The multiverse hierarchy by max tegmark department of physics mit kavli institute
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The Multiverse Hierarchy by Max Tegmark Department of Physics MIT Kavli Institute ...

The Multiverse Hierarchy by Max Tegmark Department of Physics MIT Kavli Institute

Parallel universes are now all the rage, cropping up in
books, movies and even jokes: “You passed your exam in
many parallel universes — but not this one.” However,
they are as controversial as they are popular, and it is
important to ask whether they are within the purview
of science, or merely silly speculation. They are also
a source of confusion, since many forget to distinguish
between different types of parallel universes that have
been proposed.
The farthest you can observe is the distance that light
has been able to travel during the 14 billion years since
the big-bang expansion began. The most distant visible
objects are now about 4 × 1026 meters away1, and a
sphere of this radius defines our observable universe, also
called our Hubble volume, our horizon volume or simply
our universe. In this article, I survey physics theories involving
parallel universes, which form a natural four-level
hierarchy of multiverses (Figure 1) allowing progressively
greater diversity.
• Level I: A generic prediction of cosmological inflation
is an infinite “ergodic” space, which contains
Hubble volumes realizing all initial conditions —
including an identical copy of you about 1010
• Level II: Given the fundamental laws of physics
that physicists one day hope to capture with equations
on a T-shirt, different regions of space can
exhibit different effective laws of physics (physical
constants, dimensionality, particle content, etc.)
corresponding to different local minima in a landscape
of possibilities.
• Level III: In unitary quantum mechanics, other
branches of the wavefunction add nothing qualitatively
new, which is ironic given that this level has
historically been the most controversial.
• Level IV: Other mathematical structures give different
fundamental equations of physics for that Tshirt.
The key question is therefore not whether there is a multiverse
(since Level I is the rather uncontroversial cosmological
concordance model), but rather how many levels
it has.

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  • 1. The Multiverse Hierarchy Max Tegmark Dept. of Physics & MIT Kavli Institute, Massachusetts Institute of Technology, Cambridge, MA 02139 (Dated: February 20 2005, published in “Universe or Multiverse?”, B. Carr ed., Cambridge University Press, 2007)) Parallel universes are now all the rage, cropping up in key point to remember is that parallel universes are not a books, movies and even jokes: “You passed your exam in theory, but a prediction of certain theories. For a theory many parallel universes — but not this one.” However, to be falsifiable, we need not be able to observe and testarXiv:0905.1283v1 [physics.pop-ph] 8 May 2009 they are as controversial as they are popular, and it is all its predictions, merely at least one of them. Consider important to ask whether they are within the purview the following analogy: of science, or merely silly speculation. They are also a source of confusion, since many forget to distinguish between different types of parallel universes that have General Relativity Black hole interiors been proposed. Inflation Level I parallel universes The farthest you can observe is the distance that light Unitary quantum mechanics Level III parallel universes has been able to travel during the 14 billion years since the big-bang expansion began. The most distant visi- Because Einstein’s theory of General Relativity has suc- ble objects are now about 4 × 1026 meters away1, and a cessfully predicted many things that we can observe, sphere of this radius defines our observable universe, also we also take seriously its predictions for things we can- called our Hubble volume, our horizon volume or simply not observe, e.g., that space continues inside black hole our universe. In this article, I survey physics theories in- event horizons and that (contrary to early misconcep- volving parallel universes, which form a natural four-level tions) nothing funny happens right at the horizon. Like- hierarchy of multiverses (Figure 1) allowing progressively wise, successful predictions of the theories of cosmologi- greater diversity. cal inflation and unitary2 quantum mechanics have made some scientists take more seriously their other predic- • Level I: A generic prediction of cosmological infla- tions, including various types of parallel universes. tion is an infinite “ergodic” space, which contains Let us conclude with two cautionary remarks before Hubble volumes realizing all initial conditions — delving into the details. H¨ bris and lack of imagina- u 29 including an identical copy of you about 1010 m tion have repeatedly caused us humans to underestimate away. the vastness of the physical world, and dismissing things • Level II: Given the fundamental laws of physics merely because we cannot observe them from our van- that physicists one day hope to capture with equa- tage point is reminiscent of the ostrich with its head in tions on a T-shirt, different regions of space can the sand. Moreover, recent theoretical insights have in- exhibit different effective laws of physics (physi- dicated that Nature may be tricking us. Einstein taught cal constants, dimensionality, particle content, etc.) us that space is not merely a boring static void, but corresponding to different local minima in a land- a dynamic entity that can stretch (the expanding uni- scape of possibilities. verse), vibrate (gravitational waves) and curve (gravity). • Level III: In unitary quantum mechanics, other Searches for a unified theory also suggest that space can branches of the wavefunction add nothing qualita- “freeze”, transitioning between different phases in a land- tively new, which is ironic given that this level has scape of possibilities just like water can be solid, liquid or historically been the most controversial. gas. In different phases, effective laws of physics (parti- • Level IV: Other mathematical structures give dif- cles, symmetries, etc..) could differ. A fish never leaving ferent fundamental equations of physics for that T- the ocean might mistakenly conclude that the properties shirt. of water are universal, not realizing that there is also ice and steam. We may be smarter than fish, but could The key question is therefore not whether there is a mul- be similarly fooled: cosmological inflation has the de- tiverse (since Level I is the rather uncontroversial cosmo- ceptive property of stretching a small patch of space in logical concordance model), but rather how many levels a particular phase so that it fills our entire observable it has. universe, potentially tricking us into misinterpreting our Below we will discuss at length the issue of evidence local conditions for the universal laws that should go on and whether this is science or philosophy. For now, the that T-shirt. 1 After emitting the light that is now reaching us, the most distant 2 As described below, the mathematically simplest version of quan- things we can see have receded because of the cosmic expansion, tum mechanics is “unitary”, lacking the controversial process and are now about about 40 billion light years away. known as wavefunction collapse.
