25th
January, 2013.
The quantum cat map
Reminiscing B. Chirikov and J. Ford
(Dedicated to Giulio Casati)
A short sentence
haunted me since yesterday “memories behind the memories....”. My memory failed me because I cannot remember
where I heard or read this sentence. It
sounds like something from a surrealist play, maybe it was from James Joyce Ulysses
but it could equally be something from Samuel Beckett, Eugene Ionesco or even
Marcel Proust searching for the time lost.
I suspect, in fact it must be that all this restlessness is because of
what I learned from my good friend Professor Giulio Casati about the passing
away of our elderly friend and teacher, the great Russian academician and
quantum chaos pioneer Boris Chirikov. Of
course Boris died a while ago but I was not aware of it so I am experiencing
this loss as if it just happened. Jean
Paul says memories are a paradise from which man cannot be evicted. I suspect that it could be the opposite as
well. However on this occasion my
memories wandered to a truly beautiful story which Giulio recounted to me from
a time when the three friends J. Ford, Boris Chirikov and Giulio were almost
continuously working together. Joseph
Ford was extremely interested, if not obsessed, at the time by revising quantum
mechanics and find its Achilles heel.
The context of his attempts was the new and vibrant field of quantum
chaos which was pioneered by Gutzwiller, Chirikov, Ford, Casati and Michael
Berry. Many criticized Ford that he
wanted to discredit quantum mechanics, one of the two main pillars of modern
theoretical physics using a simple and almost trivial toy model such as the cat
map, originally invented by the famous Russian mathematician V. Arnold. The three friends or three musketeers if you
want always had ongoing discussions and Ford must have felt that his close two
friends are not happy with his passion against orthodox quantum mechanics
particularly when it was based on what they perceived as whimsical
reasoning. Such persuit would inevitably
draw unwarranted criticism and opposition which would cause a distraction and
slow the pace of progress of their main work which was to build and complete
the new field they were working on, namely quantum chaos. Ford must have felt that strongly because
one day Giulio received a letter from Ford in which he spoke his mind about
this and that particularly his doubt that quantum mechanics was complete. One single sentence however convinced Giulio
and I must say also convinced me as a universal justification valid for all
people like Ford. Of course I do not
remember the exact wording of the sentence and I do not want to inconvenience
my good friend Giulio by requesting him to send me a copy of this years old
letter. At the end of the letter Ford
wrote
“I was
born to fly and even though I may not reach a destination,
at least I was
able to see very far....”.
I thought then
and still think now that the eloquency of this sentence does not stem mainly
from a literary gift of Ford nor of his mastery of the language of Shakespeare,
Milton, Keats, Yates and Shaw but it comes primarily across as superb because
it comes from a heart located on the right side and joined to a superior brain
and upright, courageous personality. I
did not know Joseph Ford personally and never met him but this is what comes
across through this sentence.
At the time when
I listened to this story in the beautiful terrace of Giulio’s home located on
the magnificent Lake Como, I never thought that I would one day be thinking
deeply on Ford’s quantum fascination let alone that I could resolve the
problem, at least for myself and my own satisfaction.
Today I feel, to
be honest I am convinced, that Ford was wrong but also right at the very same
time regarding quantum mechanics. Of
course the cat map was a simple almost trivial toy model. However this is not its drawback. The true drawback is that it is not
sufficiently simple nor sufficiently trivial.
When tackling fundamental problems searching for the very deep roots,
then one will be struck by how simple and trivial the deep roots are. The cat map has two main ingredients hiding
in it. First the Eigenvalues of the cat map
are the golden mean and its derivatives.
Second its dynamic is chaotic with Cantorian set structure. There are extremely simple random triadic
Cantor sets due to the American mathematicians Mauldin and Williams which
possess a Hausdorff dimension equal to the golden mean. The next step is that one should ask himself
what is the most amazing and slightly disturbing thing about quantum
mechanics? This is no doubt the same
thing which worried Schrödinger and Einstein more than anything else, namely
quantum entanglement or what Einstein called spooky action at distance. However this was in the past. In the meantime and following the work of Bell
and Aharenov the young Lucien Hardy gave an amazing exact solution for quantum
entanglement of two particles using orthodox quantum mechanics, a la
Dirac. He found that the probability of
quantum entanglement in this case is equal to the golden mean to the power of
5. This is almost 9.017%. This result was experimentally confirmed with
a very high degree of accuracy in many national laboratories and I showed that
Hardy’s result is generic. Thus quantum
entanglement is real and quantum mechanics is without any doubt correct and
complete. It only lacks intuitive visualization. The correct question to ask is how could we
see quantum mechanics geometrically? I
worked on this problem and realized almost immediately that the issue is
related to the so called measure zero of all Cantor sets. Let us consider a random Cantor set of the
Mauldin, William type. In constructing
the set we removed everything except of uncountably infinitely many random
Cantor end points. The sum of the length
of all these points is of course zero.
