Portals and Butterflies

This is a math island, but much of the math here is simple and speaks for itself. Ultimately, it is the inelegant that is vulnerable to reality.

As unsolved as many problems are, we are also sitting on a number of solved problems that have yet to be tied cohesively. It's in the tying that a new dimension is unlocked.

The connections I am making in this island are ongoing and subject to either refinement or removal. There are parts that go beyond the minuscule and, in fact, reach into every domain of life. I will cover some of them below.

The rotation of nature

I would call this is an accumulation-free turnover of a system that remains integrated every moment in time. The math shows this beautifully.

First, $e^x$ is special because its derivative is itself; it is growth through self-propagating change. But Euler’s formula adds $i$ to the mix, and if we take the derivative of this formula, we can see that change is turned, phase by phase, until it returns, and keeps cycling like that.

\begin{gathered} e^{ix} = \cos x + i \sin x \\[8pt] z = e^{ix} \longrightarrow z’ = iz \end{gathered}

If we assume unitary transformation, then from our dissection of time, we can see that this is dictated by $e^{-\frac{i}{\hbar}\hat{H}t}$ , which is structurally similar to $e^{i\theta}$ . And this, when applied to a closed quantum system, preserves the total probability while continuously evolving the relative phases of the components.

Applied to a composition energy eigenstates:

\begin{gathered} \frac{ c_m e^{-\frac{i}{\hbar}E_mt} }{ c_n e^{-\frac{i}{\hbar}E_nt} } = \frac{c_m}{c_n} e^{-\frac{i}{\hbar}(E_m-E_n)t} \end{gathered}

\begin{gathered} |\Psi(t)\rangle = e^{-\frac{i}{\hbar}\hat{H}t} \sum_n c_n|E_n\rangle = \sum_n c_n e^{-\frac{i}{\hbar}E_nt}|E_n\rangle \end{gathered}

Applied to a pure superposition:

\begin{gathered} \frac{\beta(t)}{\alpha(t)} = \frac{\beta}{\alpha}e^{i\omega t} \end{gathered}

\begin{gathered} H = \frac{\hbar\omega}{2}\sigma_z = \frac{\hbar\omega}{2} \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \\[8pt] U(t) = e^{-\frac{i}{\hbar}Ht} = \begin{pmatrix} e^{-i\omega t/2} & 0\\ 0 & e^{+i\omega t/2} \end{pmatrix} \\[8pt] |\Psi(0)\rangle = \alpha|0\rangle+\beta|1\rangle = \begin{pmatrix} \alpha\\ \beta \end{pmatrix} \\[8pt] |\Psi(t)\rangle = U(t)|\Psi(0)\rangle \quad\longrightarrow\quad \alpha e^{-i\omega t/2}|0\rangle + \beta e^{+i\omega t/2}|1\rangle \end{gathered}

I take this as the Whole being encoded in the coherent relations among its parts, and the next move being incorporated into the present.

Forged in Fire

Growth, by itself, is not a thing; it emerges from integrated rotation.

The Cove — interactive panoramic tour — Enter

In the cove above, it is integration $\mathcal{I}$ , not time, that turns the loop.

\begin{gathered}   U=\min(S,C),   \qquad   L=|S-C| \\[16pt]   \mathcal{I}=\frac{U}{U+L} \\[12pt] \end{gathered}

$S$ is what is available to be received; C is the capacity to receive and harness it.

$U$ is what turns through, and $L$ is what remains unmatched.

$S$ and $C$ are measured through the qualities of warmth, transformation, fluidity, continuity, movement, exchange, substance, and support. I group these into Fire, Water, Air, and Earth. These are qualities are fundamental, and in flux.

What is received in one form can return in another: Water receives the Sun’s warmth, Fire, and rises as vapour, carrying that warmth into the Air. Condensation releases the warmth; rain returns Water to Earth. What one receives becomes what another can receive, and the loop turns.

If:

\begin{gathered}   S=C>0 \quad\Longrightarrow\quad \mathcal{I}=1 \end{gathered}

... then what is received can be transformed and offered again in the next cycle.

For simplicity, we can maintain the same relationship regardless of scale:

\mathcal{I} = \frac{\sum_j U_j} {\sum_j U_j + \sum_j L_j}

Finding time

Where does time actually from? How fundamental is it, really?

