Many Discrete Worlds

Introduction

Of the dozen or so mainstream interpretations of quantum mechanics, each with its own many variants (for a list of the most important ones, see e.g., Freire, et al. [1]), Hugh Everett’s Many Worlds Interpretation (MWI) (Everett [2]) has attracted perhaps more than its share of controversy and debate. (For an excellent introduction to the whole debate, see the book edited by Saunders, et al. [3] and Saunders’ own introduction to the book (Saunders [4]) as well as many other papers including, for example, Kent [5, 6], Butterfield [7, 8], Hartle [9], Greaves and Myrvold [10], Zurek [11], Barrett [12], Tappenden [13, 14, 15], Adlam [16], Papineau and Rowe [17], Vaidman [18].). The purpose of our paper is to argue the case for a subset of MWI where our universe is one of an unimaginably large but nevertheless finite number of discrete, non-interacting, decohered Minkowski spacetimes distributed in proportion to the Born Conceptual probabilities of the quantum outcomes embedded within each of those spacetimes. This means that there are generally multiple identical copies of any given spacetime/universe/ world. We shall go on to argue that, counterintuitively, all of these universes are finite, both spatially and temporally. For brevity we refer to this picture as Many Discrete Worlds (MDW).

Decohered Histories

Our point of departure is to consider the history of our universe viewed from an arbitrary world line. This history will be defined by a series of outcomes of quantum events along the chosen world line, where, by an “event”, we mean what is often called a “measurement” but is really any interaction that results in a change of state such as a change in electron spin or momentum. These events do not generally happen directly on the world line but originate instead from anywhere within the universe: the arrival of the light cone from such a distant event impinging upon the world line might be considered as marking a corresponding branching point on the world line. (The concept of the light cone signalling such a “collapse” is not new—see, for example, Belnap [19], Bacciagaluppi [20] or Figure 8.2 of Wallace’s book (Wallace [21])—but decoherence is sometimes considered to spread out at subluminal velocities: e.g. Marvian and Lidar [22]. For our purposes, it doesn’t matter which we choose.) To make this concrete, consider Figure 1, adapted from Figure 3 of Hartle’s [23] pedagogical introduction to a “coarse-grained” picture (in effect, after decoherence is complete)—a picture that was further developed by Gell- Mann and Hartle [24]. The figure shows a wave function, ψ , expressed as the branching structure of a set of 16 alternative histories (i.e., branches), each containing three events/interactions marked by black dots. In our adaptation, we assume that the three events are mutually independent, which was unnecessary in Hartle’s example. One particular branch is highlighted with a heavier line, but in each of the 16 branches, the three events have already occurred, so that no quantum interference is explicitly portrayed in the figure—the branches are completely decohered. Any evidence of interference would only be manifest in patterns involving the alternative outcomes themselves: for example, interaction #3 has four different possible outcomes. If these four outcomes are regarded for instance as a schematic of four possible regions where an electron might land on a detecting screen after encountering a double slit, then the relative probabilities of each of the four outcomes would be a manifestation of the interference.

These branches are generally regarded as worlds or universes. While Boughn [25] and others remind us that Everett [2] himself spoke of states branching, other authors such as Wallace ([21], pp: 63), spoke of the branches being “not any less real”, a view endorsed over time by authors including DeWitt [26], Vaidman [27], Wilson [28], Hall, et al. [29]; even Everett agreed that “It looks like we would have a non-denumerable infinity of worlds” (Werner [30], pp: 95 marked “Tuesday am page 20”).

Click to enlarge

Figure 1: This figure, adapted from Hartle [23], shows the branching structure of a set of 16 alternative histories that constitute the wave function ψ , featuring three interactions. The probability amplitude associated with each interaction is αnk, where n refers to interaction #n and k is the k_th alternative outcome for that interaction. Thus, ψ is the superposition of the 16 histories, each history being the tensor product of the three vectors _nk ψ associated with the three interactions. One of the 16 alternative branches has been highlighted for discussion in the text.

