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Edited by: Ignazio Licata, ISEM-Institute for Scientific Methodology, Italy

Reviewed by: Ovidiu Cristinel Stoica, Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), Romania; Daniel Sheehan, University of San Diego, United States

*Correspondence: Eliahu Cohen

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Theoretical achievements, as well as much controversy, surround multiverse theory. Various types of multiverses, with an increasing amount of complexity, were suggested and thoroughly discussed in literature by now. While these types are very different, they all share the same basic idea: our physical reality consists of more than just one universe. Each universe within a possibly huge multiverse might be slightly or even very different from the others. The quilted multiverse is one of these types, whose uniqueness arises from the postulate that every possible event will occur infinitely many times in infinitely many universes. In this paper we show that the quilted multiverse is not self-consistent due to the instability of entropy decrease under small perturbations. We therefore propose a modified version of the quilted multiverse which might overcome this shortcoming. It includes only those universes where the minimal entropy occurs at the same instant of (cosmological) time. Only these universes whose initial conditions are fine-tuned within a small phase-space region would evolve consistently to form their “close” states at present. A final boundary condition on the multiverse may further lower the amount of possible, consistent universes. Finally, some related observations regarding the many-worlds interpretation of quantum mechanics and the emergence of classicality are discussed.

Multiverse theory (also known as Meta universe theory) is a group of models assuming that our physical reality encompasses more than one universe, i.e., there exists at least one more universe other than ours. Several types of such multiverses are known in literature [

Some of these models suggest that our physical reality comprises of infinitely many universes^{1}

One of the most common explanations of the big-bang is given by quantum fluctuation theory, which suggests that our universe began from a quantum fluctuation, and if so, it is natural to deduce that in our physical reality these fluctuations are taking place in all of our space and time dimensions (see [

The multiverse type that we shall focus on is the quilted multiverse [

The quilted multiverse provides a theoretical probabilistic approach for the existence of events before the event horizon in our physical reality. Within the quilted multiverse, the event horizon includes events that occur infinitely many times, duplicated in infinitely many universes, which might be finite or infinite. From the characterization above we deduce that there are universes within the quilted multiverse that are not only “close” at a given time (e.g., at present), that is, similar in a sense that will be defined below, but have been very “close” for a substantially large time interval. In terms of Tegmark's hierarchy [

We claim in this work (based on our preprint [

Time seems to incessantly “flow” in one direction, raising the ancient question: Why? This intensively discussed question can be answered in several ways by introducing seemingly different time arrows: thermodynamic, cosmological, gravitational, radiative, particle physics (weak), quantum, and others [

In this perspective paper we will formally treat a universe parallel to ours, having at present time a similar macrostate or even the same macrostate, yet with a slightly different microstate as a perturbation. Then we will try to apply the above thermodynamic reasoning.

Before we claim that the quilted multiverse is inconsistent with the instability of entropy decrease discussed in section 2, let us define some mathematical symbols which will be useful later on. First, suppose that we have an infinite (yet countable) number of universes, denoted by

where each universe _{j}(

Further, let us define in phase space a distance measure Δ, which quantifies the difference between the microstate of the _{j}(_{i}(_{i} = _{j} =

for

where _{0}, _{f}] is some long time interval comparable with the age of the universes.

Moreover, from the above description of the quilted multiverse, we deduce that every possible event will occur an infinite (countable) number of times. Therefore, this model suggests that there should exist a set

where ε is some threshold below which we may say that the universes are “close”, and

We now show that Equation (4) is not consistent with the thermodynamic arrow of time defined in section 2 (it will be implicitly assumed that the universe is in a non-equilibrium state). First, notice that if we have a thermodynamic system _{0} <

Two “close” universes at present time were most likely “far” in the past.

There is only a negligible probability that two close universes at present, will evolve backwards in time to two close universes in the past (see also [

Also, it is inconsistent to assume that any arbitrary change to our current universe is a valid parallel universe having the same historical source in phase space or having the same point in time of minimum entropy.

The number of possible universes can be represented by the Boltzmann relation between entropy

where _{B} is the Boltzmann constant.

Then, given the entropy of the i-th universe,

Assuming that during its 13.8 billion years of history the universe has reached a very large entropy

will be zero.

Another way to see this inconsistency is to consider the point of minimal entropy during the lifetime of our universe. When picking at random another hypothetical universe having at present the same macrostate as our universe, it is most likely to have its minimal entropy at some other time different from ours (most likely after ours). Hence, the histories of the two universes cannot be the same, unless we focus at present only on the zero measure of macrostates having their minimal entropy at exactly the same time as ours.