Take the time-dependent Schrödinger:

i\hbar\frac{d}{dt}|\Psi(t)\rangle = \hat H|\Psi(t)\rangle

It is $\hat H$ that generates change that time simply indexes, and when fixed, allows us to arrive at something that transforms an entire closed quantum system.

\begin{gathered} |\Psi(t)\rangle = e^{-\frac{i}{\hbar}\hat{H}t}|\Psi(0)\rangle \end{gathered}

\begin{gathered} |\Psi(t)\rangle = U(t)|\Psi(0)\rangle \\[12pt] i\hbar\frac{d}{dt}|\Psi(t)\rangle = \hat H|\Psi(t)\rangle \quad \longrightarrow \quad U(t) = e^{-\frac{i}{\hbar}\hat{H}t} \end{gathered}

More generally, we can say that a generator $\hat G$ produces a transformation through its exponential:

\begin{gathered} U(s)=e^{-is\hat{G}/\hbar} \end{gathered}

The time of energy

Now, there is a striking asymmetry. Momentum $\hat{p}$ is an observable and generates shifts of another observable, position $\hat{x}$ .

\begin{gathered} [\hat{x},\hat{p}] = i\hbar \\[8pt] \hat{x} \rightarrow \hat{x}+a \end{gathered}

If we assume that time and energy work identically, then:

\begin{gathered} V^\dagger \hat{H}V = \hat{H}-\varepsilon I \quad\forall\varepsilon\in\mathbb R \end{gathered}

\begin{gathered} \hat T=\hat T^\dagger,\qquad V(\varepsilon)=e^{-i\varepsilon\hat T/\hbar} \\[8pt] [\hat{T},\hat{H}] = \hat{T}\hat{H}-\hat{H}\hat{T} = i\hbar I \\[8pt] \frac{d}{d\varepsilon}\left(V^\dagger \hat{H}V\right) = \frac{i}{\hbar}V^\dagger[\hat{T},\hat{H}]V = -I \\[8pt] V(0)=I \\[8pt] V^\dagger \hat{H}V = \hat{H}-\varepsilon I \end{gathered}

But then $\hat{T}$ would generate arbitrary shifts in energy, including energies below the ground state. That cannot happen if energy has a lower bound.

\begin{gathered} E_{\min} \le \langle \psi | V(\varepsilon)^\dagger \hat{H} V(\varepsilon) | \psi \rangle = E - \varepsilon < E_{\min} \end{gathered}

\begin{gathered} \langle\psi|\psi\rangle=1,\qquad E=\langle\psi|\hat H|\psi\rangle \\[8pt] \end{gathered}

This asymmetry shows that no self-adjoint time operator generating unrestricted unitary energy translations can coexist with a bounded-below Hamiltonian.

Circle town

Still, we don't know where time comes from. $i\hbar\frac{d}{dt}|\Psi(t)\rangle = \hat H|\Psi(t)\rangle$ presupposes $t$ .

Perhaps then, time is not an independent substance, but a relationship between changes. So instead of saying $A = A(t)$ , one ultimately wants something more like:

\begin{gathered} A = A(C) \\[8pt] C=(n,\phi),\qquad A=A(n,\phi) \end{gathered}

Here, $C$ is some other physical process serving as a clock, $n$ records which cycle, and $\phi$ records where within that cycle (phase).

Pure and Whole

a four-quadrant ball in a backyard garden

There is something only present in the Whole and cannot be grasped merely by part, and I see parallels of this across life. Below is yet another parallel, but perhaps the most fundamental of them all.

Zoom out out and you'll find coherent structure among subsystems with a larger whole. This is a relational structure among parts (alternatives) within the system and they can, depending on the orientation, be harnessed to produce outcomes of varying certainty.