  • 2. 2
  • 3. 3I. LEVEL I: REGIONS BEYOND OUR COSMIC HORIZON Let us return to your distant twin. If space is infi-nite and the distribution of matter is sufficiently uniformon large scales, then even the most unlikely events musttake place somewhere. In particular, there are infinitelymany other inhabited planets, including not just one butinfinitely many with people with the same appearance,name and memories as you. Indeed, there are infinitelymany other regions the size of our observable universe,where every possible cosmic history is played out. Thisis the Level I multiverse. FIG. 2: Although an infinite universe has always been a pos- sibility, the lower limit on the size of our universe has kept growing. A. Evidence for Level I parallel universes Although the implications may seem crazy and them. Infinite models fit the data, and strong limits havecounter-intuitive, this spatially infinite cosmological been placed on the alternatives (de Oliveira-Costa et al.model is in fact the simplest and most popular one on 2003; Cornish et al. 2003). In addition, a spatially infi-the market today. It is part of the cosmological concor- nite universe is a generic prediction of the cosmologicaldance model, which agrees with all current observational theory of inflation (Garriga & Vilenkin 2001b), so theevidence and is used as the basis for most calculations striking successes of inflation listed below therefore lendand simulations presented at cosmology conferences. In further support to the idea that space is after all simplecontrast, alternatives such as a fractal universe, a closed and infinite just as we learned in school.universe and a multiply connected universe have been se- Another loophole is that space is infinite but matter isriously challenged by observations. Yet the Level I mul- confined to a finite region around us–the historically pop-tiverse idea has been controversial (indeed, an assertion ular ”island universe” model. In a variant on this model,along these lines was one of the heresies for which the Vat- matter thins out on large scales in a fractal pattern. Inican had Giordano Bruno burned at the stake in 16003), both cases, almost all universes in the Level I multiverseso let us review the status of the two assumptions (infi- would be empty and dead. But recent observations of thenite space and “sufficiently uniform” distribution). three-dimensional galaxy distribution and the microwave How large is space? Observationally, the lower bound background have shown that the arrangement of matterhas grown dramatically (Figure 2) with no indication of gives way to dull uniformity on large scales, with no co-an upper bound. We all accept the existence of things herent structures larger than about 1024 meters. Assum-that we cannot see but could see if we moved or waited, ing that this pattern continues, space beyond our observ-like ships beyond the horizon. Objects beyond cosmic able universe teems with galaxies, stars and planets.horizon have similar status, since the observable universegrows by a light-year every year as light from furtheraway has time to reach us4 . If anything, the Level I mul- B. What are Level I parallel universes like?tiverse sounds trivially obvious. How could space not beinfinite? Is there a sign somewhere saying ”Space Ends The physics description of the world is traditionallyHere–Mind the Gap”? If so, what lies beyond it? In split into two parts: initial conditions and laws of physicsfact, Einstein’s theory of gravity calls this intuition into specifying how the initial conditions evolve. Observersquestion. Space could be finite if it has a convex curva- living in parallel universes at Level I observe the exactture or an unusual topology (that is, interconnectedness). same laws of physics as we do, but with different initialA spherical, doughnut-shaped or pretzel-shaped universe conditions than those in our Hubble volume. The cur-would have a limited volume and no edges. The cosmic rently favored theory is that the initial conditions (themicrowave background radiation allows sensitive tests of densities and motions of different types of matter earlysuch scenarios. So far, however, the evidence is against on) were created by quantum fluctuations during the in- flation epoch (see section 3). This quantum mechanism generates initial conditions that are for all practical pur- poses random, producing density fluctuations described3 Bruno’s ideas have since been elaborated by, e.g., Brundrit by what mathematicians call an ergodic random field. (1979), Garriga & Vilenkin (2001b) and Ellis (2002), all of whom Ergodic means that if you imagine generating an ensem- have thus far avoided the stake.4 If the cosmic expansion continues to accelerate (currently an ble of universes, each with its own random initial con- open question), the observable universe will eventually stop grow- ditions, then the probability distribution of outcomes in ing. a given volume is identical to the distribution that you
  • 4. 4get by sampling different volumes in a single universe. In the testable prediction that we should observe no oxygenother words, it means that everything that could in prin- here, and is therefore ruled out by observation.ciple have happened here did in fact happen somewhere As a more serious example, the Level I multiverseelse. framework is routinely used to rule out theories in mod- Inflation in fact generates all possible initial conditions ern cosmology, although this is rarely spelled out explic-with non-zero probability, the most likely ones being al- itly. For instance, cosmic microwave background (CMB)most uniform with fluctuations at the 10−5 level that observations have recently shown that space has almostare amplified by gravitational clustering to form galaxies, no curvature. Hot and cold spots in CMB maps have astars, planets and other structures. This means both that characteristic size that depends on the curvature of space,pretty much all imaginable matter configurations occur and the observed spots appear too large to be consis-in some Hubble volume far away, and also that we should tent with the previously popular “open universe” model.expect our own Hubble volume to be a fairly typical one However, the average spot size randomly varies slightly— at least typical among those that contain observers. from one Hubble volume to another, so it is important toA crude estimate suggests that the closest identical copy be statistically rigorous. When cosmologists say that the 29 91of you is about ∼ 1010 m away. About ∼ 1010 m away, open universe model is ruled out at 99.9% confidence,there should be a sphere of radius 100 light-years identical they really mean that if the open universe model wereto the one centered here, so all perceptions that we have true, then fewer than one out of every thousand Hubbleduring the next century will be identical to those of our volumes would show CMB spots as large as those we ob- 115counterparts over there. About ∼ 1010 m away, there serve — therefore the entire model with all its infinitelyshould be an entire Hubble volume identical to ours.5 many Hubble volumes is ruled out, even though we have This raises an interesting philosophical point that will of course only mapped the CMB in our own particularcome back and haunt us in Section V B: if there are Hubble volume.indeed many copies of “you” with identical past lives and The lesson to learn from this example is that multi-memories, you would not be able to compute your own verse theories can be tested and falsified, but only if theyfuture even if you had complete knowledge of the entire predict what the ensemble of parallel universes is andstate of the cosmos! The reason is that there is no way specify a probability distribution (or more generally whatfor you to determine which of these copies is “you” (they mathematicians call a measure) over it. As we will see inall feel that they are). Yet their lives will typically begin Section V B, this measure problem can be quite seriousto differ eventually, so the best you can do is predict and is still unsolved for some multiverse theories.probabilities for what you will experience from now on.This kills the traditional notion of determinism. II. LEVEL II: OTHER POST-INFLATION BUBBLES C. How a multiverse theory can be tested and falsified If you felt that the Level I multiverse was large and hard to stomach, try imagining an infinite set of distinct Is a multiverse theory one of metaphysics rather than ones (each symbolized by a bubble in Figure 1), some per-physics? As emphasized by Karl Popper, the distinc- haps with different dimensionality and different physicaltion between the two is whether the theory is empirically constants. This is what is predicted by most currentlytestable and falsifiable. Containing unobservable enti- popular models of inflation, and we will refer to it asties does clearly not per se make a theory non-testable. the Level II multiverse. These other domains are moreFor instance, a theory stating that there are 666 paral- than infinitely far away in the sense that you would neverlel universes, all of which are devoid of oxygen makes get there even if you traveled at the speed of light for- ever. The reason is that the space between our Level I multiverse and its neighbors is still undergoing inflation, which keeps stretching it out and creating more volume5 This is an extremely conservative estimate, simply counting all faster than you can travel through it. In contrast, you possible quantum states that a Hubble volume can have that are could travel to an arbitrarily distant Level I universe if no hotter than 108 K. 10115 is roughly the number of protons that the Pauli exclusion principle would allow you to pack into you were patient and the cosmic expansion decelerates.6 a Hubble volume at this temperature (our own Hubble volume contains only about 1080 protons). Each of these 10115 slots can 115 115 be either occupied or unoccupied, giving N = 210 ∼ 1010 possibilities, so the expected distance to the nearest identical 6 Astronomical evidence suggests that the cosmic expansion is 115 115 Hubble volume is N 1/3 ∼ 1010 Hubble radii ∼ 1010 meters. currently accelerating. If this acceleration continues, then even 29 Your nearest copy is likely to be much closer than 1010 meters, the level I parallel universes will remain forever separate, with the since the planet formation and evolutionary processes that have intervening space stretching faster than light can travel through tipped the odds in your favor are at work everywhere. There it. The jury is still out, however, with popular models predicting are probably at least 1020 habitable planets in our own Hubble that the universe will eventually stop accelerating and perhaps volume alone. even recollapse.
