences between classical and
quantum physics.

### The Experiment

We now have the physical background needed to understand Aspect’s experiment,
which is schematically represented in Figure 4. At the center of the
experiment lies a source $S$ which prepares two ’identical’ circularly-
polarized photons $\gamma_{1}$ and $\gamma_{2}$, travelling in opposite
directions. These photons are ‘identical’ in the sense that their
polarizations are the same according to any observer who measures them (in
physics terminology, the two photons are said to be _entangled_).

Figure 4: Schematic representation of Aspect’s experiment, as described in
[1]. The red line indicates the initial orientation of $A$ and $B$. The green
dashed line indicates $A$ is oriented in the counter-clockwise direction while
the blue dashed line indicates $B$ is oriented in the clockwise direction. The
angle between $A$ and $B$ is used to calculate the probability of the outcome.

For this experiment, the observers are linear polarizers placed on opposite
sides of the source which measure the two photons’ polarizations as they pass
through. Let $A$ be the polarizer that measures $\gamma_{1}$ and $B$ be the
polarizer that measures $\gamma_{2}$. There are two possible measurements $A$
and $B$ can give for their respective photons: parallel $(+)$ or perpendicular
$(-)$. For example, if $A$ reads $(+)$, then $\gamma_{1}$’s polarization is
parallel relative to the $A$’s orientation.

Initially, $A$ and $B$ are oriented in the same direction and thus yield the
same measurement, since $\gamma_{1}$ and $\gamma_{2}$ are entangled. In this
case, the angular difference between $A$ and $B$ (denoted by $\theta$) is
zero. We thus have $P_{same}(\theta)=1$ for $\theta=0$.

In general, given the angle between $A$’s and $B$’s orientation is $\theta$,
quantum theory predicts that the probability that $A$ and $B$ have the same
measurement is $P_{same}(\theta)=\cos^{2}{\theta}$.Since the photons either
have the same or different polarizations, it follows that
$P_{diff}(\theta)=1-\cos^{2}{\theta}=\sin^{2}{\theta}$. For a mathematical
derivation of these results, see [13].

For the purposes of this experiment, we use three different orientations for
$A$ and $B$ (we’ll be referring to Figure 4 above.)

  1. (1)

Polarizer $A$ is rotated to align with green arrow while $B$ remains on the
red arrow, where the angle between the two is $\theta=22.5^{\circ}$.

  2. (2)

Polarizer $A$ remains on the red arrow while $B$ is rotated to line up with
the blue arrow to give the angle $\theta=22.5^{\circ}$.

  3. (3)

Polarizers $A$ and $B$ are aligned with the green and blue arrows
respectively, so that $\theta=45^{\circ}$.

In cases (1) and (2), we can easily see that the probability that $A$ and $B$
have the same measurement is
$P_{same}(\theta=22.5^{\circ})=\cos^{2}{(22.5^{\circ})}=.85$, rounded to the
nearest hundredth. Using the quantum game show probabilities derived in
Section 1 we obtain the following result:

$\displaystyle\textrm{(According to game show)}\qquad
P_{same}(45^{\circ})\geq.85+.85-1=0.7.$

Yet, when we use quantum mechanics to predict the angle of orientation between
$A$ and $B$, we get:

$\displaystyle\textrm{(According to quantum theory)}\qquad
P_{same}(45^{\circ})=\cos^{2}(45^{\circ})=0.5$

So who’s right: quantum mechanics, or our intuition? Both can’t be right! What
Aspect showed is that quantum mechanics wins, and intuition fails. The
probability turns out to be only $0.5$.

### The bottom line

What’s going on here? It seems that we’re missing something. Whenever
something happens that we don’t expect, that means that we’ve assumed
something which is not actually the case. But what have we assumed? It seems
that in the game-show example the only assumption we’ve made is that the laws
of probability are valid. But in fact, we’ve also assumed something else,
namely that the game-show host is unaware of the choice we are going to make,
so the prizes are set regardless of what doors (measurements) we choose.

