Hamiltonian vs Lagrangian: spacetime foliations

Different multiway paths contribute to a particular state

On grid: is this free particle propagator???

#### What is observer: branchlike hypersurface?

What is observer: branchlike hypersurface?

#### QM formalism: more local probing of branchial space.

QM formalism: more local probing of branchial space.

Quantum eigenstate: non-interacting state? No entanglement....

In multiway graph, a path that does not branch, and where nothing feeds into it.

But state can still move in branchial space?? [Relative to other states]

In multiway graph, a path that does not branch, and where nothing feeds into it.

But state can still move in branchial space?? [Relative to other states]

#### Toy Example

Toy Example

MultiwaySystem[{"A""AB","A""BA"},{"A"},5,"StatesGraph"]

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MultiwaySystem[{"A""AB","A""BA"},{"A"},5,"CausalGraph"]

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MultiwaySystem[{"A""AB","A""BA"},{"A"},5,"BranchialGraph"]

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SubstitutionSystemCausalGraph[{"A""AB","A""BA"},"A",5]

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SubstitutionSystemCausalGraph[{"A""AB","A""BA"},"AA",5]

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#### Entanglement definitions

Entanglement definitions

Tensor product state: states not interacting in multiway system: no correlation between “directions”

#### Action at distance

Action at distance

Multiway causal graph there can be connections between spatially distant events: hence “action at a distance”

Even though these are local in branch space....

Even though these are local in branch space....

#### Path integral

Path integral

State is just a state vector

Weighting of the state (aka amplitude) is the measure in the multiway geodesic bundle

Weighting of the state (aka amplitude) is the measure in the multiway geodesic bundle

#### Weighted graph Borel measure

Weighted graph Borel measure

#### Case 1: state vectors are local to branchlike hypersurfaces

Case 1: state vectors are local to branchlike hypersurfaces

### States have particular coordinate positions in multiway space

States have particular coordinate positions in multiway space

They also have a flow of causality between them defined by multiway edges

Distance between states = number of events to get from one to the other, with potentially many paths, weighted with the multiplicity of those paths

Idealization to slide the states to the same branchlike slice to compute their inner product...

< | >

#### Spacetime case: energy of flux of causal edges through spacelike hypersurfaces

Spacetime case: energy of flux of causal edges through spacelike hypersurfaces

#### There is also a flux of causal edges through branchlike hypersurfaces

There is also a flux of causal edges through branchlike hypersurfaces

Which are exactly associated with the “evolution edges” in the multiway graph

As there are more causal edges (i.e. higher H) there is more rotation in the distance

#### If there was no entanglement|interference, the MW causal graph would be a direct product of individual causal graphs

If there was no entanglement|interference, the MW causal graph would be a direct product of individual causal graphs

#### Entanglement = common ancestry

Interference = reconvergence

Entanglement = common ancestry

Interference = reconvergence

Interference = reconvergence

#### To get “expected spacetime energy” you have to average over all branchial directions the “downward” flux of causal edges

[The classical spacetime energy is on a single branch]

To get “expected spacetime energy” you have to average over all branchial directions the “downward” flux of causal edges

[The classical spacetime energy is on a single branch]

[The classical spacetime energy is on a single branch]

#### Imagine the multiway distances are given by a 2-vector [ which is (Δt, Δb) ]

Imagine the multiway distances are given by a 2-vector [ which is (Δt, Δb) ]

Number of causal edges (aka evolution edges), the larger the spread of Δb will be

Quantum “speed” is Δb/Δt : maximum is defined by entanglement cone

ArcTan[Δb/Δt]action/ℏ

#### inner product: true inner product involves looking in multiway space and taking account of path weights ;;; proxy is u. v == -1/2 dist(u,v) + u.u + v.v

inner product: true inner product involves looking in multiway space and taking account of path weights ;;; proxy is u. v == -1/2 dist(u,v) + u.u + v.v

### Compute commutators for string rewrites

Compute commutators for string rewrites

From a given state, different operators are just different edges in the multiway graph... So the commuting of operators is the extent to which it is closed.... [[ The edge sequences are geodesics in the MW graph ]]

Analog in spacetime is the fate of geodesics that start from a single point, etc. [Riemann tensor: commutator of covariant derivatives]

#### Claim is: spatial distance is encoded in the MW causal graph

Claim is: spatial distance is encoded in the MW causal graph

## Position, momentum

Position, momentum

Spacetime model:

A position observer is a detector that is spacelike extended ... and watches for causal edges slicing at a particular spatial position

Momentum observer does the same thing look for edging slicing a timelike surface

[ Energy, time case is basically this reversed ]

To get from being a position observer to a momentum observer, you have to rotate in spacetime....

