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Damian Musk

[WSC21] On CDT-based simplicial triangulation decomposition in WL

Damian Musk, Fermi National Accelerator Laboratory

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On Wolfram Language-based simplicial decomposition via the causal dynamical triangulation formalism
by Damian Musk

This project seeks to further explore spacetime discretization through Wolfram-based hypergraph transformation dynamics, in which causal sets are effectively generated algorithmically via an abstract rewriting system defined over hypergraphs—an approach to causal set theory previously outlined by J. Gorard—while motivated by the causal dynamical triangulation (CDT) formalism and the associated procedures of R. Loll and B. Ruijl's locally causal dynamical triangulations (LCDT) . This article specifically seeks to examine a BoundaryMesh-based framework for loose corresponding interpretations of Euclidean and Lorentzian space, and apply these visualizations and operators to more grounded interpretations of some of the mathematical concepts featured in CDT and LCDT.

Introduction

The construction of a unified model that explains the phenomena of both quantum mechanics and general relativity—the famed issue of quantum gravity—remains one of the most significant problems of physics and science as a whole. The most well-developed and well-known proposal is that of bosonic string theory, but string theory is a framework considered fundamentally flawed by some due to its focus on grand, somewhat arbitrary physical assumptions about the structure of the universe that better lend itself to mathematicians than scientists looking for a physical theory. Causal set theory, however, responds to this lack of physicality through its emphasis on a more general type of phenomenon familiar to every inhabitant of the Universe: that of cause and effect, which has mathematical significance at all length scales. More formally, causal set theory explores the invariance of the causal structure of Lorentzian manifold under conformal transformations, changes of coordinates

→



(σ)

that scale the metric by some function of space squared such that

μν

(σ)→

(σ)

μν

(σ)

; this allows causal set theory to concern itself with continuous spacetimes arbitrarily well-approximated by finite set of events, known as causal sets, with causal partial order relations upon them. Causal dynamical triangulation (CDT) is closely related: both CDT and causal sets attempt to model spacetime with a discrete causal structure, but while the causal set approach is relatively general, CDT assumes a more specific relationship between the lattice of spacetime events and geometry with a Lagrangian constrained by enough initial conditions as to be numerically analyzed—causal sets are far more topologically generic.This article explores a fairly straightforward version of CDT, the 1+1 case (one spatial dimension, one temporal dimension with a (+,-) metric signature), with conditions that are fairly easy to demonstrate with Wolfram's graph manipulation and generation functions. In particular, the majority of the investigation will be spent on comparing and contrasting different mathematical, physical, and visualizable properties of graphs with arbitrarily-assigned and coordinatized causal relationships: networks with arbitrarily assigned causal relationships that correspond loosely to a Lorentzian case, and coordinatized networks in Euclidean space with calculable Minkowski norms, respectively. This may give more concrete insight into the circumstances that define causal dynamical triangulation and the topological specificity that distinguishes it from the more generic causal set theory.

Arbitrary Triangulation

We will begin with a procedure that constructs triangulation by arbitrarily classifying edges as spacelike or timelike: in this way, we are forming a non-coordinatized system based on the predetermined assignment of causal relationships as opposed to any norm-based calculations.

As previously described, CDT is based upon various types of simplices, generalized triangles such that a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, et cetera. We begin with BoundaryMeshRegion[{p1,...,pn}, Line{...}] which yields a mesh defined by a boundary with coordinates p1,...,pn, as outlined by Line{...}. We can then triangulate this result—in CDT parlance, divide graph into 1-simplices—and use Graph[] and MeshConnectivityGraph[...] to obtain a triangulated graph.

We begin with a simple

BoundaryMesh[...]

type region,

ℛ

In[]:=

ℛ=BoundaryMeshRegion[{{0,0},{1,0},{1,1},{0,1}},Line[{1,2,3,4,1}]]

Out[]=

To convert this into the type of discretized region, we may define the following helper functions: GetTriangulatedPlaneEdgeList and GetTriangulatedPlaneGraph.