Consider the pure entangled state with environmental overlap $\gamma$ :

\begin{gathered} |\Psi\rangle=\alpha |A\rangle |E_A\rangle+\beta |B\rangle |E_B\rangle,\qquad|\alpha|^2+|\beta|^2=1 \\[8pt] \gamma=\langle E_B|E_A\rangle \end{gathered}

Define the path predictability (certainty) $\mathcal P$ , the local coherence $\mathcal C$ in the path basis, and the system-environment entanglement $\mathcal E$ :

\begin{gathered} \mathcal P = \left| |\alpha|^2-|\beta|^2 \right| \\[8pt] \mathcal C = 2|\alpha\beta|\,|\gamma| \\[8pt] E = 2|\alpha\beta| \sqrt{1-|\gamma|^2} \end{gathered}

This satisfies the Jakob-Bergou identity:

\begin{gathered} \mathcal P^2+\mathcal C^2+\mathcal E^2=1 \end{gathered}

\begin{gathered} \mathcal P^2 = \left(|\alpha|^2-|\beta|^2\right)^2 \quad\longrightarrow\quad 4|\alpha\beta|^2 = 1-\mathcal P^2 \\[8pt] \mathcal C^2+\mathcal E^2 = 4|\alpha\beta|^2 \left( |\gamma|^2+1-|\gamma|^2 \right) = 4|\alpha\beta|^2 = 1-\mathcal P^2 \end{gathered}

For an equal superposition:

\begin{gathered} |\alpha|=|\beta|=\frac{1}{\sqrt 2}, \quad \mathcal P=0 \\[16pt] \therefore\quad{ \mathcal C^2+\mathcal E^2=1 } \end{gathered}

\quad\mathcal C=|\gamma|, \qquad \mathcal E=\sqrt{1-|\gamma|^2} \\[12pt]

If the environment cannot distinguish the paths, the particle retains its local coherence:

\begin{gathered} |\gamma|=1 \\[8pt] \therefore\quad\mathcal C=1, \qquad \mathcal E=0 \end{gathered}

If the environment perfectly distinguishes the paths, coherence is delocalized:

\begin{gathered} |\gamma|=0 \\[8pt] \therefore\quad\mathcal C=0, \qquad \mathcal E=1 \end{gathered}

This is an entangled state. Each particle alone contains less information than the pair contains together. Specifically, no particle holds the coherence terms that hold information about the relative phase that belongs to two branches of the entangled state.

\begin{gathered} \left\{ |00\rangle\langle11|, \; |11\rangle\langle00| \right\} \subset \operatorname{terms}\!\left(\rho_{AB}^{\pm}\right) \end{gathered}

\begin{gathered} |\Phi^{\pm}\rangle = \frac{|00\rangle \pm |11\rangle}{\sqrt{2}} \\[12pt] |\Phi^{+}\rangle \neq |\Phi^{-}\rangle \\[12pt] \rho_A^{\pm} = \rho_B^{\pm} = \frac{I}{2} = \frac12 \left( |0\rangle\langle0| + |1\rangle\langle1| \right) \\[16pt] \left\{ |00\rangle\langle11|, \; |11\rangle\langle00| \right\} \subset \operatorname{terms}\!\left(\rho_{AB}^{\pm}\right) \\[12pt] \langle00|\rho_{AB}^{\pm}|11\rangle = \pm\frac12 \neq 0 ,\quad \langle11|\rho_{AB}^{\pm}|00\rangle = \pm\frac12 \neq 0 \end{gathered}

The entangled state is pure while each part is mixed, and it also cannot be factorized. $S$ is inversely correlated with purity, and it measures the von Neumann entropy.

\begin{gathered} S(\rho_{AB})=0 \\[8pt] S(\rho_A)>0, \qquad S(\rho_B)>0 \\[8pt] |\Psi\rangle_{AB} \neq |\psi\rangle_A \otimes |\epsilon\rangle_B \end{gathered}

This is how we connect purity, coherence, certainty, and entanglement.

The sphere

I mentioned earlier that relational structure among alternatives within the system (coherence) can, depending on the basis, be harnessed to produce outcomes of varying certainty (interference). Well, there is a geometric reason that certainty too is basis dependent.