  • 5. 5 A. Evidence for Level II parallel universes tio mp /me ≈ 1836 and the cosmological constant, which appears to be about 10−123 in so-called Planck units. Inflation is an extension of the big bang theory and There are models where also such non-integer parame-ties up many of the loose ends of that theory, such as ters can vary from one one post-inflationary bubble towhy the universe is so big, so uniform and so flat. A another.8 In summary, the Level II multiverse is likelyrapid stretching of space long ago can explain all these to be more diverse than the Level I multiverse, contain-and other attributes in one fell swoop (see reviews by ing domains where not only the initial conditions differ,Linde 1994 and Guth & Kaiser 2005). Such stretching is but perhaps the dimensionality, the elementary particlespredicted by a wide class of theories of elementary par- and the physical constants differ as well.ticles, and all available evidence bears it out. Much of This is currently a very active research area. The pos-space is stretching and will continue doing so forever, but sibility of a string theory “landscape” (Bousso & Polchin-some regions of space stop stretching and form distinct ski 2000; Susskind 2003), where the above-mentionedbubbles, like gas pockets in a loaf of rising bread. In- potential has perhaps 10500 different minima, may of-finitely many such bubbles emerge (Figure 1, lower left, fer a specific realization of the Level II multiverse whichwith time increasing upwards). Each is an embryonic would in turn have four sub-levels of increasing diver-Level I multiverse: infinite in size7 and filled with matter sity: IId: different ways in which space can be compact-deposited by the energy field that drove inflation. Recent ified, which can allow both different effective dimension-cosmological measurements have confirmed two key pre- ality and different symmetries/elementary articles (corre-dictions of inflation: that space has negligible curvature sponding to different topology of the curled up extra di-and that the clumpiness in the cosmic matter distribution mensions). IIc: different “fluxes” (generalized magneticuse to be approximately scale invariant. fields) that stabilize the extra dimensions (this sublevel is where the largest number of choices enter, perhaps 10500 ). IIb: once these two choices have been made, there may be a handful of different minima in the effec- B. What are Level II parallel universes like? tive supergravity potential. IIa: the same minimum and effective laws of physics can be realized in a many differ- The prevailing view is that the physics we observe to- ent post-inflationary bubbles, each constituting a Level Iday is merely a low-energy limit of a much more general multiverse.theory that manifests itself at extremely high tempera- Before moving on, let us briefly comment on a fewtures. For example, this underlying fundamental theory closely related multiverse notions. First of all, if onemay be 10-dimensional, supersymmetric and involving a Level II multiverse can exist, eternally self-reproducing ingrand unification of the four fundamental forces of na- a fractal pattern, then there may well be infinitely manyture. A common feature in such theories is that the other Level II multiverses that are completely discon-potential energy of the field(s) relevant to inflation has nected. However, this variant appears to be untestable,many different minima (sometimes called “metastable since it would neither add any qualitatively differentvacuum states”), and ending up in different minima cor- worlds nor alter the probability distribution for theirresponds to different effective laws of physics for our properties. All possible initial initial conditions and sym-low-energy world. For instance, all but three spatial di- metry breakings are already realized within each one.mensions could be curled up (“compactified”) on a tiny An idea proposed by Tolman and Wheeler and recentlyscale, resulting in an effectively three-dimensional space elaborated by Steinhardt & Turok (2002) is that thelike ours, or fewer could curl up leaving a 5-dimensional (Level I) multiverse is cyclic, going through an infinitespace. Quantum fluctuations during inflation can there- series of Big Bangs. If it exists, the ensemble of such in-fore cause different post-inflation bubbles in the Level carnations would also form a multiverse, arguably withII multiverse to end up with different effective laws of a diversity similar to that of Level II.physics in different bubbles — say different dimension- An idea proposed by Smolin (1997) involves an ensem-ality or different types of elementary particles, like tworather than three generations of quarks. In addition to such discrete properties as dimensional-ity and particle content, our universe is characterized by 8 Although the fundamental equations of physics are the samea set of dimensionless numbers known as physical con- throughout the Level II multiverse, the approximate effectivestants. Examples include the electron/proton mass ra- equations governing the low-energy world that we observe will differ. For instance, moving from a three-dimensional to a four- dimensional (non-compactified) space changes the observed grav- itational force equation from an inverse square law to an inverse cube law. Likewise, breaking the underlying symmetries of parti-7 Surprisingly, it has been shown that inflation can produce an in- cle physics differently will change the lineup of elementary parti- finite Level I multiverse even in a bubble of finite spatial volume, cles and the effective equations that describe them. However, we thanks to an effect whereby the spatial directions of spacetime will reserve the terms “different equations” and “different laws of curve towards the (infinite) time direction (Bucher & Spergel physics” for the Level IV multiverse, where it is the fundamental 1999). rather than effective equations that change.
  • 6. 6 hotels, you could infer the existence of other hotel rooms to explain the coincidence. As a more pertinent example, consider the mass of the sun. The mass of a star determines its luminosity, and us- ing basic physics, one can compute that life as we know it on Earth is possible only if the sun’s mass falls into the narrow range between 1.6 × 1030kg and 2.4 × 1030kg. Otherwise Earth’s climate would be colder than that of present-day Mars or hotter than that of present-day Venus. The measured solar mass is M ∼ 2.0 × 1030 kg. At first glance, this apparent coincidence of the habitable and observed mass values appears to be a wild stroke of luck. Stellar masses run from 1029 to 1032 kg, so if the sun acquired its mass at random, it had only a small chance of falling into the habitable range. But just as in the ho- tel example, one can explain this apparent coincidence by postulating an ensemble (in this case, a number of plan- etary systems) and a selection effect (the fact that we must find ourselves living on a habitable planet). Such observer-related selection effects are referred to as “an- thropic” (Carted 1973), and although the “A-word” isFIG. 3: Why we should not be surprised to find ourselves notorious for triggering controversy, physicists broadlyliving in 3+1-dimensional spacetime. When the partial dif- agree that these selection effects cannot be neglectedferential equations of nature are elliptic or ultrahyperbolic, when testing fundamental theories. In this weak sense,physics has no predictive power for an observer. In the re-maining (hyperbolic) cases, n > 3 admits no stable atoms and the anthropic principle is not optional.n < 3 may lack sufficient complexity for observers (no grav-itational attraction, topological problems). From Tegmark What applies to hotel rooms and planetary systems(1997). applies to parallel universes. Most, if not all, of the attributes set by symmetry breaking appear to be fine- tuned. Changing their values by modest amounts wouldble similar in diversity to that of Level II, but mutating have resulted in a qualitatively different universe–one inand sprouting new universes through black holes rather which we probably would not exist. If protons werethan during inflation. This predicts a form of a natu- 0.2% heavier, they could decay into neutrons, destabiliz-ral selection favoring universes with maximal black hole ing atoms. If the electromagnetic force were 4% weaker,production. there would be no hydrogen and no normal stars. If the In braneworld scenarios, another 3-dimensional world weak interaction were much weaker, hydrogen would notcould be quite literally parallel to ours, merely offset in a exist; if it were much stronger, supernovae would fail tohigher dimension. However, it is unclear whether such a seed interstellar space with heavy elements. If the cos-world (“brane”) deserves be be called a parallel universe mological constant were much larger, the universe wouldseparate from our own, since we may be able to interact have blown itself apart before galaxies could form. In-with it gravitationally much as we do with dark matter. deed, most if not all the parameters affecting low-energy physics appear fine-tuned at some level, in the sense that changing them by modest amounts results in a qualita- C. Fine-tuning and selection effects tively different universe. Although we cannot interact with other Level II par- Although the degree of fine-tuning is still debated (asallel universes, cosmologists can infer their presence in- exemplified in the rest of this book; see Barrow & Tiplerdirectly, because their existence can account for unex- 1986, Tegmark 1998 & Hogan (2000) for more technicalplained coincidences in our universe. To give an analogy, reviews), these examples suggest the existence of parallelsuppose you check into a hotel, are assigned room 1967 universes with other values of some physical constants.and note that this is the year you were born. What a The existence of a Level II multiverse implies that physi-coincidence, you say. After a moment of reflection, how- cists will never be able to determine the values of allever, you conclude that this is not so surprising after all. physical constants from first principles. Rather, theyThe hotel has hundreds of rooms, and you would not have will merely compute probability distributions for whatbeen having these thoughts in the first place if you had they should expect to find, taking selection effects intobeen assigned one with a number that meant nothing to account. The result should be as generic as is consistentyou. The lesson is that even if you knew nothing about with our existence.
