Correspondence between NDTMs and causal invariance
Correspondence between NDTMs and causal invariance
Could simulate NDTM with string rewriting
The tape has A,B ... The head has X,Y
TuringMachine
This is a deterministic TM:
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RulePlot[TuringMachine[{596440,2,3}]]
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We need {A,B,C} ⊗ {X,Y,_}, where _ means a tape state, and X,Y means the head is there
Rules will preserve single-headedness [need contingent causal invariance for the subspace of a single head]
With a single head, system is obviously causal invariant (“causant”).
Multiway NDTM = QTM
Multiway NDTM = QTM
NDTM : one path at a time
QTM : maintain amplitudes for all paths
QTM : maintain amplitudes for all paths
Motion
Motion
Translation of a structure unchanged to another place
Different lumps of hypergraph : with particle-like local stability, can move around
Existence of Particles
Existence of Particles
Non-ergodicity of hypergraph rewriting
Locally conserved subspaces of the graph space
Interstellar travel
Interstellar travel
Pull a few hyperedges far away ??
Can you create a bridge that persists [[ in effect, that is a local dimension perturbation ]]
When you move something somewhere, mostly it leaves no tail/tale in spacetime
If normally a dimension would diffuse diffusively ... you could potentially prevent by having its structure be full of conserved things
Exotic “matter” : features of the graph that maintain perturbed dimension as other things maintain perturbed curvature ..... [[ analogy of Einstein-Rosen bridge maintained by ρ < 0 matter ]]
When you move something somewhere, mostly it leaves no tail/tale in spacetime
If normally a dimension would diffuse diffusively ... you could potentially prevent by having its structure be full of conserved things
Exotic “matter” : features of the graph that maintain perturbed dimension as other things maintain perturbed curvature ..... [[ analogy of Einstein-Rosen bridge maintained by ρ < 0 matter ]]
Negative etc. masses
Negative etc. masses
Consider energy density
Think you’re getting ρ L^d
Imagine you have a lump; you break it in two
Chop a tube; how do you maintain energy conservation?
The dimension you don’t think about you can imagine are curled in balls
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GridGraph[{6,6,6}]
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Single object: mass ρ L^2
GIven that you think you’re in 1D, expect ρ L/2 when you split in two
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{GridGraph[{10}],GridGraph[{5,5}]}
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Graph
,VertexLabelsAutomatic
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PathGraph[Range[100,110]]
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GraphUnion
,PathGraph[Range[100,110]],PathGraph[Range[200,210]],Graph[{{110,3}}],Graph[{{23,200}}]
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In the lump, there is higher mass density than you’d expect (“it’s mass ‘hidden’ in extra dimensions”)
Regions of un-dimension-decayed spacetime from “inflation period” end up being the dark matter.....
Could the cosmic web be a reflection of a piece of hypergraph
Could the cosmic web be a reflection of a piece of hypergraph
Cosmic Web
Cosmic Web
Cosmic Censorship / Cauchy Problem
Cosmic Censorship / Cauchy Problem
When does our causal graph define a satisfactory Cauchy problem.
Are CTCs avoided in our causal graphs?
What about a “confluence diamond”?
What about a “confluence diamond”?
Origin of G
Origin of G
Proportionality between node count in space graph (~mass) and actual [physical] volume of cones in causal graph
( would suggest (spacetime volume)/mass as units of G )
G : T L^d / M [assuming mass, not mass density]
Assume mass density wrt time: M/T^3 (mass you can get to in a certain geodesic time) [time is the correct measure of graph distance]
Node density T^3 is the geodesic ball’s content
Compared to volume of spacetime cone: T L^d
T L^d = G nd T^d
G = L^d T^(1-d) / nd
I.e. nd has units of mass..... [i.e. every node has a certain amount of mass]
Fundamental point:
Assume mass density wrt time: M/T^3 (mass you can get to in a certain geodesic time) [time is the correct measure of graph distance]
Node density T^3 is the geodesic ball’s content
Compared to volume of spacetime cone: T L^d
T L^d = G nd T^d
G = L^d T^(1-d) / nd
I.e. nd has units of mass..... [i.e. every node has a certain amount of mass]
Fundamental point:
mass density : M/L^d or M T^d/c^d
G c^d has units of M T^d
Actual units of G: L^3 M^-1 T^-2
In Planck length: has c^3
Assuming cone measurement is wrt space: c^d T is cone volume
G scaling might be changing all the time as hypergraph for universe refines....
Summary
Summary
Going along an edge in the causal graph is unit of time
Projected onto the spatial graph, that is c τ
Projected onto the spatial graph, that is c τ
Geodesic spatial volume is basically τ^d [which gives the instantaneous mass]
Hypergraph Isomorphism “Distance” / Fiber Bundles for Gauges
Hypergraph Isomorphism “Distance” / Fiber Bundles for Gauges
What is Higgs field?
What is Higgs field?
What is a massless particle
What is a massless particle
Deformation that has no nodes involved ?
How do gauge particles get mass?
Why are the gauge particles massless?
Gauge Field
Gauge Field
We have a base space; we have rules being applied; these rules have different configurations related by hypergraph isomorphism
At every point in the base space there is a rule that could be applied
E.g. dimers on a grid
At every point in the base space there is a rule that could be applied
E.g. dimers on a grid
Massless particles carry information about the permutation of indices in the rewritings
You have a function which is a value in a gauge group. f(X) Move to f(X+δ) The value will effectively change just because the coordinates at that new point in the fiber bundle are different.
What quantities are independent of this effect? (E.g. curvatures)
What can we measure wrt to the internal degrees of freedom of the hypergraphs?
What quantities are independent of this effect? (E.g. curvatures)
What can we measure wrt to the internal degrees of freedom of the hypergraphs?
Each local region of edges is the content of a fiber in a fiber bundle. The choice of which edge is picked for an update is locally arbitrary
Imagine perturbing the system by changing which edges got picked for a particular rewrite. The effect of that change propagates at the speed of rewriting (i.e. speed of light). What is the structure of the change?
Make a gauge-like perturbation; this will give a gauge boson propagating out.
Claim: long-range effects of gauge bosons are analogs to rewriting order artefacts
Imagine perturbing the system by changing which edges got picked for a particular rewrite. The effect of that change propagates at the speed of rewriting (i.e. speed of light). What is the structure of the change?
Make a gauge-like perturbation; this will give a gauge boson propagating out.
Claim: long-range effects of gauge bosons are analogs to rewriting order artefacts
Accelerated Frames / Uniform Gravitational Field
Accelerated Frames / Uniform Gravitational Field
Slice causal graph at successively steeper angles.
Gravitational Covariant Derivative
Gravitational Covariant Derivative
There is a function on a space that is deformed
Change of function just because of deformation of space
Change of function just because of deformation of space
Steeper Foliations
Steeper Foliations
Fundamental point: there is one causal graph, but different foliations/different frame let you explore it in different orders.
Different inertial frame: angle on the grid
But you are always thinking of “time” as vertical wrt your spacelike hypersurface
But you are always thinking of “time” as vertical wrt your spacelike hypersurface
To get to a particular “global generation”, you experience more updating events
More updating events means more nodes, which means more mass.........
Same γ factor for mass and time. [Covariant transformation of momentum]
Same γ factor for mass and time. [Covariant transformation of momentum]
Visiting order for the causal graph
There is a canonical/optimal visiting order: aka graph embedding as downward layered graph
Event Horizons in Accelerating Frames
Event Horizons in Accelerating Frames
Critical pair converges in one step (say) of the global clock, but can take arbitrarily many in an accelerating frame
Hasse Diagrams
Hasse Diagrams