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VectorPlot[{y,-x},{x,-3,3},{y,-3,3}]
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v . p
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VectorPlot3D[{y,-x,0},{x,-3,3},{y,-3,3},{z,-3,3}]
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ResourceFunction["MultiwaySystem"][{"C""C","C""A","C""P","P""C","A""C","P""A","A""P","A""A","P""P"},"C",3,"CausalGraph"]//LayeredGraphPlot
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RotationMatrix[{u,v}]
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RotationMatrix[{{1,0,0},{0,1,0}}]
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{{0,-1,0},{1,0,0},{0,0,1}}
InfinitePlane[{0,0,0},{{
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Graphics3D[InfinitePlane[{{0,0,0},{1,0,0},{0,1,0}}]]
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Time visualization of spatial hypergraph: start from a well-foliated causal graph, and hang the spatial graphs from that.
Time visualization of spatial hypergraph: start from a well-foliated causal graph, and hang the spatial graphs from that.
Double orthogonalization:
Double orthogonalization:
E.g. for linear momentum: start with a spacelike vector defined by a geodesic in spatial hypergraph
Find the timelike vectors that are orthogonal to that spacelike vector
Find the flux through the surface defined by the family of timelike vectors.
Find the timelike vectors that are orthogonal to that spacelike vector
Find the flux through the surface defined by the family of timelike vectors.
Flux of causal edges along the direction of the “spacelike” geodesic is the rate of activity transmission along that direction
Flux of causal edges along the direction of the “spacelike” geodesic is the rate of activity transmission along that direction
Angular momentum
Angular momentum
Within the plane activity transmission that leads to no net transmission in any direction
Analogy with fluid dynamics?
Analogy with fluid dynamics?
vorticity = ∇v
Relativistic angular momentum
Relativistic angular momentum
4-momentum
Angular momentum
Pauli-Lubanski vector
Angular momentum
Pauli-Lubanski vector
Moment of mass polar vector
Quantization of Spin
Quantization of Spin
In the limit of a large spacetime hypergraph, angular momentum will be continuous.
In MWCG, can define a MW angular momentum that is defined by space+branch geodesics
Particles are localized not only in physical space, but also in branchial space
Particles are localized not only in physical space, but also in branchial space
Any spin detector must be localized in branchial space
Group theory (?)
Group theory (?)
Speculation: this is the Cayley graph of so(3)
CPT invariance
CPT invariance
T : reverse the arrows in the causal graph
T : reverse the arrows in the causal graph
P : reverse the hyperedges in the spatial hypergraph
P : reverse the hyperedges in the spatial hypergraph
C : reverse branchial edges
C : reverse branchial edges
Generalized Lorentz invariance including branchial rotation
Generalized Lorentz invariance including branchial rotation
Multiway Minkowski norm
Spinors
Spinors
SO(n)