###  8.6 Do we need to consider infinite $S$? Uncountable $S$? Infinite
$\Gamma(t)$?

In light of the remarks in Subsection 8.5 it is natural to ask what is gained,
as far as applications to foundations of quantum mechanics are at stake, from
considering infinite $S$. One important reason for considering countably
infinite sets $S$ is to be able to use the limit theorems of probability
theory, like the strong law of large numbers, that apply to idealized settings
with infinitely many random variables. Those would correspond, for instance,
to idealized sequences of experiments.

Less clear is if, for the sake of physics, there is a need for considering
uncountably large sets $S$. An argument in favor is in the fact that we
usually consider physical time to be a real number, so that we should consider
$S=\mathbb{R}$ as our fundamental setting. But is there really a reason for
thinking that physical time is not limited to rational values? And that the
real line comes in simply as a mathematical tool, providing completeness in
the mathematical sense as a convenience, but not an additional physical
reality? This is an interesting philosophical issue that will not affect the
applicability of the results in this paper in situations in which the sets
$\Gamma(t)$, $t\in S$, of interest are all finite, thanks to the results in
Section 4.

And this raises the question whether there is any reason for considering
infinite $\Gamma(t)$ in applications to foundations of quantum mechanics. In
applications of the kind proposed in Subsection 8.2, when the number of
particles in the universe is fixed (so that the configuration space is
$\mathcal{C}=\mathbb{R}^{3n}$), there should be only a finite number of sets
$R_{a}$ that are macroscopically distinguishable from each other and
meaningful to us. After all, in such a universe, there can only be a finite
number of computational devices (including human brains), each one capable of
holding some finite number of distinct computational states. Even if particles
can be created, energy considerations may limit the number of particles and
hence the number of bits that all the computers (including our brains) can
hold.

In any case, we will see in Subsection 8.15 that the partial results obtained
in case of infinite $\Gamma(t)$ and uncountable $S$ in Section 4 are
sufficient to draw the conclusion that, if we accept certain intuitive
assumptions, then events for which we compute Born-probability 0 should not
happen, even if $S$ is uncountable and $\Gamma$ is infinite. What is currently
missing in this case is the converse. So we have not ruled out that, in this
case, there could be events of positive probability that will not happen.

###  8.7 Can $F_{A}$ in Section 3 be replaced with a set $N_{A}$ that provides
more uniformity in time?

For each $A\subset\Omega$, define

$N_{A}\ =\ \left\\{\varphi\in\mathcal{H}\,:\,\mbox{for some $t_{1},...t_{k}$,
\, $p^{t_{1},...,t_{k}}_{\omega_{t_{1}},...,\omega_{t_{k}}}\varphi=0$ \, for
all $\omega\in A$ }\right\\}.$ (87)

Note that $N_{\Omega}=N$, and $N_{A}\subset F_{A}$. The difference between
$N_{A}$ and $F_{A}$ is the extra uniformity, with respect to $\omega\in A$, in
the choice of $t_{1},...,t_{k}$ in $N_{A}$. It is natural to ask if in Section
3 we could have used $N_{A}$ instead of $F_{A}$, and in particular whether in
part (b) of Theorem 2, which is directly related to interpretation, we could
replace $F_{A}$ with $N_{A}$.

The answer is that in some parts of Section 3 we can make this replacement,
but not in others that include Theorem 2. This discussion highlights some of
the technical details of the proofs in that section.

In Lemmas 1, 28 and 3 we can indeed replace $F_{A}$ with $N_{A}$, keeping the
same proofs, as the reader can check. In the case of Lemma 3, this can also be
understood more quickly by observing that from (29), we have

$\mathcal{H}_{A^{c}}\ \subset\ N_{A}\cap\mathcal{H}_{\pi}\ \subset\
F_{A}\cap\mathcal{H}_{\pi}=\mathcal{H}_{A^{c}},$

where the last step is the statement of Lemma 3.

But the proof of Lemma 4 fails if we replace $F_{A}$ with $N_{A}$. In this
proof we are using the only statement of Proposition 3 in which $F_{A}$ cannot
be replaced with $N_{A}$. For an infinite family of subsets of $\Omega$,
$\\{A_{\alpha}\\}$, $N_{\cup_{\alpha}A_{\alpha}}$ is, in general, not equal to
$\cap N_{A_{\alpha}}$, because of loss of uniformity.

The following example shows that the problem is not only with the proof, but
with the conclusion in this lemma, which is a special case of part (a) of
Theorem 2.

Suppose that $S=\\{t_{1},t_{2},...\\}$. For $i=1,2,...$, choose
$G_{i}\subset\Gamma(t_{i})$, $G_{i}\not=\emptyset$, and set
$A_{i}=\\{X_{t_{i}}\in G_{i}\\}$, $A=\cup_{i=1}^{\infty}A_{i}$. It is clear
that for each $i$, $A_{i}\in\mathcal{A}$, and hence
$A\in\mathcal{A}_{\sigma}$. Now, using (3) (as in (25)),

$\displaystyle N_{A}\ $ $\displaystyle=$ $\displaystyle\
\bigcup_{k=1}^{\infty}\,\left\\{\varphi\in\mathcal{H}\,:\,p^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}\varphi=0,\
\mbox{for all}\ (a_{1},a_{2},...)\in A\right\\}$ $\displaystyle=$
$\displaystyle\
\bigcup_{k=1}^{\infty}\,\left\\{\varphi\in\mathcal{H}\,:\,p^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}\varphi=0,\
\mbox{for all}\ (a_{1},...,a_{k})\right\\}\ =\ N.$

In the second equality, we used the fact that for any $(a_{1},...,a_{k})$
there is some $a_{k+1}\in G_{k+1}\subset\Gamma(t_{k+1})$, such that
$(a_{1},...,a_{k},a_{k+1},...)\in A_{k+1}\subset A$.

If we could replace $F_{A}$ with $N_{A}$ in the statement of Lemma 4, or part
(a) of Theorem 2, we would then have

$\mathcal{H}_{A}\ =\ N_{A}^{\perp}\ =\ N^{\perp}\ =\ \mathcal{H}_{\pi},$

where the last equality is from part (a) of Theorem 1. In particular, for
every $\varphi\in\mathcal{H}_{\pi}\backslash\\{0\\}$, we would have
$\mathbb{P}_{\varphi}(A)=||p_{A}\hat{\varphi}||^{2}=1$. This is certainly
absurd in many applications, since the sets $G_{i}$ can be very small subsets
of the corresponding $\Gamma(t_{i})$, only assumed to be non-empty above.

For a counter-example, let $\mathcal{H}=L^{2}([0,1])$, $S=\\{1,2,...\\}$,
$\Gamma(t)=\\{1,2\\}$ and $p^{t}_{1}$ be defined by
$(p^{t}_{1}\varphi)(x)=I_{[0,3^{-t}]}(x)\varphi(x)$, where (as before in this
paper) $I_{R}(x)=1$, if $x\in R$ and 0 otherwise. By necessity,
$p^{t}_{2}=I-p^{t}_{1}$. In this setting,
$\mathcal{H}^{t}_{a}\subset\mathcal{H}^{s}_{a}$, whenever $s<t$, and this
implies $p^{t}_{1}p^{s}_{1}=p^{s}_{1}p^{t}_{1}=p^{t}_{1}$. It follows that
$\mathcal{H}_{\pi}=\mathcal{H}$.

Take $G_{i}=\\{1\\}$ for each $i$. Then, for $\varphi$ defined by
$\varphi(x)=1$, we have

$\mathbb{P}_{\varphi}(A)\ \leq\
\sum_{i=1}^{\infty}\mathbb{P}_{\varphi}(A_{i})\ =\
\sum_{i=1}^{\infty}\mathbb{P}_{\varphi}(X_{i}=1)\ =\
\sum_{i=1}^{\infty}||p^{i}_{1}\hat{\varphi}||^{2}\ =\
\sum_{i=1}^{\infty}\,\left(\frac{1}{3}\right)^{i}\ <\ 1.$

It is worth pointing out that the counter-example above has $\Gamma(t)$ finite
for all $t\in S$, so that this extra assumption (as made in parts of Section
4) would not change the conclusion here.