The probability amplitude associated with each interaction in Figure 1 is αnk, where n refers to interaction #n and k is the _k_th alternative outcome for that interaction (so that, in our figure, there are four alternative outcomes for interaction #3). Following Gell-Mann and Hartle [24], the branch probability of the highlighted branch—effectively, the probability of finding yourself in a universe represented by this particular

The Frequency Distribution of Universes in MDW

branch—is:

prob. of being in universe 112232 ψ = 2 2 2 2 11 22 32 11 22 32 . . . . α α α α α α = (1) where we have used 112232 ψ for the highlighted branch.

Click to enlarge

Figure 2: The branching for the two outcomes of the 60º Stern-Gerlach experiment described in the text is shown in (a), with weightings according to the absolute squares of the probability amplitudes. The equivalent divergent/discrete picture is shown as two separate, weighted universes in (b). In (c), each of these two weighted universes is divided into multiple identical copies in numbers proportional to the weights.

In this section we propose a frequency distribution for the parallel universes over the whole MDW multiverse, which will be based on the Born rule. We go on to show, in the following section, how this distribution will lead to experimental results in any one universe which are indeed consistent with the Born rule. Note that we are not attempting to derive the Born rule here: we accept it as a given.

Consider an experiment where a particle with a vertically upward spin passes through a Stern-Gerlach magnet inclined at 60º to the vertical. The state of the particle just before it is detected may be written as 3 1 2 2 ↑+ ↓, where the spins in states ↑ and ↓ are inclined at 60º to the vertical, in alignment with the Stern-Gerlach detector; call them “spin- up” and “spin-down” respectively. Figure 2 shows the branching structure for the experiment. Figures 2(a) and 2(b) follow Figure 1(a) and Figure 1(c) of Saunders [31], who shows how the branching (or “overlapping”) and discrete (or “divergent”) pictures are equivalent. The weights of the branches are the Born probabilities for the outcomes and $|\downarrow\rangle$, namely $\frac{3}{4}$ and $\frac{1}{4}$ respectively. (For further discussion of weighted branching, see, for instance, Lewis [32], Saunders and Wallace [33], Saunders [4], Wallace [21], Wilson [34], Tappenden [15]).

To address the problem of why an observer should give more credence to being in a universe with a greater weight, we use the "branch counting" method employed in different ways by several authors: to a given universe, we assign a number of equally weighted universes in proportion to its branch weight—see, for instance, Khawaja [35], Saunders [36], and Barrett and Goldbring [37], with the latter authors using a model based on nonstandard analysis, which, in the opinion of Khawaja, may be compatible with branch counting.

Wallace [21] asks how branches can be counted when, in his words, "there is actually no such thing as the number of branches". The main issue is the preferred basis—if any rotated basis is equally valid, how can we fix on any particular basis for counting the branches? Should our summation even include the branches in all of the other possible bases? Our approach to these questions is to consider the branches only after the interactions have completely decohered—in that case, whatever it was in the environment that selected the particular basis for the observed outcome, that same process rules out all of the putative alternative bases.

In Figure 2(c) we have replaced each of the two single, weighted branches by discrete universes in numbers proportional to the weights, identical to each other within a given branch. For the illustration we have assumed that the $60^\circ$ experiment is the only event in the universe and that there are only 20 discrete universes in the multiverse. At first glance, Figure 2(c) looks as though it might represent Saunders' equi-amplitude branch-counting picture (Saunders [36]), but that would be too quick. While it is true that branch-counting in Saunders' scheme results in numbers proportional to the squares of the histories' probability amplitudes, his approach is more subtle than ours in that it accounts for all possible branch histories, whereas ours is a considerably blunter instrument, selecting only the final, completely decohered history. So, the many universes in Figure 2(c) are identical copies of either the single state (the left branch), or the single state (the right branch). Furthermore, we are not trying to derive the Born rule in any way—in MDW we simply accept that an outcome of the universal wave function is that, at complete decoherence, at the extreme coarse-grained level, there are multiple identical copies of universes in numbers proportional their respective Born probabilities.

