Physics in d dimensions

[Basic Math]

Hyperspheres

Volume:
Out[]=
d/2
π
d
r
d
2
!
Out[]=
2r,π
2
r
,
4π
3
r
3
,
2
π
4
r
2

Surface area:
Out[]=
d
d/2
π
-1+d
r
d
2
!

Differential forms

Wedge products (AKA curl etc.)
TensorWedge[]
HodgeDual[]

Chain complexes etc.

0  smooth functions  d ( functions ) [1-form]  d d ( functions ) [2-forms]  .... [ends in integer dimensions]
In[]:=
scalar=With[{d=3},Sum[x[i]^x[j],{i,d},{j,d}]]
Out[]=
x[1]
x[1]
+
x[2]
x[1]
+
x[3]
x[1]
+
x[1]
x[2]
+
x[2]
x[2]
+
x[3]
x[2]
+
x[1]
x[3]
+
x[2]
x[3]
+
x[3]
x[3]
In[]:=
With[{d=3},Grad[scalar,Array[x,d]]]
Out[]=
(1+Log[x[1]])
x[1]
x[1]
+
-1+x[2]
x[1]
x[2]+Log[x[2]]
x[1]
x[2]
+
-1+x[3]
x[1]
x[3]+Log[x[3]]
x[1]
x[3]
,Log[x[1]]
x[2]
x[1]
+x[1]
-1+x[1]
x[2]
+(1+Log[x[2]])
x[2]
x[2]
+
-1+x[3]
x[2]
x[3]+Log[x[3]]
x[2]
x[3]
,Log[x[1]]
x[3]
x[1]
+Log[x[2]]
x[3]
x[2]
+x[1]
-1+x[1]
x[3]
+x[2]
-1+x[2]
x[3]
+(1+Log[x[3]])
x[3]
x[3]

ExteriorD[pform_,coords_]:=
In[]:=
With[{d=3},SymbolicTensors`ExteriorD[Grad[scalar,Array[x,d]],Array[x,d]]]
Out[]=
SymmetrizedArray
Dimensions: {3,3}
Symmetry: Antisymmetric[{1,2}]

In[]:=
Normal[%]
Out[]=
0,0,
-1+x[3]
x[1]
+Log[x[1]]
-1+x[3]
x[1]
x[3]+
-1+x[1]
x[3]
+Log[x[3]]x[1]
-1+x[1]
x[3]
,0,0,
-1+x[3]
x[2]
+Log[x[2]]
-1+x[3]
x[2]
x[3]+
-1+x[2]
x[3]
+Log[x[3]]x[2]
-1+x[2]
x[3]
,-
-1+x[3]
x[1]
-Log[x[1]]
-1+x[3]
x[1]
x[3]-
-1+x[1]
x[3]
-Log[x[3]]x[1]
-1+x[1]
x[3]
,-
-1+x[3]
x[2]
-Log[x[2]]
-1+x[3]
x[2]
x[3]-
-1+x[2]
x[3]
-Log[x[3]]x[2]
-1+x[2]
x[3]
,0
In[]:=
With[{d=3},NestList[SymbolicTensors`ExteriorD[#,Array[x,d]]&,f@@Array[x,d],4]]
Out[]=
f[x[1],x[2],x[3]],SymbolicTensors`Tensor
(1,0,0)
f
[x[1],x[2],x[3]],
(0,1,0)
f
[x[1],x[2],x[3]],
(0,0,1)
f
[x[1],x[2],x[3]],{SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}]},SymbolicTensors`TensorSymmetrizedArray
Dimensions: {3,3}
Symmetry: Antisymmetric[{1,2}]
,{SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}],SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}]},SymbolicTensors`TensorSymmetrizedArray
Dimensions: {3,3,3}
Symmetry: Antisymmetric[{1,2,3}]
,{SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}],SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}],SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}]},SymbolicTensors`TensorSymmetrizedArray
Dimensions: {3,3,3,3}
Symmetry: Antisymmetric[{1,2,3,4}]
,{SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}],SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}],SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}],SymbolicTensors`CotangentBasis[{x[1],x[2],x[3]}]}
In[]:=
NormalSymmetrizedArray
Dimensions: {3,3,3,3}
Symmetry: Antisymmetric[{1,2,3,4}]

Out[]=
{{{{0,0,0},{0,0,0},{0,0,0}},{{0,0,0},{0,0,0},{0,0,0}},{{0,0,0},{0,0,0},{0,0,0}}},{{{0,0,0},{0,0,0},{0,0,0}},{{0,0,0},{0,0,0},{0,0,0}},{{0,0,0},{0,0,0},{0,0,0}}},{{{0,0,0},{0,0,0},{0,0,0}},{{0,0,0},{0,0,0},{0,0,0}},{{0,0,0},{0,0,0},{0,0,0}}}}
Imagine our original function is painted as a scalar on a hypergraph [every node is given a value]
After one derivative, we see every individual edge painted with the difference between the values at two ends of the edge.

General chain complex

The homomorphisms going to successive stages in the chain complex satisfy d[i+1] · d[i] = 0
But the i’s don’t have to be integers

Construction of p forms

[Have to have somewhat long geodesics to have a clear notion of orthogonality]
To make a p+1 form from a p form we are basically taking finite differences in all possible ways (following the hypergraph) relative to the numbers in the p form
In[]:=
Graph[GridGraph[{6,6}],VertexWeightTable[iRandomReal[],{i,36}],VertexLabels"VertexWeight"]
Out[]=
In[]:=
ggg=Graph[Rule@@@ResourceFunction["WolframModel"][{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},8,"FinalState"]]
Out[]=
In[]:=
HighlightGraph[ggg,NeighborhoodGraph[ggg,5,2]]
Out[]=
In[]:=
inside=VertexList[NeighborhoodGraph[ggg,5,2]];
In[]:=
outside=Complement[VertexList[ggg],inside];
In[]:=
Flatten[Outer[Last[Select[FindShortestPath[ggg,#1,#2],MemberQ[inside,#]&],Nothing]&,inside,outside]]
Out[]=
{15,28,30,30,15,15,15,39,28,42,42,46,38,30,30,52,52,15,15,15,15,39,42,42,38,42,42,46,28,28,28,30,30,28,46,30,30,52,52,52,52,22,30,30,30,30,30,30,15,15,15,15,42,15,38,39}
[ Generalized Jordan curve to prove that there is a boundary .... although it’s not a connected graph ]
Boundary of boundary:
This was an example of boundary of a boundary = 0.
We can do a generalization of this by looking at which is reached from each node from pairs of edges, not just single edges.
Instead of asking about a single geodesic coming from the outside into the inside region, asking about p geodesics doing this, and then record all p-tuples of points where these geodesics first intersect the inside region.
(Helmholtz decomposition theorem) Decompose any vector field into a divergence free and curl free part.
[ Making a p form from a collection of p edges without knowing which edge is which naturally leads to an antisymmetric object ]
In the case of a grid, we label the coordinates by real numbers
In the case of a tree, we can natural;y use p-adic numbers

Are spinors like directed edges ; vectors like undirected edges ??

Graph cohomology?

Directed edge needs to be doubled to make an undirected edge....

Classical Mechanics

Electrodynamics

Maxwell’s equations

Inverse square law

1/(area of d-dimensional sphere)

Thermodynamics

Gravitation

Black holes

Quantum Effects

Quantum Field Theory

[ Renormalization seems to be very 4 dimensional ]
[ Dimensional regularization ]

Materials

Crystallography

[ Hard to do in d dimensions]

Units

Some don’t depend on dimension; others, like G, do.