# Symmetry

Symmetry

Max Piskunov : 7de8636a9821490fcbaa2b3214a761e34d872e4c

## Transformation

Transformation

### Single hypergraph

Single hypergraph

A transformation of a hypergraph or a system of hypergraphs is a permutation of parts of its edges.

As an example, consider hypergraph:

(h1={{1,2},{2,3},{3,4},{4,5}})//WolframModelPlot

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A possible transformation could be defined as

t1[h_]:=Partition[Permute[Catenate[h],Cycles[{{1,6,4,8,7},{2,5}}]],2];

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t1[h1]//WolframModelPlot

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Note, generally transformation would not produce a hypegraph that is isomorphic to the original one.

### System of hypergraphs

System of hypergraphs

Similarly, we can consider multiple hypergraph which share some of their edges (think WolframModel evolution):

h2={{1,2},{2,3},{3,4},{4,5},{5,6}};h21=h2〚1;;3〛;h22=h2〚3;;5〛;WolframModelPlot[#,VertexLabelsAutomatic]&/@{h21,h22}

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A transformation would apply to the shared set of edges instead of individual hypergraphs in the system:

t2[h_]:=Partition[Permute[Catenate[h],Cycles[{{1,6,3,7},{2,10,9,5}}]],2];

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WolframModelPlot[#,VertexLabelsAutomatic]&/@{t2[h2]〚1;;3〛,t2[h2]〚3;;5〛}

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## Symmetries

Symmetries

### Global symmetry

Global symmetry

Now, what makes transformation a symmetry?

There are of course transformations that would turn a hypergraph to another hypergraph isomorphic to the original one. But that is not the kind of symmetry we are most interested in.

The more interesting kind of symmetry is the symmetry of the rules (i.e., a symmetry of the laws of physics). This is the kind of transformation that turns a rule to a rule that is isomorphic. Note, this is different from an automorphism, as neither part of a rule needs to be automorphic under the symmetry for the whole rule to stay the same.

For example, consider a transformation that reverses all edges (note this is a special kind of a transformation that applies identically to all edges, we will explain later how that can be generalized).

t3[h_]:=Reverse/@h

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There are rules, left- and right-hand sides of which are not automorphic under this transformation even though the entire rule is. I.e.,

RulePlotWolframModel[r3={{1,2}}{{1,3},{3,2}}],

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RulePlotWolframModel[t3/@r3],

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The fact that rule stays the same means we can apply this transformation to the initial condition, and get the evolution which is similar (and the final state can be reverted to the original final state), even though each state of the evolution is not isomorphic to its transformed version.

WolframModelPlot/@WolframModel[r3,{{1,2},{2,2}},2,"StatesList"]

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WolframModelPlot/@WolframModel[r3,t3@{{1,2},{2,2}},2,"StatesList"]

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WolframModelPlot/@t3/@WolframModel[r3,t3@{{1,2},{2,2}},2,"StatesList"]

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Note that this symmetry applies in the same way to all edges, thus it can be considered a global symmetry.

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### Local symmetry

Local symmetry

But what if we transform each edge independently. How can we tell if the rule is invariant under that transformation? How would we identify edges in the rule with edges in the state we are transforming?

The answer is that we need to verify that each application of the rule throughout the evolution can still be done with the original rule.

As an example, consider the evolution of the rule :

WolframModelPlot/@WolframModel[r3,{{1,2}},2,"StatesList"]

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Now, what if we reverse each edge independently? There are of course such inequivalent transformations, but we can choose each one of those for the sake of this argument.

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2

Generally, the rule (or more precisely the evolution) would not be symmetric under that transformation because some of the events would have inconsistent edge directions.

However, what if instead of this single rule we consider a multiway system with four rules?

RulePlot[WolframModel[r4={{{1,2}}{{1,3},{3,2}},{{1,2}}{{1,3},{2,3}},{{1,2}}{{3,1},{3,2}},{{1,2}}{{3,1},{2,3}}}],ImageSize600]

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Now, each edge can be independently reversed because each event of the multiway system can be independently mapped into one of these rules.

Thus, these rules satisfy a local symmetry with the order of , where is the number of edges in the evolution (the length of “AllExpressions”).

E

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E

We call this symmetry local because its order is not fixed but depends on the size of the evolution of the system, much like a local field transformation in a gauge theory.

### Constrained local symmetry

Constrained local symmetry

But what if we consider a subset of this system of rules?

RulePlot[WolframModel[{{{1,2}}{{1,3},{3,2}},{{1,2}}{{3,1},{2,3}}}],ImageSize300]

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Note that in this case we can still reverse some subsets of edges independently but not all of them. Specifically, each edge except for the initial one is now paired with its “sibling”, and they can only be reversed simultaneously, thus the symmetry group now has the order of , or in the case of our 2 steps evolution.

(E-1)/2

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Similarly we can consider the subset of rules

RulePlot[WolframModel[{{{1,2}}{{1,3},{3,2}},{{1,2}}{{1,3},{2,3}}}],ImageSize300]

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where one of the siblings is fixed with its parent, but the other sibling is independent.

Note that the symmetry transformation needs to be applied to “AllExpressions” (and generally on “AllExpressions” of the local multiway system) rather than on different states individually.

### Gauge symmetry

Gauge symmetry

One can imagine a constrained local symmetry where there is a phase that can be defined for different subgraphs of the hypergraph connected in some way to each other.

If each of these phase is simultaneously rotated, the symmetry is global.

However, there may be another symmetry where each of these phases can be rotated independently, however, these rotations yield perturbations in the graph that turn into gauge bosons of the effective theory.

## Algorithm

Algorithm

To determine the symmetry of the evolution in a brute-force way we can:

Compute (local) multiway evolution.

1.

For all permutations of atoms in “AllExpressions”:

2.

Are all events valid?

2.1.

Are new events not possible?

2.2.

If so, this permutation is an element of the symmetry group.

2.3.

By examining the number of elements, we can tell if it’s a local, global symmetry, or something in between.

3.

## Next

Next

We did not completely understand gauge symmetry yet, however, we have a path of constructing, or, even more interestingly, discovering it. To accomplish that we need one of two things:

We need efficient code for computing a symmetry for a given evolution. This way, for any given system, we will be able to tell whether (and how) it is globally or locally symmetric, and whether the local symmetry is trivial (exponential in the number of edges), or not (so that the order grows exponentially with a power that is the number of edges divided by a factor, or even sub exponentially).

1.

We can construct the gauge symmetry by hand. Although we should arguably only do that if the rules we come up with are simple, otherwise chances are the enumeration is going to outsmart us.

2.