Reference info: 0707.2593 [quant-ph]. Nature, 448, 23-24 (July 2007)

Download: The Nature version with nice graphics is freely available here and as PDF here. You can download the quant-ph version (with inferior graphics) here.

Comments: This paper is extremely brief because of the Nature boundary conditions. You'll find more meat in the papers described further down on this page. Also, please take a look at the fascinating Everett biography that's available here.

100 Years of Quantum Mysteries
The Scientific American paper described below gives a gentle introduction to quantum mechanics and how it may involve Level III parallel universes.
Download: Scientific American version, longer "director's cut" version
Authors: Max Tegmark & John Archibald Wheeler
Abstract: As quantum theory celebrates its 100th birthday, spectacular successes are mixed with outstanding puzzles and promises of new technologies. This article reviews both the successes of quantum theory and the ongoing debate about its consequences for issues ranging from quantum computation to consciousness, parallel universes and the nature of physical reality. We argue that modern experiments and the discovery of decoherence have have shifted prevailing quantum interpretations away from wave function collapse towards unitary physics, and discuss quantum processes in the framework of a tripartite subject-object-environment decomposition. We conclude with some speculations on the bigger picture and the search for a unified theory of quantum gravity.
Reference info: quant-ph/0101077. Scientific American, Feb. 2001, p68-75
Comments: The "director's cut" version of the Scientific American article has more text and inferior graphics.

Common objections to the MWI are discussed. It is argued that when environment-induced decoherence is taken into account, the experimental predictions of the MWI are identical to those of the Copenhagen interpretation except for an experiment involving a Byzantine form of ``quantum suicide''. This makes the choice between them purely a matter of taste, roughly equivalent to whether one believes mathematical language or human language to be more fundamental.

Publication info: quant-ph/9709032, in proceedings of UMBC workshop ``Fundamental Problems in Quantum Theory'', eds. M. H. Rubin & Y. H. Shih (1997)
The cartoon above illustrated the reader comments when my paper was featured in New Scientist (in the issue of January 24, 1998)
 
 

 
  My original paper, upon which this article was based, is click above.

1. The random number generator must be quantum, not classical (deterministic), so that you really enter a superposition of dead and alive.
2. It must kill you (at least make you unconscious) on a timescale shorter than that on which you can become aware of the outcome of the quantum coin-toss - otherwise you'll have a very unhappy version of yourself for a second or more who knows he's about to die for sure, and the whole effect gets spoiled.
3. It must be virtually certain to really kill you, not just injure you.

• The book with Hugh Everett's original Ph.D. thesis, widely lambasted but rarely read, is in my opinion an excellent pedagogical piece. Although it's sadly out of print, you can download it for free here.
• You'll find a useful set of Many-World links in the Stanford Encyclopedia of Philosophy.
• There's an excellent 2009 book on the quantum many worlds interpretation containing chapters from many of its best known advocates and critics.
• David Deutsch's popular book Fabric of Reality supports Level III.
• You'll find a fascinating free biography of Everett by Eugene Shikhovtsev here.
• Peter Byrne's 2010 Everett biography

• Tegmark 1993 describes how decoherence looks like, feels like and smells like wavefunction collapse, thereby eliminating one of the main motivations for the Copenhagen interpretation.
Indeed, one of my most exciting and memorable moments as a scientist was sitting up late one Berkeley night in the December 1991 and discovering this. It was shocking to realize that little me could discover really important things. Later, after I'd almost finished writing up the article, which I hoped would be my first paper ever, I discovered that I was scooped by many years and had rediscovered what people called decoherence. Dieter Zeh first wrote about it in 1970, and Jooh & Zeh (1986) published a table of results almost identical to my Table 2. However, this in no way diminished the fun I had - indeed, everything we ever discover on Earth, even if we get credit for discovering it first, is undoubtledly a rediscovery of something discovered on a far-away planet, so the best reason to do science is simply for the fun of it.
• Tegmark & Yeh 1994 and Tegmark & Shapiro 1994 are marginally related, giving explicit examples of the effects interaction with the environment.
• Tegmark 1996 is also marginally related, discussing how decoherence can be measured in practice.

1. Shut up and calculate, New Scientist, September 15 2007 (cover story) (popular summary of my key idea)
2. Is the Universe Actually Made of Math?, interview with me about Level IV in Discover Magazine, 6/16 2008
3. The Mathematical Universe, Founds. Phys. November 2007, 116 ("full-strength" version of my ideas)
4. Is "the theory of everything" merely the ultimate ensemble theory?, Annals of Physics, 270, 1-51 (1996 paper where I first plugged these ideas)

Which mathematical structure is isomorphic to our Universe?

Figure 1. Relationships between various basic mathematical structures. The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures - for instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.
 
