Spectral Model

Spectral Cosmological Model

Introduction

An informational static timeless universe contains all possibilities at once. A classical picture is $2^n$ configurations existing simultaneously. Time emerges as walk through the configurations.

In the Spectral model $n$ bits is replaced by static wavefunction, which is a compressed, spectral description of the information. The statistical discrete classical model emerges as statistical Born sampled dithered finite outcome. This is close analogue to canonical quantum gravity where the universe does not have an external clock. The Wheeler-DeWitt equation is famously static:

$$ \hat{H}_{\text{total}} \vert{}\Psi\rangle_U = 0 $$

The total wavefunction of the universe $\vert{}\Psi\rangle_U$ contains all possibilities at once. It doesn't evolve; it just is. This is the quantum analogue of ensemble of $2^n$ configurations existing simultaneously.

The classical model introduced time via an Ehrenfest random walk (a sequence of configurations separated by single bit-flips). See Information Theoretic Emergence of Spacetime and Matter and Entropy and Emergent Structures. The corresponding process in $\psi$ model is Page-Wootters mechanism. It splits the total universe into two subsystems: a Clock ( $C$ ) and the Rest of the Universe ( $R$ ), so that $\vert{}\Psi\rangle_U \in \mathcal{H}_C \otimes \mathcal{H}_R$.

• Instead of a sequential walk, time emerges through entanglement between the clock states $\vert{}t\rangle_C$ and the rest of the universe.
• The "state at time $t$" is the conditional wavefunction:

$$ \psi_R(t) = \langle t\vert{}_C \vert{}\Psi\rangle_U $$

Bridging to Ehrenfest Walk

To match my model, the Clock subsystem $C$ would be an operator that tracks the "entropy coordinate" or the "step number" along the minimal-change path. The Page-Wootters clock states $\vert{}t\rangle_C$ would project out a history $\psi_R(t)$ that traces exactly through the quantum superpositions of those bit-string configurations. The "minimal change" constraint (one bit difference) translates to a Hamiltonian $\hat{H}_{\text{total}}$ that only couples states that are close in Hamming distance.

From Sub-strings to "Sub-wavefunctions"

In the statistical model, we find matter by looking for minimally trivial most typical sub-strings within a massive total bitstring $U$. The quantum equivalent of a "sub-string" is a subspace factorization of the total Hilbert space, and the equivalent of "minimally trivial" is governed by Quantum Entanglement and Information Separability.

To formalize "minimally trivial sub-wavefunctions":

Step A: Defining the Subspaces

If the total universe vector space is $\mathcal{H}_U$, finding a sub-wavefunction means finding a decomposition:

$$ \mathcal{H}_U = \mathcal{H}_A \otimes \mathcal{H}_B $$

Where $\mathcal{H}_A$ is the small subsystem (the proto-particle) and $\mathcal{H}_B$ is the rest of the universe. The total wavefunction is $\vert{}\Psi\rangle_U$.

Step B: The "Minimally Trivial" Criterion (Schmidt Decomposition)

To see if a sub-wavefunction $\vert{}\psi\rangle_A$ "emerges" as a distinct structure, perform a Schmidt decomposition along that split:

$$ \vert{}\Psi\rangle_U = \sum_{k} \lambda_k \vert{}f_k\rangle_A \otimes \vert{}g_k\rangle_B $$

The coefficients $\lambda_k$ determine how intertwined the sub-wavefunction is with the rest of the universe. We can define "minimally trivial" in two ways here:

1. Maximal Autonomy (Zero Entanglement): If there is only one term in the sum ($\lambda_1 = 1$, all others $0$), the wavefunction factors perfectly: $\vert{}\Psi\rangle_U = \vert{}\psi\rangle_A \otimes \vert{}\phi\rangle_B$. Subsystem $A$ has its own pure, distinct wavefunction. It is completely independent of the background.
2. Minimal Quantum Information (Stable Trapped Entanglement): A perfectly isolated subsystem carries no relational data. A more realistic "elementary particle" might be a subsystem where the entanglement entropy $S(\rho_A) = -\text{Tr}(\rho_A \log \rho_A)$ is minimized or localized over the Ehrenfest path, meaning it maintains its structural integrity (its internal state dynamics) even as the background shifts.

Step C: Density Matrices as the "Shadow"

If we look at the sub-wavefunction of subsystem $A$, it will generally not be a pure state because it is entangled with the environment. Instead, it is described by a reduced density matrix:

$$ \rho_A = \text{Tr}_B (\vert{}\Psi\rangle_U \langle\Psi\vert{}_U) $$

Classical "minimally trivial sub-strings" are the diagonal components (the classical probabilities) of this exact density matrix $\rho_A$ when expressed in the bit-string basis.

The Synthesis

To transition this to the wave-mechanical layer:

• The Universe: A single, timeless vector $\vert{}\Psi\rangle_U$ solving $\hat{H}\vert{}\Psi\rangle_U = 0$.
• Time/Dynamics: Emerges via a Page-Wootters correlation with an internal entropy/path clock.
• Matter: Appears when looking at the total state and find specific tensor factorizations $\mathcal{H}_A \otimes \mathcal{H}_B$ where the reduced state $\rho_A$ remains highly coherent, localized, or minimally entangled with the background $B$ along the path.

In this view, particles aren't fundamental objects placed into space; they are coherent, low-entropy information structures emerging out of the global quantum state's factorization.

Emergent Structures

cosmic_psi.py --scales 4,16,20 --compositions 1:3,2:14,4:16 --parallel --t_bf_max 2

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