###  8.8 Can we eliminate the topological closure of $F_{A}$ in part (b) of
Theorem 2, or part (d) of Theorem 3?

It is natural to ask if we can replace $\overline{F_{A}}$ with $F_{A}$ in part
(b) of Theorem 2, or part (d) of Theorem 3. This is important for
interpretations of quantum mechanics, as the condition $\Psi\in F_{A}$ can
naturally be proposed to imply that if the state of the universe is $\Psi$,
then the event $A$ should not be part of our experiences. But that the
condition $\Psi\in\overline{F_{A}}$ should also have this implication is a
more delicate philosophical issue. In [31] and [32] this lead to the
consideration of a version of the superposition principle to reach this
conclusion. (See Subsection 8.9 below.)

A simple argument, though, shows that in relevant situations the closure of
$F_{A}$ is needed to make part (b) of Theorem 2 and part (d) of Theorem 3
true. As observed in Proposition 3, for any family of events
$A_{\alpha}\in\Sigma$, if $\varphi\in F_{A_{\alpha}}$, for each $\alpha$, then
$\varphi\in F_{\cup_{\alpha}A_{\alpha}}$. But
$\mathbb{P}_{\varphi}(A_{\alpha})=0$, for all $\alpha$ does not imply that
$\mathbb{P}_{\varphi}(\cup_{\alpha}A_{\alpha})=0$, unless this family of
events is countable.

###  8.9 $\overline{F_{A}}$, $\overline{N_{A}}$ and superposition of states

We recall now how in [31] and [32] the statement $\Psi\in\overline{F_{A}}$ was
translated into the statement that $\Psi$ is a superposition of states in
$F_{A}$. $\Psi\in\overline{F_{A}}$ means that there are
$\zeta_{1},\zeta_{2},...$ such that $\zeta_{i}\in F_{A}$ and
$\zeta_{i}\to\Psi$, as $i\to\infty$. Equivalently,
$\zeta_{1}+(\zeta_{2}-\zeta_{1})+(\zeta_{3}-\zeta_{2})+...$ converges to
$\Psi$. We can apply the Gram-Schmidt orthonormalization procedure (see p.46
of [26], or p.167 of [14]) to the vectors
$\zeta_{1},\zeta_{2}-\zeta_{1},\zeta_{3}-\zeta_{2},...$ to produce a sequence
of orthonormalized vectors $\eta_{1},\eta_{2},...$ that have the same closed
span, to which $\Psi$ belongs. Since $F_{A}$ is a vector space and
$\zeta_{i}\in F_{A}$, this procedure (which only involves linear operations)
gives us that also $\eta_{i}\in F_{A}$. Set
$\Psi_{i}=\langle\Psi,\eta_{i}\rangle\,\eta_{i}$, $i=1,2,...$. Then the
vectors $\Psi_{i}$ are orthogonal to each other, $\Psi_{i}\in F_{A}$ for each
$i=1,2,...$ and $\sum_{i=1}^{\infty}\Psi_{i}=\Psi$. A converse statement is
trivial, any convergent series of vectors in $F_{A}$ converges to a vector in
$\overline{F_{A}}$.

Referring to $N_{A}$, as defined by (87), since those are also vector spaces,
a similar derivation applies to $\overline{N_{A}}$.

In words, using common quantum-mechanics jargon: Belonging to
$\overline{F_{A}}$ (resp. $\overline{N_{A}}$) is the same as being a
superposition of orthogonal states in $F_{A}$ (resp. $N_{A}$).

The version of the superposition principle proposed in [31] and [32] can be
rephrased, replacing prediction with ontology, in the following fashion. Here
all the mentioned universes are supposed to be described by the same Hilbert
space $\mathcal{H}$ and group of unitary evolution operators $\\{U_{t}\\}$,
and their state is given in the Heisenberg picture by an element of
$\mathcal{H}$.

One-Sided Superposition Principle: If an event $A$ is not realized in
universes that are in states $\Psi_{i}$, $i=1,2,...$, then it is also not
realized in a universe in state $\Psi=\sum_{i=1}^{\infty}\Psi_{i}$.

The reason for the title of “one-sided superposition principle” is that if we
remove the word “not” in the two places that it appears, we obtain a statement
that is certainly false, due to interference. Superpositions cannot create new
realities, but they can eliminate realities by interference.

If we accept the idea that in a universe in a state $\Phi\in F_{A}$ (resp.
$\Phi\in N_{A}$), the event $A$ is not realized and accept also the one-sided
superposition principle, then we conclude that the same is the case in a
universe in a state $\Psi\in\overline{F_{A}}$ (resp.
$\Psi\in\overline{N_{A}}$).

In particular, if we accept the one-sided superposition principle and the idea
that in a universe in a state $\Phi\in N$ no event in $\Sigma$ is realized,
then we conclude that no event in $\Sigma$ is realized in a universe in a
state $\Psi\in\overline{N}=\mathcal{H}_{\pi}^{\perp}$.

###  8.10 Should we believe that in our universe
$\mathcal{H}_{\pi}\not=\\{0$}, and $\Psi_{\pi}\not=0$? The role of
decoherence, the “we-are-here” argument, and the ordinary nature of the
present time on a cosmological scale

In this subsection we are considering one of the particle models of Subsection
8.2 as a model for our universe. And we are considering the sort of $\pi$
discussed there, associated to a partition of the configuration space
according to sets with macroscopically meaningful descriptions. But we should
make one modification in how $\pi$ is chosen, because we are only interested
in times that, on a cosmological scale, are not too early nor too late. For
this reason we will assume $S=(t_{-},t_{+})$, where
$-\infty<t_{-}<s<t_{+}<\infty$, with $s$ being the present moment, and the
differences $s-t_{-}$ and $t_{+}-s$ being of the order of cosmological times.

It is natural to ask whether, for some values of $t_{-}$ and $t_{+}$ as above,
we should believe that $\mathcal{H}_{\pi}\not=\\{0\\}$, and more specifically
$\Psi_{\pi}\not=0$, where $\Psi$ is the Heisenberg-picture state of our model
universe.

There are three complementary ideas to discuss.

The first one is the role of environmental decoherence, [21], [40], [25],
[15], [30], [1]. The subsets $R_{a,i}$ into which $\mathcal{C}$ is partitioned
(where $a\in\Gamma$ and $i$ specifies how many particles of each kind are
present) correspond to macroscopically meaningful descriptions (labeled by
$a$), and therefore involve large numbers of particles, that should interact
with the environment producing records of the history. Now, these
environmental particles that produce records should also be described by the
vector $\Psi$. And it may happen that $\Psi$ includes components on which such
environmental memories do not form. Let us therefore leave $\Psi$ aside for
the moment, but assume that there is $\Phi\in\mathcal{H}\backslash\\{0\\}$
that supports a rich enough environment such that all events pertaining to
$\pi$ are recorded in this environment. This means that if
$t_{-}<t_{1}<t_{2}...<t_{k}\leq t<t_{+}$, then $\Phi_{t}=U_{t}\Phi$ should
decompose as

$\Phi_{t}\ =\
\sum_{a_{1},...,a_{k}}\,U_{t}\,\Phi^{t_{1},...,t_{k}}_{a_{1},...,a_{k}},$

where
$\Phi^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}\in\mathcal{H}^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}$,
for each $a_{1},...,a_{k}$. Therefore
$p^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}\Phi=p^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}U_{-t}\Phi_{t}=\Phi^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}$.
And

$\sum_{a_{1},...,a_{k}}\,p^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}\,\Phi\ =\
\sum_{a_{1},...,a_{k}}\,\Phi^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}\ =\
U_{-t}\Phi_{t}\ =\ \Phi.$

This means that $\Phi\in\mathcal{H}_{\pi}^{\prime\prime}=\mathcal{H}_{\pi}$,
where we used part (a) of Theorem 1.

