Simulation of
the Universe
By
Neil J Boucher
Synopsis
The article looks at
what it would take to simulate the universe and what such a universe might look
like. Could we tell if we were living in
such a universe??
The general conclusion
is that it is decidedly possible to simulate the universe, provided the
universe is quantized and finite, but that it would fail the Occam’s razor test in that it would not be the simplest way
to build a universe.
The Computational Task
Here we consider what
would be necessary to completely simulate the known universe. We find that it is not something that can be
achieved from within the universe.
The are about 10
particles in the universe and if we allow each to have 10
properties (which will include location, momentum, spin, charge, mass) then we
have to store 10 
pieces of
information. However looking at this a
bit more closely it becomes a bit more complex.
If, in the real world, momentum for example can take on any value then it can take on values
that include irrational values. Even to
store one such value we would need infinite storage capacity. Therefore to be computable the world must be
quantized! It also follows that all information
about the universe and its components must be quantized if we are to even
consider computerizing it. This includes the scale of the universe and so
precludes an infinite universe. The number of bits for storage of information
(as distinct from pieces of information, as one velocity for example might
require 40 bits to store it) will depend on the quantizing scale.
Since a top end PC
might have 500 gigs of memory (5 x 10
bytes) we fall a long way short and cannot even consider it
a candidate. In fact we would need to do
something rather clever to store enough information because we would need, as a
minimum, all the particles in the universe just to store the information if we
did it the way a PC does. This holds true even if we could find some way to
optimize the use of those particles (your PC might use 1000’s of atoms, or
100,000’s of particles to store 1 bit of information).
So if we were to ask
the question can we precisely model the known universe, from within the known
universe, the answer must be NO. We can
only model much simpler and smaller universes than the one that we occupy.
This eliminates the
multiple mirror image computational model; that is, we cannot have universes
that contain computers that in turn spawn new universes as models, unless those
models are significantly smaller and less complex than the original universe
that originated the model.
In an article for
Nature, June 2002, Seth Lloyd of MIT in 
bits and a CPU that would have manipulated 10
bits of information in the time since the big bang. His estimate is based on the energy
equivalent of the universe, but can be seen to be consistent with the number of
particles contained within.
The God Problem
In primitive societies
man looked about and saw that all was too complicated to explain and so he
invented a god that “explained it all”. There is however a problem with this
approach. Suppose you find a watch on a
deserted island, and you can’t understand how it came to be, so you conceive
the watch maker. Now the watch maker is
more complex that the watch so you conceive a god. The logical process of this
thinking is that there is a god of gods and so on. Each new god more complex than the one
before. This really leads nowhere. To
say that one of the god’s “always was, and did not need a creator” is simply
avoiding the question and does not explain anything. An explanation must simply the situation not
add complexity to it.
The computational
model of the universe has to face its own god problem. The computer that runs the simulation is
necessarily more complex, contains more information and is “god-like” compared
to the universe itself. You have not
solved anything at all by postulating a computational universe, but have replaced
a difficult problem about the complexity of the universe, with a yet more
complex machine to run it all in.
The Quantum Mechanical Problem
Many have looked at
quantum mechanics as a “proof” of a computational universe. This approach suffers from the same sort of
problem as the God concept. Firstly and
fore mostly no one has any idea what is going on at the quantum mechanical
level. Quantum mechanics is not a description of quantum mechanical
reality, but a mathematical construct that mimics the outward behavior of the
quantum world.
To illustrate this
lets assume someone has somehow managed to model the share market. Lets say that the model is accurate to
+/-5% (this in itself would be an
incredible achievement). Now such a
model can easily be seen as nothing more than a model, and it does not tell us
anything about you or me as investors, but simply mathematically predicts the
herd behavior. It is also not
necessarily unique. Other models based
on different premises might give equally good results. A good working stock market model will most
likely be a substitute for real knowledge rather than evidence of complete
understanding.
The masters of quantum
mechanics are aware that their mathematics is a working model and not something
based on a deeper understanding. The fact that the current quantum mechanical
model is inconsistent with Relativity suggests that it is incomplete. Its high
level of agreement with experiment to date shows it to be a very good model,
but there is reason to believe that some day researchers will find
discrepancies between the theory and measurement. After all, the model is known to be
incomplete and this must at some time show up as a disagreement with
measurements if they are precise enough.