  • 7. 7FIG. 4: Hints of fine-tuning for the parameters α and αswhich determine the strengths of the electromagnetic forceand the strong nuclear force, respectively (from Tegmark1997). The observed values (α, αs ) ≈ (1/137, 0.1) are indi-cated with a filled square. Grand unified theories rule outeverything except the narrow strip between the two verticallines, and deuterium becomes unstable below the horizontal FIG. 5: Difference between Level I and Level III. Whereasline. In the narrow shaded region to the very left, electro- Level I parallel universes are far away in space, those of Levelmagnetism is weaker than gravity and therefore irrelevant. III are even right here, with quantum events causing classical reality to split and diverge into parallel storylines. Yet Level III adds no new storylines beyond levels 1 or 2. III. LEVEL III: THE MANY WORLDS OF QUANTUM PHYSICS function. According to the Schr¨dinger equation, this o There may be a third type of parallel worlds that are state evolves over time in a fashion that mathematiciansnot far away but in a sense right here. If the fundamental term “unitary”, meaning that the wave function rotatesequations of physics are what mathematicians call uni- in an abstract infinite-dimensional space called Hilberttary, as they so far appear to be, then the universe keeps space. Although quantum mechanics is often describedbranching into parallel universes as in the cartoon (Fig- as inherently random and uncertain, the wave functionure 5, bottom): whenever a quantum event appears to evolves in a deterministic way. There is nothing randomhave a random outcome, all outcomes in fact occur, one or uncertain about each branch. This is the Level III multiverse. Al- The sticky part is how to connect this wave func-though more debated and controversial than Level I and tion with what we observe. Many legitimate wave func-Level II, we will see that, surprisingly, this level adds no tions correspond to counterintuitive situations, such asnew types of universes. a cat being dead and alive at the same time in a so- called superposition. In the 1920s physicists explained away this weirdness by postulating that the wave func- A. The quantum conundrum tion “collapse” into some definite classical outcome when- ever someone made an observation. This add-on had In the early 20th century the theory of quantum me- the virtue of explaining observations, but it turned anchanics revolutionized physics by explaining the atomic elegant, unitary theory into a kludgy, nonunitary one,realm, which does not abide by the classical rules of New- since there was no equation specifying when or how thistonian mechanics. Despite the obvious successes of the collapse occurred. The intrinsic randomness commonlytheory, a heated debate rages about what it really means. ascribed to quantum mechanics is the result of this pos-The theory specifies the state of the universe not in classi- tulate, triggering Einstein’s objection that “God doesn’tcal terms, such as the positions and velocities of all parti- play dice”.cles, but in terms of a mathematical object called a wave Over the years many physicists have abandoned this
  • 8. 8view in favor of one developed in 1957 by Princeton collapse while preserving unitarity–prevents them fromgraduate student Hugh Everett III. He showed that the seeing Level III parallel copies of themselves.collapse postulate is unnecessary. Unadulterated quan- Whenever observers are asked a question, make a snaptum theory does not, in fact, pose any contradictions. decision and give an answer, quantum effects in theirAlthough it predicts that one classical reality gradu- brains lead to a superposition of outcomes, such as “Con-ally splits into superpositions of many such realities, ob- tinue reading the article” and “Put down the article”.servers subjectively experience this splitting merely as a From the bird perspective, the act of making a decisionslight randomness (Figure 5), with probabilities in exact causes a person to split into multiple copies: one whoagreement with those from the old collapse postulate (de keeps on reading and one who doesn’t. From their frogWitt 2003). This superposition of classical worlds is the perspective, however, each of these alter egos is unawareLevel III multiverse. of the others and notices the branching merely as a slight randomness: a certain probability of continuing to read or not. B. What are Level III parallel universes like? As strange as this may sound, the exact same situation occurs even in the Level I multiverse. You have evidently Everett’s many-worlds interpretation has been bog- decided to keep on reading the article, but one of yourgling minds inside and outside physics for more than alter egos in a distant galaxy put down the magazine afterfour decades. But the theory becomes easier to grasp the first paragraph. The only difference between Level Iwhen one distinguishes between two ways of viewing a and Level III is where your doppelga¨ngers reside. In aphysical theory: the outside view of a physicist study- Level I they live elsewhere in good old three-dimensionaling its mathematical equations, like a bird surveying a space. In Level III they live on another quantum branchlandscape from high above it, and the inside view of an in infinite-dimensional Hilbert space (Figure 5).observer living in the world described by the equations,like a frog living in the landscape surveyed by the bird.9 . From the bird perspective, the Level III multiverse C. Level III parallel universes: evidence &is simple. There is only one wave function. It evolves implicationssmoothly and deterministically over time without anykind of splitting or parallelism. The abstract quantum The existence of Level III depends on one crucial as-world described by this evolving wave function contains sumption: that the time evolution of the wave functionwithin it a vast number of parallel classical story lines, is unitary. So far experimenters have encountered no de-continuously splitting and merging, as well as a number partures from unitarity. In the past few decades theyof quantum phenomena that lack a classical description. have confirmed unitarity for ever larger systems, includ-From their frog perspective, observers perceive only a ing carbon 60 buckyball molecules and kilometer-long op-tiny fraction of this full reality. They can view their own tical fibers. On the theoretical side, the case for unitarityLevel I universe, but a process called decoherence (Zeh has been bolstered by the discovery of decoherence (see1970; Giulini et al. 1996) — which mimics wave function Tegmark & Wheeler 2001 for a popular review). Some theorists who work on quantum gravity have questioned unitarity; one concern is that evaporating black holes might destroy information, which would be a nonuni-9 Indeed, the standard mental picture of what the physical world tary process. But a recent breakthrough in string theory is corresponds to a third intermediate viewpoint that could be known as AdS/CFT correspondence suggests that even termed the consensus view. From your subjectively perceived frog perspective, the world turns upside down when you stand quantum gravity is unitary. If so, black holes do not on your head and disappears when you close your eyes, yet you destroy information but merely transmit it elsewhere. subconsciously interpret your sensory inputs as though there is If physics is unitary, then the standard picture of how an external reality that is independent of your orientation, your quantum fluctuations operated early in the big bang must location and your state of mind. It is striking that although change. These fluctuations did not generate initial con- this third view involves both censorship (like rejecting dreams), interpolation (as between eye-blinks) and extrapolation (say at- ditions at random. Rather they generated a quantum tributing existence to unseen cities) of your inside view, inde- superposition of all possible initial conditions, which co- pendent observers nonetheless appear to share this consensus existed simultaneously. Decoherence then caused these view. Although the inside view looks black-and-white to a cat, initial conditions to behave classically in separate quan- iridescent to a bird seeing four primary colors, and still more dif- tum branches. Here is the crucial point: the distribution ferent to bee a seeing polarized light, a bat using sonar, a blind person with keener touch and hearing, or the latest overpriced of outcomes on different quantum branches in a given robotic vacuum cleaner, all agree on whether the door is open. Hubble volume (Level III) is identical to the distribution The key current challenge in physics is deriving this semiclas- of outcomes in different Hubble volumes within a single sical consensus view from the fundamental equations specifying quantum branch (Level I). This property of the quantum the bird perspective. In my opinion, this means that although understanding the detailed nature of human consciousness is an fluctuations is known in statistical mechanics as ergodic- important challenge in its own right, it is not necessary for a ity. fundamental theory of physics. The same reasoning applies to Level II. The process