Comments: This paper was the cover story of New Scientist - you'll find it here.  The figure to the right shows a small part of the "family tree'' of mathematical structures as described in the paper. The complete tree is probably infinitely large, so the figure way down below, at the bottom of this page where the arrows are explained, merely shows some of the most basic structures. Those complex enough to contain self-aware substructures (SASs), "observers'' such as us, are almost certainly not in this picture. Many of the highly complicated structures are likely to be dead as well, devoid of SASs. The figure below illustrates how little one needs to change some aspects of our world to make it hostile to life, and a similar figure for the dimensionality of space and time can be found here.
Level IV Links

• You might enjoy this trialog if you're interested in the question of life, the universe and everything without the equations.
• Jürgen Schmidhuber's TOE page. Here you can read about an ultimate ensemble theory that is related to what I propose in an interesting way, focusing on computations rather than mathematical structures. I have a long discussion of how I think these are related in Chapter VI of my 2007 paper.
• Light-hearted Level IV discussion
• The Level IV multiverse contains almost no information. I first argued for this "less-is-more" idea here.
• Wei Dai's "everything'' mailing list: In case you've read this far, you may be interested to know that Wei Dai has set up a mailing list for discussing this sort of ideas. Here is his message with instructions for how to join:

  Date: Thu, 15 Jan 1998
  From: Wei Dai
  Subject: ANNOUNCE: the "everything" mailing list

  You are invited to join a mailing list for discussion of the idea that all possible universes exist. Some possible topics of discussion might include:

  • What is the set of all possible universes?
  • What is a reasonable prior/posterior distribution for the universe that I am in?
  • Why do we believe that both the past and the future are non-random, but the future is more random than the past?
  • Before observing anything about the universe, should we expect it to have (infinitely?) many observers?
  • How can we/should we predict the future and postdict the past?
  Here are some papers that can serve as a basis for the discussion:
  • "Investigations into the Doomsday Argument", Nick Bostrom
  • "A Computer Scientist's View of Life, the Universe, and Everything", Juergen Schmidhuber
  • "Is ``the theory of everything'' merely the ultimate ensemble theory?", Max Tegmark
• You can surf the postings to this list tying in with my paper here. To subscribe to the mailing list, please click here.

Are multiverse theories testable?

• Is it all just philosophy?
  On Nov 3, 2003, at 14:38, Walter H.G. Lewin wrote:
  Q: Is there a way, at least in principle, that the existence of multiple universes as described by you in Sci Am, can be experimentally verified or falsified? If not, as several of my colleagues have pointed out, it falls in the realm of religion and philosophy, but not physics.
  A: Absolutely! The key point, which I emphasize in that article, is that a fundamental physical theory can be testable and falsifiable even if it contains certain entities that you cannot observe. To be testable and falsifiable, it merely nees to predict at least one thing that we can observe. A good example is the theory of eternal inflation, where our Hubble volume constitutes only an infinitesimal fraction of all space. Since this theory makes the firm prediction that Omega = 1 to an accuracy of order 10^{-5}, this model (and all those level I parallel universes with it) would have been ruled out if we had measured say Omega=0.70+0.02. Instead, our latest constraints in astro-ph/0310723 are Omega=1.01+-0.02.
• Ruling out theories
  From Erik Stengler, estengler@museosdetenerife.org, Mon Aug 11 05:38:21 2003:
  Q: How would observing that we live in an unlikely universe rule out the multiverse theory? In the event of an unlikely universe having inhabitants, they surely would find that their universe is, in fact, unlikely - should they then conclude that the multiverse theory is not correct? Why could it not be our case?
  A: It's just like with all probability: if you observe something that's a priory very unlikely, like tossing a die 10 times and getting 6 each time, you get worried. Whenever we say that a theory is ruled out at 99% confidence, we mean that something as weird as we observed would happen less than once time out of a hundred. Yet you're quite right: a small fraction of all observers do observe weird things. Some win the lottery, for instance.
  Q: It still doesn't feel right to say that if our universe turns out to be unlikely, say at 99% confidence, the multiverse theory would be ruled out. It's just this one universe that would be unlikely (i.e. one datapoint that happens to fall in the 1% tail of the distribution) - not the theory! You'd need to know many universes to see if they follow the distribution predicted by the multiverse theory, regardless of where our falls in the curve. [In the case of the dice: you'd need many series of 10 tosses to start thinking the dice have been tampered with, say all sides have six dots on them. You would not dare to accuse the dice owner of cheating with just one lucky series of tosses, even if it is the first one, would you?]
  A: This is an interesting issue, but has absolutely nothing to do with parallel universes: this is simply the standard statistical procedure of ruling things out. You may feel that there's a difference in that you're making only one observation rather than many (getting a distribution), but that difference is illusory: for instance a single cosmic microwave map data consists of millions of hot and cold spots, so you can measure their size distribution very accurately even though you have, in effect, observed only one universe.
• Why should I believe that there's at least one parallel universe? Donuts?
  From Pim van Meurs, pimvanmeurs@yahoo.com, Tue May 6 17:27:04 2003
  Q: You state in your paper "the key question is not whether parallel universes exist (Level I is the uncontroversial cosmological concordance model), but how many levels there are." Are you suggesting that the existence of level I parallel universes is not a key question? In other words, do you argue that the number of universes is larger than n=1 (the lowest number we can obtain from observation)?
  Indeed. Although it's far from obvious that n=oo, I think the astronomical evidence is very compelling that n > 1. The curvature of space measured by the cosmic microwave background is so small that if space is a (finite) hypersphere, then it is large enough to contain at least n=1000 other Hubble volumes. If space is finite by connecting back on itself like a donut, the cosmic microwave background measurements again require the donut to be large enough to contain n > 1 Hubble volumes. Of course you can always postulate that space ends abruptly right outside the cosmic horizon with a big warning sign said "MIND THE GAP", but you'll have a hard time providing an elegant mathematical formulation of that theory, let alone convincing other people of its virtues.
• So what about donuts?
  From Tomas Ensalata, zetar21@yahoo.com, April 19, 2003 3:09:04
  Q: I remember reading another piece on the BBC site which talked about the evidence for a small universe in the new satellite data. If space is finite and small will the awesome idea of parallel universes lose its current appeal?
  A: No, it would just zap level I, not levels II, III or IV. Regarding the donut universe business, you may not have noticed that the BBC story in fact talked about work I'd been involved in. There was a more detailed story on this in the New York Times, and numerous follow-up studies of this have been done by many groups. Based on what's been found so far, there's in my opinion no compelling evidence for a small universe, merely evidence that there's something funny going on that we still don't understand.