Now we see the problem. The mathematics of quantum mechanics is not reality based, but is a mathematical
construct that mimics the reality as seen in the laboratory. One should not be surprised therefore that
the mathematics looks like a model because that is all it is.
So the argument that
quantum mechanical behavior looks computational is circuitous. Quantum
mechanics as we know it, is a model of reality and nothing more. It has never claimed to be otherwise. We cannot directly infer anything about the
real nature of the universe from quantum mechanics.
A Computer for a Computational
Universe
As we have seen the
universe that we occupy cannot be computed from within the universe
itself. That is if the universe were
computational, the stuff that the computer itself is made of is not of this
universe and has no need to be related to the universe in any way. To clarify a computer that flies the space
shuttle or one that makes cars will be of the same kind and may even have the same
operating system. The computers may even happen to be identical. However the stuff that the computer is made
of is totally independent of its function and the computer makers need not know
in advance that application that the computer is intended for.
So a computer that
simulates the known universe need not, and probably would not be made of
electrons, protons and photons etc. These particles should then be regarded
as simulated and their forms
(properties) likewise can be simple mathematical constructs.
It is unlikely that
the computer would be made of the same sort of stuff that it is simulating.
Likewise the constants of the known universe (gravitational constant, the ratio
of the mass of a proton to and electron and such) can be simple mathematical
starting points. Other limitations like
the speed of light need not be applicable in any way to the computer
itself. This gives us the possibility of
allowing that the computer simulation has looked at many possibly worlds where
many possible particle properties have been tried and the successful models are
the ones that are left running.
Consciousness
A lot of people seem
to have a problem with consciousness, and I am not sure why this is. The brain is a massively parallel computer
that has many conflicting processes. For
example the fight or flight response is really a fight and flight response. The
adrenalin prepares us for both and the brain considers both; not so much
sequentially but in parallel. On this
issue and many others, the left and right hand side of the brain will come to
totally conflicting conclusions.
Therefore a higher
function of the brain has to be to manage all these inputs and decide on a
course of action. It seems straight
forward enough to allow that this management center would start as a fairly basic
bit of brain hardware, but as the brain evolves the demands on the management
system would rapidly increase. The
management center in the human brain would be called on to make many varied and
complex decisions.
Consciousness in this
light is the evolution of the management function from one that simply resolves
conflicting “reports” from other brain centers to one whereby the management
system can reflect on its own performance (that is it begins to manage itself). The management system is now not only aware
of the other brain functional centers but becomes aware of itself. This argument is consistent with the point of
view that all brains except the most simple would have evolved some form of
self consciousness and that the level of consciousness would increase with the
brain complexity.
Consciousness in this
view has a purpose. A management system
that does not review its own performance would be doomed to repeat the mistakes
of the past. However one that does have
a review process, can allow the subordinate centers to show a degree of folly
and ignore it.
Suppose the management
center were not self reviewing and was simply an automatic “judge”. It simply weighs the inputs from all the
competing brain sections and makes a decision.
In order to avoid repeating mistakes then since the management is not
self reviewing, the subordinate centers need to get feedback on the result of
there advice and to modify their advice for the next similar event
accordingly. While this may happen in
some instances, there is a lot of evidence from clinical studies that the
subordinate parts of the brain do not have good feedback paths. In split brain experiments one part of the
brain will consistently deny a truth known to the other part of the brain. Conflict over something as simple as whether
or not the person likes a certain food can exist between the two halves of the
brain. Another instance of this is a
person with brain damage who could recognize his mother readily when she called
in by phone, but would insist that she was an imposter when she appeared in
person.
From the experiments
above (and others) I think we can conclude that the subordinate parts of the
brain are only minimally conscious and are not especially good at communicating
to each other. The management system is
thus the bit that becomes self aware. As
a caution here, when speaking of parts of the brain the meaning should be taken
as “functional parts of the brain”. It
is not suggested that the brain is physically partitioned into a conscious and
an unconscious part, but rather that this partition is functional.
So to be truly
efficient the brain needs to be self-appraising on some level and it appears
that this is the highest level. If
consciousness requires a considerable computing overhead it would make sense to
centrally locate it. The importance of
this is that were the consciousness to be distributed it would not need to be
particularly “strong” because at each place that it occurred it would be
specialized. By centralizing consciousness
we are giving consciousness (which is merely initially at least the ability for
self-performance evaluation) a chance to evolve into something very powerful
indeed.
