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Rotational Symmetry and Lorentz Symmetry

Posted 7 months ago
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Improved illustrations of the interweaving grid-graphs:

An example of how to achieve a graph from which a 2D space with Euclidean distance emerges at a large scale, and therefore rotational also symmetry:

Euclidean Plane Graph

Analogous example for a 2D SpaceTime with Lorentz symmetry:

Minkowski Plane Graph

I noticed that Wolfram said in a live-stream, that you need to have random-looking connections as opposed to a grid-like structure in order to achieve the Euclidean non-Manhattan distances. But I found that you could instead just use a very large number of interweaving grids that are connected to each other in such a way that they each represent different angles of rotation while maintaining the same scale, which can actually be achieved by simple local rules on graphs. The long range graph distances would approximate Euclidean distance as the number of grids A,B,C,D... is growing. Analogous constructions can be made for a Minkowski space time, where the different grids represent frames of reference of different boosts. A very large number of such interweaving grids (called A,B,C...) then analogously leads to approximate Lorentz distance on the large scale and hence also to Lorentz symmetry. They can be arranged as a tree of girds or as a grid of grids. Note that this leads to a density of nodes that could be an astronomical number per plank cube, while the length of the edges could just be the plank length itself. Such 2D spaces that already contain rotational and Lorentz symmetries could then be used as building blocks to generate a 4D space time, but this would be more complicated and I would not be surprised if quantum phenomena emerge from these complications.

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For our 3+1-dimensional space time, the Lorentz-group is six-dimensional: three dimensions for rotations + three dimensions for velocities = 6 dimensions. I found that there are extremely simple local rules based on which a very large number of simple 4-dimensional gird-graphs can be interlaced with each other in order to form a graph, such that the geodesic-distance of this graph perfectly approximates the Lorentz distance in 4D, resulting in the emergence of isotropy from discrete Wigner-rotations. Each of the grids corresponds to a point in the Lorentz-group. These points are arranged such that they completely fill out the six-dimensional Lorentz-group homogeneously. For an example of such a simple construction see my archived post: https://web.archive.org/web/20200922042112/https://community.wolfram.com/groups/-/m/t/1953906 This concept should find its way into the Wolfram physics project at some point.

Probably one can save the isotropic hard limitation on the speed of light, making resistances associates with timelike grid edges negative.

Maybe the isotropic hard limitation of the speed of light can be saved, if resistances associated with spacelike grid edges are positive, and resistances associated with timelike edges are negative.

I leave you a post that you might be interested in that might have a connection with your work. https://community.wolfram.com/groups/-/m/t/2027996

Blockquote If you want macroscopically isotropic measure on the square grid, it's simple. Imagine that your graph is a network of identical electrical resistances, and define the distance between two points A and B as a resistance between two leads connected to these points. When the points are far enough from each other, the resistance becomes a function of the Euclidean distance.

Yes, but then you wouldn't get the isotropic hard limitation on the speed of light, or how would you solve that otherwise?

Posted 4 months ago

That would obviously be right for a small number of grids, but the point of my post is that it would work with a very large number of interweaving grids, all pointing in different directions, constructed by simple local rules.

Posted 7 months ago

Given the transportation rule for distance, no gridlike representations of empty space would work. Instead a graph should be "sufficiently irregular" to produce isotropy on large scale.

Posted 7 months ago

Another guess: a wave equation modelled on a square grid looks more or less isotropic. Its finite-difference scheme can be seen as a sort of local update rule (and maybe combined with branching). However it implies storing values on graph nodes, and possibly has relativity issues.

Oh, so that's probably equivalent to Jonathan's model

If you want macroscopically isotropic measure on the square grid, it's simple. Imagine that your graph is a network of identical electrical resistances, and define the distance between two points A and B as a resistance between two leads connected to these points. When the points are far enough from each other, the resistance becomes a function of the Euclidean distance.

What do you think of my 4D version from the link above? Its rotational invariance emerges accidentally from boosts. Also do you think it could it be possible that its exponentially growing Hausdorff dimension (of sorts) is what provides the many parallel worlds that enable large molecules to be in multiple states/places at once?

Neat idea, but I'm not sure that I like the concept of having hypergraphs with such large edge density! (not least because it leads to an unbounded Hausdorff dimension for space...)