In conclusion: the assumption that we have enough decoherence in our universe
that such a $\Phi$ as above exists implies $\mathcal{H}_{\pi}\not=\\{0\\}$.
And $\mathcal{H}_{\pi}$ should be the set of elements in $\mathcal{H}$ that
support lasting environmental memories of all events associated to $\pi$.

The second idea, to which we turn now, will explain why we may believe in a
statement that is related to, but weaker than $\Psi_{\pi}\not=0$. To explain
what this weaker statement is, we define, for $s\in(t_{-},t_{+})$, the
following coarsening of $\pi$:

$\pi(s)\ =\ \\{p^{t}_{a}\,:\,t_{-}<t\leq s,\,a\in\Gamma(t)\\}.$

Clearly $\mathcal{H}_{\pi(s)}$ decreases to $\mathcal{H}_{\pi}$, as $s$
increases to $t_{+}$. We will also use the self-explanatory notation
$\Omega(s)$, $\Sigma(s)$ and

$N(s)\ =\ \left\\{\varphi\in\mathcal{H}\,:\,\mbox{ for some
$t_{1},...,t_{k}\in(t_{-},s]$, \
$p^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}\varphi=0$ \ for all
$a_{1},...,a_{k}$}\right\\}.$

Suppose that $s$ is the present moment. It seems very reasonable to believe
that if $\Psi\in N(s)$, then no $A\in\Sigma(s)$ would be realized. After all,
in this case there are $t_{1},...,t_{k}\in(t_{-},s]$ such that
$\Psi\perp\mathcal{H}^{t_{1},...,t_{k}}_{a_{1},...,a_{k}}$, for each
$a_{1},...,a_{k}$. And this should mean that each event
$\\{X_{t_{1}}=a_{1},...,X_{t_{k}}=a_{k}\\}$ should not be realized. But the
union of these events over $a_{1},...,a_{k}$ is $\Omega(s)$ and so contains
any $A\in\Sigma(s)$.

Now, if $\Psi_{\pi(s)}=0$, then
$\Psi\in\mathcal{H}_{\pi(s)}^{\perp}=\overline{N(s)}$, where we used part (a)
of Theorem 1, applied to $\pi(s)$. If we accept the one-sided superposition
principle stated in Subsection 8.9, we should conclude, as explained at the
end of that subsection, that all our experiences up to the present time would
not be realized. And since we are here, an have one experience or another, we
should believe that $\Psi_{\pi(s)}\not=0$. This is the weaker statement
alluded to above.

Should we upgrade this belief to the belief that $\Psi_{\pi}\not=0$? For this
we apply a third idea. One argument in this direction evokes the absence of
anything special about the present moment, as compared to other times that are
of the same order of magnitude in a cosmological scale. If $\Psi$ includes a
component with a rich enough environment to allow for $\Psi_{\pi(s)}\not=0$,
when $s$ is the present moment, we should expect
$\Psi_{\pi(s^{\prime})}\not=0$ for at least $s^{\prime}$ larger than $s$ by a
cosmological extension of time. And this perhaps is the most that we can argue
for and believe. And it is certainly good enough for practical purposes.

What would it mean if $\Psi_{\pi(s^{\prime})}=0$ at some future time? The one-
sided superposition principle would then imply that we (in the way we
understand ourselves, with the kind of possible experiences labeled by
$\Gamma$) would not be part of this universe after time $s^{\prime}$. And
perhaps this is the way things will be in a cosmological time in which the
universe will look very different from its present state.

In this scenario, whatever event in $\Sigma$ we would have predicted not to
happen after time $s^{\prime}$ will indeed not happen. But if we assumed
$\Psi_{\pi}\not=0$ when making predictions, we would have erred in the
opposite direction, incorrectly predicting events to happen that will actually
not happen. In this scenario, as our own existence would not go beyond time
$s^{\prime}$, we would not be there to realize that we were wrong.

###  8.11 A minimalistic ontology for non-collapse quantum mechanics

The mathematical results in this paper support a minimalistic ontology for
non-collapse quantum mechanics, that conforms with our experiences, including
our perceptions of apparent collapses of the wave function according to Born’s
rule. This ontology is also compatible with the one-sided superposition
principle of Subsection 8.9.

In this subsection we will build the theory based only on the mathematical
results in Section 2. In the next two, we will further elaborate on this
theory, using also notions from Section 7. And in Subsection 8.14 we will see
how this ontology relates to the results in Sections 3 and 4.

In this ontology, the primary physical reality is limited to a vector $\Psi$
which belongs to a Hilbert space $\mathcal{H}$ and a strongly continuous group
of unitary operators on $\mathcal{H}$, $(U_{t})_{t\in T}$, where
$T=\mathbb{R}$, or $T=\mathbb{Q}$. In the latter case we are assuming that
only rational times have physical meaning, as discussed in Subsection 8.6. We
could also entertain the idea of assuming $T=\epsilon\mathbb{N}$, where
$\epsilon$ is a time interval shorter than anything that we can (presently)
measure.

At this point it is important to explain what we mean by “primary reality” and
how it differs from the broader use of “reality” below. For a good
illustration of the distinction consider the concept of cellular automata, as
the well known Game of Life, [20]. The primary reality is limited to a grid, a
deterministic updating rule in discrete time and an initial configuration of
alive and dead cells of the grid. (Those are analogous, respectively, to our
$\mathcal{H}$, $(U_{t})_{t\in T}$ and $\Psi$.) But in addition to this primary
reality, there are patterns of alive and dead cells that develop and propagate
in time. And this is actually the reason for the interest in the model. In
particular because these propagating patterns can produce the same
computations as a Turing machine. In our terminology, such patterns are
elements of the “derived reality”, or simply “reality” of the system. If the
grid and the deterministic updating rule are fixed, we may regard the patterns
that develop and propagate in time as features of the initial configuration.

We need to propose a theory about the nature of our experiences, compatible
with the primary quantum ontology proposed above, and with the fact that these
experiences are well described by textbook quantum mechanics (with collapse
according to Born’s rule). Theorem 1 and the analogy above suggest an answer:
Our experiences are in one-to-one correspondence with a class of patterns in
$\Psi$.

We should think of each possible $\pi$, with $S\subset T$, as a tool for
analysing the features of $\Psi$. For this reason, we will call each such
$\pi$ an “analyser”. Given such an analyser, we have its associated sets
$\Omega$ and $\Sigma$. And Theorem 1 provides us with
$\\{p_{A}:A\in\Sigma\\}$, which is a projection valued measure on
$\mathcal{H}_{\pi}$. Given $A\in\Sigma$, we say that $A$ is a $\pi$-pattern in
$\Psi$ if $p_{A}\Psi\not=0$.

The proposal is to regard any $\pi$-pattern $A$ for any analyser $\pi$ as part
of the reality defined by (or derived from) $\Psi$. The corresponding
postulate is:

Ontological Postulate: For any analyser $\pi$ and any $A\in\Sigma$,

$\mbox{$A$ is part of reality}\ \ \ \Longleftrightarrow\ \ \ p_{A}\Psi\not=0,$
(88)

And the idea is that our experiences are in one-to-one correspondence with the
$\pi$-patterns of $\Psi$ for an appropriate $\pi$. In short: that our
experiences are $\pi$-patterns of $\Psi$, for a certain $\pi$.

There are several interesting aspects of such a theory.

First, it satisfies the one-sided superposition principle of Subsection 8.9:

If $p_{A}\Psi_{i}=0$, for $i=1,2,...$, then
$p_{A}\left(\sum_{i}\Psi_{i}\right)=0$.