The violation of Bell’s inequality demonstrates a similar hidden assumption in
our analysis of Aspect’s experiment. Table 1 assumes that there are four
definite, distinct possibilities for the two prepared photons. In other words,
the photons have “already decided” what they are going to do before the
experimenter decides which measurement to make. By doing so, we assume the
state they are in doesn’t depend on which measurements are actually made.
Mathematically, we describe this by saying that the photon states prior to
measurement are _independent_ of which measurements are subsequently made.

Based on our experience, this assumption is ‘obviously’ true. How could a
measurement of a photon reach back into the past and change how the photon was
created? But let us recall Sherlock Holmes’ famous dictum: “When you have
eliminated the impossible, whatever remains, however improbable, must be the
truth.” [3]

##  3 The process space proposal

In the previous section, we suggest that perhaps we have to take another look
at how we envision reality. One way to develop a new perspective is to look
for possible analogies with situations that we’re more familiar with. This is
exactly what Einstein did when he viewed gravitation in terms of the “curving”
or “bending” of spacetime, which can be compared to the curving of a rubber
sheet in three-dimensional space.

In this section we’ll attempt to explain the “quantum game show” by taking the
same approach. First we’ll describe a commonplace situation, and then proceed
to develop possible connections between this well-understood occurrence and
the puzzling situation we see with quantum reality. In the next section, we’ll
compare this approach with another popular explanation of the “quantum game
show”, namely the so-called the “many-worlds interpretation”. See which you
think is better!

### A pebble in a pond

Suppose a pebble is thrown into an otherwise still pond. The splash from the
pebble produces waves that form concentric circles, propagating outward from
the location of the splash (see Figure 5). Eventually the waves reach the
pond’s shore. At any given instance, each wave touches the shore at a single
point, which appears to be running down the shore with (nearly) constant
velocity. This creates the impression of an ‘object’ moving along the
shore–whereas in fact there is no ‘object’ at all, only a moving location
where the wave contacts the shoreline.

Figure 5: Waves from the pebble in a pond. Each wave contacts the shoreline at
a point (in red), and the point of contact moves down the shore to the right.

Neither the original splash nor the emanating circular waves are apparent at
the shoreline, which just sees a running point of contact.

Is it possible that physical ‘objects’ (like photons) are actually something
like this? Is a photon in some sense a “point of contact” between spacetime
and something that happens outside of spacetime? In the next section, we
develop this idea a bit further by constructing a simplified conceptual model.

### Introducing the ‘process dimension’

One way to develop a more thorough understanding of a situation is to build a
model. Models can be physical (like an architect’s scale model of a building),
or they can be visual or conceptual, like a diagram (which requires more
imagination to appreciate). Model-building always involves making some
simplifications, in order to reduce unnecessary complications and bring out
essential details of the situation being modeled.

Now we’re trying to model a situation where events that occur outside of space
and time “collide” with our universe. But it’s difficult enough to visualize
four-dimensional spacetime, let alone events that occur somewhere outside! To
get a hold of this idea to begin with, we should try to simplify as much as
possible, without excluding the essential details. In doing this, we may once
again cite the example of Einstein, who is often quoted as saying “Everything
should be made as simple as possible, but no simpler.” 111The exact quote from
Einstein is longer: “…the supreme goal of all theory is to make the
irreducible basic elements as simple and as few as possible without having to
surrender the adequate representation of a single datum of experience.” [14]

So let’s consider a simplified “universe” in which there is only one space
dimension and one time dimension. Then we can represent spacetime as a 2
dimensional plane, where the $x$ and $y$ axes represent space and time
respectively. We now introduce into the picture a third dimension, which we’ll
call the _process dimension_ , which is _outside_ of space and time. We
suppose that events occur in this higher-dimensional “process space”, and the
influence of any event propagates down along the process dimension.

Figure 6 shows an event in process space that occurs outside of spacetime, but
nonetheless influences both the creation and subsequent measurements of two
photons. For this perspective, it appears that nothing is actually
“travelling” between the creation and detection of the photons: there’s only
the illusion of movement, which comes from the fact that both the creation and
detection originate from a common source.

Figure 6: Representation of photon creation and measurement in the
“superspace” picture. The process (which occurs outside of spacetime)
influences the photon creation and measurements: while in spacetime, we only
see the creation and measurements.