Multiverse philosophy

• Will I rob a gas station?
  From Glenn Statler, gstatler@telusplanet.net, July 14, 2003 7:42:58
  Q: I stumbled upon your website last fall via the 'sanitized' (dumbed-down) multiverse article in Scientific American. The Scientific American version left out enough to be confusing and I found your website version to be much more understandable. I think that the layperson is more intelligent than we give credit for and so the article should have been edited less.
  The personally troubling aspect of the multiverse theory, which, fortunately and unfortunately, seems quite plausible, is that---if every conceivable universe exists---that means that your similar being, and mine, somewhere out there is (....to be gentle as I can be) an axe-murder and (not so gentle) worse! That is hard to accept for me in this universe, even if true. I guess the good news is that it isn't really me...but pretty darn close to me since if every iteration is plausible then somewhere our "clone" has seen typed this email and then went out and robbed the local gas station, 7-11, next closest gas station, etc. etc.! Then again, our clones are also the James Bonds, Elvis Presleys or Ghandis and Mother Teresas somewhere else. Wow! My autograph is worth something somewhere!
  I have played with the number of earthlings times the number of thoughts times the number of earths and galaxies and the results are small relative to even the number of protons in our Hubble  volume---even after several beers or summertime gin and tonics to add some extra insight!
  A: I too wish it were possible to publish less diluted stuff in Sci Am, but as you know, it's the editors and not us authors that set the rules. Things inconsistent with the laws of physics will never happen - everything else will. However, to cheer you up: even if some of your twins hold up gas stations, most of your twins certainly don't, given what I already know about your personality; it's important to keep track of the statistics, since even if everything conceivable happens somewhere, really freak events happen only exponentially rarely.
  
• Will I run over a squirrel?
  From Mike Sanders, mike.sanders@pearsoned.com, Apr 6 2004 at 14:37
  Q: Within the context of the multiverse, doesn't every conceivable physical possibility occur? If I'm driving my car and stop abruptly to keep from hitting a squirrel, don't I purposely run over that same squirrel in an alternate universe. And if so, isn't the number of universes that follow each outcome approximately the same?
  A: No - and that's the crux. The laws of physics and your behavior evolved through natural selection create much regularity across the multiverse, so you'll try to spare that squirrel in the vast majority of all parallel universes where "you" are pretty similar to the copy reading this email (just as regards the above-mentioned gas station robbery). The fractions only split close to 50-50 for decisions that you perceive as a very close call.
  
