Every time I've written ten mainstream papers, I allow myself to indulge in writing one wacky one, like my Scientific American article about parallel universes. This is because I have a burning curiosity about the ultimate nature of reality; indeed, this is why I went into physics in the first place. So far, I've learned one thing in this quest that I'm really sure of: whatever the ultimate nature of reality may turn out to be, it's completely different from how it seems. So I feel a bit like the protagonist in the Truman Show, the Matrix or the 13th Floor trying to figure out what's really going on.

Parallel Universe Overview (Levels I-IV)
I think that there are at least 4 different kinds of parallel universes lurking out there, summarized in the figure down below. In this universe, I've published a series of articles about these 4 multiverse levels (I'd recommend starting with the the first one): If you're feeling puzzled by the notion of parallel universes, I'd recommend starting with the first one, which is the most read article I've written so far.

The Four Multiverse Levels
The figure above is explained in my above-mentioned Scientific American review article. There I survey physics theories involving parallel universes, and the bottom line is that they form a natural four-level hierarchy of multiverses allowing progressively greater diversity. To me, the key question is not whether parallel universes exist (Level I is the uncontroversial cosmological concordance model), but how many levels there are. I discuss how multiverse models can be falsified and argue that there is a severe "measure problem" that must be solved to make testable predictions at levels II-IV.
Level II: Anthropic evidence
Interesting hints of a Level II multiverse come from the observation that many constants of nature appear fine-tuned for life, having values in the narrow range allowing our existence (if they vary across the multiverse, we'll find ourselves in one of those places where we can exist, and there's no embarrassing fluke coincidence to explain). To check whether there really is fine-tuning, it's therefore interesting to compute what would happen if various constants were different, and I've looked at these effects: Some of my colleagues will foam at the mouth if you mention the "A-word", anthropic, and grumble about this not being science. I think most of the rhetoric is caused by the pro and con crowd meaning different things and talking past each other. Including so called selectrion effects is clearly not optional: a study ignoring relevant selection effects can come to totally incorrect conclusions, just like phone pollsters ignoring the fact that young voters are more likely to lack a land line. On the other hand, I dislike the popular term "anthropic principle", since "principle" makes including anthropic selection effects sound some-how optional.
Level I & II further reading
If you're looking for a books about this subject, Martin Rees' book Our Cosmic Habitat provides an excellent equation-free account of cosmological evidence for Level I and II multiverses, and Alex Vilenkin's Many Worlds in One is my favorite equation-free introduction to inflation and Level II. If you don't mind math, then there's a nice (technical) paper on Level I by Garriga & Vilenkin who, as opposed to this guy, have so far avoided being burnt at the stake for it.


There may be a third type of parallel worlds that are not far away but in a sense right here. If the equations of physics are what mathematicians call unitary, as they so far appear to be, then the universe keeps branching into parallel universes as in the cartoon below: whenever a quantum event appears to have a random outcome, all outcomes in fact occur, one in each branch. This is the Level III multiverse. Although more debated and controversial than Level I and Level II, I've argued that, surprisingly, this level adds no new types of universes. Below are a series of papers of mine discussing these parallel universes in more detail.

Many lives in many worlds
When Everett's theory as it celebrated its 50th anniversary in 2007, Nature invited me to write an assessment of its state of health. I argue that accepting quantum mechanics to be universally true means that you should also believe in parallel universes.

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.

What goes at the top?

The Interpretation of Quantum Mechanics: Many Worlds or Many Words?

This was my first paper arguing for quantum parallel universes. The part that has attracted the most interest is the quantum suicide experiment I mention at the end. I remember feeling pretty flabbergasted when I came up with this idea. Most interesting ideas are had by many people independently, and other people have independently come up with similar experiments.
Download: My paper. If you prefer non-technical articles, this paper of mine was covered by New Scientist and the the Guardian.
Abstract: As cutting-edge experiments display ever more extreme forms of non-classical behavior, the prevailing view on the interpretation of quantum mechanics appears to be gradually changing. A (highly unscientific) poll taken at the 1997 UMBC quantum mechanics workshop gave the once all-dominant Copenhagen interpretation less than half of the votes. The Many Worlds interpretation (MWI) scored second, comfortably ahead of the Consistent Histories and Bohm interpretations. It is argued that since all the above-mentioned approaches to nonrelativistic quantum mechanics give identical cookbook prescriptions for how to calculate things in practice, practical-minded experimentalists, who have traditionally adopted the "shut-up-and-calculate interpretation'', typically show little interest in whether cozy classical concepts are in fact real in some untestable metaphysical sense or merely the way we subjectively perceive a mathematically simpler world where the Schrodinger equation describes everything - and that they are therefore becoming less bothered by a profusion of worlds than by a profusion of words.

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.