In a sense the interval which we use to construct this Cantor set does
not exist in the naive physical sense yet, and maybe amazingly so, this physically
nonexistent point set has a respectable finite Hausdorff dimension comparable
with the initial dimension of a line.
The initial dimension of the line of which we designed the Cantor set
was unity. The dimension of the
classically nonexistent point set is on the other hand, equal to the golden
mean, namely 0.618033989... . The
dimension could be regarded in a sense as the spirit of a body which long
departed from the here to the hereafter.
In fact one could say that the dimension and equally the Cantor point
set are the halo of a line or a unit interval which is no longer there. The important question to ask now is what is
the distance between the different Cantor set points? This is of course something varying between
one third of the original unit length and zero in the limit of infinite
iteration. On the other hand suppose our
original continuous line was representing a one dimensional spacetime geometry,
then our entire spacetime would have in this case no length at all although it
has a dimension. The distance between
all points of this Cantorian geometry of a Cantorian spacetime would be
zero. Classically of course this makes
no sense. Points separated by zero
distance are just another way of saying that these points are not separated at
all. In this sense we found something
marvellous indeed. We have infinitely
many points which have a dimension equal to a positive finite value, namely
0.618033989... and never the less we can see the same set as a single point
albeit a Cantorian point. Imagine life
in such a space. Although you have
uncountably infinitely many points in this world there is no meaning what so
ever for the concept of spatial separation.
This is very strange classically speaking but this is the essence of
quantum entanglement which in turn is the essence of quantum mechanics. Neither Einstein nor of course Schrödinger
knew anything about Cantor sets.
Einstein’s way of looking at things was to follow his teacher Herman
Minkowsky’s geometrical thinking.
Einstein could not envisage any geometry supporting such phenomena as
quantum entanglement and that is why he rejected quantum mechanics. My guess is that had Einstein known anything
about Cantor sets or at least presented with the theory of noncommutative
geometry he probably would have changed his mind and accepted quantum
mechanics.
From the above
we must conclude that the golden mean’s theoretical and experimental results of
Hardy’s quantum entanglement is a proof that the preceding mental picture of a
random Cantor set geometry for quantum spacetime is correct.
Joseph Ford was
born to fly indeed and although he did not reach a definite destination he saw
an important aspect of reality because he was able to see very far.
The story could
have ended at least for me at this point but it did not. Dark energy which in a sense is a halo of
ordinary energy wanted a continuation for Joseph Ford’s surge into space. The WMAP cosmological measurement as well as
the analyses of certain supernova events revealed with very high accuracy that
only 4.5% of all matter and energy presumed to be contained in the universe are
there. This result means that either
Einstein’s relativity is rather inadequate when applied to the entire cosmos or
alternatively that 95.5% of energy and matter in the universe is in a
mysterious form completely different from anything we have seen or
experienced. I looked very carefully
at this problem which is considered one of today’s main challenges in cosmology
and physics. It turns out that the problem
is connected yet again to quantum entanglement and that Einstein’s famous
formula E = mc2 should be extended to a quantum relativity formula E
=
mc2 ≈ mc2/22. That means Einstein’s formula must be
multiplied with half of Hardy’s quantum entanglement or what is approximately
equivalent to be divided by 22. The result in this case gives with astonishing
accuracy the correct measurement of WMAP and similar tests which use the entire
universe as a laboratory. The 2011 Nobel
Prize in Physics went to these measurements.
Although I made
these calculations myself, I am still in a state of disbelief that the laws of
nature could be so simple and so beautiful.
Eddington, Dirac, Ford and many similarly inclined scientists seem to
have had the right intuition all along.
(Dedicated to my
friend Giulio Casati in memory of Joseph Ford and Boris Chirikov.)
Mohamed S. El Naschie
25th January, 2013.
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