• Multiverse ethics
  From Gerald, Oct 5, 2003, 14:31
  Q: Doesn't the multiverse theory completely trivialize existence? It puts the burden for individual responsibility on the shoulders of the universe. Why do anything? If you decide to be a lazy slug, that just means that your particle clone elsewhere will be the one who wins the Nobel prize. And vice versa. Similar destructive arguments can be applied to morality. If the theory is correct, "wrongdoing" doesn't exist. Ultimately I've realized one almost has to believe in fantasies, in theories that only could be possible but probably aren't. Otherwise, one cannot make meaningful decisions to advance their own survival or to aid anyone else.
  A: I'm not convinced that the existence of parallel universes implies that I should dramatically alter my behavior. Yes, some near-clones of me indeed win the Nobel prize, but only a very small fraction of them! As in the gas station question above, it's important to keep track of the statistics, since even if everything conceivable happens somewhere, really freak events happen rarely, in an exponentially small fraction of all parallel universes. It's these statistics that make existence complex and interesting rather than trivial.
• Seductive circularity?
  From Richard Reeves, valueprint@earthlink.net, April 18, 2003 14:23:31
  Q: As a layperson I have a great deal of difficulty grasping the concept of Infinity, in fact I suppose Kant would say it's an outright impossibility since we have no conceptual template for doing such a thing. But it always worries me when I hear theoretical scientists or philosophers tossing around the idea that "something must exist in reality because it is conceivable" (in this case in the context of the discussion of infinity and its implications for parallel universes). It reminds me of that venerable philosophical canard, the Ontological Argument, which states: If God is the greatest conceivable Being then He must necessarily exist in reality, otherwise He wouldn't be the greatest conceivable being. Versions of this argument have been knocking around for a millenium, a tribute to its seductive nature. I can't help wondering if the arguments for parallel universes are equally circular and seductive.
  I certainly wouldn't claim that "something must exist in reality because it is conceivable"; the point of the article is merely that it can exist, and that we shouldn't be so dismissive of big ideas just because they seem weird.
  
• Multiverse theology?
  From Ernesto Viarengo, viarengo@asl19.asti.it, Jun 9, 2003, at 19:20
  Q: Is it in your opinion possible to imagine a "scientific theology", based on the assumption that in an infinite universe all is possible and even necessary, also the evolution of some intelligent life until a level that we usually consider typical of God? A: An interesting question. I certainly believe the laws of physics in our universe allow life forms way more intelligent than us, so I'd expect that they have evolved (or been built) somewhere else, even at Level I. I think many people wouldn't be happy to call them "God", though, since they would be outside of our cosmic horizon and thus completely unable to have any effect on us, however smart they are (assuming there are no spacetime wormholes). However, perhaps they can create their own "universe", for instance by simulating it, playing God to its inhabitants in a more traditional sense. And perhaps we ourselves live in such a created/simulated universe...
• A digital universe?
  From Ninad Jog, ninad@wam.umd.edu, Jul 21, 2003, at 2:09,
  I believe that self-aware-substructures can arise in spacetimes with fewer than 3 space dimensions (n < 3) despite the absence of gravity. These SAS will evolve from what are currently known as Artificial Life forms or Digital Organisms that reside in habitable universes such as the Avida and Tierra artificial life software platforms. DOs can evolve only on specialized platforms with minimum-length instruction sets, so that any arbitrary mutation in an organism's genome (instruction) results in a different legitimate instruction from the set. [...] The cyber universe is qualitatively different from our own, but does that mean it's a separate type of universe (another level), or is it part of the level-II multiverse? I'll be most interested in your comments. Yes, the n<3 argument applies only for universes otherwise identical to ours, not to the sort you are simulating, which need indeed not have any meaningful dimensionality. I would term the DO "Cyber Universe" you simulate as part of our own, since we can interact with it even though the DO's, if they were complex enough to be self-aware, would as you say be unaware of our existence. They would derive that their universe obeyed "laws of physics" that were simply the rules that you had programmed. My guess is that the Level IV multiverse also contains such a cyber universe existing all on its own, without it being simulated on a "physical" computer. It's DO/SAS inhabitants couldn't tell the difference, of course. However, such a cyber universe could have an infinite implementation space and an infinite number of evolution steps; I suspect that any DO we can simulate on our current computers is way too simple to be self-aware in any interesting sense, and this would require a much larger implementation space to allow greater DO complexity.
• Are we a computer simulation?
  Perhaps, but I suspect not. First of all, even if we are, there's presumably at least one physical reality that is not simulated by a computer in some other reality. (The one in which we're simulated, or one simulating that one, or one simulating that one, ...). This means that simulations alone don't solve the problem of explaining physical existence. Assuming that this pre-existing physical reality is mathematical (described by some equations, or more rigorously, isomorphic to some mathematical structure as per this paper, this suggests that mathematical existence and physical existence are but one and the same thing. Because every possible computer simulations corresponds to a mathematical structure, this means that all such realities would exist regardless of whether someone simulated them on a computer or not. You'll find a much more detailed discussion of this here.
• Dogs and physical intuition
  From Harry W. Hickey, Arlington, VA, hwhickey@starpower.net, Fri Apr 11 23:30:16 2003
  Q: Having read your article in May "Scientific American", where you touch on this point: "the tight correspondence between the worlds of abstract reasoning and of observed reality", I would like to raise the question. Could our powers of what we think of as abstract reasoning perhaps be conditioned by the empirical physical nature of the universe we have evolved in? Consider, for example, the Euclidean geometry that seems so obvious to us. Perhaps we are evolved to function in Euclidean 3-space. In which case, other animals besides ourselves should have some instinctive sense of geometry. "A straight line is the shortest distance between two points." I think dogs know that. Hold a dog-biscuit out to a dog and see if it doesn't advance in a straight line to the biscuit.
  A: Yes, I agree that what we call our physical intuition is evolved for the particular world we find ourselves in. If we all lived in a 4-dimensional world, I bet the both 4D people and 4D dogs would find 4D geometry completely natural. Yet I suspect that even hypothetical mathematicians living in 4D or any other type of universe would uncover the exact same mathematical structures that we do.
• Isn't there quantum randomness?
  From Benjamin Dozier, May 10, 2003 13:36:06
  Q: When describing "Level III Multiverses", you state that on a quantum scale, all of the possible outcomes of a specific event actually happen, though each possibility occurs in a different parallel universe. We percieve that only one of the possibilities really occured, as we are observers in only one universe. In this way, there is no randomness to the outcome of an event, as all possible outcomes happen. But is it randomness that determines in which multiverse I, as a concious observer, will perceive the event in?
  A: That's a good question with a good answer: no, a different "you" will perceive (different outcomes of) the event in each of the parallel universes. Suppose a quantum measurement can produce outcomes 0 or 1. Then after the measurement, there's two parallel universes, each with a "you" with all memories you had before the measurement, one with the added memory of measuring "0" and one with the added memory of measuring "1". There's nothing random about this. Don't ask "how do I know which of the two guys is me?" - they both are.