In[]:=

GetTriangulatedPlaneEdgeList[triangulatedGraph_]:=DirectedEdge@@#&/@EdgeList[triangulatedGraph];GetTriangulatedPlaneGraph[region_]:=MeshConnectivityGraph[TriangulateMesh[region]];GenerateTriangulatedMeshAdjaceny[region_]:=Module[ {triangulatedPlaneGraph,triangulatedPlaneVertexList, triangulatedPlaneEdgeList}, triangulatedPlaneGraph = GetTriangulatedPlaneGraph[region]; triangulatedPlaneEdgeList=GetTriangulatedPlaneEdgeList[triangulatedPlaneGraph]; triangulatedPlaneVertexList=VertexList[triangulatedPlaneGraph]; Graph[triangulatedPlaneEdgeList,VertexLabels->Placed["Name",Tooltip]]]triangulatedGraph = GetTriangulatedPlaneGraph[ℛ];triangulatedPlaneEdgeList = GetTriangulatedPlaneEdgeList[GetTriangulatedPlaneGraph[ℛ]];triangulatedPlaneVertexList=VertexList[GetTriangulatedPlaneGraph[ℛ]];

Column[{triangulatedGraph,GenerateTriangulatedMeshAdjaceny[ℛ]}]

Out[]=

We can now create a function,

randomTimelikeAssigner

, that takes the edge list of the triangulated region and creates a randomly generated association corresponding approximately half of the edges with a timelike value ("T") and the other half with a spacelike value ("S").

In[]:=

randomTimeLikeAssigner[triPlaneEdgeList_]:=Association[With[ {randomSubset=RandomSample[triPlaneEdgeList,Floor[Length[triPlaneEdgeList]/2]]}, If[ MemberQ[randomSubset,#], #->"T", #->"S" ]&/@triPlaneEdgeList]]updateGraphEdgeAssoc[vertex_]:=Association[ If[ MatchQ[Head[#], UndirectedEdge], #->"S", #->"T" ]&/@vertex];

The resulting association informs most of the visualization employed for the triangulated BoundaryMesh and how causal relationships between different nodes of the network are defined. One part of the former is undirecting edges assigned spacelike, implemented with UndirectIfSpacelike—as well as, for the sake of completion, UndirectIfSpacelikeOrLightlike.

In[]:=

undirectIfSpacelike[edgeAssoc_,vertex_]:=If[ StringMatchQ[edgeAssoc[vertex], "S"], UndirectedEdge@@vertex, vertex];undirectIfSpacelikeOrLightlike[edgeAssoc_,vertex_]:=If[ StringMatchQ[edgeAssoc[vertex],"S"]||StringMatchQ[edgeAssoc[vertex],"L"], UndirectedEdge@@vertex, vertex];

We now approach the issue of causality in CDT: specifically, that locally causal dynamical triangulations should exhibit well-behaved vertices strictly surrounded by four timelike edges. This can be achieved fairly straightforwardly via IncidenceList:

In[]:=

countNonspacelikeEdges[edgeList_,vertex_]:=Length[Select[IncidenceList[edgeList,vertex],Head[#]==DirectedEdge&]];does11CDTCaseObeyCausalityLocally[edgeList_]:=FreeQ[countNonspacelikeEdges[edgeList,#]==4&/@VertexList[edgeList],False];

We should now be able to graph 3-dimensional graphs that present this locally-causally acceptable behavior through a visualization that incorporates three key pieces of causal information: (1) whether connections are spacelike, lightlike, or timelike, denoted through colors set in the original arbitrarily-assigned association; (2) preserved arbitrarily defined causal directions defined through the original convention with the all-directed BoundaryMesh; and (3) whether vertices are causally appropriate or not in Loll's 1+1 CDT local causality condition, for which points are green if the rule is violated and blue if they are appropriate.