Second,

$p_{A}\Psi\not=0\ \ \ \Longleftrightarrow\ \ \ p_{A}\Psi_{\pi}\not=0,$ (89)

so that, for the relevant $\pi$, our experiences are only affected by
$\Psi_{\pi}$, not by what $\Psi-\Psi_{\pi}$ may be. As a consequence, we have
no information, through our experiences, of what $\Psi-\Psi_{\pi}$ is. For us,
it is as if the Heisenberg-picture state of the universe were $\Psi_{\pi}$,
rather than $\Psi$. And since we have experiences, it must be the case that
$\Psi_{\pi}\not=0$.

Third, Theorem 1 implies that, if $\Psi_{\pi}\not=0$, then (88) is equivalent
to

$\mbox{$A$ is part of reality}\ \ \ \Longleftrightarrow\ \ \
\mathbb{P}_{\Psi_{\pi}}(A)\not=0.$ (90)

Now, which $\pi$ is relevant in describing our human experiences? We will
start with a broad proposal, then scrutinize it and settle for a very precise
instance of that proposal as our $\pi$.

The natural starting point is to assume that the setting is one of the
particle models of Subsection 8.2. And that $\pi$ is as defined there, with
each $p_{a}$ associated to a subset $R_{a}=\cup_{i}R_{a,i}$ of the
configuration space $\mathcal{C}$, which admits a macroscopically meaningful
description to us, labeled by $a\in\Gamma$, and where $i$ gives the number of
particles of each type present in each component $R_{a,i}$ of $R_{a}$. To
assure that $\Psi_{\pi}\not=0$, the set $S$ may need to be limited to an
interval $(t_{-},t_{+})\cap T$, for some $t_{-}$ that is finite and
significantly smaller than the present time on a cosmological scale, and some
$t_{+}$ that is finite and significantly larger than the present time on a
cosmological scale, as explained in Subsection 8.10. And, as emphasized in
that subsection, the physical phenomenon responsible for $\Psi_{\pi}\not=0$ is
decoherence. And the meaning of “macroscopic” in the definition of $\pi$
relates to our sets $R_{a}$ being defined by the positions of large numbers of
particles that allow for environmental decoherence to happen.

We can now theorize that our experiences are the events $A$ that are
$\pi$-patterns of $\Psi$, for a $\pi$ as defined in the last paragraph.

This works well in that (90) implies that human experiences are precisely the
ones that have positive probability according to standard Born-collapse
quantum mechanics. Here it is important to stress that we perceive the
Heisenberg-picture state of the universe as $\Psi_{\pi}$, rather than $\Psi$,
as $\Psi-\Psi_{\pi}$ does not affect us. And that the “effective state of the
universe for us”, $\Psi_{\pi}$, is what we use in computing the Born
probabilities. It is also important to understand that if
$\mathbb{P}_{\Psi_{\pi}}(A)>0$ and also $\mathbb{P}_{\Psi_{\pi}}(A^{c})>0$,
then both are human experiences, but in the Everettian sense that humans
branch, and along each branch only perceive one of these two events. What is
accomplished here is to eliminate the naive, but important, criticism of non-
collapse quantum mechanics, according to which if no collapse happens, then
every event $A\in\Sigma$ would happen. Events with
$\mathbb{P}_{\Psi_{\pi}}(A)=0$ have $p_{A}\Psi=0$ and, according to the
postulate above, do not happen in any branch!

There is a way to explain the last statement above that may be helpful in
convincing skeptical readers, who would insist that without collapses every
$A\in\Sigma$ would happen. Consider a fictitious model universe, in which
nature picks an infinite collection of independent realizations of one of the
stochastic processes $(x_{t})_{t\in\mathbb{R}}$, defined in Subsection 8.4,
that satisfy (85) with $\varphi=\Psi_{\pi}$. In this model universe, we have
infinitely many trajectories $x_{t}$ in configuration space, and we can think
of this ensemble of independent trajectories as an ensemble of different
worlds that do not interact with each other. In each one there are humans
whose experiences are identical to those predicted by Born-collapse quantum
mechanics, in a universe in state $\Psi_{\pi}$. And since there are infinitely
many of these worlds, each event $A\in\Sigma$ that has
$\mathbb{P}_{\Psi_{\pi}}(A)>0$ is experienced in some of them (actually in
infinitely many of them). But no event $A\in\Sigma$ that has
$\mathbb{P}_{\Psi_{\pi}}(A)=0$ is experiences in any of them. The assumption
that our experiences correspond to events that are $\pi$-patterns of $\Psi$ is
equivalent, thanks to Theorem 1, to the statement that our experiences are
identical to those of the humans in this model-infinite-ensemble universe.
Now, imagine that in this model-infinite-ensemble universe a group of
scientists is performing a sequence of identical independent quantum
experiments that may each time result in outcome 1 with Born-probability 0.9,
or outcome 2 with Born-probability 0.1. In each experiment there are worlds in
which outcome 1 happens and worlds in which outcome 2 happens. But in no world
does the frequency of outcomes 1 converge to a number different from 0.9.
Having every possible outcome in each single experiment in this universe does
not imply having every possible outcome in infinite series of experiments.

Another issue that may be raised is whether (90) captures all the ways in
which Born’s-rule-collapse probabilities are used in standard collapse quantum
mechanics. Argumentation answering this question in the affirmative is
presented in Sections 3 and 6 of [31].

Now to some essential criticism of the kind of choice above of $\pi$. There is
a substantial amount of subjectivity involved. What is “meaningful”? Would we
all agree that a certain family of sets $R_{a}$ correspond to each label
$a\in\Gamma$ that we describe in a certain way? How fine can the partition of
the configuration space be to still allow decoherence to happen? Fortunately,
there is a good way to solve these issues, if we accept the (currently
standard) view that our perceptions are encoded in the physical state of our
brains, and that our mental processes are in one-to-one correspondence with
computational processes produced as our brains behave according to the same
physical laws that apply to everything else.

Before returning to humans, it is helpful to consider a computer of the kind
that we build with silicon. In this context, we can introduce the relevant
analyser $\pi$ by partitioning the configuration space into the sets $R_{a}$
labeled by the computational states of the computer, including a set in which
the computer is not present in the universe and a set in which the
computational state of the computer may not be well defined. Instead of
“experiences that it has”, we should talk of “computations that the computer
performs”. If occasionally the computer receives bits of input that correspond
to outcomes of quantum experiments, the computer will branch in an Everettian
sense, with each branch continuing to compute separately from the others. If
we accept (88), and suppose that, due to decoherence (expected, since each bit
of information is encoded by the state of a very large number of particles)
$\Psi_{\pi}\not=0$, then, by (90), events $A\in\Sigma$ (which now pertain to
the computational processes of this computer) will be part of the collective
reality of the branching versions of the computer when and only when
$\mathbb{P}_{\Psi_{\pi}}(A)>0$. The important point that we want to emphasize
is that here the partition of the configuration space into the parts $R_{a}$
is objective!