How many parallel universes are there?

• Why must we have duplicates?
  From Richard Reeves, valueprint@earthlink.net, April 18, 2003 14:23:31
  Q: Given infinity, why isn't it equally plausible that the worlds within it would express infinite variety, rather than repetition
  The answer is that there are only a finite number of possible states that a Hubble volume can have, according to quantum theory. Even classically, there are clearly only a finite number of perceptibly different ways it can be.
  
• How rigorous is this?
  From Bert Rackett, bertrckt@pacbell.net, Sat Apr 19 22:22:13 2003
  Q: I very much enjoyed reading your Scientific American and Science and Uitimate Reality papers, but I am entirely befuddled about your estimates for likely distance of an identical environment. You claim that the volume may be completely defined by a (very long) list of binary values denoting the presence or absence of a proton, but this of course oversimplifies things.
  A: Although classical physics allows an infinite number of possible states that a Hubble volume can be in, it's a profound and important fact that quantum physics allows only a finite number. The numbers I mentioned in the article, like 10^10^118 meters, were computed using the exact quantum-mecanical calculation, and the classical stuff about counting protons in a discrete lattice arrangement was merely thrown in as a pedagogical example to give a feel where the numbers come from, since that turns out to give the same answer.
• Why must all regions have duplicates, not just one?
  From Jeffery Winkler, jeffery_winkler@mail.com, Oct 13, 2003, at 0:58
  Q: Just because something is infinite, does not mean that all possibilities are realized. The number pi is infinitely long, pi = 3.14159... and in that case, all combinations of digits are realized. However, the number 1/3, converted into a fraction, is also infinitely long, 1/3 = .33333... and in that case, all combinations of digits are not realized.
  A: That's correct: infinite space alone guarantees only that SOME Hubble volume will have a duplicate, not that our own will. However, if (as in the current cosmological standard model) the cosmic density fluctuations originate from quantum fluctuations during inflation, their statistical properties DO guarantee that our (and indeed every) Hubble volume has a duplicate.
• Is there a countable or uncountable infinity of universes?
  From Alex Filippenko, April 15, 2003 18:33:19
  Q: In your calculation of the average distance between you and another copy of you, did you take into account the uncertainty principle and its effect on the number of possible states? I've always explained away "copies" of myself by saying that the Universe may be infinite in size, but *countably infinite*. The number of possible states is *uncountably infinite*, on the other hand.... so any particular state only occurs once, on average.
  A: Let's first ignore the important complication of past history and ask how many physically distinct states N there are in a volume V. In classical physics, N is infinite (indeed uncountably infinite) as you say, since even specifying the position of a single particle requires infinitely many decimals. In quantum mechanics, however, N is finite: if the temperature never exceeds T, we of course have N < ln S, where S is the entropy of the thermal state with temperature T (I'm taking Bolzmann's constant k=1). Interestingly, the number of states appears to be finite even when taking general relativity into account, which is closely related to the holographic principle: the entropy is maximized if all the matter in V is in a single black hole, in which case, as you know, the Bekenstein-Hawking formula says that N is of order the surface area measured in Planck units. So yes, I see your reasoning, and find it quite striking that quantum mechanics, uncertainty principle and all, contrary to what one might expect, gives fewer states than classical physics. In the limit V->oo, quantum mechanics therefore gives a countable rather than uncountable infinity of states.
• Is it countable even with continuous wave functions?
  From David Fotland, fotland@smart-games.com, August 3, 2003 21:09:49
  You argued that the total number of possible states in a universe is finite, so if the total of all universes is infinite, then every possible universe must exist. I understand that quantum states have discrete vales, but the wave function is a continuous function. Can't the probabilities that give the possible locations of particles have any real value?
  Interestingly, they can't: you can prove that in a finite volume, there's only a discrete number of allowed quantum wavefunctions. If the energy is finite, it's even a finite number.
• But even a hydrogen atom has infinitely many states!
  From Attila Csoto, csoto@matrix.elte.hu, Wed Mar 17 12:59:29 2004
  Q: You say in your papers that the number of possible quantum states within the Hubble-volume is finite. I understand your argument, but there is a problem which puzzles me. If we single out one hydrogen atom in our Hubble volume, it has itself an infinite number of different bound states. So one could imagine a Hubble sphere next to ours which is the same as ours except that this hydrogen atom iis not in its ground state but in the next excited state, and in the next sphere in the next higher state, etc. These universes differ from each other by a tiny amount of energy but I don't think that this should matter. So, my question is: how can we have a finite number of possible quantum states in our sphere, if one hydrogen atom already has an infinite number of possible bound states?
  A: There's infinitely many bound states if only space is truly infinite. There's in fact a beautiful old paper by Erwin Schrödinger deriving the exact solutions for a hydrogen atom in a closed finite Universe, showing that in this case, the number of bound states is finite.
• How many histories are there?
  From Alex Filippenko, April 15, 2003 19:16:39
  Q: There is also, however, the issue of *past history* -- one can achieve the Universe with n particles in a number of different ways depending on their motion (and these will have different futures), so the number of possibilities increases dramatically. Add to this quantum effects (identical particles, non-deterministic trajectories, etc.) - doesn't this suggest that there aren't any duplicates of ourselves?
  A: First of all, to consider whether there are copies of ourselves, we need only consider spacetime regions of time-like extent less than 100 years or so, since we don't live forever. To be really conservative, one could consider counting how many discernibly different histories a 4D box of size L x L x L x L/c (in comoving coordinates x conformal time) could have with temperatures never exceeding some large value T, say taking L = Hubble radius. This would simply replace 10^{10^118} in the calculation I gave by 10^{10^118*(4/3)} ~ 10^{10^157} as you go from 3D to 4D counting, i.e., still a finite number. I think this is overly conservative, since whatever the true laws of physics turn out to be, we do have some sort of causality whereby the history is determined by what happens on a 3D spacelike hypersurface. I fully agree with you that quantum mechanics makes it impossible for us to compute our future from such initial data, but all that matters for your question about whether we have duplicates is of course how many possible histories there are.
  