In[]:=

minkowskiColorer[edgeAssoc_,vertex_]:=Which[ StringMatchQ[edgeAssoc[vertex],"T"], vertex->Red, StringMatchQ[edgeAssoc[vertex],"L"], vertex->Yellow, StringMatchQ[edgeAssoc[vertex],"S"], vertex->Purple];violationColorer[edgeList_,vertexList_]:=If[ countNonspacelikeEdges[edgeList,#]==4, #->Green, #->Blue]&/@vertexList;

We proceed to define the following Module for

In[]:=

GetRefinedGraph[triPlaneGraph_,triPlaneEdgeList_,type_]:=Module[ class, randomlyAssignedEdgeAssoc,undirectedIfSpacelike,undirectedIfSpacelikeOrLightlike, minkowskiColoredEdges,violationColoredvertices,undirectedAssignedEdgeAssoc, undirectedIfSpacelikeOrLightlikeAssignedEdgeAssoc,undirectedIfSpacelikeAssignedEdgeAssoc, class=ToLowerCase[type]; randomlyAssignedEdgeAssoc=randomTimeLikeAssigner[triPlaneEdgeList]; undirectedIfSpacelike=undirectIfSpacelike[randomlyAssignedEdgeAssoc,#]&/@triPlaneEdgeList; undirectedIfSpacelikeOrLightlike=undirectIfSpacelikeOrLightlike[randomlyAssignedEdgeAssoc,#]&/@triPlaneEdgeList; undirectedIfSpacelikeAssignedEdgeAssoc = updateGraphEdgeAssoc[undirectedIfSpacelikeOrLightlike]; undirectedIfSpacelikeOrLightlikeAssignedEdgeAssoc = updateGraphEdgeAssoc[undirectedIfSpacelikeOrLightlike]; violationColoredvertices=violationColorer[triPlaneEdgeList, VertexList[triPlaneGraph]]; Which[ StringMatchQ[class,"original"], GraphPlot3D[ triPlaneEdgeList, DirectedEdges->False, VertexLabels->Placed["Name",Tooltip] ], triPlaneEdgeList, StringMatchQ[type,"timelike"], GraphPlot3D[ Graph[undirectedIfSpacelike, VertexLabels->Placed["Name",Tooltip], EdgeStyle->(minkowskiColorer[undirectedIfSpacelikeAssignedEdgeAssoc,#]&/@undirectedIfSpacelike) ] ],undirectedIfSpacelikeAssignedEdgeAssoc,undirectedIfSpacelike, StringMatchQ[class,"timelikelightlike"], GraphPlot3D[ undirectedIfSpacelikeOrLightlike, VertexLabels->Placed["Name",Tooltip], EdgeStyle->(minkowskiColorer[undirectedIfSpacelikeOrLightlikeAssignedEdgeAssoc,#]&/@undirectedIfSpacelikeOrLightlike) ], undirectedIfSpacelikeOrLightlikeAssignedEdgeAssoc,undirectedIfSpacelikeOrLightlike, StringMatchQ[class,"timelikelightlikewviolations"], GraphPlot3D[ undirectedIfSpacelikeOrLightlike, VertexLabels->Placed["Name",Tooltip], DirectedEdges->False, EdgeStyle->(minkowskiColorer[undirectedIfSpacelikeOrLightlikeAssignedEdgeAssoc,#]&/@undirectedIfSpacelikeOrLightlike), VertexStyle->violationColoredvertices ],undirectedIfSpacelikeOrLightlikeAssignedEdgeAssoc, undirectedIfSpacelikeOrLightlike, True, "Input a valid type!\n" ]]

Column[{GetRefinedGraph[triangulatedGraph,triangulatedPlaneEdgeList,"timelike"][[1]],GetRefinedGraph[triangulatedGraph,triangulatedPlaneEdgeList,"TimelikeLightLike"][[1]],GetRefinedGraph[triangulatedGraph,triangulatedPlaneEdgeList,"TimelikeLightLikeWViolations"][[1]]}]

Out[]=

As can be seen in the final graph, there are a number of events that violate this condition of local causality, thereby only making the network valid should these points be removed as to still form a triangulated surface consisting of these elemental (1,1) Minkowski triangles in accordance to local gluing rules; basic functions for such approaches are below.