This same picture can be generalized to describe any phenomenon that involves
cause and effect. Suppose an event in our spacetime dimension occurs which
seemingly “causes” another event. In our experience, the events happen
sequentially. Yet, in the process dimension there is an unfolding process that
directly influences both the “causing” event and the ‘caused” event, which
produces the correlation between the two.

##  4 Another view: the “Many Worlds” alternative

Another theory of reality is the ‘many-worlds interpretation’ (MWI), which
describes reality as having multiple alternative timelines that branch out at
different events. A fanciful popular representation of this idea can be found
in the “Terminator” movie series, in which time-traveling agents move along
different timelines to try and change past events (unfortunately, time travel
is not actually a part of MWI!)

MWI was first described as a possible explanation of quantum mechanics by Hugh
Everett in 1957 in his Ph.D. thesis at Princeton University[6] (it is
interesting however that Erwin Schrödinger, one of the founders of quantum
theory, had similar ideas previously[8]). Everett was trying to solve a major
puzzle: it seemed that quantum particles like electrons and protons take the
form of vague probability clouds, which instantaneously collapse suddenly
every time a measurement is made. This so-called “collapse of the wave packet”
is not part of the equations themselves, but is what evidently happens in our
experience. In our quantum game show, this phenomenon is reflected in the fact
that the choice of one door seems to instantaneously influence the result
behind a different door, which is seemingly impossible.

Everett addressed this problem by suggesting that there is no “collapse” at
all–instead, when a measurement is made all possible results do occur, but on
different branches of reality. I only measure one result because the “I” that
I’m aware of is following one of the branches. The different branches have
different “weights”, which explains why different outcomes appear to have
different probabilities.

Despite its weirdness, the MWI has experienced a resurgence in popularity, and
several prominent physicists are now enthusiastic supporters. Some well-known
examples include David Deutsch, Sean Carroll and Lev Vaidman, all of whom have
popularized the idea in articles[24], books[2], and/or videos
([4],[21],[22],[23][5]).

##  5 Which explanation is simpler?

We have presented two very different explanations for the quantum game show.
Which explanation is right? Or are they both wrong?

This may not be the right question to ask. In fact, in physics there has never
(so far) been an explanation that was “right”. For a long time, Newton’s Three
Laws were regarded as absolute truth. But around the beginning of the 20th
century, it was found that these laws fail miserably outside of a very limited
range of phenomena (which just so happened to be the phenomena that physicists
had previously been able to study systematically). The very same thing has
happened to every other famous equation in physics. We have no reason to
believe that this situation will ever change–the weight of evidence points to
the conclusion that we will never have a “right” equation in physics.

So instead of ‘Which is right?’, perhaps we should ask other questions.
Following Einstein’s original quote in [14], two questions we should be asking
are: ‘Which agrees better with experience?’ and ‘Which is simpler?’
Unfortunately, the question about experience is unhelpful in deciding between
the process model and many worlds, because _so far_ , both predict the same
thing. This situation may change however: David Deutsch claims there may be an
experimental test of MWI [8], and the process model should lead to tiny
deviations from the quantum wavefunction [19].

So for the present, we will focus on the question, “Which explanation is
simpler?”. Certainly such judgements are subjective, and individuals will have
different predilections. A “simpler” theory should be easier to represent. So
let us compare representations of the two theories.

Figure 7 depicts a 0-space + 1-time dimensional “many worlds” universe. The
diagram is full of splitting timelines, where each split represents a
spacetime event that has two possible outcomes (red or green). All of these
timelines together constitute “reality”. The number of world lines increases
exponentially as time progresses. “I” experience being on one of these
branches, but alternate “I”’s have different experiences on different
branches. The events in “my” past and future are correlated because the
current red/green state depends on previous branchings.

Figure 7: 0-space + 1-time dimensional universe, according to the many worlds
picture.

In contrast, Figure 8 represents the process space model for a 0+1 dimensional
universe. Here we see ‘bubbles’ in two dimensional process space, which
represent the formation of cause-and-effect pairs. Instead of the wild
branching of MWI, the physical spacetime universe appears as a single cross
section taken from a larger picture.