• Can a finite universe grow infinite?
  From Mark Tranchant, mtranch2@ford.com, Tue Apr 15 03:41:08 2003
  Q: Your Level I calculations rely on two assumptions:
  1. A Big Bang approximately 14 billion years ago
  2. Infinite space to play with
  How do you reconcile these two - how do we get from a tiny expanding sphere to infinite space in finite time? Incidentally, congratulations for writing the first article I've read to use a number larger than a googolplex...
  A: If space started out finite (curved up like a hypersphere, say), then it always remains finite. Conversely, if it is infinite now, it was infinite from the very beginning. For more details on this, check out Ned Wright's cosmology FAQ. A interesting exception to this is inflation (see next question), which can create something infinite out of something finite.
  
• How can you make an infinite universe in a finite bubble?
  From David Coule, David.Coule@port.ac.uk, November 29, 2003 19:39:41
  Q: In the Level I case, where does it say that a spatially infinite universe is a prediction of inflation? There is suposedly a geodesic incompletness theorem of Borde, Guth and Vilenkin, gr-qc/0110012, that shows that only a finite time worth of inflation could have happened.So if the initial patch is finite there is still at present only a finite volume created. Garriga and Vilekin's philosophical mussings contradicts this result!
  A: The trick is that you generically obtain an infinite universe even after a finite abount of inflation. The subtle trick involves the t=constant spatial hypersurvaces perceived by observers curving upwards in spacetime towards the infinite future time direction. Loosely speaking, the infinite future time direction gets warped into an infinitete space. Please see Garriga & Vilenkin 2001, Phys. Rev. D 64, 043511 and references therein for details.
  
• Why 2^n and not n factorial?
  From Peter Lindner, Mon Apr 21 10:58:03 2003
  Q: On page 43 of the Scientific American article, subtitled "How Far Away is a Duplicate Universe?" isn't there a mathematical error? It says 2 to the 4th or 16 possible arrangements, but there are actually 4 factorial or 4x3x2x1 = 24 possible arrangements. If it were n possible arrangements, the correct answer would be n!, not 2^n.
  A: Let call n=10^118. If each proton where different, then there would indeed be n! ways possible states: you'd have n choices for where to put the first one, n-1 choices for where to put the second one, etc. However, a deep principle of quantum physics is that protons are indistinguishable particles, so that interchanging the positions of two protons gives you back exactly the same physical state. This is why there's only 2^n states, not n!.
  