For a pure state represented by Bloch vector $\mathbf r$ , measured along axis $\mathbf n$ :

\begin{gathered} \rho = \frac{1}{2} \left( I+\mathbf r\cdot\boldsymbol{\sigma} \right) \end{gathered}

\begin{gathered} \text{Three perpendicular axes of the Bloch sphere} \\[8pt] Z\text{-axis} \longleftrightarrow |0\rangle,\ |1\rangle \\[8pt] X\text{-axis} \longleftrightarrow |+_x\rangle,\ |-_x\rangle \\[8pt] Y\text{-axis} \longleftrightarrow |+_y\rangle,\ |-_y\rangle \\[15pt] \rho = \frac{1}{2} \left( I+\mathbf r\cdot\boldsymbol{\sigma} \right) \\[8pt] \Pi_{+\mathbf n} = \frac{1}{2} \left( I+\mathbf n\cdot\boldsymbol{\sigma} \right) , \quad \Pi_{-\mathbf n} = \frac{1}{2} \left( I-\mathbf n\cdot\boldsymbol{\sigma} \right) \\[8pt] P(\pm_{\mathbf n}) = \operatorname{Tr} \left( \rho\Pi_{\pm\mathbf n} \right) = \frac{1}{4} \operatorname{Tr} \left[ \left( I+\mathbf r\cdot\boldsymbol{\sigma} \right) \left( I\pm\mathbf n\cdot\boldsymbol{\sigma} \right) \right] \\[8pt] \operatorname{Tr}(I)=2, \qquad \operatorname{Tr}(\sigma_i)=0, \qquad \operatorname{Tr}(\sigma_i\sigma_j)=2\delta_{ij} \\[12pt] P(+_{\mathbf n}) = \frac{1+\mathbf r\cdot\mathbf n}{2}, \qquad P(-_{\mathbf n}) = \frac{1-\mathbf r\cdot\mathbf n}{2} \end{gathered}

Therefore, if the state is pure, then we can know that the degree of alignment between $\mathbf r$ and $\mathbf n$ directly informs the certainty of measurement outcomes.

\begin{gathered} \mathbf r=\mathbf n \quad\Longrightarrow\quad P(+_{\mathbf n})=1 \end{gathered}

\begin{gathered} \mathbf r=-\mathbf n \quad\Longrightarrow\quad P(-_{\mathbf n})=1 \\[8pt] \mathbf r\cdot\mathbf n=0 \quad\Longrightarrow\quad P(+_{\mathbf n}) = P(-_{\mathbf n}) = \frac12 \end{gathered}

Purity belongs to the state, while certainty belongs to the relation between the state and the way it is met.

Creating space

Space in its totality, from the space between thoughts to the space between islands, is not what we think it is.

In The Gut Universe, I said that I concentrate the entire globe into a mouthful. What I was trying to say there was: a local system can bring together contributions from widely separated histories .... without preserving those histories as a widely separated arrangement. Put another way, the physical distance between things need not be all that relevant when it enters a vortex of sorts. One may call the gut a blackhole of gastronomy.

Compress and record

Our discovery of relational time in Circle town, $A = A(C)$ , moves us to the beginning of compression territory.

Suppose there are $N$ distinguishable cycle counts and $M$ distinguishable phases within a cycle:

\begin{gathered} B_{\text{count}} = \left\lceil \log_2 N \right\rceil \\[10pt] B_{\text{reading}} \geq \left\lceil \log_2(NM) \right\rceil \end{gathered}

Because of the structure of C, we can reliably distinguish [time] readings, distributed over an even range and resolution, and we can do this with relatively and exponentially less information (space) than the sheer number of possible distinguishable positions.

This gives us a question to carry into physical space: how many distinguishable states can a region support, and what determines that number?

Enter the blackhole

This part is really a sequel to Pure and Whole.

Derived from the Bekenstein–Hawking entropy formula ...

\frac{S_{\mathrm{BH}}}{k_B} = \frac{1}{4}\frac{A}{\ell_P^2}

Here, $S_{BH}$ is the blackhole entropy and $A$ is its horizon area. The entropy measure is one quarter of the horizon area expressed in Planck-area units. There is now a relationship between our information measure in Compress and record and an actual geometric area.

But the resulting count scales with area not volume, that is, $R^2$ not $R^3$ . So one may ask: are we assigning too many independent possibilities to a gravitational region?

There is a distinction from ordinary data compression. An arbitrary collection of $n$ independent bits has $2^n$ alternatives. No lossless encoding into fewer than $n$ binary bits can preserve all of them ... that is simply counting.

But suppose all components are not independently specifiable ... then many nominal combinations may not be physical possibilities in the first place? One may then simply avoid counting possibilities that were never independent. So then, what are the genuinely independent degrees of freedom?

