Ark & Tony Smith on cellular automata for physics

[John G]

This continues a discussion Archaea and I were having here:

http://cassiopaea.org/forum/index.php/topic,40894.msg632058.html

Ark from his blog (via Google translate):

I mentioned in the previous note, that our suspicions in recent years are beginning to focus on cellular automata. Perhaps this is a better model, better testing platform for both physics and biology, better platform than eg. Linear differential equations... The simplest cellular automaton is a machine built with cells arranged in a line. In the simplest machine, each cell can only be in one of two states. Let's call it 0 and 1... I look left, look right,... What can I see? What arrangements? such:

110 101 100 111 011 010 001 000

We agree on just such a sequence. Eight opportunities. The rule says what is to be my state after each of these eight possibilities. It will be 0 or 1... Therefore, there are so many rules as there are eight digit numbers in the binary system, that is 2 to the power of eight, that is, 256...

One question, however, arises immediately, thanks to the experience of the previous notes about the blurring and sharpening images: Of the 256 different machines, if evolution is reversible and how much irreversible?...

Because this is a reasonable number, namely 256... In previous posts BJAB suggested dealing with two-dimensional slot machines. The first version of the environment was a cross 5-membered. United ambient stitch is 2 5 = 32. Slot 2-dimensional cross stitches, with two states is 2 32 = 4,294,967,296. In these discern it would be much difficult...

Edward Fredkin (b. 1934) - a scientist, a professor at Carnegie Mellon University (CMU) and an early pioneer of digital physics. The greatest merit won in the field of research reversible account computer and cellular automata... Well Fredkin had the idea to force the machine to the reversibility of telling him to extend the memory of the extra step backward... Brilliant! My future depends not only on my present, but from my recent past...

So that the "yes" and "no" can be true at once. I reach out to the logic of trivalent Lukasiewicz... Dialectic like. Bows idealist Hegel: Truth and falsehood fuse with each other, are not assertive opposites... Only in Euclidean metric subspace and its orthogonal complement have trivial intersection. When, dialectically, we describe the "becoming", the starting point should be the phase space. Metric in phase space is symplectic, and its form symplectic subspace and its orthogonal complement can have a non-trivial intersection! So I'm at home. I like Hermitian metrics, and the imaginary part of the hermitian metric is a form of symplectic. And at Kreidika each object is part of the material and the spiritual part. A part of the spiritual is the imaginary unit, the square root of -1. Everything is correct! Hence the composite wave function. Quantum mechanics nobody understands, which does not mean that it is useless...

Somebody, don't remeber who, once said it's a good thing the last bit in the universe doesn't radically change physics since there would be no hope of being able to understand it. There has to be something that limits the size of the monad to something reasonable like 28=256. Adding memory (perhaps even of the future) and trivalent (law of three-like) logic and other things like symmetry breaking at low energy compared to high energy can make things difficult to understand again but that's still much better than having a ridiculously huge monad to go through (like string theory's 10500 anthropic landscape vacuum states).

So what's a monad? For a discrete rather than continuous spacetime, it could be what's at a spacetime vertex. Loop quantum gravity (LQG) is the most well known discrete spacetime model. Tony and LQG's John Baez once agreed that non binary 26 degrees of freedom E6/F4 would be nice at a vertex but it wasn't "foamy" so you couldn't build it into a spacetime. "Binary" vertices like in Feynman Checkerboards might make better spacetimes.

So what do Ark and Tony like for a monad? From Ark:

http://cassiopaea.org/forum/index.php/topic,27044.msg333558/topicseen.html#msg333558
http://cassiopaea.org/forum/index.php/topic,27044.msg335018/topicseen.html#msg335018

Back to the fundamental groupoid - the atom or monad of the Universe: There are also four arrows... Our groupoid, consisting of a couple of points and of connecting them arrows, is just a frame, a scaffolding, and, as such, not very attractive to our minds. Therefore, in order to make our structure more “habitable”, we will cover it with a smooth structure – much like Bartholdi covered the skeleton of the Statue of Liberty with copper. Instead of copper we use another smooth material, this time of a mathematical nature, that is with numbers. Copper was the right material for the sculptor Frédéric-Auguste Bartholdi. We will use the right material for our construction complex numbers. The details of the construction are in Appendix. Here it is enough to say that our mathematical sculpture, based on the fundamental groupoid skeleton, when finished, has the shape of the algebra of 2x2 complex matrices:

[a b]
[c d]

where a,b,c,d are complex numbers, each having a real and an imaginary part. WE have four complex numbers, that is eight real numbers. Our complete construction spans therefore an eight-dimensional space!