  • 9. 9of symmetry breaking did not produce a unique outcomebut rather a superposition of all outcomes, which rapidly h g 0 ? G g 0went their separate ways. So if physical constants, space- Quantumtime dimensionality and so on can vary among parallel g General Field 0 g Relativity h Theoryquantum branches at Level III, then they will also vary g 8 G 0 camong parallel universes at Level II. Nonrelativistic In other words, the Level III multiverse adds nothing Special Quantum Nuclear Particle Relativity Physics phenomenonew beyond Level I and Level II, just more indistinguish- Mechanicsable copies of the same universes–the same old story lines h g 0playing out again and again in other quantum branches. Electro- Classical AtomicThe passionate debate about Everett’s theory therefore magnetism Mechanics Physicsseems to be ending in a grand anticlimax, with the dis-covery of less controversial multiverses (Levels I and II) Solid State Statistical Chemistrythat are equally large. Physics Mechanics Needless to say, the implications are profound, andphysicists are only beginning to explore them. For in- Electric Mechanics Hydro- Thermo- propertiesstance, consider the ramifications of the answer to a long- of solids of solids dynamics dynamicsstanding question: Does the number of universes expo- Astrophysicsnentially increase over time? The surprising answer isno. From the bird perspective, there is of course only Electrical Mechanical Earth Evolut Engineering Engineering Scienceone quantum universe. From the frog perspective, what matters is the number Computerof universes that are distinguishable at a given instant– Science Biologythat is, the number of noticeably different Hubble vol-umes. Imagine moving planets to random new locations, Psychology Medicinimagine having married someone else, and so on. At thequantum level, there are 10 to the 10118 universes with Sociologytemperatures below 108 kelvins. That is a vast number,but a finite one. FIG. 6: Theories can be crudely organized into a family tree From the frog perspective, the evolution of the wave where each might, at least in principle, be derivable from morefunction corresponds to a never-ending sliding from one fundamental ones above it. For example, classical mechanicsof these 10 to the 10118 states to another. Now you are can be obtained from special relativity in the approximationin universe A, the one in which you are reading this sen- that the speed of light c is infinite. Most of the arrows aretence. Now you are in universe B, the one in which you less well understood. All these theories have two components:are reading this other sentence. Put differently, universe mathematical equations and words that explain how they areB has an observer identical to one in universe A, except connected to what we observe. At each level in the hierarchy of theories, new words (e.g., protons, atoms, cells, organisms,with an extra instant of memories. All possible states cultures) are introduced because they are convenient, cap-exist at every instant, so the passage of time may be turing the essence of what is going on without recourse toin the eye of the beholder — an idea explored in Greg the more fundamental theory above it. It is important to re-Egan’s 1994 science-fiction novel Permutation City and member, however, that it is we humans who introduce thesedeveloped by physicist David Deutsch of the University of concepts and the words for them: in principle, everythingOxford, independent physicist Julian Barbour, and oth- could have been derived from the fundamental theory at theers. The multiverse framework may thus prove essential top of the tree, although such an extreme reductionist ap-to understanding the nature of time. proach would of course be useless in practice. Crudely speak- ing, the ratio of equations to words decreases as we move down the tree, dropping near zero for highly applied fields such as medicine and sociology. In contrast, theories near the top D. Two world views are highly mathematical, and physicists are still struggling to understand the concepts, if any, in terms of which we can The debate over how classical mechanics emerges from understand them. The Holy Grail of physics is to find whatquantum mechanics continues, and the decoherence dis- is jocularly referred to as a “Theory of Everything”, or TOE,covery has shown that there is a lot more to it than just from which all else can be derived. If such a theory existsletting Planck’s constant h shrink to zero. Yet as Fig- ¯ at all, it should replace the big question mark at the top of the theory tree. Everybody knows that something is missingure 6 illustrates, this is just a small piece of a larger puz- here, since we lack a consistent theory unifying gravity withzle. Indeed, the endless debate over the interpretation of quantum mechanics.quantum mechanics — and even the broader issue of par-allel universes — is in a sense the tip of an iceberg. In theSci-Fi spoof “Hitchhiker’s Guide to the Galaxy”, the an-swer is discovered to be “42”, and the hard part is finding
  • 10. 10the real question. Questions about parallel universes may equations are equally real. This is the Level IV multi-seem to be just about as deep as queries about reality can verse. First we need to digest two other ideas, however:get. Yet there is a still deeper underlying question: there the concept of a mathematical structure, and the notionare two tenable but diametrically opposed paradigms re- that the physical world may be one.garding physical reality and the status of mathematics,a dichotomy that arguably goes as far back as Plato andAristotle, and the question is which one is correct. A. What is a mathematical structure? • ARISTOTELIAN PARADIGM: The subjec- tively perceived frog perspective is physically real, Many of us think of mathematics as a bag of tricks that and the bird perspective and all its mathematical we learned in school for manipulating numbers. Yet most language is merely a useful approximation. mathematicians have a very different view of their field. They study more abstract objects such as functions, sets, • PLATONIC PARADIGM: The bird perspec- spaces and operators and try to prove theorems about tive (the mathematical structure) is physically real, the relations between them. Indeed, some modern math- and the frog perspective and all the human lan- ematics papers are so abstract that the only numbers guage we use to describe it is merely a useful ap- you will find in them are the page numbers! What does proximation for describing our subjective percep- a dodecahedron have in common with a set of complex tions. numbers? Despite the plethora of mathematical struc-What is more basic — the frog perspective or the bird tures with intimidating names like orbifolds and Killingperspective? What is more basic — human language fields, a striking underlying unity has emerged in theor mathematical language? Your answer will determine last century: all mathematical structures are just specialhow you feel about parallel universes. If you prefer the cases of one and the same thing: so-called formal sys-Platonic paradigm, you should find multiverses natu- tems. A formal system consists of abstract symbols andral, since our feeling that say the Level III multiverse rules for manipulating them, specifying how new stringsis “weird” merely reflects that the frog and bird perspec- of symbols referred to as theorems can be derived fromtives are extremely different. We break the symmetry by given ones referred to as axioms. This historical devel-calling the latter weird because we were all indoctrinated opment represented a form of deconstructionism, since itwith the Aristotelian paradigm as children, long before stripped away all meaning and interpretation that hadwe even heard of mathematics - the Platonic view is an traditionally been given to mathematical structures andacquired taste! distilled out only the abstract relations capturing their In the second (Platonic) case, all of physics is ulti- very essence. As a result, computers can now prove the-mately a mathematics problem, since an infinitely intel- orems about geometry without having any