The number of discrete, decohered universes in the multiverse, $N_0$, is fixed—and also finite, as we shall argue later—and the occurrence of an event does not mean that the number of universes increases to contain the event's outcomes. The distribution of these universes is tenseless and determinate. Each universe is discrete and doesn't split: they are simply part of the structure of the multiverse. Note also that the fixed number of universes means that the MDW multiverse is not amenable to analysis such as that applied by Short [38] to discrete many-worlds models, where his analysis exposes contradictory values for probabilities$^1$.

As we shall discuss in more detail in (Section: Spatiotemporally Finite Universes in a Finite Multiverse), the number of events in the multiverse cannot be greater than the number of universes, $N_0$, needed to contain all of the different outcomes of each of these events. Since $N_0$ is finite, we can identify individual events with a label, $n$. Of course, no universe in the multiverse will contain every event: the configurations of individual universes—meaning the complete history of all of the event outcomes that the universes contain, along with the spacetime locations of the outcomes from beginning to end—are determined by the unitarily evolving state from which the complete decohered history may in principle be extracted. We shall use $N_0$ for the number of events in the $n^{\text{th}}$ universe with a given configuration of outcomes. Since a given universe/branch can contain only one outcome of the $n^{\text{th}}$ event, then the number of different universes containing a specific event has to be at least as great as the total number of possible outcomes of that event. For a given event, a list of the different possible outcomes may be found within the sum of the vectors for all possible outcomes of the event:

$$\sum_{k_n} |\psi_{ink_n}|$$

where $k_n$ is the $k^{\text{th}}$ outcome of the $n^{\text{th}}$ event in the $i$th universe.

1 In effect, Short considers a multiverse containing two universes, $|0\rangle + |1\rangle$ so that the probability of being in either one is $\frac{1}{2}$. Then he applies a transformation that transforms $|1\rangle$ into $|1\rangle + |2\rangle$, leaving the $|0\rangle$ intact, so that there are now three universes, $|0\rangle + |1\rangle + |2\rangle$. By the axioms in the paper, this means that the probability of being in any one of the final three universes is $\frac{1}{4}$, which is a contradiction. However, in MDW, there is no contradiction, and in this scenario there would be four universes at the outset, $|0_1\rangle + |0_2\rangle + |1_1\rangle + |1_2\rangle$, with a probability of $\frac{1}{4}$ of being in any one, making a total probability of $\frac{1}{4}$ of being in $|0\rangle$, or $|1_2\rangle$, are still $\frac{1}{4}$, $\frac{1}{4}$, and $\frac{1}{4}$ respectively, and there is no contradiction.

For illustration, the sum of the two possible outcomes for interaction #1 in Figure 1 would be

$$\left( |\psi_{i11}| + |\psi_{i12}| \right)$$

where “$1$” and “$2$” refer to the outcomes represented by $\alpha_{11}$ and $\alpha_{12}$, respectively in Figure 1 (and, for instance, “$2$” and “$2$” would refer the outcomes of the event in state $|\psi_{i2}|$ represented by $\alpha_{21}$ and $\alpha_{22}$ respectively).

The complete state for any universe in the multiverse may be found as a unique branch within the collection of all possible combinations of the outcomes of all $N_i$ events contained in each branch, where the product sign is to be understood as the tensor product:

$$\prod_{n=1}^{N_i} \sum_{k_i} |\psi_{i n k_i}|$$

Remember that the events with their outcomes are assumed here to be unentangled, i.e., independent. As we shall discuss in (Section: Entanglement in MDW), entangled events are considered as a single event, but the arithmetic doesn’t change otherwise.

Using (2), and continuing with the same illustration, the collection of all possible combinations of the outcomes of interactions #1 and #2 in a given universe, $i$, where there are no other events, is:

$$\left( |\psi_{i11}| + |\psi_{i12}| \right) \otimes \left( |\psi_{i21}| + |\psi_{i22}| \right)$$

$$= |\psi_{i11}| |\psi_{i21}| + |\psi_{i11}| |\psi_{i22}| + |\psi_{i12}| |\psi_{i21}| + |\psi_{i12}| |\psi_{i22}|$$

where the tensor product sign has been dropped. These represent the four possible branches in Figure 1 containing interactions #1 and #2.

