A New Kind of Science: The NKS Forum > Pure NKS > Feynman checkerboard as a model of discrete space-time
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Ed Hanna
retired/self employed
Florida, USA

Registered: Jun 2006
Posts: 10

Feynman checkerboard as a model of discrete space-time

I've read Stephen Wolfram's NKS book, and it is *very* impressive.

I took particular note of two passages from the book.

In chapter 8, section 1 (Issues of Modeling), page 365, he writes:
"It is usually a good sign on the other hand if a model is simple, yet still manages to reproduce, even quite roughly, a large number of features of a particular system. And it is an even better sign if a fair fraction of these features are ones that were not known, or at least not explicitly considered, when the model was first constructed."

And in chapter 9, section 5 (Ultimate Models for the Universe), page 469:
"But it also means that if one once discovers a rule that sufficiently many features of the universe, then it becomes extremely likely that this rule is indeed the final and correct one for the whole universe."

As Tom Lehrer once said, "I have a modest example here", that I think meets both of Stephen Wolfram's criteria quoted above. (This is a conceptual paper that may be just a crackpot theory, or it may be a useful model of discrete space-time, depending on your point of view.)

The model has two basic rules:
1. Space-Time is a lattice of discrete quanta. (A checkerboard is an example of a 2D space-time with 1 dimension of space and 1 dimension of time.)
2. In each row (tick of time), a unit particle (one filled cell) moves to the next cell (or not) depending on an assigned probability. (If we assign a probability of 50%, we can use normal dice, where an even number means a move and an odd number means no move.)

The following example illustrates seven rolls of the dice with a standard checkerboard, starting at the top left corner, and then making seven dice rolls: 6, 9, 8, 4, 5, 7, 7.

..|_1_|_2_|_3_|_4_|_5_|_6_|_7_|_ 8_| Position
1 |_x_|___|___|___|___|___|___|___|
2 |___|_x_|___|___|___|___|___|___|
3 |___|_x_|___|___|___|___|___|___|
4 |___|___|_x_|___|___|___|___|___|
5 |___|___|___|_x_|___|___|___|___|
6 |___|___|___|_x_|___|___|___|___|
7 |___|___|___|_x_|___|___|___|___|
8 |___|___|___|_x_|___|___|___|___|
Time

That's about as complicated as it gets, so it seems to meet the criteria of NKS page 365, quoted above. There is certainly no explicit consideration of Special Relativity or Quantum Mechanics. And yet, even though the model is very simple, it can exhibit complex behavior that seems to mimic / duplicate: (1) a maximum speed of light, (2) which is a constant from any frame of reference, (3) the Lorentz equation for the addition of velocity, (4) Heisenberg's uncertainty principle, and (5) wave / particle duality.

These 5 emergent properties may not qualify as "sufficiently many features", but I would propose that they are important *key* features of both Special Relativity and Quantum Mechanics, and thus seems to meet the criteria of NKS page 469, quoted above.

Revised paper with two additional figures (and link to computer simulation) has been posted on arXiv at http://arxiv.org/abs/cs.CE/0607018

Regards,
Edward Hanna
stq50@yahoo.com

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07-13-2006 07:13 PM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Regular lattices do not readily reproduce the observed invariance of physical laws with respect to reference frame. They have preferred directions. In your example, with diagonal movement allowed, a particle that moves constantly along the diagonal effectively goes farther than one that alternates movements along the x and y axes. On a hex grid the divergence from uniformity is less, but still present.

This is one of the reasons Wolfram argues in the book that no simple, direct translation of CAs will correctly capture fundamental physics. CAs on regular lattices have too much built in structure. Hence the shift to networks, which can reproduce different effective curvatures and let them vary with position, and need not have any preferred directions.

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07-13-2006 10:41 PM
Ed Hanna
retired/self employed
Florida, USA

Registered: Jun 2006
Posts: 10

Dear Jason,

Thank you for your reply. I understand what you are saying, and I may have a satisfactory answer for you.

The problem of anisotropy is sometimes stated that a continuum has no preferred directions, while a regular lattice surely does. Or that particles in a lattice should go farther (or faster) in certain preferred directions compared to others, which is not observed in nature.

I also want to state for the record that nature is whatever it is, and it is up to us to try to determine what it chose to be. If nature chose a space-time continuum, then I am a crackpot and doomed to failure. If nature chose a discrete space-time lattice, then it would be very cool to figure this out.

The short answer to the question of anisotropy is that the maximum speed in different directions within a discrete grid is always one unit of position per unit of time. After say, three ticks of time, a particle with constant motion will always be three cells away from its origin, so by definition it has traveled the same distance (no farther) at the same speed (no faster) in all directions, so I don't think the anisotropy question really has to do with distance or speed per se.