Because the management
center makes its decisions in a different way to the rest of the brain it would
be important when weighing the inputs that it was aware that its own input
(essentially memories of its past
decisions) are of a different kind and need to be treated differently to the
subordinate inputs. Thus from an early
evolutionary stage it would be an advantage to distinguish its own inputs from
the others. It is already starting to
become self aware!
Really therefore
consciousness is not of itself a problem for a computational universe. The computer needs to be massively parallel
with a management system that can itself evolve.
Quantum Computers
Much has been made of
the potentially staggering computing power of a quantum computer. However with today’s limited knowledge of quantum
computers there is no certainty at all that large quantum computers could ever
be built. The problem is that the
quantum nature only reveals itself at very small scales. Once a large number of particles are
assembled the quantum nature fades away to make way for classical behavior. The
result of this is that a 10 particle quantum computer is still a thing of the
future and that its capabilities, even when built, would not match that of a
$3.00 handheld calculator. Current
theory is not capable of predicting upper limits to the size of a quantum
computer, simply because it is not understood what it is that governs the
transition to classical behavior.
We also need to be
careful to distinguish computational power with storage capabilities. Potentially, the quantum computer might be
the ultimate number cruncher, but it does not offer an equivalent boost in
storage capabilities. Additionally what gives the quantum computer its power is
its massive parallel computing power; the downside however is that once we
extract one of the solutions we lose the others.
Quantum computers
could come in handy however for generating true randomness; which is something
that a macro computer is unable to do.
The quantum computer need not be especially powerful to do this and such
a device is within the feasibility of today’s technology.
Non Computable Things
Many things are not
computable. For example irrational
numbers like pi cannot be represented digitally on a computer. It is
approximately 3.14159 or more closely 3.14159 26535 89793 23846 26433 83279
50288 41971 69399 37511 (the value to 50 decimal places), but no matter how
many digits we calculate it to we will never have it precisely. As the true
value of pi is infinite in length then it would require an infinite storage
capacity to hold it. However in a
quantized world, this is not a problem as there is a “smallest” dimension of
space. Therefore the concept of a
perfect circle is a fiction and any real circle will in fact be a polygon with
a finite number of sides. In such a world pi is merely and abstraction.
So this is not an
argument for or against a computable universe, rather it is one against the
fiction of the mathematical concept of pi.
In a computable universe the value of pi, like all other values would be
truncated to a finite precision.
The Virtually Non-computable Real
World
Mathematicians and
some physicists have convinced themselves and many others that the world is
essentially a mathematically dominated domain.
Mathematicians only study problems that lead to neat mathematical
solutions and for them it is a case of not seeing the woods for the trees. Physicist proceed to eliminate the clutter
(the real world) and concentrate on the essence (the part of the system that they
are studying, isolated from outside influences). So studying one atom in isolation might lead
to some rather neat mathematics. Two atoms are decidedly messy and anything
more is quite frightful.
A good example of the
consequences of this is the study of ballistics. The path of a ball (lets say a cannon ball)
through a vacuum under the influence of gravity that does not go too high, has
a neat mathematical solution. It is so
neat and beloved by the physicist that even today school children are taught to
calculate the path of a cannon ball fired through a vacuum. The fact that
probably no cannon has ever been fired in a vacuum is not mentioned. Most texts dismiss the atmosphere as
something that will cause some drag on the cannon ball but for the purposes of
the study can be ignored. The truth is
that allowing for the drag is very messy and the neat mathematics dissolves
when account is taken of this parameter.
Without the drag the vertical and horizontal motions can be regarded as
occurring independently and out pops a simple solution. Allow for the drag and this independence
disappears. The motions become
correlated and no neat solution is possible (the solution requires a computer
to do a lot of approximate calculations over many small paths). But here’s the punch line…the error that
results from ignoring the drag of the air is about 50% (the actual value varies
according to the drag coefficient of the projectile and its speed). This huge error is conveniently ignored! Even
the famous conclusion that the angle of launch for maximum range is 45 degrees
turns out to be wrong when the drag it taken into account (41 degrees or
thereabouts is more like it).
