In[]:=

VectorPlot[{y,-x},{x,-3,3},{y,-3,3}]

Out[]=

v . p

In[]:=

VectorPlot3D[{y,-x,0},{x,-3,3},{y,-3,3},{z,-3,3}]

Out[]=

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}]

In[]:=

RotationMatrix[{{1,0,0},{0,1,0}}]

Out[]=

{{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[]=

In[]:=

Graphics3D[InfinitePlane[{{0,0,0},{1,0,0},{0,1,2}}]]

#### 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)