Combining branchial space with ordinary space....

#### Foliating the MWCG

Foliating the MWCG

Wave-particle duality

Image you started a geodesic bundle in the MWCG. The bundle will spread in both spacelike (light cone) and branchlike (entanglement cone) directions.... In the MWCG, can mix spacelike/branchlike according to foliation

In spacetime, it follows a standard geodesic path

In branchtime, the thing may spread out

In spacetime, it follows a standard geodesic path

In branchtime, the thing may spread out

Wave of a certain frequency vs. particle of a certain energy....

Particle with given energy: flux of causal edges through spacelike surfaces

Frequency is associated with rate of phase change of e^(i H t) : its version of energy....

Wavepacket in space: has a spatial falloff; in branchial space, also has a falloff...

Particle with given energy: flux of causal edges through spacelike surfaces

Frequency is associated with rate of phase change of e^(i H t) : its version of energy....

Wavepacket in space: has a spatial falloff; in branchial space, also has a falloff...

#### Scaling all cross-section of bundles

Scaling all cross-section of bundles

Look at all path counts reaching a certain slice; now consider what fraction are the ones you care about....

( solid angle)

( solid angle)

#### Uncertainty Principle

Uncertainty Principle

Rotation in spacetime (going from measuring position to measuring timelike flux)

Claim: rotation in spacetime involves rotation in branchtime

Claim: rotation in spacetime involves rotation in branchtime

x, p uncertainty relation

First this, then that. How do you operationally go from one to the other?

Rotating an apparatus with n gradations takes O(n^2) operations [[ takes n time steps ]]

At each step, there is an x-direction update, and a p-direction update that could be done....,

The x, p measurement sequence induces dispersion in branch space....

Measured x using a fine toothed comb.... (with O(n) teeth)

Now measure p by having transported their comb, but actually their comb fuzzed out in branchial space.....

Rotating an apparatus with n gradations takes O(n^2) operations [[ takes n time steps ]]

At each step, there is an x-direction update, and a p-direction update that could be done....,

The x, p measurement sequence induces dispersion in branch space....

Measured x using a fine toothed comb.... (with O(n) teeth)

Now measure p by having transported their comb, but actually their comb fuzzed out in branchial space.....

#### Commutator failure might be thought of as an angle mismatch

Commutator failure might be thought of as an angle mismatch

### Propagation of wave packets

Propagation of wave packets

Measure branchial dispersion : angle ℏ/τ

If there is a bundle of MW geodesics, some fraction of them will get “dispersed”... Assume they are all branched.

Number of geodesics for energy E is essentially E/(branchial area)

Transition amplitude is the overall measure of the beginning and end of the geodesic bundle onto their states....

If they are zero distance apart the amplitude will be 1

As they go further, there is more dispersion....

Event density aka causal edges determines number of branches

Number of geodesics for energy E is essentially E/(branchial area)

Transition amplitude is the overall measure of the beginning and end of the geodesic bundle onto their states....

If they are zero distance apart the amplitude will be 1

As they go further, there is more dispersion....

Event density aka causal edges determines number of branches

Measuring the dispersion going down the MWCG. Look at an angle because it doesn’t depend on the foliation....

[ Angles are complicated because of metric on branchial space ;;; and increasing dimension of branchial graph with time ]

[ Angles are complicated because of metric on branchial space ;;; and increasing dimension of branchial graph with time ]

#### Correspondence between conformal structure [ cones ... ]

Correspondence between conformal structure [ cones ... ]

#### Quantum Hamiltonian : defines flux of edges across branchlike hypersurfaces in the multiway causal graph

Quantum Hamiltonian : defines flux of edges across branchlike hypersurfaces in the multiway causal graph

Lagrangian: not measuring flux across timelike-transverse hypersurfaces, but instead measuring overall flux in any direction.....

global hypersurface [[[ from a given point ]]]

global hypersurface [[[ from a given point ]]]

Lagrangian : flux of edges in the multiway graph

Action is the multiway volume integral of the Lagrangian .... Lagrangian is like the Ricci scalar......Integrating to get the action is ∫L Transporting around a loop the change in angle is R area....

g

#### Lagrangian relative to the “vacuum Lagrangian”

Lagrangian relative to the “vacuum Lagrangian”

If you didn’t prepare the quantum state ..... there will be some behavior.

Now include your preparation....

?? normalization of states in Hilbert space .......

Now include your preparation....

?? normalization of states in Hilbert space .......

<Refractive index for geodesics>

#### Single path going through region of changed Ricci scalar

Single path going through region of changed Ricci scalar

Density of alternate branches.....

Graphics[Line[AnglePath[RandomReal[{0,2Pi},20]]]]

In[]:=

Out[]=

Angle affects the probability of hitting the right spot to contribute to the measure of a particular outcome....

[[In high-dimensional space, probably only the angle matters....]]

In a changing of dimensions, need to use angles.....

In a changing of dimensions, need to use angles.....

“Hitting the target” is simply the projection of its angle on the final surface / aka sin/cos