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Feynman diagrams as causal graphs
Feynman diagrams as causal graphs
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Vertices in Feynman diagrams are like events; edges in Feynman diagrams are like causal edges
In Feynman diagrams there are two “multiway-like” attributes: 1. different diagrams, 2. different places in space where vertices occur [~ momentum integration]
For the spatial part, we could imagine a “tiny universe” with a small number of positions/nodes. E.g. 0-dimensional field theory == diagram counting
For Feynman diagrams we’re basically distinguishing the interacting particles from the “background spacetime”
“Integrating over momenta/positions” is event equivalencing ... the result is a “pure diagram”
Structure of a diagram:
initial causal edges from “initial events” (“creation events”) (“big bang events”)
eventually: final causal edges
initial causal edges from “initial events” (“creation events”) (“big bang events”)
eventually: final causal edges
Initial state can be formed by a bunch of spacelike separated creation events
Where do the probabilities come from? Imagine all edges of the multiway graph have weight 1. Different initialfinal transformations have different path counting ... which gives a weight/amplitude for each
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ResourceFunction["SubstitutionSystemCausalPlot"][ResourceFunction["SubstitutionSystemCausalEvolution"][{"BA"->"AB","AB"->"BA"},"ABBABBAA",4,{"Random",1}],EventLabels->False,CellLabels->True,CausalGraph->True]
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Multiway graph for transpositions
Multiway graph for transpositions
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alltrans[list_]:=Table[SubsetMap[Reverse,list,{i,Mod[i+1,Length[list],1]}],{i,Length[list]}]
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NestGraph[alltrans,{{1,1,0,0,0}},3,VertexLabels->Automatic]
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LayeredGraph[NestGraph[alltrans,{{1,1,0,0,0}},3,VertexLabels->Automatic]]
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Momentum/energy
Momentum/energy
Flux of causal edges through spacelike hypersurfaces ∝ energy
Another thing (if there’s a background space) is the length of the causal edge
Mass is related to causal edges that “stay in the same place” [ mass is related to causal connections that do not span space ]
Families of singleway graphs are Feynman diagrams
Families of singleway graphs are Feynman diagrams
[ After event equivalencing : one can get loop diagrams ]
"AB"->"BA"->"AB"->"BA"
Coupling constant may be connected to the branching
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ResourceFunction["MultiwaySystem"][{"AB"->"BA","BA"->"AB"},ResourceFunction["StringTuples"]["AB",4],2,"EvolutionCausalGraph"]
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For each initial and final state, there are a collection of causal graphs that lead from one to the other; there is also a multiway system ;
[Feynman diagrams correspond to the “independent slices” of the multiway graph, after equivalencing]
[Feynman diagrams correspond to the “independent slices” of the multiway graph, after equivalencing]