If the story above involving a computer is understood and we accept the
hypothesis that our mental processes are manifestations of computational
processes in our brains, then there is no relevant difficulty in replacing the
computer by the family of humans. We should partition the configuration space
$\mathcal{C}$ according to the computational states of our brains, including a
set in which there are no humans present and (possibly) sets in which some
humans have undefined states of mind. More precisely, we should define the
sets $R_{a}$ so that two points of the configuration space, $x$ and $y$,
belong to the same $R_{a}$ if and only if in these two configurations we have
the same humans present and each one has the same state of mind in $x$ as in
$y$. The rest of the story is the same as that of the computer, with one
complication. We should account for any possible number of humans at any time.
And, if our particle-model universe admits creation of particles, in principle
there is no limit for this number, so that $\Gamma$ will be infinite. We will
use the notation $\pi_{H}$ for this analyser, and use the notation
$\Omega_{H}$ and $\Sigma_{H}$ accordingly. If we accept the (standard)
premises made above, about how our mental processes relate to the physical
universe, then $\pi_{H}$ has been defined in an objective way. And therefore
also $\Phi=\Psi_{\pi_{H}}$ has an objective definition. And the fact that
$\Phi\not=0$ is once more due to decoherence, which is expected to happen if
each computational bit in our brains is encoded by the state of a large number
of particles. And all our perceptions are the same that would happen if the
state of our universe were $\Phi$ instead of being $\Psi$. And in making
predictions about our future perceptions, we should compute Born probabilities
$\mathbb{P}_{\Phi}(A)$, $A\in\Sigma_{H}$ and deem as impossible to happen
events that have probability 0. And this is what we indeed do, using for
$\Phi$ approximate inferences based on our previous and current perceptions.

One can make the case that in interpreting quantum mechanics all we have to
accomplish is to produce a coherent, logically consistent, theory and predict
our human perceptions correctly from it. (See, e.g., Chapter 9 of [30] and
references therein.) The theory presented here, with $\pi_{H}$ as the relevant
analyser from the human perspective, fulfils these requirements!

One should not misunderstand the statement that $\pi_{H}$ is the relevant
analyser for the purpose of predicting human perceptions, with the incorrect
idea that in the theory only $\pi_{H}$-patterns are part of reality. The
postulate above applies to every analyser $\pi$, as stated. And there are good
philosophical reasons for thinking about what we can learn from considering
other related analysers. This is the subject of the next subsection.

###  8.12 Refinements of $\pi_{H}$ and coarsenings of refinements of
$\pi_{H}$. The realm of classical physics $\pi_{C}$

Among the partitions of the configuration space into sets with
“macroscopically meaningful descriptions”, an important class is that of
refinements of $\pi_{H}$. It is true that we may sometimes disagree on the
precise borders of the sets $R_{a}$, but often we all agree with several of
the relevant macroscopic descriptions, as for instance with the description of
a measuring device pointing to a certain result, or with the letters that are
printed on a piece of paper. And this allows us to consider various
interesting refinements of $\pi_{H}$. It is worth looking at this in some more
detail. The analyser $\pi_{H}$ corresponds to a partition of the configuration
space into sets $R_{a}$, where each $a$ corresponds to a given state of mind
for each human present. And this means that in producing this partition, we
are concerned with the location of the particles in the brains of the humans.
To refine $\pi_{H}$, we break each $R_{a}$ into parts, according to the
location of many other particles that form all sorts of other things we may
include and that have non-controversial macroscopic descriptions for us. This
could include all the other particles forming the bodies of these humans,
forming other animals, like cats (alive and dead), computers, books (including
all the letters and digits printed on each page), the moon (even if no one is
looking at it), …

The effective state of the universe based on our perceptions is the vector
$\Phi=\Psi_{\pi_{H}}$. And when we compute Born-rule probabilities we are
using the best information we have to approximate $\Phi$ (or at least how
$\Phi$ looks inside our lab). When we consider a $\pi$ that refines $\pi_{H}$,
as in the previous paragraph, we may wonder if $\Psi_{\pi}$ would not be
substantially different from $\Phi$. But we should not worry about it when the
refinement of $\pi_{H}$ is, as above, based on “macroscopic descriptions”,
once again because of decoherence. The point is that we are aware of the
phenomenon of environmental decoherence, and have proposed ways in which it
happens in our universe, based on our knowledge about the state of the
universe that we perceive, namely $\Phi$. From this knowledge, we see that
$\Phi$ includes a rich enough environment to assure enough decoherence
affecting cats, books, the moon, etc, that
$\Phi\in\mathcal{H}_{\pi}^{\prime\prime}=\mathcal{H}_{\pi}$, as explained in
Subsection 8.10. Therefore, since
$\mathcal{H}_{\pi}\subset\mathcal{H}_{\pi_{H}}$, we have
$\Psi_{\pi}=p_{\pi}\Psi=p_{\pi}p_{\pi_{H}}\Psi=p_{\pi}\Psi_{\pi_{H}}=p_{\pi}\Phi=\Phi$.

It is interesting to compare a typical event $A\in\Sigma_{H}$ with its
refinement $A^{\prime}\in\Sigma$, corresponding to a $\pi$ that refines
$\pi_{H}$ in the fashion described above. (For the definition of $A^{\prime}$
see Section 7. The intuitive meaning of $A^{\prime}$ is that it provides all
the ways in which $A$ could happen in terms of descriptions based on $\pi$.)
$A$ could, for instance, be the event that “at time $t_{1}$ Jane saw three
moons of Jupiter, and between times $t_{2}$ and $t_{3}$ Hui heard a meow
sound”. In comparison, depending on what $\pi$ is, $A^{\prime}$ could also
include a description of the telescopes that Jane could have used, the hats
she possibly had on, the possible expressions on her face, …, and the cat that
produced the sound that Hui heard, or the person who was imitating the sound
of a cat, or …. Using Theorem 8 and the fact that $p_{\pi}\Phi=\Phi$, we have
$\mathbb{P}_{\Phi}(A^{\prime})=||p_{A^{\prime}}\hat{\Phi}||^{2}=||p_{A}p_{\pi}\hat{\Phi}||^{2}=||p_{A}\hat{\Phi}||^{2}=\mathbb{P}_{\Phi}(A)$.
So that $A^{\prime}$ is part of reality if and only if $A$ is. We can also
consider $B\subset A^{\prime}$ that specifies a certain telescope, a certain
hat, and a certain cat. In the view presented here, these are parts of the
reality derived from $\Psi$, provided that $p_{B}\Psi\not=0$, which from (89)
and what we saw in the last paragraph amounts to $p_{B}\Phi\not=0$, or
equivalently $\mathbb{P}_{\Phi}(B)>0$. Now, if in $C\subset A^{\prime}$ Jane
was looking at Jupiter, at time $t_{1}$, with naked eyes, then the laws of
physics would entail $p_{C}\Psi=p_{C}\Phi=0$ and, in particular, we would have
$\mathbb{P}_{\Phi}(C)=0$. And as much as Jane had branched in her life, this
event would not be part of her reality along any branch. The point of all of
this is that human perceptions and their causes “out there in the world” are,
in the view presented here, possible parts of reality, and physics laws
restrict them and relate them in the proper way. (In this paragraph we abused
notation in the following way. Theorem 1 provides a probability measure
$\mathbb{P}_{\Phi}$, on $(\Omega_{H},\Sigma_{H})$ associated to $\pi_{H}$ and
a different one, on $(\Omega,\Sigma)$, associated to its refinement $\pi$. We
should have distinguished them in the notation used, but did not do it, since
when we write $\mathbb{P}_{\Phi}(A)$, with $A\in\Sigma_{H}$, or
$\mathbb{P}_{\Phi}(C)$, with $C\in\Sigma$, it should be clear which
probability measure we mean. We will continue to abuse notation in this way,
when no confusion is possible.)

Of course, there should also be refinements of $\pi_{H}$ that specify the
mental state of other animals. The only reason we did not include this feature
in $\pi_{H}$ itself was lack of necessity, when our task was to account for
human experience.

It is also natural to consider coarsenings of $\pi_{H}$, or of the refinements
of $\pi_{H}$ discussed above. Those can naturally be obtained by focusing on a
subset of the humans and then lumping together the $R_{a}$ according to the
state of mind of the humans in this subset, regardless of the state of mind of
the other humans. For instance, at the risk of being called solipsistic, an
individual may consider only the states of his or her mind in defining $\pi$,
and perhaps refine it to include only certain aspects of the world of his or
her interest. In another example, as we make predictions for our foreseeable
future, we may cap the number of humans that we distinguish in $\pi$ to a
large but finite upper bound, lumping together in one $R_{a}$ all the
configurations with a larger number of humans. Assuming that each human brain
can only encode finitely many different computational states, we see that the
corresponding $\Gamma$ is now finite.