As we mentioned, not every event in the multiverse appears in every universe. For instance, suppose that an event described by a state $|\psi_{i4k}|$ is triggered by the interaction #1, but only if the outcome is $|\psi_{i11}|$ and not $|\psi_{i12}|$, so that, in universes where the outcome is $|\psi_{i12}|$, there is no state $|\psi_{i4k}|$. If there are three possible outcomes to the event in state $|\psi_{i4k}|$, written $|\psi_{i41}|$, $|\psi_{i42}|$, $|\psi_{i43}|$, then, from (2), there are 8 possible configurations of universes, as shown by the following combinations of vectors:

$$|\psi_{i11}| |\psi_{i21}| |\psi_{i41}| + |\psi_{i11}| |\psi_{i21}| |\psi_{i42}| + |\psi_{i11}| |\psi_{i21}| |\psi_{i43}|$$

$$+ |\psi_{i12}| |\psi_{i21}| |\psi_{i41}| + |\psi_{i11}| |\psi_{i21}| |\psi_{i42}| + |\psi_{i11}| |\psi_{i21}| |\psi_{i43}|$$

where the last two terms do not contain the state $|\psi_{i4k}|$ by the supposition.

We assign a unique identifier to each universe with a given configuration of event outcomes, so that, even although two states for a universe may be otherwise identical, they are nevertheless distinguished by the value of the label, $i$. The number distribution of such universes, identical but for their labels, is determined by the Born rule. The configuration of outcomes in a given universe within the collection of universes in (2) may be specified thus:

$$\prod_{n=1}^{N_i} |\psi_{i n k_i}|$$

where only one outcome for each event $n$ is selected, namely outcome $k_n$. An example would be the triplet of vectors in the second term of (3), namely, $|\psi_{i11}| |\psi_{i21}| |\psi_{i42}|$. In general, the otherwise identical universes with this particular configuration may be distinguished by their label, $i$, as in this expression which uses the summation to allocate a different value of $i$ to each configuration:

$$\sum_{n=1}^{N_i} \prod_{n=1}^{N_i} |\psi_{i n k_i}|$$

where $|\alpha_{nk_i}|^2$ is the absolute square of the probability amplitude for the $k_n$th outcome of event $n$. The upper limit of the sum in (5) is the number of identical universes in the multiverse with the configuration in (4), as derived from the Born rule:

$$N_0 \prod_{n=1}^{N_i} |\alpha_{nk_n}|^2$$

Because we have used the Born rule to arrive at this number distribution, then, as we shall see later, the Born rule re-emerges experimentally within individual universes.

Continuing the illustration, if we assume, say, $\sqrt{\frac{3}{4}}, \frac{1}{\sqrt{2}}$, and $\sqrt{\frac{3}{5}}$ for the probability amplitudes $\alpha_{11}$, $\alpha_{21}$, and $\alpha_{42}$ respectively, then we see from (6) that the number of universes in the multiverse having the above configuration,

$$|\psi_{i11}| |\psi_{i21}| |\psi_{i42}|$$ is:

$$N_0 \prod_{n=1}^{N_i} |\alpha_{nk_n}|^2 = N_0 |\alpha_{11}|^2 |\alpha_{21}|^2 |\alpha_{42}|^2 = 40 \times \frac{3}{4} \times \frac{1}{2} \times \frac{3}{5} = 9$$

where there is no $|\alpha_{3k_n}|^2$ in the product because it doesn’t appear in (3). We have assumed for the illustration that there are just 40 universes in this multiverse, including multiple copies of the possible universes. Of course, nothing subtle is going on here; essentially, all we are saying is that, for a given configuration of outcomes, the number of identical universes in the multiverse with that configuration is proportional to the product of the probabilities of the individual outcomes according to the Born rule.

Now we check that MDW gives the same result as (1) for the probability of the highlighted branch in Figure 1. The probability amplitudes for the outcomes of the three events in the branch are α_11, α22, and α_32 respectively.