In[]:=

AmendViolatingIncidenceList[edgeList_]:=With[{firstFourEdges=edgeList[[1;;4]]},Join[#->Red&/@firstFourEdges,#->Purple&/@Complement[edgeList,firstFourEdges]]];

In[]:=

AmendViolatingIncidenceList[edgeList_]:=With[{firstFourEdges=edgeList[[1;;4]]},Join[#->Red&/@firstFourEdges,#->Purple&/@Complement[edgeList,firstFourEdges]]];

In[]:=

UpdateEdgeAssoc[edgeAssoc_,vertex_]:= adjacentEdges=IncidenceList[Keys[edgeAssoc],vertex]; If[ countNonspacelikeEdges[adjacentEdges]!=4,  Scan[(edgeAssoc[#]="T")&][adjacentEdges[[1;;4]]]; Scan[(edgeAssoc[#]="S")&][Complement[adjacentEdges,adjacentEdges[[1;;4]]]]  ];AmendViolating[edgeAssoc_]:= With[ {vertexList=VertexList[Keys[edgeAssoc]]}, UpdateEdgeAssoc[edgeAssoc,vertexList[[1]]] ]

Coordinatized Triangulation

Instead of an arbitrarily-assigned Lorentzian space that defines direction through relationships defined independently of continuum volumes, we will proceed with a coordinatized Euclidean space that allows us to calculate and classify Minkowski norms between edges with our (+,-) signature in the 1+1 case.

After having investigated the CDT-based properties of edge-vertex networks, we shall now proceed to examine Minkowski norm-motivated systems of establishing these networks. To accomplish this, we shall first obtain coordinate values for each of the vertices of our triangulated plane through AnnotationValue[graph_, VertexCoordinates], for which we interpret vertex coordinates as being in the form (x,t) in 1+1 Minkowski space. Then, by obtaining all edge combinations and then taking the interval of each event pair, we take the Minkowski norm

||(x,t)||=

: if this norm is negative (the square of the temporal distance is greater than that of the spatial distance), events are timelike and will be causally connected by an edge alongside lightlike events with

||(x,t)||=0

; spacelike events with

||(x,t)||>0

, meanwhile, will be left as points unjoined by any edges.

In[]:=

normedList[coord_]:=(coord->(First[First[coord]-Last[coord]]^2-Last[First[coord]-Last[coord]]^2));getMinkowskiNormedGraph[triPlanedGraph_]:=Module[ {points=Tuples[AnnotationValue[triPlanedGraph,VertexCoordinates],2]}, Graph[ DirectedEdge@@First[#]&/@Select[normedList/@points,Last[#]<0&], VertexLabels->Placed["Name",Tooltip] ]]pruneIncidenceLists[graph_]:=Flatten[Table[ With[ {adjacentEdges=IncidenceList[graph,VertexList[graph][[i]]]}, If[ Length[adjacentEdges]>4, adjacentEdges[[1;;4]], adjacentEdges ] ], {i,Length[VertexList[graph]]}]]

In[]:=

reducedGraph=getMinkowskiNormedGraph[GetTriangulatedPlaneGraph[ℛ]]prunedGraph=Graph[Nest[pruneIncidenceLists,reducedGraph,5]]

Out[]=

Graph

Vertex count: 334

Edge count: 55604



Out[]=

With this procedure, we can examine the graph as it grows increasingly refined through repeatedly applying pruneIncidenceLists[].