Figure 8: 1-dimensional spacetime as a slice in 2-dimensional process space

Which is simpler? Let the reader decide.

##  6 Re-envisioning “reality”

Suppose that the process proposal is true, and the universe is a “seashore”
for a larger “ocean” in which inaccessible processes take place. What are the
consequences of this new outlook? How does it affect how we look at the
universe, at physics, and at ourselves? The following subsection describes
some of the conceptual consequences.

### Ideas to consider

  * #1:

_Spacetime is a cross-section, not a process._ We typically think of reality
as a process that develops over time, and the sciences of physics, chemistry,
biology etc. as describing processes in time. In virtually all equations in
science that involve time, time is the independent variable and other
quantities are functions of time (Einstein’s equations in general relativity
are the only exception that I can think of). Of course we do this because we
experience the universe as an unfolding process in time. But this is a
terrible reason! We also experience the earth as stationary–and we all know
where that led us.

In the process space model, time is no longer singled out as special, and
instead is placed on the same level as dimensions of space . Time seems
different to us only because of the way we experience the universe. For
example, if you’re riding a train things appear to be moving in one direction.
But that’s just because we’re riding the train. If we were riding a different
train, things would look like they’re moving in a different direction. In the
same way, things seems to develop in time just because of how we humans are
‘travelling’ through spacetime.

From this angle, we may see that the complicated branching of MWI in Figure 7
can be traced back to the presumption that the universe is a process in time.
If we attribute this process to another dimension, then the complication
disappears.

Unfortunately, the process space model does not show us how to predict the
long-term future or access the distant past. It can’t help us build a time
machine or a faster-than-light transporter. We’re stuck in our universe, and
the picture doesn’t show us how to jump off into the process dimension to see
the processes developing. But on the other hand, it doesn’t say that we
_can’t_ jump off!

  * #2

_There is no collapse of the wave packet._ The conundrum posed by the so-
called “collapse of the wave packet”, was one of the primary motivations for
MWI. But in the process theory, the conundrum disappears. The so-called wave
packet is depicted as a statistical description of what goes on in the process
dimension “behind the scenes”, which sets the stage for what we see in
spacetime. The process evolves in wavy fashion, but gives rise to sharp,
specific physical outcomes in spacetime.

  * #3:

_Entanglement is uncovered, not produced._ Einstein’s “spooky action at a
distance” is a consequence of quantum entanglement, in which particles or
systems become mysteriously and invisibly “linked” due to interactions. The
process space model shows that interactions do not “produce”
entanglement–rather, both the interaction and the measurements of entangled
particles are vestiges of the same extra-universal process.

  * #4:

_There are no causes without effects._ In the process space model, what we
typically call “causes” and “effects” share a common origin outside of the
universe. When the Hubble space telescope detects light from galaxies, the
detection was not a “afterthought”, but was “predestined” together with the
light’s emission thousands of years ago. This idea is actually not new–it has
been proposed before, by Wheeler and Feynman[25] and earlier by Tetrode[15].
However, they did not envision the cause-effect connection as the culmination
of a process that takes place outside of the universe.

  * #5:

_There is no “reality” apart from observation._ We typically think of an
“experiment” as an observer looking at some phenomenon or physical system from
the outside, and making impartial, passive observations and measurements. This
concept was already roughed up by quantum mechanics, which showed that quantum
measurements are never passive, and always disturb the system they measure.
The process space model goes one step further, and proposes that this is
because there is no “objective” reality “out there” to measure. The division
between “phenomenon” and “measurement” is somewhat arbitrary–both are aspects
of a larger whole within process space.

It is interesting that there are several similar examples within physics of
unifications between apparently disparate phenomena. There is for example the
unification of electric and magnetic fields in classical electromagnetism; the
gauge field representations of matter and electromagnetism in quantum
electrodynamics (see the wooden fish example in [16]); and “Grand Unified
Theories” and “Theories of Everything” in modern particle physics.)

One may raise the objection that reality must exist independent of
observation, since different observers always observe the same reality.
However, this is a magic trick that is due to the common origin of _all_
measurements within process space.