Again we should wonder if $\Psi_{\pi}$ could be substantially different from
$\Phi$ in case, say, that only the state of mind of some of the humans is used
in the definition of $\pi$. Here the equality $\Psi_{\pi}=\Phi$ seems to be
justified by invoking the assumption of “homogeneity of the scales on which
decoherence operates”. As we understand it, decoherence affects our brains by
means of phenomena that operate in homogeneous fashion in a very large scale,
including the electromagnetic background radiation that fills the universe. It
is true that the phenomena that we are aware of as producing decoherence are
accounted for by $\Phi$, and we do not know anything about $\Psi-\Phi$. But it
seems reasonable to think that that component of $\Psi$ shares this sort of
large scale homogeneity feature, since it is part of the same natural
phenomenon, namely our universe. This assumption implies that the component of
$\Psi$ on which the brain of each one of us decohers should be the same one.
And therefore it should be the common $\Phi=\Psi_{\pi_{H}}$ that we infer from
our perceptions.

We can now summarize what we propose for our universe, based on the ideas
presented above. We suppose that it is well modeled by a particle model of the
kind described in Subsection 8.2, with the appropriate particles. Those, as
far as we currently understand it, are the photons, the quarks, the leptons,
etc, from what is called the “standard model”. The corresponding Hilbert space
$\mathcal{H}$ is the appropriate Fock space. The nature of the Hamiltonian and
of the Heisenberg-picture state vector $\Psi\in\mathcal{H}$ are such that they
allow for an analyser $\pi_{C}$, which is a substantial refinement of
$\pi_{H}$ based on the objects that we describe in classical physics, and has
the properties that yield what we call our “classical world(s)”, or the “realm
of classical physics” (hence the “C” in $\pi_{C}$). These properties include
the assumption that $\Psi_{\pi}=\Phi$ is the same for a wide range of
coarsenings $\pi$ of $\pi_{C}$. The idea is that $\Phi$ provides enough
decoherance, and $\Psi-\Phi$ does not add to it across the relevant scales, to
assure this constancy of $\Psi_{\pi}$. Here $\pi$ could be as fine as
$\pi_{C}$ itself, or as coarse as only describing a piece of dust, say
(anything that is still well described by classical physics would fit here).
Other examples of allowed $\pi$ would be $\pi_{H}$ and the other analysers
mentioned above in this subsection, or the $\pi$ associated to the
computational states of a computer, from the previous subsection. How fine can
the partition corresponding to $\pi_{C}$ be? This is the question of
understanding how small something can be to still be the subject of enough
decoherence in $\Phi$ to fit in the definition of $\pi_{C}$ (and hence be
called “macroscopic”). Reflecting on this question makes clear that there is a
certain fuzziness in the definition of $\pi_{C}$, and the answer may depend on
the cosmological time scale considered, for the reasons presented in
Subsection 8.10. This question is nothing but the question of determining the
limits of the realm of classical physics, which certainly has a fuzzy
boundary, which can and should be investigated experimentally. Another
question is what we can say about $\Psi-\Phi$. The answer suggested by all the
considerations so far is nothing. This vector could be 0, it could be
comparable to $\Phi$ in norm, or it could be much larger than $\Phi$. In any
case it would not affect our experiences in the theory proposed here, based on
the Ontological Postulate from Subsection 8.11, and the idea that our mental
processes correspond to $\pi_{H}$-patterns of $\Psi$, or equivalently,
$\pi_{C}$-patterns of $\Psi$. (To see this last equivalence, let
$A\in\Sigma_{H}$ and $A^{\prime}$ be its refinement to $\pi_{C}$. Then Theorem
8 tells us that $p_{A^{\prime}}\Psi=p_{A}p_{\pi_{C}}\Psi=p_{A}\Phi=p_{A}\Psi$,
so that $A$ is a $\pi_{H}$-pattern of $\Psi$ iff $A^{\prime}$ is a
$\pi_{C}$-pattern of $\Psi$. Note that, as in the example involving Jane and
Hui, here we also have $\mathbb{P}_{\Phi}(A)=\mathbb{P}_{\Phi}(A^{\prime})$.)
Since $\Psi-\Phi$ has no effect on our experiences, and is therefore
inaccessible to us via experiments, it is tempting to assume that $\Psi=\Phi$,
i.e., that $\Psi\in\mathcal{H}_{\pi_{C}}$. This assumption is nevertheless not
needed and may be criticized as being anthropocentric. For this reason we take
an agnostic position on whether $\Psi\in\mathcal{H}_{\pi_{c}}$, or not.

The analyser $\pi_{H}$ and the other ones discussed so far in this subsection
involve “macroscopic” objects (we are supposing that each computational bit in
our brains is also encoded by the state of a large number of particles). The
relevance of this assumption was emphasized repeatedly above, in connection to
the need of decoherence. But in a refinement of $\pi_{H}$, could we include
microscopic phenomena too? For instance, if we are performing a double slit
experiment with one electron, can we refine $\pi_{C}$, by partitioning the
sets $R_{a}$ according to the location of this electron while in flight?
Nothing prevents us from doing it. We lose the assurance that
$\Psi_{\pi}\not=0$. But in case we still have $\Psi_{\pi}\not=0$, we can even
associate probability
$\mathbb{P}_{\Psi_{\pi}}(A)=||p_{A}\Psi||^{2}/||\Psi_{\pi}||^{2}$ to an event
$A$ involving the location of the electron. And this probability will be
positive if and only if the event $A$ is part of reality. If $\Psi_{\pi}=0$,
then also $p_{A}\Psi=0$, and $A$ is not part of reality. Now, it should be
pointed out that these considerations have no operational meaning for us
humans. When for an event $B\in\Sigma_{H}$ we compute $\mathbb{P}_{\Phi}(B)$,
the meaning is that $B$ is part of our perceptual reality if and only if this
probability is positive. So that we can predict that if this probability is 0,
$B$ will not happen. But in the case of the event $A$ mentioned above, whether
it is or is not part of reality has no implication for our perceptions, unless
those are also accounted for by an event $B\in\Sigma_{H}$. From a pragmatic
point of view, this is in agreement with the quantum recipe of the textbooks,
that tell us to only associate probabilities to outcomes of experiments, not
to events like the location of the electron in the double slit experiment,
before it hits the screen. In the view presented here, it is not that such
probabilities cannot be defined and related to the ontology of the theory.
They can, when $\Psi_{\pi}\not=0$. The issue is that they are related to
aspects of that ontology that are not amenable to experimental scrutiny by us.