In[]:=

Manipulate[Graph[Nest[pruneIncidenceLists,reducedGraph,i]],{i,0,10,1},SaveDefinitionsTrue]

Out[]=

In[]:=

examineIncidenceLists[graph_]:=Length/@Table[IncidenceList[graph,VertexList[graph][[i]]],{i,Length[VertexList[graph]]}]

In[]:=

Manipulate[examineIncidenceLists[Graph[Nest[pruneIncidenceLists,reducedGraph,i]]],{i,0,10,1},SaveDefinitionsTrue]

Out[]=

{336,326,174,290,286,288,260,328,322,208,222,290,316,302,322,284,308,204,222,322,270,246,306,

322,352,322,298,306,320,486,286,204,178,238,310,352,230,330,264,264,338,324,236,174,322,

196,246,214,262,178,208,254,270,282,292,352,244,244,266,304,338,330,186,298,322,342,386,

340,354,392,392,372,368,452,380,350,394,394,344,346,320,338,492,328,460,350,362,368,340,

338,334,360,394,438,350,370,438,344,350,344,356,388,432,406,422,450,484,370,458,346,434,

478,372,388,418,420,434,366,398,332,450,334,396,422,470,456,458,350,400,434,358,364,348,

396,364,268,210,242,366,412,414,474,320,310,330,276,298,226,336,250,298,300,308,348,364,

402,256,430,408,254,242,312,274,294,356,374,362,312,336,336,326,290,180,294,316,332,214,

286,300,320,324,314,184,266,284,256,264,212,246,266,228,288,324,306,310,310,318,326,310,

318,282,298,262,280,294,184,242,318,262,286,198,304,262,218,224,314,252,276,328,230,310,

244,252,216,224,246,278,300,258,320,312,194,224,236,254,338,328,330,334,328,324,178,204,

302,306,282,332,328,260,256,302,322,254,340,484,334,340,342,448,338,392,366,346,342,402,

402,376,348,454,362,366,352,396,352,418,440,436,416,478,384,358,350,348,342,340,354,342,

332,356,364,352,358,394,406,382,438,436,482,432,472,426,458,410,358,390,394,466,354,380,

396,388,374,356,390,394,396,440,350,410,404,414,440,342,370,454,378,372,344,416,468,462,

368,374,330}

Analysis and Physical Assessment

A major term of mathematical interest is the two-dimensional Einstein-Hilbert action in the continuum,

S=κ∫

|g|

(R-2Λ)

, for the inverse of Newton's constant,

; the cosmological constant,

; and the determinant of the Lorentzian spacetime metric,

μν

The path integral and its regularized counterpart in terms of dynamical triangulations can be transformed to read

∫[g]

-iΛ∫

|g|

→

∑

C(T)

iλV

(T)

in which the formal integration over diffeomorphism equivalence classes

[g]

of metrics on the lefthand side has been replaced by a sum over inequivalent triangulations

with the usual dynamical triangulation measure involving the order

C(T)

of the automorphism group of

. The previous sections describing BoundaryMesh-based implementations of CDT-type structures in spaces under different computational circumstances then suggest a discrete path integral implementable in this same system by taking an ensemble of TriangulateMesh[]-type triangulations consistent with defined causal relationships on the manifold, since the CDT integral entails integration over all possibilities concerning the partition function on that integral. This similarly opens possibilities for the exploration of other quantities, particularly dimensional estimators, methods designed to determine the dimensionality

of a continuum Lorentzian manifold

(ℳ,g)

using only the discrete information provided by a causal network faithfully embedded into

(ℳ,g)

. (An injective mapping of a causal set in such a manifold is faithful if it forms a uniform distribution for some characteristic spacetime discreteness scale; in causal set theory methods, this is typically ensured through statistical methods, but will we be simplified here for the sake of physical exploration.) One such method is the Born-Surya-Versteegen estimator, which succeeds in defining a one-parameter family of truly discrete spatial distance functions on inextensible antichains in causal sets by assessing the volume