WOLFRAM NOTEBOOK

Conjugate variables are associated with orthogonal directions in multiway space

If operators O1, O2 correspond to conjugate variables, then O1, O2 yield a nontrivial branch pair
And by virtue of their branch, they are orthogonal
Position is a feature that is visible in the MW CG...
In MW CG, every event has a {t, x, b} coordinate

1/Δx elements in the x measuring device; reconfiguring these for momentum leads to O(1/Δx) dispersion in the branchial direction [ which leads to a spread of everything ]
Everything turns into a ball in multiway space; of a certain size....

Local similarities

Entanglement monotone

Manifold of manifolds

( Like fiber bundle ? )
Branching and merging of multiway edges ↔ string field theory

AdS/CFT

Boundary of region in MW CG

Units

Causal edge:
time extent : τ
spatial extent : c τ
energy : [ ϵ : unit of energy ]
Node of hypergraph:
​
Edge of hypergraph:
​
​
Contribution of a single edge to energy:
Minimum energy ~ h c / radius of universe
In[]:=
h
c
//UnitConvert
Out[]=
4.52×
-52
10
kg
2
m
/
2
s
In[]:=
energyunit=UnitConvert[%147,"ElectronVolt"]
Out[]=
2.82×
-33
10
eV
Contribution of single edge to curvature:
1 + r^2 R
In[]:=
ΔR==1
^2
Out[]=
ΔR
1
/
2
radius of the visible universe
Gravitational constant is a scaling between spatial curvature and energy density
Energy density of a single edge : ϵ/(c^3 τ^4)
G / c^4 ϵ/(c^3 τ^3) ==
1/Ru^2
In[]:=
G/c^4*hc/Ru/(c^3τ^3)1/Ru^2
Out[]=
Gh
6
c
Ru
3
τ
1
2
Ru
In[]:=
Solve[%,τ]
Out[]=
τ
1/3
G
1/3
h
1/3
Ru
2
c
,τ-
1/3
(-1)
1/3
G
1/3
h
1/3
Ru
2
c
,τ
2/3
(-1)
1/3
G
1/3
h
1/3
Ru
2
c
In[]:=
1/3
G
1/3
h
1/3
Ru
2
c
/.{G->
,h->
,Ru->
,c->
}
Out[]=
1
3
radius of the visible universe
3
G
3
ℏ
/
2
c
In[]:=
UnitConvert[%]
Out[]=
1.622×
-23
10
s

Take 2 : use (c τ) as the curvature scale

In[]:=
G/c^4*hc/Ru/(c^3τ^3)1/(cτ)^2
Out[]=
Gh
6
c
Ru
3
τ
1
2
c
2
τ
In[]:=
Solve[%,τ]
Out[]=
τ
Gh
4
c
Ru
In[]:=
%/.{G->
,h->
,Ru->
,c->
}
Out[]=
τ
1
G
ℏ/(radius of the visible universe
4
c
)
In[]:=
UnitConvert
1
G
ℏ/(radius of the visible universe
4
c
)
Out[]=
1.98×
-105
10
s
In[]:=
te=%170;

Planck units: take 2

What is an energy h c / Ru in Planck units?
In[]:=
UnitConvert
,"PlanckEnergy"
Out[]=
2.31×
-61
10
E
P

Using Planck units

In[]:=
energyunit
Out[]=
2.31×
-61
10
In[]:=
*%151//UnitConvert
Out[]=
1.24×
-104
10
s
In[]:=
%
​

Alternative derivation (black hole based)

Spacetime energy density per causal edge : ϵ /
energy of universe =

Previous version

spacetime energy density associated with causal edge : ϵ / (τ ( c τ)^d)
G t t T ~ L L R : both dimensionless
T ~ (energy / unit volume)
Imagine the contribution of a single edge to
(1 +
R ~ 1/( macrolength^2 )
Ru size of universe
G τ^2 T / c^4 == (c τ)^2 / Ru^2
Is it c^4 or c^(d+1) ???
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.