I think the question of an anisotropic space-time model is really about the shape of a light-sphere, or in a flat drawing, a light-circle.

This requires a visual aid (I love visual aids). The easiest piece of a discrete space-time lattice to draw here on the NKS Forum would be a 2-D slice of space-time showing two dimensions of space. A 2-D flatland can be tiled with triangles, squares, or hexagons, and it's easiest to draw squares, so let's discuss 2-D anisotropy in terms of a lattice of tiny squares, even though 2-D nature would probably have chosen a more symmetrical lattice of hexagons, rather than squares.

Figure 1. Deterministic light circle
...|_1_|_2_|_3_|_4_|_5_|_6_|_7_|_8_|_9_ X-axis
1 |___|___|___|___|___|___|___|___|___|
2 |___|_x_|_x_|_x_|_x_|_x_|_x_|_x_|___|
3 |___|_x_|___|___|___|___|___|_x_|___|
4 |___|_x_|___|___|___|___|___|_x_|___|
5 |___|_x_|___|___|_o_|___|___|_x_|___|
6 |___|_x_|___|___|___|___|___|_x_|___|
7 |___|_x_|___|___|___|___|___|_x_|___|
8 |___|_x_|_x_|_x_|_x_|_x_|_x_|_x_|___|
9 |___|___|___|___|___|___|___|___|___|
Y-axis

If a unit particle originates at position (5,5), shown as an "o" in the lattice above, and if it were allowed to move in any direction, including diagonally, then after three deterministic moves it would be in one of the cells marked with an "x". This appears at first to be obviously not much of a circle, and hardly isotropic.

The paper on arXiv proposes that motion is probabilistic rather than deterministic. This results in a probability mapping of the final results, rather than a deterministic result as shown above. A probabilistic light circle (see Figures 2 and 3 below) will be a lot fuzzier than a deterministic light circle (see Figure 1 above).

If a unit particle originates at position (5,5), shown as an "o" in the lattice below, and if it is allowed to move probabilistically to the (R)ight, or (D)own, or (B)oth, with equal probability, i.e., in all directions with equal probability, then after three probabilistic moves it would have followed one of 27 possible paths:

01. RRR
02. RRD
03. RRB
...
25. BBR
26. BBD
27. BBB

Figure 2. Plotting these 27 positions gives the probability map of a quarter light circle.
...|_1_|_2_|_3_|_4_|_5_|_6_|_7_|_8_|_9_ X-axis
1 |___|___|___|___|___|___|___|___|___|
2 |___|___|___|___|___|___|___|___|___|
3 |___|___|___|___|___|___|___|___|___|
4 |___|___|___|___|___|___|___|___|___|
5 |___|___|___|___|_o_|___|___|_1_|___|
6 |___|___|___|___|___|___|_3_|_3_|___|
7 |___|___|___|___|___|_3_|_6_|_3_|___|
8 |___|___|___|___|_1_|_3_|_3_|_1_|___|
9 |___|___|___|___|___|___|___|___|___|
Y-axis

This is very different from the deterministic light circle shown in Figure 1. Note that any particles that happen to go straight down will wind up at position (5,8) with 100% probability, but that any particles that happen to move along the diagonal (down and to the right) are likely to wind up at (7,7) about 86% of the time, and at (8,8) only 14% of the time. Note also that the vertical distance from (5,5) to (5,8) is 3.00 units, while the diagonal distances from (5,5) to (7,7) and (8,8) are about 2.83 and 4.24 respectively. This means that the average distance traveled in the diagonal direction will be 86% of 2.83 + 14% of 4.24 = about 3.03 units - very nearly the same as the 3.00 units in the vertical direction!

If we mark these 100% vertical and 86% diagonal cells for all four quadrants, the probabilistic results begin to look much more like a real circle:

Figure 3. Highlights of a probabilistic light circle.
...|_1_|_2_|_3_|_4_|_5_|_6_|_7_|_8_|_9_ X-axis
1 |___|___|___|___|___|___|___|___|___|
2 |___|___|___|___|_x_|___|___|___|___|
3 |___|___|_x_|___|___|___|_x_|___|___|
4 |___|___|___|___|___|___|___|___|___|
5 |___|_x_|___|___|_o_|___|___|_x_|___|
6 |___|___|___|___|___|___|___|___|___|
7 |___|___|_x_|___|___|___|_x_|___|___|
8 |___|___|___|___|_x_|___|___|___|___|
9 |___|___|___|___|___|___|___|___|___|
Y-axis

If we keep in mind that the space-time lattice is likely to be at the Planck scale (as derived in the paper posted on arXiv), and that the lattice may be hexagons rather than squares, and that any measurement we can perform in the real world would involve a light circle many, many, many orders of magnitude larger than these grids, then it would become exceedingly difficult to detect the anisotropic deviation from a perfect circle.

