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Branchlike lattice coordinates arbitrariness

Posted 1 year ago
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I was reading and studying the paper Some quantum mechanical properties of the Wolfram Model but at page 27 when the author define the branchial geometry it isn't explained how the coordinates of the $\mathbb{Z}^{1,n}$ are assigned to every state of the branchial hypersurface.

In particular, if we are going to define a norm, so a "metric" on this lattice using this coordinate as argument of the calculation, I presume there is a specific and not arbitrary system to map the coordinate in the hypersurface.

If this system is missing in the formulation then all the successive computation are based on an arbitrary coordinate set that can be changed at will, obtaining the type of norm desired from time to time.

After a search in the paper on relativistic properties of the model I noticed this inaccuracy is present even for the Minkowski lattice on page 13.

How are these coordinates assigned? On what basis? How do the rules of evolution can act on the coordinates of the points?

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In the relativity paper, I say "If one now considers performing a layered graph embedding of a causal graph..."

Layered digraph embedding (at least for acyclic digraphs) is a well-defined optimization problem, in which all edges are given by monotonic-downwards curves, and crossings between edges are minimized, so a metric constructed on the basis of such an embedding into a discrete lattice is a well-defined metric. Am I missing something here?

I think this problem has been mentioned several times now, so maybe a more detailed elaboration would be nice.

I honestly don't see how you can define any meaningful embedding in a space like, say, $\mathbb{Z}^{1,3}$. For one thing, any finite graph can always be (even polygonally if you want) embedded in an infinite number of different ways into $\mathbb{R}^3$ (that is, without any crossings between vertices). Adding a temporal dimension to this space would inevitably add even more ambiguity.

By the way, let's say we have a graph and choose a certain embedding into $\mathbb{Z}^{1,3}$ by assigning four coordinates to each vertex $(t,\vec{x})$. It is always possible to obtain a new embedding by the transformation (which is just one simple example of many): $$(t,\vec{x})\mapsto (t,\lambda \vec{x})\qquad \lambda \neq 0$$

This could easily turn space-like distances between vertices into time-like distances, deeming these terms useless. Am I wrong here?

Thanks a lot, this was exactly the problematic point I was trying to explain. It is not a trivial procedure but is overlooked as taken for granted in the papers, causing many problems of understanding.

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