Expression (6) gives the number of universes containing these three events with these particular outcomes:

2 2 2

0 11 2 2 3 2 . . N α α α Therefore, the probability of being in such a universe is, trivially,

N which, of course, is the same result that we found in equation (1) using the branching model of Gell-Mann and Hartle [24]. It is also in the same format as given by Saunders [36], except that he still grants some uncertainty in the precise numbers of universes because of any residual interference, no matter how small, whereas, in MDW, we are considering the universes only after any interference between event outcomes has been irreversibly wiped out.

Everett’s universal wave function is at the heart of MDW just as it is in MWI and other interpretations of quantum mechanics, and this leads to the same mathematics and the same observed quantum-mechanical outcomes in MDW as in the other interpretations. The agreement between the results of equations (1) and (7) just emphasizes the equivalence of the overlapping picture in Figure 2(a) and the MDW picture in Figure 2(c).

Experimental Verification of the Born Rule In MDW

We recognise that such a straightforward notion of probability in equation (7) is frequentist, and may be challenged, particularly as we make the case in Section: Spatiotemporally Finite Universes in a Finite Multiverse that _N_0 is finite. Saunders [31, 40] analyses the fundamentals of probability in quantum mechanics, and self-locating uncertainty is discussed by many authors, often as an adjunct to deriving the Born rule (see, for instance, Vaidman [27, 41, 42], Wilson [28], Sebens and Carroll [43], McQueen and Vaidman [44], Dawid and Friederich [45]). However, we shall avoid that rabbit hole: rather than try to derive the Born rule, we shall simply acknowledge that it is intrinsic to the structure of MWI and MDW.

We wish to show that, if the numbers of discrete universes in the MDW multiverse are distributed in proportion to the Born rule as in expression (6), then quantum outcomes in any individual universe, including our own, will also be according to the Born rule. It is by no means obvious that such a numbers distribution in the multiverse will indeed lead to the quantum probabilities that we observe in our universe, and we must be careful how we investigate this. Crucially, we shall not follow the routes normally chosen to demonstrate quantum probabilities, namely (1) the frequentist approach—Saunders [4] gives an introduction to the problems with frequentism—or (2) Bayesian and decision-theoretic strategy: see for instance Deutsch [46], Wallace [47], Greaves and Myrvold [10]. While these paths are well trodden and intuitive, they are not without their obstacles (see, for instance, Barnum, et al, [48]), and, instead, we shall follow Everett:

“Imagine a very large series of experiments made by an observer. [… For] a ‘typical’ branch, the frequency of results will be precisely what is predicted by ordinary quantum mechanics.” (Werner [30] pp: 96, marked “bottom of page 20”) So, we ask a simple question of the MDW structure: if we repeat a given quantum experiment many times in succession, and then aggregate all of the measured outcomes of each of the experiments in the series, do they converge towards the result predicted by the Born rule?

Click to enlarge

Suppose that the 60º experiment is repeated for a large number of times so that the total number of experiments in the series is N_E, which we substitute for _Ni in expression (6). There are only two possible outcomes for each experiment, namely ↑ and ↓, with Born probabilities ¾ and ¼ respectively. From (6), the number of universes containing p spin-up and (

$$ \left(N _ {\mathrm {E}} - p\right) s $$

spin-down outcomes in any particular ( ) E p N p N $$ _ {0} \left(\frac {3}{4} \uparrow\right) ^ {p} \left(\frac {1}{4} \downarrow\right) ^ {\left(N _ {E} - p\right)} $$

0 3 1 4 4 order of outcomes is , where we have included the spin direction for clarity. The number of combinations of that number of spin-up and spin-down E N p       outcomes is the binomial coefficient, . Hence, the total number of universes in which there are p spin-up outcomes $$ \text {a n d} \left(N _ {\mathrm {E}} - p\right) $$

spin-down outcomes in a series of NE

experiments is ( ) E E 0 3 1 number of universes with outcomes 4 4 p N p N p N p −     ↑ = ↑ ↓          (8) In Figure 3 we show equation (8) plotted as histograms for the outcomes of three different series of 60º experiments, containing, in each universe, _N_E = 100, 1,000 and 10,000 experiments respectively. We have rescaled each of these three histograms so that their peaks are all the same height.