  * #6

_Spacetime is becalmed._ Conventional quantum field theory takes the view that
(quoting Lawrence Krauss in [11]) “Empty space is a boiling, bubbling brew of
virtual particles that pop in and out of existence in a time scale so short
that you can’t even measure them.” The process space model interprets this
supposed storm of activity as a mathematical trace of the unfolding process
located outside of accessible spacetime.

  * #7:

_Free will returns!_ There is a pervasive “scientific” idea (championed by Sam
Harris [9], among others) that past physical processes dictates our present
and future. This is an illusion, says the process space model. You are not
determined by your past, nor merely the product of circumstances. Both you and
previous circumstances come from somewhere else. There are hidden processes
that gave rise to make you who you are in this world, but there are not
physical processes in this universe.

One might suggest that the process theory just displaces determinism into
another dimension, and that we are still the outcome of processes outside of
ourselves. Not so fast. In the process space model, the meaning of “self” is
no longer so clear. How far do “I” extend into the process dimension? To say
that “I” am confined to spacetime is like saying that an iceberg is only the
visible part that sticks above the water.

  * #8:

_The quest for an ultimate “Theory of Everything” is a rainbow chase._
Consider the evolution of the conception of the physical world in Western
intellectual history. Greek philosophers conceived of solid objects as space-
filling entities having extension and rigidity as fundamental properties. Much
later, objects were re-envisioned as consisting of ensembles of jiggling
atoms, held together by electromagnetic forces. Later, atoms themselves became
mostly empty space, consisting of much smaller particles (with electrons
having no size at all). Then these particles turned into quanta, which are a
strange amalgam of particle and wave. Later, quanta turned into resonances of
a field, and different particles corresponded to different components of a
multdimensional field. Then we had vibrating strings, then membranes. This
process is clearly not going to end.

By now, it should be abundantly clear that every physical theory is simply
convenient conceptualization of our current level of interaction with the
universe. We are never getting any closer to “true reality”, which is a
concept that should be banished from the scientific/philosophical vocabulary.
The best we can do is provide a series of analogies, supported by mathematical
substructures. These are all temporary structures, and will always be
superseded. In a related context, Ludwig Wittgenstein gave the analogy of a
ladder, which is used to climb up to a higher level and is then discarded as
unnecessary.

  * #9:

_Science explains almost nothing._ Physical science is concerned with
relationships between events, phenomena, and conditions in spacetime. But the
process space model shows that spacetime itself is only a single slice of
process space, and, as such, has no direct access to the bulk of processes
that rule the universe. As such, science can only offer explanations and
predictions that are highly generalized or tightly localized in space and
time. Consider for example the famous “baby picture of the universe” (see
[7]), which shows variations in the temperature of the microwave background
radiation of the universe on the order of 200 microkelvins. It is indeed a
great triumph that the cosmic inflation theory predicts the observed
blotchiness. However, it cannot even hope to explain the location or extent of
a single blotch. At the small end of the scale, lattice quantum chromodynamics
(QCD) simulations of a single neutron require computing power equivalent to
6,000 laptops operating continuously for a century [10].

  * #10:

_Consciousness may be extra-physical._ There are phenomena within the physical
world that do not have their origin within the physical world, and thus cannot
be constructed by physical procedures. Consciousness may be one such
phenomenon (and angels?). So we may not be able to build conscious computers.

### More in depth

Mathematical and theoretical aspects of the process theory for ordinary
quantum mechanics (not quantum field theory) has been developed in a series of
papers ([18],[20]). The second paper also explains physical consequences of
the theory that differ from conventional quantum mechanics, which could
conceivably be used to test the theory.

A video series that presents the theory on a more popular level has been
placed on YouTube [17]. Note however this series does contain some
inaccuracies.