• Doesn't our apparent free will imply infinite possibilities?
  From David Taub, gonzo@bredband.net, Jun 15, 2003
  Q: This has to do with the finite number of states of our universe, N~10 to the 10 to the 118. Doesn't my "illusion of free thought" produce a contradiction? Shouldn't it be theoretically impossible then to conceive of the number N+1? What I mean is let's say I was asked to pick a random number between 1 and infinity... it seems to me I could pick just about any number, including N+1 and higher, which seems to cause a problem with the finite number of states idea.
  A: That's a clever and very cute argument - I like it! Some huge integers larger than N are easy for us to describe, say N+1 or 10^N. However, almost all integers are so "generic" that they would take more than 10^118 bits to describe, and therefore can't be described by any man, beast or supercomputer in our Hubble volume. The proof is simple: Given some computer language, let the P(n) be the shortest computer program that produces the integer n as output. P(n) is simply a string of bits, and its length is known as the algorithmic complexity of n. We can think of the bit string P(n) simply as being an integer written out in binary. Consider the set of integers {P(n)} for n < N. No two integers in this set can be identical, since the programs producing different integers must be different. Therefore the largest integer in this set is >= N, proving that you can describe at most N different numbers in our Universe regardless how clever your computer language is. Alternatively, you can take P(n) to be a file containing a math book describing the integer n - the conclusion is, of course, the same.
• How many universes are there at the different levels?
  From Chris Fraas, Las Vegas, cpfraas@yahoo.com, Tue Sep 9 17:18:03 2003
  Q: Is a Level III multiverse simply a doppleganger of a Level I multiverse? If so, does this mean that if there are infinite, or innummerable, Level I multiverses, there are just as many Level II, III and IV multiverses?
  A: If space is finite, there'll be only finitely many at Level I but still infinitely many at Level III. If space is infinite, there's infinitely many at both Level I and Level III, and indeed just as many distinguishably different Level III universes as Level I ones (or, if Level II exists, as many as at Level II).
• Might there not be only a handful of parallel universes?
  From Douglas Scott, dscott@astro.ubc.ca, Apr 30, 2005
  Q: Imagine that you want to be a party-pooping skeptic (like me, say) and you would like not to believe in any more parallel universes than you have to. Then presumably I can take the smallest possible consistent closed (or doughnut) universe, which isn't nearly big enough that there's another copy of me, etc. And I don't need to believe in Level II either, since that depends on the specific inflationary model or whatever ("Landscape" is the new braneworld jargon I gather). Then Level III is just interpretation, and I can no longer say "it doesn't add any more universes than I already had at Levels I and II, so you may as well accept them too". So I just imagine that quantum mechanics is hard to grasp, and the "many worlds" idea is just another of our pathetic attempts to understand quantum reality. Oh, and Level IV is just plain crazy. So I'm left with no need for any parallel universes. Have you ever thought about this from the "pessimistic" rather than "optimistic" point of view?
  A: Excellent question. I certainly have, since as you know, I tend to be on the skeptical fringe when it comes to many observational claims of other sorts. The smallest allowed closed FRW space indeed contains only about 1000 Level I universes, and you could get by with even fewer in a doughnut space. Rather, I think the strongest evidence comes not from observation alone, but from observation coupled with theory. If you're willing to accept that there is no fundamental mathematical theory underlying all the regularity we've uncovered so far (but that physical reality is merely a social construct or somesuch --- what I'd call a "many words interpretation" of physics), then I agree with you: the only convincing evidence for the existence of a parallel universe would then be if you could directly observe it, which you by definition can't.
  However, if you do accept the notion that there's some fundamental mathematical theory that we just haven't found yet, then I think it's highly likely that there really are parallel universes, since it's proven amazingly hard to find models predicting what we observe without also predicting parallel universes as part of the package. Inflation, quantum mechanics and string theory are three examples of this --- let me elaborate briefly on each one.
  Inflation: Inflation generically predicts the Level I multiverse, i.e., infinite space and an ergodic random field for the initial conditions. Alex Vilenkin, Alan Guth and Andrei Linde have been going around saying this for a long time now, and I've never seen any serious objection to their argument. This happens essentially independently of the shape of the inflaton potential.
  Quantum mechanics: I disagree that the distinction between Everett and Copenhagen is "just interpretation". The former is a mathematical theory, the latter is not. The former says simply that the Schrödinger equation always applies. The latter says that it only applies sometimes, but doesn't given an equation specifying when it doesn't apply (when the so-called collapse is supposed to happen). If someone were to come up with such an equation, then the two theories would be mathematically different and you might hope to make an experiment to test which one is right.
  String theory: I agree with you that we don't yet know for sure whether there really is a string landscape or whether string theory even has anything to do with physics. But I find it interesting and amusing that even this most ambitious attempt to date to kill the parallel universe explanation for fine tuning (by deriving all the fundamental physical constants from pi and e and pure mathematics) is now suggesting that this one fundamental theory gives more than a googol effective theories. It's another striking example of how hard it is to get uniqueness, to make one and only one universe. If there is indeed some sort of "landscape", then inflation would populate it, making some distant Hubble volumes have other effective laws of physics than ours. If not, inflation will still make a Level II multiverse --- it's just that all its elements will have the same effective laws of physics.
  Level IV is certainly crazy. But before dismissing it completely, please do give me an alternative explanation of why we keep uncovering more and more mathematical regularity in physical world.