From Tony:

www.valdostamuseum.com/hamsmith/SES02.pdf

Complex Clifford Periodicity
Cl(2N;C) = Cl(2;C) x ...(N times tensor product)... x Cl(2;C)
Cl(2;C) = M2(C) = 2x2 complex matrices
...
where the Real Clifford Periodicity is
Cl(N,7N;R) = Cl(1,7;R) x ...(N times tensor product)... x Cl(1,7;R)
...
Cl(1,7) is 2^8 = 16x16 =256-dimensional

So they both have an interest in a 28=256-dim monad besides just 256 being a reasonable number to work with. There's a Clifford algebra 8-dim periodicity. This means for example that Cl(8)xCl(4)=Cl(12) but Cl(7)xCl(5)≠Cl(12). Both Ark and Tony work with a conformal gravity that can access the imaginary part of a complex spacetime perhaps related to Ark's "So I'm at home. I like Hermitian metrics, and the imaginary part of the hermitian metric is a form of symplectic... part of the spiritual is the imaginary unit, the square root of -1". But can this imaginary unit structure fit with a 28=256-dim monad? Does the 28=256 of cellular automata even fit well with the 28=256 of Clifford algebra?

 

[Archaea]

I watched a couple of videos on Youtube explaining cellular automata, specifically Wolfram's 256, 1 dimensional basic CA's, as well as the game of life. The play list is here:

https://www.youtube.com/playlist?list=PLRqwX-V7Uu6YrWXvEQFOGbCt6cX83Xunm

I mentioned in the previous note, that our suspicions in recent years are beginning to focus on cellular automata. Perhaps this is a better model, better testing platform for both physics and biology, better platform than eg. Linear differential equations...

Is it possible to model physics with CA's? For example, can we model a basic Newtonian gravitational system using CA's? And if so, what's the simplest CA that can do that? Also is it possible to model waves and interference patterns with CA's? And if so, is the simplest CA that can do that simpler than the one that can model Newtonian stuff?

So that the "yes" and "no" can be true at once. I reach out to the logic of trivalent Lukasiewicz... Dialectic like. Bows idealist Hegel: Truth and falsehood fuse with each other, are not assertive opposites... Only in Euclidean metric subspace and its orthogonal complement have trivial intersection. When, dialectically, we describe the "becoming", the starting point should be the phase space. Metric in phase space is symplectic, and its form symplectic subspace and its orthogonal complement can have a non-trivial intersection! So I'm at home. I like Hermitian metrics, and the imaginary part of the hermitian metric is a form of symplectic. And at Kreidika each object is part of the material and the spiritual part. A part of the spiritual is the imaginary unit, the square root of -1. Everything is correct! Hence the composite wave function. Quantum mechanics nobody understands, which does not mean that it is useless...

What's Ark saying here?

I still haven't learned anything about Clifford algebra yet, I want to watch some Youtube lectures and maybe read the Wikipedia page. After that, due to my natural talent, I should be an expert... :P

My last question is: how do basic symmetries like rotations and reflections tie in with CA's? Are they just operations on an n dimensional unit hypercube? Or is it something else?

 

[John G]

The Feynman Checkerboard is probably the best place to start as far as seeing what cellular automata can do.

www.valdostamuseum.com/hamsmith/USGRFckb.html

The starting point is the observation that the left |-> and right |+> going states of the 1+1 dim checkerboard model can be labeled by complex numbers

|-> ---> (1 + i)

|+> ---> (1 - i)

...

this makes it look very natural to identify the imaginary unit appearing in the sum over paths with the "generator" of kinks in the path. To generalize this to higher dimensions, more square roots of -1 are added, which gives the quaternion algebra in 1+3 dimensions. The two states |+> and |-> from above, which were identified with complex numbers, are now generalized to four states identified with the following quaternions (which can be identified with vectors in M^4 indicating the direction in which a given path is heading at one instant of time):

(1 + i + j + k)

(1 + i - j - k)

(1 - i + j - k)

(1 - i - j + k)

...