Regards,
Edward Hanna
stq50@yahoo.com
www.geocities.com/stq50

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07-14-2006 11:09 PM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Yes, I got it the first time, but it doesn't deal with the problem. When you smear out the motions, you spread away from the maximum possible speed. Motions along the preferred lattice directions can be as low probability as you like, but they will still appear superluminal when your idea of light speed comes from the emergent circle instead of the micro-lattice. You can take your pick. You either predict preferred directions of motion or low probability but possible superluminal speeds in preferred directions. And neither is observed. Nature didn't do it that way, and the idea just doesn't work.

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07-15-2006 04:08 AM
Joshua
N/A
Chicago,IL

Registered: Jun 2006
Posts: 10

Relationship

That is EXACTLY what I did in my paper! I took QED and used it as a Feynman checkerboard. :-) I had to rewrite paths in Feynman diagrams into CA but it was worth it.

Attachment: qed-ca-1.pdf
This has been downloaded 1450 time(s).

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07-16-2006 04:51 AM
Ed Hanna
retired/self employed
Florida, USA

Registered: Jun 2006
Posts: 10

Dear Jason,

Hmmm, I see what you are saying, and I agree up to a point - I'll have to ponder on this.

(I am also wondering if a light-circle might look different to an observer inside the discrete space-time, than it does to an observer like us looking at it from outside.)

Sorry for the delayed response.

Regards,
Ed Hanna

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07-18-2006 02:01 AM
Ed Hanna
retired/self employed
Florida, USA

Registered: Jun 2006
Posts: 10

Dear Jason,

While still pondering the serious obstacle of isotropy, I have a somewhat unrelated question.

Are there very many 2D or 3D CA that you know of where the CA grid itself bends or curves in response to the evolution of the CA? (Rather than acting as a passive stage where all the CA action is taking place.)

The reason I ask, is that I'm investigating a 3D CA model of rhombic dodecahedral unit cells (as predicted in NKS page 930). This model seems to mimic / duplicate the real universe where the curvature of space has to do with mass and gravity.

Regards,
Ed

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07-20-2006 04:39 PM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Rhombic dodecahedrons don't bend or change at all, any more than cubes do, in reaction to anything. They are just a different crystal structure for the cells, but a fixed one. With their own anisotropies, of course.

If instead you make the connections or regions dynamic, it isn't a CA anymore, but something else. Cellular Automaton is not a free-floating signifier for any discrete model, but a specific type, and in that type the structure of connection is fixed, while the contents within it change state. The network rewrite systems Wolfram proposes for fundamental physics change their connection - but they aren't CAs, they are networks.

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07-20-2006 06:12 PM
Ed Hanna
retired/self employed
Florida, USA

Registered: Jun 2006
Posts: 10

Dear Jason,

Thank you for being patient, and continuing to reply to my posts - I appreciate the feedback.

I don't want to appear pig-headed, and I do understand what you are saying about the advantage of network systems over a rigid cell structure, but I am following a rather tenuous trail of my own with an interesting payoff.

And I don't mean to be a pest, but what I had in mind when I posed the question about CA that bend was something much simpler.

To take the simplest example I can think of, suppose that you ran CA rules 50, 222, and 84, as shown in NKS pages 55-56, with the added requirement that whenever a certain configuration occurs, say two adjacent black squares, then the common edge between the two black squares creases by one degree. Meaning that the two black squares no longer lay flat at 180 degrees, but lay with a 179 degree angle between them. (The squares themselves do not bend or change at all.)

At the end of 100 steps, rule 50 space-time will still be completely flat, since there are never two adjacent black cells.

At the end of 100 steps, rule 222 space-time will be quite curved, with 100 tiny creases totaling 100 degrees.

At the end of 100 steps, rule 84 space-time will be mostly flat, with a 1 degree "furrow" moving off to the right, as if a 2-square pseudo-particle were carrying a bit of curved space-time around with it.

This allows us do some interesting things reminiscent of a quote on http://en.wikipedia.org/wiki/John_Archibald_Wheeler "Matter tells space how to curve. Space tells matter how to move."

Regards,
Ed

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07-22-2006 04:22 PM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Since angle in whatever sense you are talking about has no effect on the evolution of the CA - which is on its own completely one dimensional, incidentally - it will evolve just as it always does. You can total any feature of it you like and ascribe any meaning you like to such totals, but they won't feed back into the actual evolution of the CA in any way, because there isn't any room for them to do so in the CA's rules. Nor do black cells correspond to "matter" in any sense, since they aren't conserved etc. Wheeler was simply describing some basic features of general relativity in English words. I see no "payoff", nor even any coherent proposal or thought, in what you describe.