In the series of 10,000 experiments per universe, we see that in virtually all of the _N_0 universes, the number of spin-up outcomes in a universe is almost exactly ¾ of the total number of experiments in that universe (that is, in this case, 7,500). Taking the width of the peak to be proportional to the percentage standard deviation of the plot, namely,

, it will be seen that, as the number of experiments in the series, _N_E, increases, so the percentage of spin-up outcomes more closely matches the ideal result of 75% predicted by the Born rule for a series containing an indefinitely large number of 60º experiments in virtually every one of the parallel universes in which the series takes place. This echoes Saunders’ observation (Saunders [31]) that “branches recording relative frequencies of outcomes of repeated measurements at variance with the Born rule have, collectively, much smaller amplitude than those that do not (vanishingly small, given sufficiently many trials)”.

In summary, following Everett’s thought experiment, we have shown that, if the numbers of universes with given outcomes are apportioned across the MDW multiverse according to the Born rule for those outcomes as in expression (6), then, in a series of experiments within any single one of these universes, the results will generally be consistent with Born rule predictions.

For clarity, we repeat the assertion in the first paragraph of Section: The Frequency Distribution of Universes in MDW we have not attempted to derive the Born rule in any way— we have assumed the Born rule, and then we have shown that experiments in any given universe to demonstrate the Born rule will be successful if the numbers of universes in the multiverse are distributed according to the Born outcome probabilities for those experiments.

Entanglement in MDW

Suppose that two particles are emitted in opposite directions along the y-axis in an entangled singlet state where the total spin is zero, and that the spin of each particle is measured with a Stern-Gerlach detector set at an angle θ to the other, as measured in the x-z plane. In MDW, we consider the two separate measurements of two entangled particles as a single event. Pictorially, in any given universe, the two measurements are embedded at two different spacetime locations, and may be spacelike separated. The probability amplitudes for all four possible (mutually exclusive) outcomes may be written

1 sin 2 2 θ ↑↑; 1 cos 2 2

θ ↑↓; 1 cos 2 2

θ ↓↑ and

1 sin 2 2 θ ↓↓.

where the first arrow always represents the spin of the first particle and mutatis mutandis for the second arrow.

Hence, the absolute square of the probability amplitude for a match, (m), where( ) (

$$ (m) = \left(\left| \uparrow \uparrow \right\rangle o r \left| \downarrow \downarrow \right\rangle\right), \mathrm {i s} $$ $$ \left(\frac {1}{2} \sin^ {2} \frac {\theta}{2} \uparrow \uparrow + \frac {1}{2} \sin^ {2} \frac {\theta}{2} \downarrow \downarrow\right) = \sin^ {2} \frac {\theta}{2} (m). $$ .

Similarly, the absolute square of the probability amplitude for a non-match, (n), where ( ) (

$$ (n) = \left(\left| \uparrow \downarrow \right\rangle o r \left| \downarrow \uparrow \right\rangle\right), $$ is ( ) 2 cos 2 n θ . Here, the brackets around “m” and “n” mean that the terms are used purely indicatively and have no numerical value in this position.

If we put θ = 120º, the probabilities for a match and a non-match are, respectively, ¾ and ¼. Therefore, in a series of _N_E entanglement experiments, from expression (6), the number of universes containing $m$ matched and $(N_E - m)$ non-matched outcomes in any particular order of outcomes

is $N_0 \left( \frac{3}{4}(m) \right)^m \left( \frac{1}{4}(n) \right)^{(N_E - m)}$, so that, taking account of all possible combinations, the total number of universes in which there are $m$ matches and $(N_E - m)$ non-matches in a series of $N_E$ entanglement experiments is

$$\text{number of universes with } m \text{ matched outcomes} = N_0 \left( \frac{N_0}{m} \right) \left( \frac{3}{4}(m) \right)^m \left( \frac{1}{4}(n) \right)^{(N_E - m)}.$$