So consider a lattice in Minkoswki space generated by a unit cell spanned by the four (Clifford) vectors... Now consider a "Clifford algebra-weighted" random walk along the edges of this lattice... Next, assume that this coupled system of differential equations is solved by a spinor Q

Q = Q' (1+y0)(1+iE)/4

E = (y2 y3 + y3 y1 + y1 y2)/sqrt(3)

...

This ansatz for solving the above system by means of a single spinor Q is, as I understand it, the central idea... For sure, every Q that solves the system of coupled differential equations that describe the amplitude of the random walk on the hyper diamond lattice also solves the Dirac equation...

My proposal to answer the question raised by Urs Schreiber Does every solution of the Dirac equation also describe a HyperDiamond Feynman Checkerboard random walk? uses symmetry.

The hyperdiamond random walk transformations include the transformations of the Conformal Group:

rotations and boosts (to the accuracy of lattice spacing);
translations (to the accuracy of lattice spacing);
scale dilatations (to the accuracy of lattice spacing): and
special conformal transformations (to the accuracy of lattice spacing).

Therefore, to the accuracy of lattice spacing, the hyperdiamond random walks give you all the conformal group Dirac solutions, and since the full symmetry group of the Dirac equation is the conformal group, the answer to the question is "Yes".

So you can model path integrals, the Dirac equation, and conformal group transformations (which give you gravity via the MacDowell-Mansouri action). Tony also gives tree level particle masses and force strengths via diffusion equations on the Checkerboard. The problem though is that an electron doesn't actually go through every possible path on the Checkerboard assigning imaginary units every time it zig-zags. It's too much math even for humans with computers. Computer generated particle masses/force strengths aren't any more accurate than Tony's simplified "tree level" diffusion equations. One can sort of get around the idea of electrons needing to do tons of calculating if you think of all information as a pre-existing mosaic that includes path integral-like patterns.

That trivalent, Hagel fused truth and falsehood logic includes a quantum-like 0 and 1 at the same time. It also includes kind of the real and the imaginary (math-wise) at the same time and you can see this via Checkerboard phases with the imaginary units or via the conformal transformations (which can access the imaginary part of complex spacetime). It gets spiritual/teleological in the sense of the future drawing the past or even communicating with the past. Path integral calculations include paths that zig-zag through the future.

Yeah you certainly get into the detailed math calculations more than I do. Following you tends to involve lots of Wikipedia reading for me. Like I mentioned before I tend to just stick to the general group theory structure. I even mostly think of Clifford Algebra in terms of group theory root system relationships. I can do say Cl(8) x Cl(4) to get Cl(12) in terms of the graded algebra dimensions but this is a rather simple matrix multiplication-like operation.

So how does elementary cellular automata help? It like the Feynman Checkerboard and its Cl(8) vertices is a 28 thing but it's not obvious how it relates. To me it seems like it might be useful for showing how the 8 Clifford Algebra basis vectors pair up to form the four axes of the D4 Lie algebra root system. Maybe it can do more but that's way beyond me and my mostly root system based approach.

 

[Archaea]

The Feynman Checkerboard is probably the best place to start as far as seeing what cellular automata can do.

www.valdostamuseum.com/hamsmith/USGRFckb.html

I'll give this a read.

I watched a Youtube lecture on Clifford algebra and I read some of the Wikipedia page, and alas, I'm not an expert I think I missed a few steps because if you look at the construction of quaternions on the Wikipedia page they say the Clifford product is:

But as far as I can see nowhere on the page do they define what the product of two vectors vw and wv is, which seems like an oversight to me.

Anyway, I've got some studying to do if I want to understand this stuff, which is probably going to take a while.

I've been thinking a fair bit about CA's as well, I might post some of my thoughts later.

