Cellular Automata: how I discovered the topic
In 1987 we were finishing second year of physics. Boring days, so the
classroom started to play dots and strips. I went to the library to look
for some recipes to win the squares, and I drew the bible of gaming,
"Winning Ways for your Mathematical Plays", from Conway et al.
Of course I already know of computing and Life. But we were surprised by
the variety of this CA and specially by the particles, the sliders, whose
speed is (inversely) proportional to their mass. I asked to my geometry
teacher -who plainly told he was too old to understand it, and suggested
me to keep notes for the future- and I discussed with my friends in the
classroom. Of course Life is not conservative, but for sure there were
another.
Some months later we surrender. Nobody was able to find out how a
continuous limit of a cellular automata could be defined. And you need
it, seriously. Ask the people of lattice gauge theory about fermions, for
instance. Or ask Zeno about changing reference frames. Or Haag about
locality. Or... well, you need the continuous limit, seriously.
Mr. Feynman speaks about someone trying to open a locked box. People
comes and ask, "what about trying 20-54-56?". And well, perhaps you are
trying to proceed systematically. Perhaps there is a reason ruling out
this combination. Perhaps it is not a combination box, and the casual
bypasser has not noticed it. And perhaps we have already tried it.
I found that we have already tried it, or at least the simple
combinations. Forget Wolfram, who makes one wondering about the criteria
of the Physical Review referees. But take Filkenstein and his Space-Time
codes. Take the effective action approach, the lattice theorists and all
the believers in a fundamental cutoff of the Theory. Take Feynmann
himself: when he was student, he tried to approach discrete derivatives
by a summation of spinors (or so; never published anyway). Take all the
people trying to relate discrete derivatives with quantum mechanics; even
with, and this is better, with deformed calculus.
In the mean time I finished the graduate. I become interested on Local
Quantum Physics. C* Algebras
In Karpatz I learned of a discrete model of space, the two-points non
commutative geometry. A mix between continuous and discrete. In Leipzig I
heard of the full theory of geometry, but I was unable to write two lines
in my notebook
A friend come to Zaragoza to explain us the Connes-Lot model. There was
possible to work at the same time continuous and discrete, and best of
all, the discrete part was coding particles!
In Barcelona a short workshop joined Mandelbrot, Connes and Asthekar.
Someone pushed Connes about the discrete model which should be got in the
high energy limit. He asked tautologically, "it could be a quantum
logic".
In Les Houches (1995) Alain told us about the groupoid joining
infinitesimal calculus and finite differences. And he show us that the
continuity condition in the groupoid topology was exactly the
quantization of phase space.
In Vietri, we were surprised by a hopf algebra in the Renormalization
Group. At that time I returned to speculate about fermions and
differential systems. Now we know there are four fermions; it is a point
unknown to the founding fathers, and even to the young Feynmann.
One year later we learnt that the Hopf algebra in RG was the same that
the Hopf Algebra in Runge-Kutta integration. So perhaps there was a link
between QFT and discrete calculus, after all!
But then the sea of string theory started to rise, pervading everything,
sucking everything.
Do no mistake me; a CA-like system, a logic, a discrete system, could be
the basis of the Theory; it is not ruled out. But the observations must
be formulated in the continuous space, the one we are able to perceive
"with the eyes of reason".
But if your perception fails, look it in this way: the Theory must include
quantum theory. If there are a quantum regime, there are a continuous
limit: the one you get taking h->0. Classical fields. Or classical
mechanics, if you take, before, the one-particle, low energy, limit. And
even without the limit, the quantised theory has a lot of properties from
the continuum. So there are at least a way to get a continuos theory, and
if we are to start from a CA, such way is the definition of a continuous
limit for a CA.
My current guess is that some theory is renormalized, then a renormalization
scale appears, h, and a fixed point does exists, the classical limit,
when h goes to zero. You can call Cellular Automata to the bare cut-off
theory, if you want. You cal also call it discrete mechanics, or even
runge-kutta mechanics. But the real prize will be to describe the scaling
and renormalization process. To know how to open the box is more
difficult that just guessing the key.
Sirs, if you want, there are work to be done!