In particular,

In particular,

Graphics[{Table[Arrow[Line[Reverse@{{i,0},{0,i}}]],{i,10}],Style[Table[Point[{x,y}],{x,0,10},{y,0,10}],Orange]}]

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Graphics[{Table[Arrow[Line[Reverse@{{i,0},{0,2/3i}}]],{i,10}],Style[Table[Point[{x,y}],{x,0,10},{y,0,10}],Orange]}]

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Every updating event is an opportunity to spread in branchial space...

It both represents moving of effect in configuration [motion] space, and spreading in branchial space.... because it gives the opportunity for a branching event.

It both represents moving of effect in configuration [motion] space, and spreading in branchial space.... because it gives the opportunity for a branching event.

Every event is an opportunity for branching

Graphics[{Table[Arrow[Line[Reverse@{{i,0},{0,i}}]],{i,10}],Style[Table[Point[{x,y}],{x,0,10},{y,0,10}],Orange]}]

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<< We should make a picture showing the arrows going in branchial space, including one whose projection into spacetime is 90° away from the px arrow. >>

As you decrease δx, aka increase the number of same points, you increase the number of unresolved critical pairs ~1/δx ... therefore you go a distance [[[ looking at the branchial graph, you could hop from one element one pair to another etc. .... ]]]

#### The full story of a family of observers is a complete slice of the foliation

The full story of a family of observers is a complete slice of the foliation

#### In QM formalism, we look at local “measurements” in slices of the multiway graph

In QM formalism, we look at local “measurements” in slices of the multiway graph

In our quantum formalism, we have global states

| S1 > + | S2 > XXXXXX

#### A measurement measures the magnitude of a projection onto a certain state

A measurement measures the magnitude of a projection onto a certain state

A zero time analog of how much |A> evolves to |B>

t,

<

b

#### Linear superposition of states corresponds to a measure in branchial space

Linear superposition of states corresponds to a measure in branchial space

#### Could have a partial multiway graph, that only talks about degrees of freedom for some part of the universe

[nontrivial to disentangle]

Could have a partial multiway graph, that only talks about degrees of freedom for some part of the universe

[nontrivial to disentangle]

[nontrivial to disentangle]

Wave function of the universe: amplitude for every point in the branchial graph of the universe

### Plan B:

Plan B:

#### Each event is a quantum operator

Each event is a quantum operator

A critical pair represents different outcomes from a measurement

#### One operator is the time evolution operator

One operator is the time evolution operator

Which may or may not generate branches

#### The states in the multiway system are eigenstates; a slice of a geodesic bundle is a superposition of states [aka a measure on branchial space is a superposition of states]

The states in the multiway system are eigenstates; a slice of a geodesic bundle is a superposition of states [aka a measure on branchial space is a superposition of states]

Measurement of an eigenstate is application of a replacement operation to an eigenstate, which generally produces many outcomes , on many different branches

To an observer, all the states in their slice are their universe (and are equivalent) [[[ all in a superposition ]]]

#### An idealized measurement is a fake time evolution, with specific replacement operations

An idealized measurement is a fake time evolution, with specific replacement operations

#### We’ve colored our branchlike slice with multiplicities...

We’ve colored our branchlike slice with multiplicities...

### Observer chooses particular measurement operations based on the sculpting of their foliation of multiway space

Observer chooses particular measurement operations based on the sculpting of their foliation of multiway space

Decoherence : all operations eventually happen....