Since this is in the same form as equation (8), then the results of series of entanglement experiments can be represented by Figure 3 where we substitute "matching" for "spin-up" in the text for the two axes. So we come to the equivalent conclusion, that, if we continue to repeat the above entanglement experiment a sufficiently large number of times in any single, given universe, then it is increasingly likely that we shall be able to demonstrate that we are in a universe where the proportion of matching outcomes approaches the result of $\frac{3}{4}$ predicted by quantum mechanics for this entanglement experiment.

In treating the two measurements of the entangled particles as a single event, we are recognizing that the outcomes are distributed across the multiverse just as described in Section: The Frequency Distribution of Universes in MDW, so that, in any given universe (our own, for instance), the detection of one of the pair has no effect on the detection of the other. This echoes the comment of Boughn [49]:

"[...] the magic of the correlations of Bell’s entangled system is a direct consequence of the quantum behavior of a single spin $\frac{1}{2}$ particle."

Spatiotemporally Finite Universes in a Finite Multiverse

We used angles of $60^\circ$ and $120^\circ$ to illustrate the above thought experiments because the outcome probabilities are rational. This means that the number of identical universes with a given configuration can be integer, from expression (6). However, probabilities predicted by quantum mechanics are generally irrational. This raises the question of whether or not the multiverse is finite with regard to the number of universes in it. If it is not uncountably infinite, then quantum mechanics will turn out to be only an ideal approximation (no matter how accurate) to the actual structure of the multiverse according to MDW.

To answer the question of finitude, consider, for the moment, weighted universes as in Figure 2(b) rather than the multiple universes of Figure 2(c), and recall that our use of the term "universe" refers to its complete spatiotemporal extent after decoherence. Suppose that such a weighted universe contains $m$ mutually independent quantum events: for example, $m$ non-entangled events, each of which is located at a spacelike distance from all the others. Since each of these events must have at least two possible outcomes, then there are at least $2^m$ possible combinations of outcomes, with at least $2^m$ configurations of weighted universes containing these outcomes.

Recalling from Section: Beyond the Cosmological Particle Horizon that events in a universe are not limited to the finite volume within the particle horizon, we begin by supposing that $m$ is countably infinite, which would mean an uncountable infinity (at least $2^m$) of weighted universes in the multiverse.

In any universe, if we trace the cause of any event successively backwards in time, we shall ultimately arrive at the same root event that is common to all universes. In the multiverse of our supposed uncountable infinity of universes, the number of universes containing an outcome of an event directly caused by the root event must also be uncountably infinite. In turn, the number of universes containing a succeeding outcome will also be uncountably infinite. Proceeding thus, we see that every outcome of every event in such an uncountably infinite multiverse will be contained in an uncountably infinite number of universes.

In such a multiverse, the intuition is that it is not meaningful to speak of the probability of being in a universe with any particular configuration of outcomes. If so, then demonstrating the Born rule by conducting a series of $60^\circ$ experiments within a given universe would not work, because it would no longer be meaningful to say that in nearly every universe the percentage of spin-up outcomes matches the 75% predicted by the Born rule within a specified limit. Nevertheless, if we regard each weighted universe with a given percentage of spin-up outcomes as the sole element of a set, then might the problem be amenable to a measure such as the Lebesgue measure?

Discussion

The MDW picture of discrete universes diverging from each other rather than branching at every interaction is essentially that of modal realism described by Lewis [32]. However, the motivation for our proposal stems ultimately from the need to account for quantum uncertainty, whereas Lewis developed his thesis from a different philosophical perspective, lying outside of the ambit of this paper. The case for discrete, diverging universes is also considered by Saunders and Wallace [33], Saunders [4], Wallace [21], Wilson [34]. None of these authors, though, promotes the notion of a finite number of discrete, diverging universes, with Wallace saying that you shouldn’t think of each universe as an individual unit (his picture is of “slabs” of universes) and Wilson being agnostic on whether the number of universes is either finite or infinite. Deutsch [59] argues for a continuous infinity of worlds. No author uses their model to deduce that the universes in MWI are spatially finite.