physical intu-ligent mathematician given the fundamental equations of ition whatsoever about what space is like.the cosmos could in principle compute the frog perspec- Figure 7 shows some of the most basic mathematicaltive, i.e., compute what self-aware observers the universe structures and their interrelations. Although this fam-would contain, what they would perceive, and what lan- ily tree probably extends indefinitely, it illustrates thatguage they would invent to describe their perceptions there is nothing fuzzy about mathematical one another. In other words, there is a “Theory of They are “out there” in the sense that mathematiciansEverything” (TOE) at the top of the tree in Figure 6 discover them rather than create them, and that contem-whose axioms are purely mathematical, since postulates plative alien civilizations would find the same structuresin English regarding interpretation would be derivable (a theorem is true regardless of whether it is proven byand thus redundant. In the Aristotelian paradigm, on a human, a computer or an alien).the other hand, there can never be a TOE, since one isultimately just explaining certain verbal statements byother verbal statements — this is known as the infinite B. The possibility that the physical world is aregress problem (Nozick 1981). mathematical structure Let us now digest the idea that physical world (specif- IV. LEVEL IV: OTHER MATHEMATICAL ically, the Level III multiverse) is a mathematical struc- STRUCTURES ture. Although traditionally taken for granted by many theoretical physicists, this is a deep and far-reaching no- Suppose you buy the Platonist paradigm and believe tion. It means that mathematical equations describe notthat there really is a TOE at the top of Figure 6 — and merely some limited aspects of the physical world, but allthat we simply have not found the correct equations yet. aspects of it. It means that there is some mathematicalThen an embarrassing question remains, as emphasized structure that is what mathematicians call isomorphicby John Archibald Wheeler: Why these particular equa- (and hence equivalent) to our physical world, with eachtions, not others? Let us now explore the idea of mathe- physical entity having a unique counterpart in the math-matical democracy, whereby universes governed by other ematical structure and vice versa. Let us consider some
  • 11. 11 GENERAL RELATIVITY QUANTUM FIELD THEORY LORENTZ SU(2) (3+1-dimensional pseudo-Riemannian manifolds with tensor fields obeying PDEs of, say, (Operator-valued fields on R 4obeying certain Lorentz- invariant PDEs and commutation relationships, structure would not correspond to a single frame of it, GROUP Specialize Einstein-Maxwell theory with perfect fluid component) acting on an abstract Hilbert space) but to the entire videotape. LIE LIE DIFFERENTIAL MANIFOLDS WITH METRIC Given a mathematical structure, we will say that it has GROUPS ALGEBRAS OPERATORS TENSOR FIELDS MANIFOLDS physical existence if any self-aware substructure (SAS) ALGEBRAS LINEAR OPERATORS TENSOR SPACES REAL MANIFOLDS COMPLEX MANIFOLDS HILBERT SPACES DISTRI- BUTIONS within it subjectively, from its frog perspective, perceives TRIPLE Define linear vector mappings Define linear Define charts and atlases Add inner product Define linear functionals itself as living in a physically real world. What would, FIELDS Add 4th functions of n vectors Rn Cn BANACH SPACES REAL FUNCTIONS mathematically, such an SAS be like? In the classical binary op. VECTOR etc. DOUBLE SPACES REAL Define . =, + & COMPLEX COMPLEX physics example above, an SAS such as you would be Add new ckass Add ordering, FIELDS (vectors), define vector sum & scalar product least upper bound NUMBERS property for pairs NUMBERS Add FUNCTIONS a tube through spacetime, a thick version of what Ein- Add 3rd norm & binary operation, etc. ABELIAN Define =, + & . Dedekind cuts for complete- ness MEASURABLE SPACES stein referred to as a world-line. The location of the tube FIELDS RATIONAL ABELIAN Add commutativity NUMBERS METRIC SPACES Add measure would specify your position in space at different times. Define =, + & . for pairs GROUPS FIELDS Add multiplicative INTEGERS Add metric Within the tube, the fields would exhibit certain complex Add commutativity GROUPS RINGS inverse Define =, + & . for negative numbers TOPOLOGICAL SPACES behavior, corresponding to storing and processing infor- Add inverse Specialize Add commutativity and 2nd assoc. & distrib. binary operation NATURAL NUMBERS Add open set ABSTRACT mation about the field-values in the surroundings, and SEMI- axioms DODECA- GROUPS Add number theory symbols & axioms GEOMETRIES at each position along the tube, these processes would Add symbols HEDRON GROUP Add = & associative binary operation with identity LOWER and axioms for PREDICATE =, in, union SETS Add RELATIONS give rise to the familiar but mysterious sensation of self- CALCULUS and intersection cartesian Add quantifiers products awareness.From its frog perspective, the SAS would per- BOOLEAN ceive this one-dimensional string of perceptions along the ALGEBRA Add symbols & axioms for brackets, not, and, etc. tube as passage of time. MODELS FORMAL Although our example illustrates the idea of how our SYSTEMS physical world can be a mathematical structure, this par- ticular mathematical structure (fields in four-dimensionalFIG. 7: Relationships between various basic mathematical space) is now known to be the wrong one. After real-structures (Tegmark 1998). The arrows generally indicate izing that spacetime could be curved, Einstein doggedlyaddition of new symbols and/or axioms. Arrows that meet searched for a so-called unified field theory where the uni-indicate the combination of structures — for instance, an al- verse was what mathematicians call a 3+1-dimensionalgebra is a vector space that is also a ring, and a Lie group pseudo-Riemannian manifold with tensor fields (top cen-is a group that is also a manifold. The full tree is probably ter in Figure 7), but this failed to account for the ob-infinite in extent — the figure shows merely a small sample served behavior of atoms. According to quantum fieldnear the bottom. theory, the modern synthesis of special relativity theory and quantum theory, the universe (in this case the Level III multiverse) is a mathematical structure known as anexamples. algebra of operator-valued fields (top right in Figure 7). Here the question of what constitutes an SAS is more A century ago, when classical physics still reigned subtle (Tegmark 2000). However, this fails to describesupreme, many scientists believed that physical space black hole evaporation, the first instance of the Big Bangwas isomorphic to the mathematical structure known as and other quantum gravity phenomena, so the true math-R3 : three-dimensional Euclidean space. Moreover, some ematical structure isomorphic to our universe, if it exists,thought that all forms of matter in the universe cor- has not yet been found.responded to various classical fields: the electric field,the magnetic field and perhaps a few undiscovered ones,mathematically corresponding to functions on R3 (ahandful of numbers at each point in space). In this C. Mathematical democracyview (later proven incorrect), dense clumps of matter likeatoms were simply regions in space where some fields Now suppose that our physical world really is a math-were strong (where some numbers were large). These ematical structure, and that you are an SAS within it.fields evolved deterministically over time according to This means that in the Mathematics tree of Figure 7, onesome partial differential equations, and observers per- of the boxes is our universe. (The full tree is probablyceived this as things moving around and events taking infinite in extent, so our particular box is not one of theplace. Could, then, fields in three-dimensional space few boxes from the bottom of the tree that are shown.)be the mathematical structure corresponding to the uni- In other words, this particular mathematical structureverse? No, since a mathematical structure cannot change enjoys not only mathematical existence, but physical ex-— it is an abstract, immutable entity existing outside of istence as well. What about all the other boxes in thespace and time. Our familiar frog perspective of a three- tree? Do they too enjoy physical existence? If not, theredimensional space where events unfold is equivalent, from would be a fundamental, unexplained ontological asym-the bird perspective, to a four-dimensional spacetime metry built into the very heart of reality, splitting mathe-where all of history is contained, so the mathematical matical structures into two classes: those with and with-structure would be fields in four-dimensional space. In out physical existence. As a way out of this philosophicalother words, if history were a movie, the mathematical conundrum, I have suggested (Tegmark 1998) that com-