 

[John G]

Where Wikipedia said "This formulation uses the negative sign so the correspondence with quaternions is easily shown" seems to be a redefinition to me but via this link:

https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction

and using their equations for Clifford multiplication, dot product, and wedge product gives:

a*b + b*a = 2(a.b)

 

[Archaea]

Where Wikipedia said "This formulation uses the negative sign so the correspondence with quaternions is easily shown" seems to be a redefinition to me but via this link:

https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction

and using their equations for Clifford multiplication, dot product, and wedge product gives:

a*b + b*a = 2(a.b)

That article was helpful, thanks. The vector product makes sense now.

FWIW, the Wikipedia page for the Exterior product is here.

 

[Archaea]

Just an update, I'm still studying this stuff, just finished watching some lectures on Lie algebra on Youtube and I found this pdf on Google which I want to skim through:

Hans Samelson: Notes on Lie Algebras

Next thing to study after this is external algebra, and then Clifford algebra. It'll probably take a while, I'll try to give updates every now and then.

 

[John G]

So how does elementary cellular automata help? It like the Feynman Checkerboard and its Cl(8) vertices is a 28 thing but it's not obvious how it relates. To me it seems like it might be useful for showing how the 8 Clifford Algebra basis vectors pair up to form the four axes of the D4 Lie algebra root system. Maybe it can do more but that's way beyond me and my mostly root system based approach.

I created an elementary cellular automata rule partitioning and wrote a paper based on pairing up the Clifford Algebra basis vectors (aka cellular automata bits) to form the root system axes (aka Cartan subalgebra).

http://vixra.org/pdf/1611.0030v2.pdf

 

[Archaea]

I gave up on lie algebra, it was too hard. I had a bit of trouble understanding the point of it all, and how the algebra relates to the groups in a meaningful way. So I still don't really understand any of this stuff, but it looks like it might be similar to Dr Paul A LaViolette's model of a reaction-diffusion ether. I'd be interested to know if cellular automata is understandable in terms of general systems theory and if there exists a cellular automata that does the same thing as Dr LaViolette's model.

Also, just a random question, you said you did some work with cellular automata, do you know if they can be used to solve problems? So, given an input string and a set of rules can a cellular automata give a meaningful output?

 

[John G]

I gave up on lie algebra, it was too hard. I had a bit of trouble understanding the point of it all, and how the algebra relates to the groups in a meaningful way. So I still don't really understand any of this stuff, but it looks like it might be similar to Dr Paul A LaViolette's model of a reaction-diffusion ether. I'd be interested to know if cellular automata is understandable in terms of general systems theory and if there exists a cellular automata that does the same thing as Dr LaViolette's model.

Also, just a random question, you said you did some work with cellular automata, do you know if they can be used to solve problems? So, given an input string and a set of rules can a cellular automata give a meaningful output?

The Lie group and Lie Algebra are very closely related to the point where I tend to think of them as the same thing.

https://en.wikipedia.org/wiki/Logarithm_of_a_matrix
https://en.wikipedia.org/wiki/Lie_group%E2%80%93Lie_algebra_correspondence

You can have a Lie group/algebra for gravity and a Lie group/algebra for the Standard Model and then you can look for a bigger Lie group to handle both and then think about what else from the large group exists when you do a symmetry break down to the smaller groups. The ether/zero point energy/Compton radius vortex particles-type things are for me conformal group (D3 Lie algebra) things. With the conformal group, you have to think of spacetime as having complex not just real dimensions so unusual compressible ether kinds of things are possible. For physics, cellular automata would be a Clifford algebra thing aka the large Lie group E8 can be a symmetry break from the elementary cellular automata Cl(8).

That Cl(8) is at the nodes of a Feynman Checkerboard which has the conformal group as a symmetry group as well as the Standard model groups and the other structures that come with the symmetry break like spinor fermion creation/annihilation operators and position/momentum operators. It's similar to U(N) gauge theory on branes.

At IBM I worked with cellular automata for image processing for optical test of raw circuit boards. Image processing is probably still the main use for cellular automata.

https://arxiv.org/ftp/arxiv/papers/1407/1407.7626.pdf

One think I'm trying to do recently is to think of the Feynman Checkerboard's cellular automata in terms of elementary cellular automata. Wolfram in general has tried to think of cellular automata on links between nodes of a network spacetime.

http://blog.stephenwolfram.com/2015/12/what-is-spacetime-really