Quantum immortality
The quantum suicide argument
above raises the intriguing question of whether the quantum many-worlds interpretation implies subjective immortality more generally, and I've been getting lots of emails about this. Here's an email response I wrote on the subject:

From max@sns.ias.edu Sat Nov 28 13:20 EST 1998
To: everything-list@eskimo.com, max@sns.ias.edu
Subject: Quantum immortality

Hi guys,

Here's a brief comment on the issue of whether the MWI implies subjective immortality. This has bothered me for a long time, and a number of people have emailed me about it after the Guardian and New Scientist articles came out. I agree that if the argument were flawless, I should expect to be the oldest guy on the planet, severely discrediting the Everett hypothesis. However, I think there's a flaw. After all, dying isn't a binary thing where you're either dead or alive - rather, there's a whole continuum of states of progressively decreasing self-awareness. What makes the quantum suicide work is that you force an abrupt transition. I suspect that when I get old, my brain cells will gradually give out (indeed, that's already started happening...) so that I keep feeling self-aware, but less and less so, the final "death" being quite anti-climactic, sort of like when an amoeba croaks. Do you buy this?

I think a successful quantum suicide experiment needs to satisfy three criteria:
  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.
Most accidents and common causes of death clearly don't satisfy all three. However, if only criterion 1 is violated, you'd still feel immortal as long as there is a Level I multiverse with surviving copies of you. In other words, it appears that this type of macabre experiment can show whether there is a multiverse more generally. Moreover, if you rig it so that it would kill either both you and a friend or neither of you in each round, then you would have someone to talk with afterwards who'd also be a multiverse believer.

Quantum Level III multiverse further reading
This site also contains the latest versions of some closely related papers of mine: The Level IV mathematical multiverse
The initial conditions and physical constants in the Level I, Level II and Level III multiverses can vary, but the fundamental laws that govern nature remain the same. Why stop there? Why not allow the laws themselves to vary? Welcome to the Level IV multiverse. You can think of what I'm arguing for as Platonism on steroids: that external physical reality is not only described by mathematics, but that it is mathematics. And that our physical world (our Level III multiverse) is a giant mathematical object in the Level IV multiverse of all matematical objects. I first started having ideas along these lines back in grad school around 1990, and have written several papers about it over the years.
  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)
The easiest place to start is with 1 or 2. Paper 3 largely supercedes 4, finally finished after 11 years of procrastination. You'll find some more detail about these papers below.

Which mathematical structure is isomorphic to our Universe?

The Mathematical Universe
Download: arXiv:0704.0646
Abstract: I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters and initial conditions to broader issues like consciousness, parallel universes and Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel's sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.
Reference info: The Mathematical Universe, Founds. Phys. November 2007, 116
Warning: Sections III, IV and the appendix of this paper are quite technical, so if you're among the 99.99% who don't have a Ph.D. in physics, perhaps skip those sections. The shorter and non-technical version I wrote for fNew Scientist is easier to read - it's here. The older paper below is slightly less technical. A much easier read covering related questions, certified 100% equation free, is the 3-way food fight here that I co-authored with Piet Hut and Mark Alford. Section IV of my parallel universe article here and the multiverse FAQ here may also be more accessible.
The arrows indicate the close relations between mathematical structures, formal systems, and computations. The question mark suggests that these are all aspects of the same transcendent structure (the Level IV multiverse, including our world), and that we still have not fully understood its nature.

Comments: I think of this paper as the sequel to one below that I wrote in 1996, clarifying and extending the ideas described therein, and including related ideas that I had fun thinking about in the interim but never got around to writing up.

The original 1996 paper: Is "the theory of everything" merely the ultimate ensemble theory?
Download: gr-qc/9704009
Abstract: We discuss some physical consequences of what might be called "the ultimate ensemble theory'', where not only worlds corresponding to say different sets of initial data or different physical constants are considered equally real, but also worlds ruled by altogether different equations. The only postulate in this theory is that all structures that exist mathematically exist also physically, by which we mean that in those complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically "real'' world. We find that it is far from clear that this simple theory, which has no free parameters whatsoever, is observationally ruled out. The predictions of the theory take the form of probability distributions for the outcome of experiments, which makes it testable. In addition, it may be possible to rule it out by comparing its a priori predictions for the observable attributes of nature (the particle masses, the dimensionality of spacetime, etc) with what is observed.

Reference info: gr-qc/9704009. Annals of Physics 270, 1-51 (Received November 19, 1996; published November 20, 1998)

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

Max' multiverse FAQ: frequently asked questions
I've received hundreds and hundreds of emails asking excellent multiverse questions, and will attempt to answer the most frequent ones here in this multiverse FAQ. I've taken the liberty to reproduce the questions from some of you, occasionally slightly abbreviated or edited - if you'd like yours removed, just let me know.

Are multiverse theories testable?

Multiverse philosophy

How many parallel universes are there?