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Why Hexagonal Lattices

Graphene has an energy-momentum relation that obeys a massless Dirac equation. So I moved on from the square model? to the kind of hexagonal lattice that defines graphene. Here, the lattice is not to be thought to be embedded in a surrounding space. The issue here is not describing physical graphene but using it as inspiration for an abstract structure that naturally describes the Dirac equation.

It seems the analogy between the behavior of graphene and the 2D Dirac equation arises from the honeycomb lattice giving rise to two essentially different vertices, one represented as "-<" and the other as ">-". These play the role of spin up and down in the spin basis (Z; directions in the lattice called X and Y). The general idea is the Dirac equation emerges as time derivatives of occupancy of a given site (say a "-<" site we'll call spin up in Z) relate to the space derivatives of the opposite spin as the particle moves.

Graphene seems complementary to the Feynman Checkerboard in that graphene seems to describe motion only in the two directions orthogonal to spin, whereas the Feynman Checkerboard, modeling the 1D Dirac equation that does not describe spin, does not describe those dimensions, which link opposite spins.

Another reason I have chosen to focus on hexagonal lattices: a trivalent network is the minimum structure that can express any information. (See my FQXI paper). Any vertex with greater valance can be built from loops of trivalent vertices. And a hexagonal lattice has trivalent vertices everywhere, always the same.

Why a regular lattice

One common argument against regular lattices (such as square cell cellular automata) describing space on a fundamental level is that there appears to be no favored direction in physical space, or anything like an "artifact" of a lattice structure. That in General Relativity spacetime can be curved suggests that a connected network model also not have a fixed background. Wolfram's network approach has focussed on irregular networks. Some (such as the Quantum Gravity Research Institute) explore quasicrystal models to remove drawbacks of regular lattices .

For one, the more regular lattices explored in these wiki pages are an exploratory exercise from which something could be learned to build on in the future. Second, many of the models with irregular backgrounds still have particles emerge as structures in those networks. To me it seems surprising that such particles would have the same behavior and properties no matter where physically they were located, no matter how different the background structure.

I should make it clear that I am imagining physics as a superposition of such lattices. Perhaps the nature of the superposition would remove some dependence of physics on the lattice structure itself.

Paper draft on the models I was experimenting with in 2016.

I'm still building on some ideas I had at that time, but the derivation of the Dirac equation presented in that draft now seems very clunky and not intuitive at all.

Some notes:

  • General note: I find myself coining a lot of neologisms. Apologies! I also go on detours to explore how discrete models could incorporate more known physics. But I need to pull back to emphasize the main goal of a better understanding of spinors and the Dirac equation.
  • Though these are two dimensional lattices, the intention was for a third dimension to emerge holographically. I now think I didn't pay enough attention to how this was implemented. I used projected coordinates. Sometimes I've thought: the way that spin works quantum mechanically is that spin up in Z also has projections onto spin up and down in X and Y. This means that if you try to use a fermion's spin to "point" in a direction, say, Z, you can't avoid also pointing to a lesser extent also in X, -X, Y, and -Y directions. This is not a rigorous argument, but I thought it was suggestive of this basic fact that in the 2D lattice, trying to code 3 dimensions would lead to some sometimes overlapping. Still, it would seem the positive Z direction might overlap more with positive X than negative X, for example.
  • In my guiding philosophy the changes to the lattice are to be global reversible substitutions. I imagined that the particle could be represented by some structure on a vertex that would then move about the lattice. The problem with this is that the reversible substitutions would have to be of the form: move every vertex in a certain direction. But what would define that direction unambiguously? In the Feynman Checkerboard for example, we say: either move in the same direction as before, or randomly change direction. Now with a 2D lattice and not the 1 spatial dimension of the Checkerboard, when it changes direction, does it move to the Left or the Right? There is no definite direction Left or Right that can be defined locally. One could be viewing the 2D lattice from one side or the other for example.
  • One possibility that I abandoned was making the structure of the vertex something like this, and imagining it had moved to its present location through the edge pointing to the left:

This unambiguously distinguishes the 3 outgoing lines. However, to move in a coordinate direction, one would make a substitution that would move this structure to a neighboring site and reorient it. To move through a hexagonal lattice in a coordinate direction, you must move alternating turning Left and Right. And this can't be done with only attention to the local structure. If you copy this vertex to the one at the lower right, copy the "notch" so it again points to where it came in, and the notch within the notch now to direct it to move to the right, you still need to invoke the global structure of the regular hexagonal lattice in order to define which is Left and Right. One can always imagine twisting about an edge so that Left and Right are reversed.

  • So what I ended up doing was not using a strict hexagonal lattice, but making the following change:

There are many points in the construction of a structure to work on where the construction seems a bit arbitrary!

I've learned that just as graphene is an example of a physical hexagonal lattice, there is a class of materials known as "zeolites" that show more general shapes, including one similar to this one, called a Type AFI zeolite, with a common variety known as SAPO-5.

Here is a page from a searchable database of zeolites on this kind of zeolite.

Assume the particle is some further structure on the hexagon (which was the vertex in before the substitution). The ambiguity of which direction to turn, relative to the incoming direction, is not completely removed in all cases. It goes along with the ambiguity over whether the "double edges" emerging from the hexagon above, are twisted or not. One can have a global substitution rule which says: flatten out the double-edges emerging from the hexagonal "vertex", and then turn Left.

The problem is that whether the double-edges are flattened is not a local condition. The issue is whether the network is a planar graph. The criterion that must be satisfied is known as Kuratowski's theorem

Imagine representing the pseudohexagonal network on a plane. If the double-edges that cross over are odd around a loop, then it will be nonplanar.

So twists are like a gauge field in that they represent a local convention, and what is unambiguous is not whether a particular double-edge is twisted, but whether a loop is planar. This is similar to the role of Wilson loops in gauge theory.

This large hexagon with an even number (2) of twists can be seen to be planar; an outgoing double-edge has simply been bent inwards.

There are some subtleties here. If you look at a loop, cutting off all external connections, whether it is planar depends on whether there are an even (planar) or odd (nonplanar) number of twists. When I write twists I mean that in a particular planar representation the lines of a double-edge cross over. When dealing with a hexagonal loop, even twists are equivalent to even non-twists. But were there to be loops of 3 (or any odd number) instead of 6, the symmetry between twists and non-twists is broken. Even twists around a loop still make it planar.

For purposes of planarity, changing the twists on all 3 double-edges meeting at a small hexagon has no effect. Every 2 out of the 3 are on the perimeter of one large hexagon. Changing all 3 will change the twists of each 2 around a large hexagon. If both are twisted, neither will become twisted, preserving the even or odd number of twists. If neither is twisted, both become twisted. If only one is twisted, now the other will be twisted, keeping the evenness or oddness around that loop.

This leads to an analogy between twists and Pauli matrices?

Related concepts, variants

A lot of neologistic terms, apologies!