  • 12. 12plete mathematical democracy holds: that mathemat- This argument also makes assumption 2 more appeal-ical existence and physical existence are equivalent, so ing, since it implies that any conceivable parallel uni-that all mathematical structures exist physically as well. verse theory can be described at Level IV. The Level IVThis is the Level IV multiverse. It can be viewed as a multiverse, termed the “ultimate Ensemble theory” inform of radical Platonism, asserting that the mathemat- Tegmark (1997) since it subsumes all other ensembles,ical structures in Plato’s realm of ideas, the Mindscape therefore brings closure to the hierarchy of multiverses,of Rucker (1982), exist “out there” in a physical sense and there cannot be say a Level V. Considering an en-(Davies 1993), casting the so-called modal realism the- semble of mathematical structures does not add anythingory of David Lewis (1986) in mathematical terms akin to new, since this is still just another mathematical struc-what Barrow (1991; 1992) refers to as “π in the sky”. If ture. What about the frequently discussed notion thatthis theory is correct, then since it has no free parame- the universe is a computer simulation? This idea occursters, all properties of all parallel universes (including the frequently in science fiction and has been substantiallysubjective perceptions of SASs in them) could in princi- elaborated (e.g., Schmidthuber 1997; Wolfram 2002).ple be derived by an infinitely intelligent mathematician. The information content (memory state) of a digital com- puter is a string of bits, say “1001011100111001...” of great but finite length, equivalent to some large but fi- D. Evidence for a Level IV multiverse nite integer n written in binary. The information pro- cessing of a computer is a deterministic rule for chang- We have described the four levels of parallel universes ing each memory state into another (applied over andin order of increasing speculativeness, so why should we over again), so mathematically, it is simply a function fbelieve in Level IV? Logically, it rests on two separate mapping the integers onto themselves that gets iterated:assumptions: n → f (n) → f (f (n)) → .... In other words, even the most sophisticated computer simulation is just yet an- • Assumption 1: That the physical world (specif- other special case of a mathematical structure, and is al- ically our level III multiverse) is a mathematical ready included in the Level IV multiverse. (Incidentally, structure iterating continuous functions rather than integer-valued ones can give rise to fractals.) • Assumption 2: Mathematical democracy: that A second argument for assumption 2 is that if two en- all mathematical structures exist “out there” in the tities are isomorphic, then there is no meaningful sense same sense in which they are not one and the same (Cohen 2003). In a famous essay, Wigner (1967) argued that “the This implies assumption 2 when the entities in questionenormous usefulness of mathematics in the natural sci- are a physical universe and a mathematical structure de-ences is something bordering on the mysterious”, and scribing it, respectively. To avoid this conclusion thatthat “there is no rational explanation for it”. This argu- mathematical and physical existence are equivalent, onement can be taken as support for assumption 1: here the would need to argue that our universe is somehow madeutility of mathematics for describing the physical world of stuff perfectly described by a mathematical structure,is a natural consequence of the fact that the latter is but which also has other properties that are not describeda mathematical structure, and we are simply uncover- by it. However, this violates assumption 1 and impliesing this bit by bit. The various approximations that either that it is isomorphic to a more complicated math-constitute our current physics theories are successful be- ematical structure or that it is not mathematical at all.cause simple mathematical structures can provide good The latter would be make Karl Popper turn in his grave,approximations of how a SAS will perceive more complex since those additional bells and whistles that make themathematical structures. In other words, our successful universe non-mathematical by definition have no observ-theories are not mathematics approximating physics, but able effects whatsoever.mathematics approximating mathematics. Wigner’s ob- Another appealing feature of assumption 2 is that itservation is unlikely to be based on fluke coincidences, provides the only answer so far to Wheeler’s question:since far more mathematical regularity in nature has been Why these particular equations, not others? Having uni-discovered in the decades since he made it, including the verses dance to the tune of all possible equations alsostandard model of particle physics. resolves the fine-tuning problem of Section II C once and A second argument supporting assumption 1 is that for all, even at the fundamental equation level: althoughabstract mathematics is so general that any TOE that is many if not most mathematical structures are likely todefinable in purely formal terms (independent of vague be dead and devoid of SASs, failing to provide the com-human terminology) is also a mathematical structure. plexity, stability and predictability that SASs require, weFor instance, a TOE involving a set of different types of course expect to find with 100% probability that weof entities (denoted by words, say) and relations between inhabit a mathematical structure capable of supportingthem (denoted by additional words) is nothing but what life. Because of this selection effect, the answer to themathematicians call a set-theoretical model, and one can question “what is it that breathes fire into the equationsgenerally find a formal system that it is a model of. and makes a universe for them to describe?” (Hawking
  • 13. 131993) would then be “you, the SAS”. A. Future prospects There are ample future prospects for testing and per- E. What are Level IV parallel universes like? haps ruling out these multiverse theories. In the com- ing decade, dramatically improved cosmological measure- The way we use, test and potentially rule out any the- ments of the microwave background radiation, the large-ory is to compute probability distributions for our future scale matter distribution, etc., will test Level I by fur-perceptions given our past perceptions and to compare ther constraining the curvature and topology of space andthese predictions with our observed outcome. In a mul- will test level II by providing stringent tests of inflation.tiverse theory, there is typically more than one SAS that Progress in both astrophysics and high-energy physicshas experienced a past life identical to yours, so there is should also clarify the extent to which various physicalno way to determine which one is you. To make predic- constants are fine-tuned, thereby weakening or strength-tions, you therefore have to compute what fractions of ening the case for Level II. If the current world-wide effortthem will perceive what in the future, which leads to the to build quantum computers succeeds, it will provide fur-following predictions: ther evidence for Level III, since they would, in essence, be exploiting the parallelism of the Level III multiverse • Prediction 1: The mathematical structure de- for parallel computation (Deutsch 1997). Conversely, ex- scribing our world is the most generic one that is perimental evidence of unitarity violation would rule out consistent with our observations. Level III. Finally, success or failure in the grand challenge of modern physics, unifying general relativity and quan- • Prediction 2: Our future observations are the tum field theory, will shed more light on Level IV. Either most generic ones that are consistent with our past we will eventually find a mathematical structure match- observations. ing our universe, or we will bump up against a limit to the unreasonable effectiveness of mathematics and have • Prediction 3: Our past observations are the most to abandon Level IV. generic ones that are consistent with our existence.We will return to the problem of what “generic” means B. The measure problemin Section V B (the measure problem). However, onestriking feature of mathematical structures, discussed in There are also interesting theoretical issues to resolvedetail in Tegmark (1997), is that the sort of symmetry within the multiverse