Authors who consider an uncountable infinity (a continuum) of universes include Deutsch [60], Boström [61], Holland [62], Poirier [63], Parlant, et al. [64], Schiff and Poirier [65]. (Others, such as Wilson ([34], pp: 178) accept the possibility of a continuum of universes, but are open to the prospect that the number may yet be finite.) Exceptionally, Hall, et al [29], Vaidman [42] and Sebens [66], among a few others, are unconvinced by the notion of a continuum of universes, the latter two authors making the case that regions containing infinite densities of worlds cannot be compared in assessing probabilities of being in a particular world.

Returning for a moment to the two papers of Hall, et al [29] and Sebens [66], their authors having arrived independently at essentially the same quantum interpretation, the key idea is to apply the quantum potential of the de Broglie-Bohm interpretation not just to particles in a single universe but to particles in a configuration space containing a finite number of discrete universes. However, in contrast to the universes in MDW, these universes appear effectively to interact, ultimately as a result of how the de Broglie-Bohm quantum potential determines the configurations of universes in this interpretation. This interaction is most clearly manifest as a mutual repulsion in configuration space, which increases with increasing similarity between the worlds. This would effectively rule out universes being identical, or indeed identical for any significant part of their trajectories, whereas such identical universes are integral to the MDW interpretation.

Boström [61] maintains that it is legitimate to use the Lebesgue measure over a continuum of universes, effectively pointing out that, with universes that are continuous in configuration space, the Lebesgue measure can be used as a measure of the proportions of universes contiguously adjacent to each other sharing a particular state. However, while his argument would be correct for universes containing, for instance, only a series of 60º experiments, that contiguity is broken by the uncountable number of unrelated outcomes, which is another way to view the above argument that the Lebesgue measure cannot cope with an uncountably infinite number of pairwise disjoint sets of universes.

MDW shares some features of the many-threads picture described by Barrett [67]. Each thread is the history—the “experience”—of one trajectory through the dendritic tracery of MWI. In contrast to MDW, however, where two or more identical universes are allowed, there is apparently only one thread for each trajectory upwards through the branches, and there is an infinite number of threads in its multiverse.

Conclusion

We have presented the case for a fixed, finite number of discrete, non-splitting, non-interacting, spatiotemporally finite, decohered spacetimes emerging from Everett’s Universal Wave Function. The MDW approach gets round the problem of why an observer should give more credence to being in universes with greater branch weights by using the “branch counting” method. This divides universes with given branch weights into multiple identical universes in numbers proportional to the branch weights. As expected, probabilities derived from branch weights according to Gell Mann and Hartle [24] are equal to those calculated for the corresponding discrete universes in the MDW picture.

We showed that each universe should be considered as containing all decohered outcomes over the entire extents of their spacetimes, including events/interactions occurring beyond any cosmological particle horizon. We also showed that a series of experiments as prescribed by Everett within a universe will demonstrate consistency with the Born rule. It turns out that a countably infinite number of interactions needs an uncountably infinite number of universes, and we showed why measures such as the Lebesgue measure would fail in that case, with the result that the Born rule would not be demonstrable. This leads us to conclude that the number of universes in the multiverse must be finite and, as a corollary, that the universes are finite in space and duration.

In summary, the scope of Many Discrete Worlds is the same as that of the Many Worlds Interpretation, with Everett’s Universal Wave Function evolving according to Schrödinger’s equation, accounting for indeterminism (through self-locating uncertainty), while the branch-weight problem of MWI is avoided (because in MDW all universes are equally weighted). In addition, if the assumption is made that the multiverse and its component universes are essentially mathematical structures, then identical universes are nevertheless distinguishable as distinct substructures of the multiverse.

While one or more of the above features of MDW are to be found in other interpretations of quantum mechanics, the force of the arguments in this paper strongly suggests to us that, uniquely, these features are combined in our multiverse of Many Discrete Worlds.