We shall try to approach the problem from a different perspective now, beginning with some qualitative considerations. One should note two extreme distance scales between universes in a multiverse. When two universes are extremely close (that is, different but virtually indistinguishable so that 0 < Δ ≪ 1) at some point in time, they may have a non-negligible probability evolving backwards to extremely close initial states, thereby creating no inconsistency. However, having infinitely many universes which are identical to ours for all practical purposes is not too interesting. On the other hand, if two universes are far apart right now, stability (which corresponds to small perturbation) again plays no role. But this is not the case we wish to rule out.

Between these two contingent cases, lie the problematic distances to which instability considerations can be applied. This may pose a constraint on the distribution of universes within a multiverse—there might be infinitely many universes which are very far from each other and an infinite number of universes which are extremely close, but we do not expect too many universes to be intermediately close when we demand consistency over long times.

Let us examine now for concreteness a 6_{j}, _{j}) on the grid, where _{j}/_{j} encapsulates the three position/momentum vectors, respectively, of each particle in this universe. We now start to gradually fill the hypercube with more and more distributions. We begin with those having a slight overlap (or no overlap at all) with the original one and with each other, thus corresponding to universes which are very different. As this process continues, we will have to fill the finite phase space with more and more distributions, closer to each other, until a point (let us denote it by Δ =

We now apply similar arguments to those appearing at the end of the previous section. It seems that in a countably infinite phase space (allowing a countably infinite number of parallel universes) and a finite point in time

To resolve this apparent shortcoming of the quilted multiverse we must pose a condition on the possible distance between the universes, and eventually on their density. In case that

for some threshold 0 < ^{2}

It could be interesting to apply the above considerations to other kinds of multiverses. However, when the values of physical constants, and moreover, physical laws themselves, in other universes become different from those we know now in our universe, the distance between our universe and others might be very large at present (and furthermore vary with time). Therefore, it is not obvious how to apply stability considerations to these kinds of multiverse.

On the other side of the multiverse scale, there is the many worlds interpretation of quantum mechanics (also known as the quantum multiverse). In previous works [

These past results hint that the multitude of universes proposed by the many-worlds interpretation may not be needed in order to account for our empirical observations in a time-symmetric manner. Other kinds of multiverse can be handled the same way, and indeed, posing both initial and final boundary conditions on a multiverse should dramatically lower the measure of possible universes within it: Regardless of the dynamics, when the final state of the multiverse is evolved backwards in time, it must be compatible with any earlier state. As noted in Aharonov and Reznik B [

Multiverse theory has various models that describe different structures of the physical reality. One of these models is the quilted multiverse, which postulates that every possible event is occurring infinitely many times in nature, thus there are infinitely many universes resembling ours. At first glance, this model seems to be self-consistent. However, we have shown that this model negates basic thermodynamic principles. The difference between microstates in two “close” universes cannot be ϵ small at each point in time, or even along a finite, sufficiently large time interval. Therefore, any possible type of multiverse would better not assume such a relation between two universes. Moreover, every universe must have its unique past and future in the sense that there is no other universe with the same, or even very close, state over a substantial part of its life time. We therefore have to limit ourselves only to those universes whose macrostates at present time evolve backwards to the same point in time of minimal entropy such as ours. These obviously reside in a very small fraction of phase space and may evolve in a consistent way. Further constraints on the number of possible universes may arise when augmenting this analysis with a final boundary condition on the multiverse.

These findings corroborate previous ones of our group [

YA: Initiated the work; YA, EC, and TS: Developed the presented ideas; EC and TS: Wrote the manuscript with comments from YA.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We wish to thank Avshalom C. Elitzur and Daniel Rohrlich for helpful comments. YA acknowledges support of the Israel Science Foundation Grant No. 1311/14, of the ICORE Excellence Center “Circle of Light” and of DIP, the German-Israeli Project cooperation. EC was supported by ERC AdG NLST. TS thanks the John Templeton Foundation (Project ID 43297) and from the Israel Science Foundation (grant no. 1190/13). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of any of these supporting foundations. A earlier version of this work is available as an arXiv preprint [

^{1}In the context of this work we shall assume a discrete phase-space, meaning that all infinities are countable.

^{2}In the quilted multltiverse, the number (or commonness) of universes does not correspond to probability/Born rule, but in contrast, for the many-worlds-type multiverse we would have to employ a different logic as presented in section 5.