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What if number of nodes = mass?
What if number of nodes = mass?
Mass density
Mass density
Size of particle related to e
Size of particle related to e
What if particle is fixed clump of nodes?
What if particle is fixed clump of nodes?
Phase space vs. ℏ
Phase space vs. ℏ
Position is given by location in hypergraph
Momentum (flux) is number of edges crossing a surface per unit “time”
Position is given by location in hypergraph
Momentum (flux) is number of edges crossing a surface per unit “time”
Momentum (flux) is number of edges crossing a surface per unit “time”
Somehow branching sets scale of granularity
Or related to e.g. spin?
Somehow branching sets scale of granularity
Or related to e.g. spin?
Or related to e.g. spin?
nodes / volume
events / volume
nodes / volume
events / volume
events / volume
Constant of proportionality in Vr related to mass?
Constant of proportionality in related to mass?
V
r
Elementary light cone
Is there an elementary energy-momentum cone
Elementary light cone
Is there an elementary energy-momentum cone
Is there an elementary energy-momentum cone
Multiway causal graph contains everything
Multiway causal graph contains everything
Multiway has neither cones nor balls, really
volume of causal cone
volume of causal cone
(c T)^d = spacetime volume
Causal graph as tree = no interaction
Causal graph as tree = no interaction
Can we measure the dimension of the multiway graph?
Can we measure the dimension of the multiway graph?
Is the multiway graph exponential (as opposed to the causal graph)
Is the multiway graph exponential (as opposed to the causal graph)
Exponent in multiway graph ~ quantum decoherence time
Two pieces of spacetime with the same dimension, and curvature... but different constants of proportionality [cf conformal invariance]
Two pieces of spacetime with the same dimension, and curvature... but different constants of proportionality [cf conformal invariance]
G determines the actual mass, based on node density
G determines the actual mass, based on node density
In[]:=
Out[]=
G ~ L^3 m^-1 t^-2
1/G ~ m t^2 L^-3
1/G ~ m t^2 L^-d
1/G ~ m t^2 L^-d
Newton law of gravitation:
E ~ G m^2/L^(d-2)
E ~ G m^2/L^(d-2)
1/r^(d-1) is force
[Charge relates distance to gauge degrees of freedom] <events per spatial hypergraph volume??>
[Charge relates distance to gauge degrees of freedom] <events per spatial hypergraph volume??>
How many events can happen in this given hypergraph volume
[Given a particular updating order, (roughly) how many events would occur in a generation in that volume]
[Given a particular updating order, (roughly) how many events would occur in a generation in that volume]
Vertex degree is independent degree of freedom from dimension
Vertex degree is independent degree of freedom from dimension
Gauge fields: ? isomorphism in hypergraph
In a rewrite, many ways to align the rewrite with the actual graph
But if you do one alignment locally, there is some connection to neighboring updates (?)
Gauge fields: ? isomorphism in hypergraph
In a rewrite, many ways to align the rewrite with the actual graph
But if you do one alignment locally, there is some connection to neighboring updates (?)
In a rewrite, many ways to align the rewrite with the actual graph
But if you do one alignment locally, there is some connection to neighboring updates (?)
Imagine you have a grid, and your rewrite is triples of nodes in a line
Imagine you have a grid, and your rewrite is triples of nodes in a line
In[]:=
GridGraph[{10,10}]
Out[]=
Covariant derivative:
∂ - i (e / ℏ) A
∂ - i (e / ℏ) A
{{1,2},{2,3}}{{1,2},{2,3}}
Causal graph looks at shared edges
In[]:=
HypergraphPlot/@WolframModel[{{1,2},{2,3}}{{1,2},{2,3}},List@@@EdgeList[GridGraph[{5,5}]],3,"StatesList"]
Out[]=
In[]:=
HypergraphPlot/@WolframModel[{{1,2},{2,3}}{{1,2},{2,3}},List@@@EdgeList[GridGraph[{5,5}]],3,"CausalGraph"]
Out[]=
In[]:=
HypergraphPlot/@WolframModel[{{1,2},{2,3}}{{1,2},{2,3}},List@@@EdgeList[GridGraph[{5,5}]],3,"CausalGraph","EventOrderingFunction""Random"]
Out[]=
Tensors
Tensors
From the outside, can paint values on tuples of hyperedges
In GR, the tensor is obtained by looking at the structure of the network by “geodesic probing” [network structure tensor]
Another type of tensor would measure things about “orientation” of events / updatings [event probing tensor]
Fact about “geodesic probing tensor”
Fact about “geodesic probing tensor”
Divergence of geodesic bundle vanishes in “vacuum” (i.e. uniform node density)
In fiber bundle
In fiber bundle
Base space is collection of nodes
Fibers are the collections of edges of each node
Fibers are the collections of edges of each node
Lie group for local gauge transformations
Lie group for local gauge transformations
Relates to interchange between hyperedges at a node
Minimally there is a permutation that relates hyperedges
Elements of gauge group map between different updatings, where each update is defined by a subset of hyperedges at a node
Elements of gauge group are transformations between subsets
G: powerset(fibers)->powerset(fibers)
Elements of gauge group are transformations between subsets
G: powerset(fibers)->powerset(fibers)
In[]:=
Permutations[Range[4]]
Out[]=
For a given rule, what is the actual transformation?
In reality,
G:possible LHS assignments -> possible LHS assignments
G:possible LHS assignments -> possible LHS assignments
G is the hypergraph automorphism group
Imagine the automorphism group for a coarse-grained update involving many nodes
Individual updates on larger hypergraphs
Lie group comes from subgroup of group of infinite permutations....
Group can be represented by : a certain set of generators which are permutations....
What are the limit automorphism groups of grids?
graphical regular representation of the automorphism group
graphical regular representation of the automorphism group
Can be like the graph itself
Claim: nilpotency / non-Abelianness
Claim: nilpotency / non-Abelianness
Aggregated hypergraph rewrite
Aggregated hypergraph rewrite
Do n rewrites; how can we do that as a single bigger rewrite?
For a given rewrite, what does it do to a series of hypergraphs?
What were the effective LHSs of multiple rule applications....
Grothendieck limit of symmetric groups
Grothendieck limit of symmetric groups
Putting units on graphs
Putting units on graphs
Spatial graph
Spatial graph
Each edge is an elementary length
Each node has a certain elementary mass
Edges also have a certain energy [relation between length and energy is like cosmological constant]
Edges also have a certain energy [relation between length and energy is like cosmological constant]
Causal graph
Causal graph
Each edge is a spacetime interval:
the edge has unit of time
the edge has unit of time
Each node in a causal graph has units of charge ???
Multiway graph
Multiway graph
Each edge has units of time
If we embed a branchlike hypersurface then the “branchial graph” edges are in units of energy
If we embed a branchlike hypersurface then the “branchial graph” edges are in units of energy
[Each edge of the branchial graph is a critical pair]
Is the volume element in the multiway graph an elementary action?
Edge has units of energy ???
cf https://en.wikipedia.org/wiki/Energetic_space
We can also compute the branchial graph of the multiway causal network