We end this subsection with a further discussion and clarification of the role
of some of the different probability measures provided by Theorem 1, part (b).
Strictly speaking, for each analyser $\pi$ we have a distinct measurable space
$(\Omega,\Sigma)$, and then, for each
$\varphi\in\mathcal{H}_{\pi}\backslash\\{0\\}$ we have a distinct probability
measure $\mathbb{P}_{\varphi}$ on this space. This is a very large set of
probability measures! Now, the proposal in the previous subsection and in this
one is that the most relevant analysers, from our perspective, are $\pi_{H}$
and its refinement $\pi_{C}$, that both share $\Phi=\Psi_{\pi}$ and that the
relevant probabilities are $\mathbb{P}_{\Phi}(A)$, either for
$A\in\Sigma_{H}$, or $A\in\Sigma_{C}$ (abusing notation as explained before).
The probabilities $\mathbb{P}_{\Phi}(A)$, for $A\in\Sigma_{C}$ are the ones
that the quantum mechanics textbooks tell us to compute, to make predictions.
(In doing it, they sometimes state that a microscopic system must interact
with a macroscopic measuring device, for a “potentiality” to become a
“reality”. The role of decoherence removes the need for such mysterious
statements.) For instance, in a double slit experiment with a single electron,
in which we are using an old fashioned photographic plate as the screen,
$A_{i}$ may be the event in $\Sigma_{C}$ that a pixel $i$ is sensitized.
$B_{i}$ may be the event in $\Sigma_{H}$ that Carla sees the pixel $i$ as a
white dot, as she looks at the plate after developing it. And $B_{i}^{\prime}$
will denote the refinement of $B_{i}$ to $\Sigma_{C}$. The textbooks tell us
how to compute $\mathbb{P}_{\Phi}(A_{i})$ (precisely in the same way that
Theorem 1 does), and our understanding of what happens when one looks at a
developed photographic plate tells us that $B_{i}^{\prime}=A_{i}$, and
therefore
$\mathbb{P}_{\Phi}(A_{i})=\mathbb{P}_{\Phi}(B_{i}^{\prime})=\mathbb{P}_{\Phi}(B_{i})$.
(For the justification of the last equality, one can use Theorem 8, as in the
example involving Jane and Hui.) Suppose now that $\pi$ is a refinement of
$\pi_{C}$ based on the position of the electron at a certain time $t$, before
it hits the screen. And let $A_{i}^{\prime\prime}$ denote the corresponding
refinement of $A_{i}\in\Sigma_{C}$ to the sigma-algebra $\Sigma$ associated to
$\pi$. Lack of decoherence of the electron’s location in $\Phi$, i.e., lack of
recording of the electron’s position at time $t$ in the environment provided
by $\Phi$, means that
$\Phi\not\in\mathcal{H}_{\pi}^{\prime\prime}=\mathcal{H}_{\pi}$ (see
Subsection 8.10). And this implies that $\Psi_{\pi}=\
p_{\pi}\Psi=p_{\pi}p_{\pi_{C}}\Psi=p_{\pi}\Phi=\Phi_{\pi}\not=\Phi$. In case
$\Phi_{\pi}\not=0$, the probability
$\mathbb{P}_{\Psi_{\pi}}(A^{\prime\prime}_{i})=\mathbb{P}_{\Phi_{\pi}}(A^{\prime\prime}_{i})$
is well defined, but is not related to the relevant probability
$\mathbb{P}_{\Phi}(A_{i})=\mathbb{P}_{\Phi}(B_{i})$ in any simple way.

###  8.13 The relevance of $\\{p_{A}:A\in\Sigma\\}$ being a projection valued
measure for the validity of our logical reasoning about our perceptions, and
for the computational aspect of life

In the theory of Subsection 8.11, our perceptions are events $A\in\Sigma_{H}$
that satisfy $p_{A}\Psi\not=0$. The fact that $\\{p_{A}:A\in\Sigma_{H}\\}$ is
a p.v.m. explains then the validity of our use of some basic rules of logic in
thinking about these perceptions. Below go some basic instances:

If $A\subset B$ and we believe that $B$ will not be one of our perceptions
(along any branch of our existence), then we reason that also $A$ will not be
one of our perceptions. And indeed, (PVM7) implies that if $p_{B}\Psi=0$, then
$p_{A}\Psi=0$.

If we believe that each one of the disjoint $A_{1},...,A_{n}$ will not be one
of our perceptions, then we reason that also $A=\cup_{i=1}^{n}A_{i}$ will not
be one of our perceptions. And indeed, (PVM4) implies that if
$p_{A_{i}}\Psi=0$, $i=1,...,n$, then $p_{A}\Psi=0$.

If we believe that one among $A_{1},...,A_{n}$ will not be one of our
perceptions, then we reason that also $A=\cap_{i=1}^{n}A_{i}$ will not be one
of our perceptions. And indeed, (PVM8) implies that if $p_{A_{i}}\Psi=0$, for
some $i=1,...,n$, then $p_{A}\Psi=0$.

If we believe that each one of the disjoint $A_{1},A_{2},...$ will not be one
of our perceptions, then we reason that also $A=\cup_{i=1}^{\infty}A_{i}$ will
not be one of our perceptions. And indeed, (PVM2) implies that if
$p_{A_{i}}\Psi=0$, $i=1,2,...,$, then $p_{A}\Psi=0$.

More formally, the examples above and others can be derived from the
observation that the set $\\{A\in\Sigma_{H}:p_{A}\Psi=0\\}$ is a sigma-ideal,
i.e., a family of elements of $\Sigma_{H}$ that has the following three
properties: It contains $\emptyset$, is closed with respect to taking subsets
and with respect to taking countable unions.

But we do make mistakes: If we believe that each one of the disjoint, but
possibly uncountably many, $A_{\alpha}$, will not be one of our perceptions,
then we (sometimes) reason that also $A=\cup_{\alpha}A_{\alpha}$ will not be
one of our perceptions. This is not justified, and counter-examples are not
hard to find. For instance, consider an infinite sequence of independent
identical experiments that may result in one of two outcomes, each with a Born
probability that is positive. Each possible sequence of outcomes has Born
probability 0, and therefore will not be one of our perceptions (along any
branch). But the union of all the individual outcomes is $\Omega$, which has
$p_{\Omega}\Psi=p_{\pi}\Psi=\Psi_{\pi}\not=0$.

Such mistakes are of little consequence for the survival of a species that
usually only needs to deal with finite sets of $A_{i}$ at a time. So the
persistence of such mistakes, not having been eliminated by natural selection
is not a surprise.

It is very interesting to observe that the structure of quantum mechanics, in
a rich enough universe like ours, provides for the existence of patterns in
the state vector that embed rules of logic and therefore can instantiate
classical computations. Computations performed by DNA-based, or RNA-based
wetware that are essential for life as we know it, animal brains and silicon
based computers are possible thanks to this structure and this richness. (See
also the concept of IGUS in [15].)

###  8.14  Reducing the Ontological Postulate to more intuitive statements

The Ontological Postulate can be justified by its success. It produces a
theory that makes the same predictions of Born’s-rule-collapse quantum
mechanics, without the collapses. But one may wonder if it can be reduced to
more intuitive statements.

This can be done as we relate the theory proposed in Subsection 8.11 to the
results in Sections 3 and 4 and the ideas from [31] and [32] presented in
Subsection 8.9. Suppose that we accept the following two premises:

  * (P1)

If $\Psi\in F_{A}$, then $A$ is not part of reality.

  * (P2)

The one-sided superposition principle.

Then, as explained in Subsection 8.9, we conclude that when
$\Psi\in\overline{F_{A}}$, $A$ should not be part of reality. Combining this
with part (a) of Theorem 3 we have

$p_{A}\Psi=0\ \ \ \Longrightarrow\ \ \ \Psi\in\overline{F_{A}}\ \ \
\Longrightarrow\ \ \ \mbox{$A$ is not part of reality}.$ (91)

This reduces half of our postulate to (P1) and (P2) above.

Now, suppose we go further and also accept a third premise:

  * (P3)

If $A$ is not excluded from reality by (P1) and (P2) above, then $A$ is part
of reality.

Since $F_{A}\subset\overline{F_{A}}$ and $\overline{F_{A}}$ is closed with
respect to taking superpositions, we have then

$p_{A}\Psi=0\ \ \ \Longleftrightarrow\ \ \ \Psi\in\overline{F_{A}}\ \ \
\Longleftrightarrow\ \ \ \mbox{$A$ is not part of reality},$ (92)

where the leftmost implication to the left can currently only be justified by
Theorem 2 if $S$ is countable (e.g., $S\subset\mathbb{Q}$, or
$S\subset\epsilon\mathbb{N}$), or by Theorem 3 if $\Gamma$ is finite. This
reduces our postulate to (P1), (P2) and (P3), in these cases.