#### States of the system which are individual hypergraph configurations

“quantum observation frame” (which contains many states; like a spacetime frame contains many spatial points) [an observation frame is a branchlike hypersurface]

quantum amplitudes of states in the qof

States of the system which are individual hypergraph configurations

“quantum observation frame” (which contains many states; like a spacetime frame contains many spatial points) [an observation frame is a branchlike hypersurface]

quantum amplitudes of states in the qof

“quantum observation frame” (which contains many states; like a spacetime frame contains many spatial points) [an observation frame is a branchlike hypersurface]

quantum amplitudes of states in the qof

Then there is < particular state | qof with measure >

There is no zero-time formalism; all projection is evolution

Probability is: what fraction of geodesic bundle intersects where you are going?

#### Post-measurement time evolution is the aggregate evolution of “the metric”

Post-measurement time evolution is the aggregate evolution of “the metric”

### Assume the vector in Hilbert space is the position {t, b}

Assume the vector in Hilbert space is the position {t, b}

Tangent vector is the generator of time translations

#### Wavepacket = geodesic bundle in multiway graph

Wavepacket = geodesic bundle in multiway graph

Each individual geodesic ends up after a certain interval at a different b value

The more updating events, the wider the dispersion in b values can be.... [[[ more updating events = more causal edges = more energy = more dispersion ]]]

Claim: phase angle refers to dispersion

e ^ ( i H t )

Along each path e ^ ( i S / ℏ )

### Sculpting an observation QOF

Sculpting an observation QOF

Can you get lots of critical pairs to converge? The more you can, the more you wind up with definite basic state

### Entanglement cone

Entanglement cone

How fast can I get away from coherence?

### Have to follow partial ordering for a meaningful notion of time

Have to follow partial ordering for a meaningful notion of time

## Basic states are ?orthonormal basis states for Hilbert space

Basic states are ?orthonormal basis states for Hilbert space

Superpositions are then vectors in the Hilbert space [ rays because of normalization ]

Each basic state is at some position in multiway space: vector= {t, b}

Claim: vectors for different basic states are orthogonal v1 . v2 = 0 (not necessary)

distance between tips of the vectors: (v1 - v2)^2 = v1^2 + v2^2 - 2 v1 . v2

#### [ Don’t really need to discuss Hilbert space ... ]

[ Don’t really need to discuss Hilbert space ... ]

When a geodesic branches, it changes its angle

Product of Exp[i S] is sum of the angles

Assume every edge in the multiway graph goes a fixed distance in multiway space.....

But at some angle....

But at some angle....

i S gives you the probability that there is a branching in unit time

The total amount of branching is determined by energy: number of branches that are generated

Come into a state; it either branches or it doesn’t; into k branches

The total amount of branching is determined by energy: number of branches that are generated

Come into a state; it either branches or it doesn’t; into k branches

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

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Draw our multiway system laid out in branchial space; then it will look all wiggly like this....

The Lagrangian, AKA action, is defining the propensity to branch....

Lagrangian mechanics is in the multiway system; Hamiltonian in causal graph ??

#### Is QFT the mean field theory of the multiway system?

Is QFT the mean field theory of the multiway system?

The Lagrangian of QFT is stated in terms of particle fields.... Complicated transformation of underlying hypergraph....

#### Are particle-like results general enough to apply to arbitrary LHSs? [I.e. LHSs as particles....]

Are particle-like results general enough to apply to arbitrary LHSs? [I.e. LHSs as particles....]

#### Spin-statistics: commuting or non-commuting

Spin-statistics: commuting or non-commuting

## Distance vs. Displacement etc.

Distance vs. Displacement etc.

{Δx, Δt}

Distance measure is a machine for turning pairs of points into numbers

Distance has certain properties....

Distance has certain properties....

{,,}

Δ

1

Δ

2

Δ

3

https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

Quadratic form

Using a i is too much; therefore remove dof from projectivity

{x1,x2,x3,t1,t2,XXXX}

An orthogonal matrix is applied to all the coordinates

We want to leave a quadratic form invariant....

Sum[ki^2,XXXXX]

t^2-x^2+ib^2

You are coding which kind of distance you have by using a number......

x^2 + i t^2

# Metatheory of abstraction....

Metatheory of abstraction....

Resolution of critical pairs as dictionary/abstraction.....

Resolution of all critical pairs .... everything is connected........