In[]:=
VectorPlot[{y,-x},{x,-3,3},{y,-3,3}]
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v . p
In[]:=
VectorPlot3D[{y,-x,0},{x,-3,3},{y,-3,3},{z,-3,3}]
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In[]:=
ResourceFunction["MultiwaySystem"][{"C""C","C""A","C""P","P""C","A""C","P""A","A""P","A""A","P""P"},"C",3,"CausalGraph"]//LayeredGraphPlot
Out[]=
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},{{
In[]:=
Graphics3D[InfinitePlane[{{0,0,0},{1,0,0},{0,1,0}}]]
Out[]=

Time visualization of spatial hypergraph: start from a well-foliated causal graph, and hang the spatial graphs from that.

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.

Flux of causal edges along the direction of the “spacelike” geodesic is the rate of activity transmission along that direction

Angular momentum

Within the plane activity transmission that leads to no net transmission in any direction

Analogy with fluid dynamics?

vorticity = ∇v

Relativistic angular momentum

4-momentum
Angular momentum
Pauli-Lubanski vector
Moment of mass polar vector

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

Any spin detector must be localized in branchial space

Group theory (?)

Speculation: this is the Cayley graph of so(3)

CPT invariance

T : reverse the arrows in the causal graph

P : reverse the hyperedges in the spatial hypergraph

C : reverse branchial edges

Generalized Lorentz invariance including branchial rotation

Multiway Minkowski norm

Spinors

SO(n)