A related example is
the many body problem. It is relatively
easy to model mathematically two bodies that are isolated from the real world
that orbit around one another. Make it
three (for example as in the earth, moon sun) and the task becomes most
formidable. Go to four or more and it is
virtually impossible. And yet most solar systems are more complex than a four
body situation particularly if the
asteroids are included.
Another seeming simple
problem is the solution to a detector circuit as seen in figure 1 below. In
fact this circuit could have come out of a high school physics book.
The circuit consists
of a signal source, a diode and a load with a filtering capacitor. If enough simplifying assumptions are made it
is relatively easy to extract mathematical solutions for this circuit. However if we look into the situation a bit
further we find the situation is definitely too complex for high school
consideration.
Lets accept that the
diode equation is I=Io.e
Where e= the charge of
an electron
V= the voltage across the diode
K=Boltzmann’s
constant
T= the absolute temperature.
Io= the reverse leakage current of
the diode (a constant)
I= the current through the diode.
which of itself is a simplification
of the diode behavior so we are already filtering some of the clutter already.
We can simplify this even more by noting the e/kT=40
(approximmately) so we can model the diode as
I=Io.e
…eq 1
Lets ignore the
capacitor for now (more clutter that makes the problem harder) and just
consider the load resistance.
The voltage loop
around the circuit would give us
Vs=V+I.R …eq 2
where R is the load resistance
Vs is the generator voltage
(instantaneous value).
So here we have a
fairly simply pair of equations that define the circuit mathematically. However
the simplicity is deceptive. We can
conclude that
Vs=V+ Io.e.R but to solve this
for V is anything but easy.
These are but a few
examples of the enormous difficulty of actually using mathematics to model the
real world. Most real world problems are
too complex to yield to mathematics.
This is not to say that all is lost.
We can use approximate numerical methods to model all of the above
examples to a very high degree of accuracy.
However the solutions are not neat mathematical ones but involve
iterative methods that start with a guess and then home in on the solution. Numerical methods are widely used and consume
much computer time but they are evidence of how poorly the real world conforms
to mathematics.
In summary, if you
simplify the problems by removing the real world clutter, you should not be
surprised that you get neat mathematical solutions, because that was why you removed
the clutter in the first place.
Chaos
Chaos theory tells us
that the outcome of any chaotic event is highly dependent on the initial
state. If the initial state itself is
uncertain (as it must be in quantum mechanics) then the outcome is unpredictable
and any initial uncertainty will multiply over time. If computers were to mimic
nature they would need to quantize everything. Quantizing errors are additive
and in a chaotic system even small errors can lead to very different outcomes
for the same original state.
However such
graininess exists in the real world. A world system that brings the weather
into being can be resolved down to molecular parts and at this level the
patterns would differ from those of a continuum. So the real world becomes digital in that
sense. A computer with sufficient resolution would not be distinguishable from
the real thing.
Conclusion
If the universe is a
simulation of if it is simulable then the following
is true.
- There is no place in such a universe for
irrational numbers like
and e and any
equation containing these numbers will be an approximation. - In a computed or computable universe the
equations of physics will be digital.
That is the only permitted solution will be integer multiples of
the small units (of space, .time, mass etc).
- The universe and everything in it will be
finite. This would apply also to
quantum states
- A simulated universe made of the same
stuff as the universe would be inefficient and we should expect that such
a computer would be made of other “stuff”, which we can call computrons. Computrons
would not be subject to the laws of the known universe.
As an afterthought on decoherence,
the computability requirement would leave its mark on this too. Because computable quantum states would need
to be finite in number, then so too would entangled ones. Assuming that the total permitted number of
states (the finite number) is a constant, entangled quantum states would become
more and more restrictive. The most
improbable states would become increasingly improbable (there are more of these
states so that if the total is fixed the more improbable states would be the
big losers). So a collection of quantum
objects would not suddenly decohere as is the current
expectation, but they would slowly become less and less quantum-like as the
number of entangled objects increased.
Decoherence by interaction with a non quantum object would
simply be the result of an entanglement of a quantum object with one that has
massively restricted options (the non-quantum object) When the entanglement
occurs the quantum object becomes part of the “non-quantum” object and the
restriction that the total number of
probabilities be a constant would rob the quantum object of its quantum
nature.
Thus a computable
world would be detectable. The
finite-ness of it would give it away.