## References

  * [1] Alain Aspect, Jean Dalibard, and Gérard Roger.  Experimental test of bell’s inequalities using time-varying analyzers.  Physical review letters, 49(25):1804, 1982.
  * [2] Sean Carroll.  Something deeply hidden: quantum worlds and the emergence of spacetime.  Dutton, 2019.
  * [3] Arthur Conan Doyle.  The Sign of the Four.  Penguin Classics, London, 2014.
  * [4] David Deutsch.  Apart from universes.  Vimeo, 2009 [Online].  https://vimeo.com/5490979.
  * [5] David Deutsch.  Multiple worlds and our place in them.  Youtube, 2021 [Online].  https://www.youtube.com/watch?v=b_6vYwCkIpc.
  * [6] Hugh Everett.  “relative state” formulation of quantum mechanics.  In B. DeWitt and N. Graham, editors, The Many Worlds Interpretation of Quantum Mechanics, pages 141–150. Princeton University Press, 2015.
  * [7] Megan Gannon.  New ‘baby picture’ of universe unveiled.  Space.com, 2012 [Online].  https://www.space.com/19027-universe-baby-picture-wmap.html.
  * [8] John Gribben.  The many-worlds theory, explained.  MIT Press, 2020 [Online].  https://thereader.mitpress.mit.edu/the-many-worlds-theory/.
  * [9] Sam Harris.  Free will.  Simon and Schuster, 2012.
  * [10] Glenn Roberts Jr.  Supercomputers provide new window into the life and death of a neutron.  Berkeley Lab, 2018 [Online].  https://newscenter.lbl.gov/2018/05/30/supercomputer-simulations-new-window-lifetime-death-neutron/.
  * [11] Lawrence Krauss.  Lawrence krauss on ‘a universe from nothing’.  National Public Radio, 2012 [Online].  https://www.npr.org/2012/01/13/145175263/lawrence-krauss-on-a-universe-from-nothing.
  * [12] Lorenzo Maccone.  A simple proof of bell’s inequality.  American Journal of Physics, 81(11):854–859, 2013.
  * [13] Frank Rioux.  A brief description of aspectś experiment.  LibreTexts, 2020.  https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Tutorials_(Rioux)/Quantum_Teleportation/354%3A_A_Brief_Description_of_Aspect’s_Experiment.
  * [14] Andrew Robinson.  Did einstein really say that?  Nature, 557(30), 2018 [Online]. doi: https://doi.org/10.1038/d41586-018-05004-4.
  * [15] Hugo Martin Tetrode.  On the causal connection of the world, an extension of classical dynamics.  Zeitschrift fur Physik, 10:317–328, 1922.
  * [16] Chris Thron.  God is light, sin is entropy: Physical analogies for biblical concepts.  Fifth Interdisciplinary Conference on Science, Technology and Religious Ideas, Kentucky State University, 1994 [Online].  https://crosspollen.neocities.org/sin_entropy.htm.
  * [17] Chris Thron.  Is cause and effect an illusion? arguments from physics.  Youtube, 2015 [Online].  https://www.youtube.com/playlist?list=PL2uooHqQ6T7PgUzsM95xUWfBuyeGBUXzx.
  * [18] Chris Thron and Johnny Watts.  A signal-processing interpretation of quantum mechanics.  The African Review of Physics, 8, 2013.
  * [19] Christopher Thron.  An accumulative model for quantum theories.  arXiv preprint arXiv:1507.03944, 2015.
  * [20] Christopher Thron.  An accumulative model for quantum theories.  Electronic Journal of Theoretical Physics, 12(33), 2015.  http://www.ejtp.com/ejtpv12i33.html (see also https://arxiv.org/abs/1507.03944.
  * [21] Lev Vaidman.  From bell inequalities to many worlds interpretation.  YouTube, 2014 [Online].  https://www.youtube.com/watch?v=jKGuGptafvo.
  * [22] Lev Vaidman.  In favor of the many worlds interpretation.  CRNS-Grenoble, 2020 [Online].  https://www.youtube.com/watch?v=CE3rzGHUjqI.
  * [23] Lev Vaidman.  In the many-worlds interpretation.  Harvard University, 2021 [Online].  https://www.youtube.com/watch?v=1Yx-wQYODiA.
  * [24] Lev Vaidman.  Many-worlds interpretation of quantum mechanics.  Stanford Encyclopedia of Philosophy, 2021 [Online].  https://plato.stanford.edu/entries/qm-manyworlds/.
  * [25] John Archibald Wheeler and Richard Phillips Feynman.  Interaction with the absorber as the mechanism of radiation.  Reviews of modern physics, 17(2-3):157, 1945.