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 IIV)
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):
 OVERVIEW:
 LEVEL III (Quantum Multiverse):
 LEVEL IV (Mathematical Multiverse):
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 abovementioned
Scientific American review article.
There I survey physics theories involving parallel universes, and the bottom line is that they form a natural fourlevel hierarchy of multiverses allowing progressively greater diversity.
 Level I: A generic prediction of cosmological inflation is an infinite ergodic universe, which contains Hubble volumes realizing all initial conditions  including an identical copy of you about 10^{1029} meters away.
 Level II: In many models, inflation can produce multiple Level I multiverses that have different effective physical constants, dimensionality and particle content.
 Level III: In unitary quantum mechanics, other branches of the wavefunction add nothing qualitatively new, which is ironic given that this quantum parallel universes have historically been the most controversial.
 Level IV: Other mathematical structures give different fundamental equations of physics.
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 IIIV.
Level II: Anthropic evidence
Interesting hints of a Level II multiverse come from the observation that many constants of nature appear finetuned 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 finetuning, it's therefore interesting to compute what would happen if various constants were different,
and I've looked at these effects:
 The effect of changing the dimensionalities of space and time: astroph/9702052
 The effect of changing the CMB fluctuation amplitude Q~10^{5}: astroph/9709058
 The effect of changing neutrino masses: astroph/0304536
 The effect on changing the dark matter density, dark energy density and CMB fluctuation
amplitude: astroph/0511774
 The effect of changing the masses of elementary particles: arXiv:0903.1024
Some of my colleagues will foam at the mouth if you mention the "Aword",
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 somehow 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 equationfree account of cosmological evidence for Level I and II multiverses,
and Alex Vilenkin's Many Worlds in One
is my favorite equationfree 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 [quantph].
Nature, 448, 2324 (July 2007)
Download:
The Nature version with nice graphics is freely available
here
and as PDF here.
You can download the quantph 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
subjectobjectenvironment decomposition.
We conclude with some speculations on the bigger picture
and the search for a unified theory of quantum gravity.
Reference info:
quantph/0101077.
Scientific American, Feb. 2001, p6875
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 nontechnical articles, this paper of mine was covered by
New Scientist and the the
Guardian.
Abstract:
As cuttingedge
experiments display ever more extreme forms of nonclassical 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 alldominant 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 abovementioned approaches to nonrelativistic
quantum mechanics give identical cookbook prescriptions for how to calculate
things in practice, practicalminded experimentalists, who have traditionally
adopted the "shutupandcalculate 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 environmentinduced
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:
quantph/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 manyworlds 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: everythinglist@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 selfawareness. 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 selfaware, but less and less so, the final
"death" being quite anticlimactic, sort of like when
an amoeba croaks. Do you buy this?
I think a successful quantum suicide experiment needs to satisfy three criteria:
 The random number generator must be quantum, not classical (deterministic),
so that you really enter a superposition of dead and alive.
 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 cointoss  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.
 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:

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 faraway 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.
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.
 Shut up and calculate, New Scientist, September 15 2007 (cover story) (popular summary of my key idea)
 Is the Universe Actually Made of Math?, interview with me about Level IV in Discover Magazine, 6/16 2008
 The Mathematical Universe, Founds. Phys. November 2007, 116 ("fullstrength" version of my ideas)
 Is "the theory of everything" merely the ultimate ensemble theory?, Annals of Physics, 270, 151 (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 nontechnical 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 3way food fight
here that I coauthored 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: grqc/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 selfaware 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: grqc/9704009. Annals of Physics 270, 151 (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 selfaware 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.
 Lighthearted Level IV discussion
 The Level IV multiverse contains almost no information. I first argued for this "lessismore"
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 nonrandom, 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:
 You can surf the postings to this list tying in with my paper
here.
To subscribe to the mailing list, please click
here.
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?

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 astroph/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 followup 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' (dumbeddown)
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 thatif every conceivable
universe existsthat means that your similar being, and mine, somewhere
out there is (....to be gentle as I can be) an axemurder 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, 711, 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 volumeeven 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 abovementioned gas station robbery).
The fractions only split close to 5050 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 nearclones 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 selfawaresubstructures 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
minimumlength 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 levelII 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 selfaware, 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 selfaware 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 preexisting 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 3space.
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 dogbiscuit 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 4dimensional
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 quantummecanical
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 BekensteinHawking 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@smartgames.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 Hubblevolume 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,
nondeterministic 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 timelike 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:
 A Big Bang approximately 14 billion years ago
 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, grqc/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, n1 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 partypooping
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 socalled
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 socalled branes
(threedimensional 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.
 The measure punishes complexity.
 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.
 Only Gödelcomplete (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 Gödelundecidable
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 Gödelundecidable one.