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07-22-2006 08:12 PM
Ed Hanna
retired/self employed
Florida, USA

Registered: Jun 2006
Posts: 10

Dear Jason,

I shouldn't have implied any similarity to a real CA. To a non-expert such as myself, a CA is anything with a regular lattice of unit cells that evolves over time according to set rules.

Once you adopt a 3D lattice of unit cells, add a rule that conserves black cells, add a rule that appropriately bends/folds (i.e., curves) the 3D lattice of unit cells (the unit cells themselves do not bend or change shape at all, as in the 1D example given above), and add a rule about moving along the geodesic of the resulting curved space, then it would probably not be a CA per se, but I believe it yields an interesting payoff in terms of a few key emergent properties that mimic / duplicate general relativity, where the curvature of space has to do with mass and gravity.

I am currently describing these 3D results in a companion paper to the one that started this thread. That first paper purports to show that a simple model of discrete space-time can exhibit complex emergent behavior that that mimics / duplicates a few key features of both special relativity and quantum mechanics.

The (crackpot) implication is that SR, GR, and QM might all be emergent properties of a single underlying discrete space-time structure with perhaps six (by my count) very simple rules.

Of course, if nature chose to make space-time a continuum, then any efforts along these lines are doomed to failure. If, on the other hand, nature chose to make space-time discrete, this would be a very cool thing to figure out.

Still considering the serious issue of isotropy,
Ed

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07-25-2006 12:59 PM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Obviously if you want curvature in any meaningful sense, yes it changes the cells, which no longer remain ordinary cubes in r3. Lengths change etc.

And no, your scheme won't fit if nature is discrete and fail if it isn't, it will also fail if nature is discrete but isn't using distorted cubes as its primitives.

It is too naive a picture. It has been considered long since and found wanting. That is the reason Wolfram instead proposes a network, not a CA - see in particular the section starting on page 475. It is easy to get a curvature tensor along with discreteness - but it is a regular cubic lattice that goes.

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07-25-2006 02:33 PM
Ed Hanna
retired/self employed
Florida, USA

Registered: Jun 2006
Posts: 10

Dear Jason,

Thank you for your reply and continued patience.

I had previously thought that a model of discrete space-time could flex / fold *between* the unit cells, while the unit cells themselves could remain unchanged, but I see your point about "curvature in any meaningful sense", and I now understand the advantages of a network, over a lattice of unit cells. (Even though I might visualize it in terms of unit cells, - connecting the midpoints of those unit cells should result in the corresponding network.)

And no, I didn't mean to imply that the proposed model is the only way to address discreet space-time. When I wrote "If, on the other hand, nature chose to make space-time discrete, this would be a very cool thing to figure out", I meant to figure out in any sense, using anyone's model, or any combination of models.

I agree that the model is very naive (I would have used the word "simplistic"), but I am nevertheless stunned by the ability of the model to mimic / duplicate those five key features from *both* special relativity and quantum mechanics, even in spite of the issues of isotropy that you brought up in your first reply. (Still pondering on that.) The model also suffers from being presented only in terms of conceptual issues. Wolfram captured this issue very neatly on page 1043 of NKS ( http://www.wolframscience.com/nksonline/page-1043 ) when he wrote "particularly in making contact with existing physics it is almost inevitable that all sorts of technical formalism will be needed".

The companion paper I am writing is based on a 3D geometric model of discrete space-time. The proposed uncurved state is not a "regular cubic lattice" as you mentioned in your last post, but rather a lattice of rhombic dodecahedra, one of the five unit cells suggested on page 929 of NKS ( http://www.wolframscience.com/nksonline/page-929h-text ). This uncurved state then flexes / bends at logically appropriate places, resulting in a few key emergent properties that mimic / duplicate general relativity, where the curvature of space has to do with mass and gravity. In terms of our discussion here, this curved state would be best represented by the corresponding network of nodes obtained by joining the midpoints of the rhombic dodecahedra.

Regards,
Ed

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08-07-2006 01:44 PM
Lawrence J. Thaden

Registered: Jan 2004
Posts: 350

Which is it: Network or CA?

In this thread Jason Cawley wrote:

“Wolfram … proposes a network, not a CA”.

In another section of this forum Gershom Zajicek M.D. of Hebrew University of Jerusalem at Jerusalem, Israel
wrote:

“Neural nets are still poor models for processes encountered in the organism. We are left with CA.”

Is it unreasonable to expect that one type of model will be sufficient for both physical and living phenomena?

So which will it be?

__________________
L. J. Thaden

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08-07-2006 06:21 PM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

Network does not equal neural network. It is just the generalization of locality of connection, allowing what is next to any location to vary from location to location instead of being a fixed pattern repeated throughout.

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08-07-2006 06:57 PM

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