Connection to the quantum state

This has a concrete counterpart in holographic physics. Firstly, below, "boundary" refers to the lower-dimensional quantum system where the state is defined, while the "bulk" is the higher-dimensional spacetime geometry that the quantum system encodes.

In certain holographic models, the Ryu–Takayanagi relation connects the entropy of a boundary region to the area of a surface extending through the interior.

- \operatorname{Tr}\!\left(\rho_R \ln \rho_R\right) \simeq \frac{\operatorname{Area}(\gamma_R)}{4\ell_P^2}

Here we consider an interior with three spatial dimensions and one time dimension, whose geometry is static. $\rho_{\mathcal R}$ is the reduced quantum state of a boundary region $\mathcal R$ ; the expression measures its von Neumann entropy.

When the full boundary state is pure, this entropy measures entanglement between $R$ and its complement. $\gamma_{\mathcal R}$ is the appropriate minimal-area surface extending through the gravitational bulk and anchored on the region’s edge.

Notice where the two quantities live: a quantum-state property in the boundary description corresponds to a geometric area in the bulk description.

Spatial geometry may itself be something emergent rather than just sitting there waiting to be filled.

But I want to see how far we can really go with this, and this goes right back to The Gut Universe, and I will briefly outline the connections I see below.

Spacelike Gastronomy

Because we already know entanglement is a thing, what really interests me is: could entanglement between separated systems correspond to a geometric connection that their ordinary separation does not reveal?

It has been demonstrated that entanglement can be remain for particles that are spacelike separated, that is, particles outside each other's light cones.

\begin{gathered} |\Psi_L\rangle = \bigl(U_A(L)\otimes U_B(L)\bigr)|\Psi_0\rangle \\[8pt] \boxed{ S\bigl(\rho_A(L)\bigr)=S\bigl(\rho_A(0)\bigr)=1, \qquad S\bigl(\rho_{AB}(L)\bigr)=0 } \\[12pt] \end{gathered}

\begin{gathered} |\Psi_L\rangle = \bigl(U_A(L)\otimes U_B(L)\bigr)|\Psi_0\rangle \\[8pt] \rho_{AB}(L)=|\Psi_L\rangle\langle\Psi_L| \\[8pt] \rho_A(L) = \operatorname{Tr}_B\rho_{AB}(L) = U_A(L)\rho_A(0)U_A^\dagger(L) \\[8pt] \boxed{ S\bigl(\rho_A(L)\bigr)=S\bigl(\rho_A(0)\bigr)=1, \qquad S\bigl(\rho_{AB}(L)\bigr)=0 } \\[12pt] L>c|\Delta t| \quad\Longleftrightarrow\quad \text{spacelike-separated measurement events} \end{gathered}

Here $U_A(L)$ and $U_B(L)$ describe that ideal transport to separation $L>c|\Delta t|$ , where $c$ is light speed and $Δt$ is the difference between the measurement times in one inertial frame.

For a suitable two-sided holographic black hole with a four-dimensional gravitational bulk, the leading semiclassical relationship is:

S(\rho_{LR}) = 0, \qquad S(\rho_L) = S(\rho_R) \simeq \frac{A_h}{4\ell_P^2 \ln 2} > 0

Here $L$ and $R$ label the two quantum systems; $\rho_{LR}$ is their joint state, while $\rho_L$ , $\rho_R$ describe either side alone. $A_h$ is the horizon area, and $\ell_P$ the Planck length. The factor $\ln2$ expresses the entropy in bits.

So, I am willing to entertain the possibility that geometry expresses relationships belonging to the joint quantum state. This takes us right back to my comments about the gut. What is all this? It is—literally—an emergent space that remains a compelling exploration for me, and I leave it at that for now.

Finally, butterflies

What is love? Not what it represents or what it feels like ... but at it's core ... what is love?

Love is a force that motivates integration.

When I deeply value someone for their own sake ... say I delight in their existence, want to know them, and care about what becomes of them .... then their good becomes part of what I live for ... and I seek a way for both of us to live inside of a virtuous cycle.

The sentences above are worth re-reading (especially for myself). And there is a lot more to uncover about love, as love ... can easily grow to become the most potent force in a person's life. But I think that exploration belongs in another island so ... I'll leave it at that.