theories, first and foremost theand invariance properties that are responsible for the sim- measure problem.plicity and orderliness of our universe tend to be generic, As multiverse theories gain credence, the sticky issue ofmore the rule than the exception — mathematical struc- how to compute probabilities in physics is growing from atures tend to have them by default, and complicated ad- minor nuisance into a major embarrassment. If there areditional axioms etc. must be added to make them go indeed many identical copies of you, the traditional no-away. In other words, because of both this and selec- tion of determinism evaporates. You could not computetion effects, we should not necessarily expect life in the your own future even if you had complete knowledge ofLevel IV multiverse to be a disordered mess. the entire state of the multiverse, because there is no way for you to determine which of these copies is you (they all feel they are). All you can predict, therefore, are prob- V. DISCUSSION abilities for what you would observe. If an outcome has a probability of, say, 50 percent, it means that half the We have seen that scientific theories of parallel uni- observers observe that outcome.verses form a four-level hierarchy, in which universes be- Unfortunately, it is not an easy task to compute whatcome progressively more different from ours. They might fraction of the infinitely many observers perceive what.have different initial conditions (Level I), different effec- The answer depends on the order in which you counttive physical laws, constants and particles (Level II), or them. By analogy, the fraction of the integers that aredifferent fundamental physical laws (Level IV). It is ironic even is 50 percent if you order them numerically (1, 2,that Level III is the one that has drawn the most fire in 3, 4, ...) but approaches 100 percent if you sort themthe past decades, because it is the only one that adds no digit by digit, the way your word processor would (1, 10,qualitatively new types of universes. 100, 1,000, ...). When observers reside in disconnected Whereas the Level I universes join seemlessly, there are universes, there is no obviously natural way in which toclear demarcations between those within levels II and III order them. Instead one must sample from the differ-caused by inflating space and decoherence, respectively. ent universes with some statistical weights referred to byThe level IV universes are completely disconnected and mathematicians as a “measure”.need to be considered together only for predicting your This problem crops up in a mild and treatable mannerfuture, since “you” may exist in more than one of them. at Level I, becomes severe at Level II (see Tegmark 2004
  • 14. 14for a detailed review), has caused much debate at Level entire set can be generated by quite a trivial computerIII (de Witt 2003, Mukhanov 2005), and is horrendous at program, whereas a single number can be hugely long.Level IV. At Level II, for instance, Linde, Vilenkin and Therefore, the whole set is actually simpler.others have published predictions for the probability dis- Similarly, the set of all solutions to Einstein’s fieldtributions of various cosmological parameters. They have equations is simpler than a specific solution. The for-argued that different parallel universes that have inflated mer is described by a few equations, whereas the latterby different amounts should be given statistical weights requires the specification of vast amounts of initial dataproportional to their volume (e.g., Garriga & Vilenkin on some hypersurface. The lesson is that complexity in-2001a). On the other hand, any mathematician will tell creases when we restrict our attention to one particularyou that 2 × ∞ = ∞, so there is no objective sense in element in an ensemble, thereby losing the symmetry andwhich an infinite universe that has expanded by a factor simplicity that were inherent in the totality of all the el-of two has gotten larger. Moreover, a finite universe with ements taken together.the topology of a torus is equivalent to a perfectly peri- In this sense, the higher-level multiverses are simpler.odic universe with infinite volume, both from the mathe- Going from our universe to the Level I multiverse elimi-matical bird perspective and from the frog perspective of nates the need to specify initial conditions, upgrading toan observer within it. So why should its infinitely smaller Level II eliminates the need to specify physical constants,volume give it zero statistical weight? After all, even in and the Level IV multiverse eliminates the need to spec-the Level I multiverse, Hubble volumes start repeating ify anything at all. The opulence of complexity is all in(albeit in a random order, not periodically) after about the subjective perceptions of observers (Tegmark 1996)10 to the 10118 meters. — the frog perspective. From the bird perspective, the If you think that is bad, consider the problem of assign- multiverse could hardly be any statistical weights to different mathematical struc- The complaint about weirdness is aesthetic rather thantures at Level IV. The fact that our universe seems rel- scientific, and it really makes sense only in the Aris-atively simple has led many people to suggest that the totelian worldview. Yet what did we expect? Whencorrect measure somehow involves complexity. we ask a profound question about the nature of real- ity, do we not expect an answer that sounds strange? Evolution provided us with intuition for the everyday C. The pros and cons of parallel universes physics that had survival value for our distant ancestors, so whenever we venture beyond the everyday world, we So should you believe in parallel universes? We have should expect it to seem bizarre. Thanks to clever in-seen that this is not a yes/no question — rather, the most ventions, we have glimpsed slightly beyond the frog per-interesting issue is whether there are 0, 1, 2, 3 or 4 levels spective of our normal inside view, and sure enough, weof multiverses. Figure 1 summarizes the evidence that have encountered bizarre phenomena whenever depart-we have discussed for the different levels. The principal ing from human scales in any way: at high speeds (timearguments against them are that they are wasteful and slows down), on small scales (quantum particles can bethat they are weird. at several places at once), on large scales (black holes), The wastefulness argument is that multiverse theories at low temperatures (liquid Helium can flow upward), atare vulnerable to Occam’s razor because they postulate high temperatures (colliding particles can change iden-the existence of other worlds that we can never observe. tity), etc..Why should nature be so wasteful and indulge in such A common feature of all four multiverse levels is thatopulence as an infinity of different worlds? Yet this ar- the simplest and arguably most elegant theory involvesgument can be turned around to argue for a multiverse. parallel universes by default. To deny the existence ofWhat precisely would nature be wasting? Certainly not those universes, one needs to complicate the theory byspace, mass or atoms–the uncontroversial Level I multi- adding experimentally unsupported processes and ad hocverse already contains an infinite amount of all three, so postulates: finite space, wave function collapse, ontologi-who cares if nature wastes some more? The real issue cal asymmetry, etc. Our judgment therefore comes downhere is the apparent reduction in simplicity. A skeptic to which we find more wasteful and inelegant: manyworries about all the information necessary to specify all worlds or many words. Perhaps we will gradually getthose unseen worlds. more used to the weird ways of our cosmos, and even But an entire ensemble is often much simpler than one find its strangeness to be part of its charm.of its members. This principle can be stated more for-mally using the notion of algorithmic information con- Acknowledgements: The author wishes to thanktent. The algorithmic information content in a number Anthony Aguirre, Aaron Classens, Marius Cohen, Angel-is, roughly speaking, the length of the shortest computer ica de Oliveira-Costa, Alan Guth, Shamit Kachru, An-program that will produce that number as output. For drei Linde, George Musser, David Raub, Martin Rees,example, consider the set of all integers. Which is sim- Harold Shapiro, Alex Vilenkin and Frank Wilczek forpler, the whole set or just one number? Naively, you stimulating discussions and George Musser for exten-might think that a single number is simpler, but the sive editing and improvement of the text of a shorter
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