###  8.15 A close relationship between Born’s rule and the one-sided
superposition principle

From (11) in part (d) of Theorem 1 we know that

$p_{A}\Psi=0\ \ \ \Longleftrightarrow\ \ \ \Psi_{\pi}=0\ \ \mbox{or}\ \
\mathbb{P}_{\Psi_{\pi}}(A)=0.$

Therefore, if we accept the assumptions (P1) and (P2), from Subsection 8.14,
then (91) implies

$\mathbb{P}_{\Psi_{\pi}}(A)=0\ \ \ \Longrightarrow\ \ \ \mbox{$A$ is not part
of reality}.$

And if we also accept (P3) and assume $S$ countable or $\Gamma$ finite, then
(92) implies

$\mathbb{P}_{\Psi_{\pi}}(A)>0\ \ \ \Longleftrightarrow\ \ \ \mbox{$A$ is part
of reality}.$

(In the left-hand side, the assumption $\Psi_{\pi}\not=0$ is implicit, since
otherwise $\mathbb{P}_{\Psi_{\pi}}(A)$ would not be defined.) These
consideration show how closely related Born’s rule is, in the context of non-
collapse quantum mechanics, to the one-sided superposition principle,
expanding on the thesis of [31] and [32].

It is natural to ask if one can have a version of non-collapse quantum
mechanics that does not satisfy Born’s rule, but instead satisfies some other
probability rule. This is the case in the context of the particle models of
Subsection 8.2, if nature chooses at each time $t\in\mathbb{R}$, independently
of anything else, a point $x_{t}$ from the configuration space $\mathcal{C}$
with probability $\mathbb{P}^{\prime}_{\Psi_{\pi}}(x_{t}\in\cup_{i}R_{a,i})$
proportional to $||p^{t}_{a}\hat{\varphi}||^{\alpha}$, $\varphi=\Psi_{\pi},$
with some $\alpha\not=2$, where $\Psi$ is the Heisenberg-picture state of the
universe and $\pi$ is some special analyser, that satisfies
$\Psi_{\pi}\not=0$.

###  8.16 Choosing between pilot-wave theories and the minimalistic ontology
of Subsection 8.11

As we observed in Subsection 8.11, the minimalistic ontology proposed there
cannot be distinguished through experiments from the alternative proposal that
we live in an infinite-ensemble universe, in which each world in the ensemble
is an independent realization of a process $(x_{t})_{t\in S}$ that satisfies
(85) with $\varphi=\Psi_{\pi}$, where the partition of the configuration space
into the sets $R_{a}$, $a\in\Gamma$ and the set $S$ correspond to the analyser
$\pi=\pi_{H}$. The same is true if $\pi$ is an appropriate refinement of
$\pi_{H}$, as described in Subsection 8.12, including $\pi_{C}$.

Experiments would also not distinguish our experiences under these proposals
from those in a universe with a single, or any finite number of independent
realizations of such a process $(x_{t})_{t\in S}$.

Choosing among these theories seems to be a pure matter of personal taste.

If the issue is only experimental adequacy, also pilot-wave theories that
satisfy (86), but do not satisfy (85), are alternatives. But here there are
already serious manifestations of discontent in the literature: in Section
10.2 of [22] and Section 5 of [23], and in Section 5 of [2], because in such
theories our memories and records do not have to correspond to our true past.

One should try to show that (under appropriate conditions on the Hamiltonian,
perhaps) a process $(x_{t})_{t\in S}$ that satisfies (85) can have very nice
properties, including continuous paths interrupted by jumps at creation and
annihilation of particles, Markovianity, being described by a differential
equation, etc.

If this turns out to be the case, such a pilot-wave theory would probably be
very attractive. On the other hand, for people who prefer less baggage in a
metaphysical theory and see abstraction as no obstacle (perhaps even an
advantage) the minimalistic ontology of Subsection 8.11 will probably still be
a better choice.

In the discussion above the choice between the minimalistic ontology of
Subsection 8.11 and the alternatives is only about what is considered to be in
the primary ontology of the theory. Even in the minimalistic ontology, the
processes $(x_{t})_{t\in S}$ that satisfy (85) exist in the derived ontology
(as mathematical constructs). And an infinite ensemble of independent versions
of such processes exists as well in the same sense. And as we saw in
Subsection 8.11, such an infinite ensemble helps us understand the meaning of
the Ontological Postulate introduced there. This postulate is equivalent to
the statement that the only events in $\Sigma$ that are realized are those
that are realized in this infinite-ensemble of independent processes.
Therefore one can think of these processes as aids to the visualization of the
realities encoded in the primary ontology given simply by $\mathcal{H}$,
$(U_{t})_{t\in T}$ and $\Psi$.

A person choosing a metaphysics in which a realization of a process
$(x_{t})_{t\in S}$ is also part of the primary ontology will be faced with the
question of why to prefer one single realization of such a process with this
status rather than infinitely many independent ones. In other words, between a
pilot-wave theory with a single realization and one with an infinite ensemble,
why prefer one to the other? (The remaining choice of a finite number, larger
than 1, of realizations will probably be discarded in comparison with those,
for lack of motivation, or even on aesthetic grounds.)

In favor of a single realization one can argue that it is more economic and at
the same time sufficient. (But then, the minimalistic ontology is even more
so.) Perhaps one would add a strong preference for having a theory without
human branching. (But humans would still be branching in the wave function,
even if only one branch would be considered to be real. And this raises the
question of what to make of the humans in the branches of the wave function
that evolve and behave like you and me, without being “real”.)

In favor of an infinite ensemble one can argue that it preserves the symmetry
among the branches of the wave function. For instance, at the end of the
infamous experiment involving a cat, there will be two branches that evolve
quite differently (for the cat at least). Regardless of the way a single
process $(x_{t})_{t\in S}$ goes at the end of this experiment, both branches
evolve according to $(U_{t})_{t\in T}$ in ways that encode coherent subsequent
stories. Why would nature produce all of this and only realize one of these
stories?

The debate above, between one or an infinite ensemble of realizations of
$(x_{t})_{t\in S}$ may be taken as an argument in favor of the minimalistic
ontology. That ontology avoids the choice, by considering any number of such
processes with the same status of derived realities. All very useful for our
understanding of how the universe evolves, of our place in it, and why we can
use textbook quantum mechanics to predict our future and describe our past.

One should not think that by being placed in the derived ontology the pilot-
wave processes $(x_{t})_{t\in S}$ become less important. Their existence, as
mathematical objects, shows that it is possible to have the particles of the
universe move in physical space, between their creation and annihilation, in
ways that are compatible with our experiences and our records and memories of
these experiences. This negates the very common statements according to which
nothing like this could be done. See for instance Chapters 1, 2 and 7 of [7],
for an extensive criticism of such statements.

The reader should have inferred from the discussion in this subsection and
previous ones what my own preferences for interpretations of quantum mechanics
are. Among those discussed here, they are ordered as follows. The minimalistic
ontology first, followed by an infinite-ensemble-pilot-wave model that
satisfies (85), followed by a single-pilot-wave model that satisfies (85). The
option of a pilot-wave model that satisfies (86), but does not satisfy (85)
does not seem plausible to me, for the reasons (admitting incorrect records
and memories) presented before.

Readers are invited to come to their own conclusions and to possibly apply the
theorems in this paper in different ways that may further shed light on issues
in the interpretation of quantum mechanics. They are also invited to expand
and elaborate on the proposals in this paper, and to possibly settle
mathematical issues left open here, as the conjecture raised after Theorem 3,
and the questions posed in Subsection 8.4.

Acknowledgements: It is a pleasure to thank Marek Biskup, Christopher de
Firmian and Jim Ralston for enlightening conversations. Jim Ralston also
deserves many thanks for carefully reading the paper and making several
constructive suggestions. Thanks are also given to Jean Bricmont and Shelly
Goldstein for enlightening conversations on pilot-wave theories, especially
Bohmian mechanics.

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