See my comment on this related post (https://community.wolfram.com/groups/-/m/t/1950834), or read my Q&A answer (https://www.wolframphysics.org/questions/spacetime-relativity/why-do-you-get-a-euclideanriemannian-metric-as-opposed-to-a-taxicab-metric-induced-on-your-hypergraphs/) for an explanation of how we actually derive Riemannian (and, indeed, Lorentzian) metrics in the context of our models - it is, unsurprisingly, not just the combinatorial distance metric. Or have a look through my relativity paper (http://wolframphysics.org/technical-documents/) for some more complete mathematical details :)

Blockquote Are you sure that the underlying topology is still a 2d grid. If you tie infinitely many grids together in that way. You loose the property of dimensionality as Wolfram defined it since a single step adds infinitly many neighbors making the dimension undefined.

Depending on how exactly you connect the grids, the number of direct neighbors per node can go towards infinity but it can also stay a small number despite the number of grids going towards infinity. But anyways, as far as I remember the way Wolfram defined dimensionality is based on how the volume of a ball grows w.r.t. its radius, where ball is a set of nodes that are at most at the graph distance of the radius and the volume is the number of nodes in the ball. But this is just one kind of definition. You could instead take a set of random nodes and measure all the distances between them. Based on this set of long range distances you can see how many dimensions there are. It's like a complicated trigonometry exercise. A simple example: take four points on the graph, measure the 6 distances. From this information construct a tetrahedron in a real 3D space. If the tetrahedron looks flat, then the graph probably represents a 2D space.

The ratios such as 1:3 indicate how Two girds are connected to each other such as A and B. Since arcsin(1/3) is irrational w.r.t pi you could just keep on using 1:3 to connect grid C to B and so on to keep the rotation going.

Ahh, i thought you varied the ratio to cover the space. But doing it with the same ratio would work too.

If you choose random starting points that aren't on the same grid you can't walk in parallel at maximal precision but neither can you move real elementary particles at sub-plack precision. You can however take a path through multiple different grids until you find a grid that approximately points in the direction you want to move which you can use for approximate parallel motion.

Are you sure that the underlying topology is still a 2d grid. If you tie infinitely many grids together in that way. You loose the property of dimensionality as Wolfram defined it since a single step adds infinitly many neighbors making the dimension undefined.

the number of edges you have to walk over

This part was not clear. As it also could have been weighted in some manner.

The ratios such as 1:3 indicate how Two girds are connected to each other such as A and B. Since arcsin(1/3) is irrational w.r.t pi you could just keep on using 1:3 to connect grid C to B and so on to keep the rotation going. Due to the irrationality, the orientations of the grids would approximately fill out all the angles from 0 to 2pi. But this is just an simple example, you can also let the ratio change as you noted.

In my examples only a minority of nodes is shared between grids for simplicity. But a further option would be to let the grids share half of their nodes or even the majority and you could in principle still achieve the desired distances and symmetries but it's more difficult to visualize.

If you choose random starting points that aren't on the same grid you can't walk in parallel at maximal precision but neither can you move real elementary particles at sub-plack precision. You can however take a path through multiple different grids until you find a grid that approximately points in the direction you want to move which you can use for approximate parallel motion.

In graph theory the distance is the length of the shortest path measured in the number of edges you have to walk over. But the Lorentz distance on the other hand is the length of the Longest time-like path.

Will each node be connected to a node in a 1:2, 1:3, 1:4, 1:5, ... , 2:3, 2:5, .... grid?

If not, why not? Wouldn't that imply that the measurement system is not invariant under parallel transport?

How can a distance be a path? A distance is the results of applying a metric to two or a norm to one or a difference objects. Metric and norms map by definition into the positive real numbers. Paths are not real numbers as far as i am concerned.

Yes it is integer valued, think of very large integers. If you use this distance to for instance measure the 4 sides and 2 diagonals of a very large Quadrilateral, then the ratios between these 6 lengths will approximate what you'd expect from Euclidean geometry, if you let the size of the Quadrilateral as well as the number of grids goes towards infinity.

The Spacetime is a directed acyclic graph (DAG), so there exists a longest path along the arrows. Paradoxically the Lorentz distance is the longest path between two nodes. This longest path is the one where the largest amount of time passed between two events, which is also a geodesic of this flat spacetime.

Am I understanding it correctly that your proposed Euclidean distance between nodes measures the number of hops between nodes?

So your Euclidean distance results in Integers only?

What is the Lorenz distance you talked about?

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