Miscellaneous

• Other Big Bangs
  From Jim Morford, II, Columbus, MS, JaimeZX@aol.com, May 20, 2003, at 4:18
  Q: If these various types of multiverses (I, II, III, and IV) exist, how does the Big Bang fit in with the creation of each?
  A: Our Level I and III multiverses were created in one and the same Bang. Our Level II multiverse was created by one and the same "extended Bang", or whatever you want to call the eternal mess of inflating bubbles corresponding to stochastic eternal inflation. Other parallel universes at Level IV, however, have nothing to do with our own Big Bang, existing completely separately from the space and time of our universe. There can also be other Level I, II and III multiverses coming from separate Bangs.
• Superluminal motion?
  Q: How can material be greater than 42 billion lightyears away if the universe is only 14 billion years old? Wouldn't that require superluminal motion?
  A: First of all, although nothing can move though space faster than light, there's no speed limit on how fast space itself can expand, so to distant galaxies can recede from each other faster than the speed of light. In other words, the strict speed limit in special relativity gets relaxed in general relativity. Second, if space is infinite now, it was infinite to start with --- for more details on this, check out Ned Wright's cosmology FAQ.
• Superluminal travel?
  From Shane King, shane.king@consultant.com
  Q: Some civilizations could be billions of years more advanced than us. Now that is a cool prospect. However, if they were billions of years ahead, technically, then surely they would be able to travel everywhere instantly (laws of known physics permitting) so we should see them. Same as the fact that time travel is not possible because where are all the tourists?
  A: Indeed, I thinks that's a good argument for physics not permitting travel faster than the speed of light!
• Doppelgängers?
  From Matti Seitsonen, saviomassa@hotmail.com, June 24, 2003 14:13:09
  Q: I couldn't quite grasp the concept of parallel universe. Are they all variations or versions of our universe, or are there completely different other universes that do not contain doppelgängers of you and me, but completely different objects than our universe?
  A: The latter. Level I contains variations of our universe as you say, whereas Level II and Level IV both add many universes which are so different that they won't contain any doppelgängers.
• Is there a parallel universe just millimeters away?
  From Richard Sylrich, sylrich@impulse.net, Tue Jun 24 18:50:04 2003
  Q: Ran across this article the other day. I compared it to your estimation of "distance" of other potential parallel universes. If I read it correctly, another parallel universe exists less than one millimeter away. If I sit down to the table to eat my dinner, is the other "me" sitting in my lap? Hopefully, if you want to respond, you will not go to advanced Math...that is not my dimension at this time.
  A: Although this idea of so-called branes (three-dimensional universes that are quite literally parallel to ours, a short distance away in another dimensions) is indeed taken seriously by the scientific community, I don't feel that it's quite fair to call them "parallel universes". The reason is that we can interact with them via gravity. For instance, if a star on a parallel brane would pass through our solar system, it would disrupt the orbit of our planet in a very noticeable way. For this reason, most people believe that the physics on such parallel branes (if they exist) is sufficiently different from the physics here that stuff like stars, planets and people eating dinner can't exist there.
• Measure? Gödel?
  From Alex Vilenkin, vilenkin@cosmos.phy.tufts.edu, Apr 30, 2005
  Q: If all mathematical structures (MS) are given equal weight, and there is an infinite number of them, then shouldn't we expect to find ourselves in some incredibly complicated MS? Also, what about Gödel? If the physical world is isomorphic to a MS, then what is the physical counterpart of Gödel's undecidable propositions?
  A: I think this "measure problem" is unsolved at Level IV and, as you know, even at Level II. At Level IV, I can envision three resolutions.
  1. The measure punishes complexity.
  2. We actually live in a very complicated mathematical structure, but perceive very little of this complexity. We've repeatedly been surprised to discover new layers of complexity as our experiments got better (atoms, elementary particles, relativity, quantum mechanics, etc.) -- perhaps there's a vast number of additional layers.
  3. Only Gödel-complete (fully decidable) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity. Let me remind you that although we conventionally use a Gödel-undecidable mathematical structure (including integers with Peano's recursion axiom, etc.) to model the physical world, it's not at all obvious that the actual mathematical structure describing our world actually is a Gödel-undecidable one.