Wave packets from the spectrum

This paper demonstrates that by transforming the Fock basis, any Hamiltonian can be reinterpreted as a local lattice theory with wave packet propagation, where the system's dispersion relation and integrability are determined by its energy spectrum, offering a potential framework for quantum mereology.

The Gist

Imagine you have a giant, chaotic box of Lego bricks. Inside this box, every single brick is connected to every other brick in a completely random, tangled mess. If you try to push one brick, the whole box shakes violently. To an outside observer, this looks like a system with no structure, no order, and no "local" rules (where a brick only affects its immediate neighbors).

This is essentially what the authors of this paper are looking at: a quantum system defined by a "Hamiltonian" (a mathematical rulebook for how energy moves) that looks completely random and chaotic. In the language of physics, this is like a "random matrix" where every part of the system talks to every other part instantly.

The Big Question:
Is it possible that this chaotic mess is actually hiding a simple, orderly structure underneath? Could we just be looking at the Lego box from the wrong angle?

The Solution: Changing the "Camera Angle"
The paper argues that the way we usually look at these systems (the "Fock basis") might be the wrong perspective. It's like trying to describe a beautiful, organized city by looking at a photo of it through a kaleidoscope. The photo looks like a jumbled mess of colors, but if you rotate the kaleidoscope (change your mathematical perspective), the city suddenly snaps into focus.

The authors developed a new algorithm—a set of mathematical steps—to "rotate" the view of the quantum system. Here is how they did it, using a simple analogy:

1. Finding the "Quiet Neighbors"

Imagine you are in a crowded, noisy room where everyone is shouting. Your goal is to find one person who is relatively quiet and mostly isolated from the noise.

• The authors' algorithm scans the chaotic quantum system to find a single "qubit" (a tiny quantum bit, like a switch that can be on or off) that interacts as little as possible with the rest of the system.
• They call this a "minimum interaction" qubit. It's the person in the room who is whispering while everyone else screams.

2. Peeling the Onion

Once they find that one quiet person, they "peel them off" the system. They then look at the remaining crowd and find the next quietest person who is isolated from the new remaining group.

• They repeat this process over and over, peeling off one quiet layer at a time, until the entire chaotic system is broken down into a stack of individual, quiet layers.

3. The Magic Transformation

Here is the surprising part: When they reassemble the system using these newly found "quiet layers," the chaos disappears.

• Before: The system looked like a random mess where everything affected everything else instantly (highly non-local).
• After: The system looks like a neat, 1D line of dominoes or a string of beads. In this new view, a particle (a wave packet) can sit in one spot and then travel down the line to the next spot, just like a wave moving through water or a runner on a track.

The "Wave Packet" Discovery
The paper demonstrates this with a specific example. They started with a Hamiltonian that was generated completely at random (like rolling dice to decide how every part of the system connects).

• The Result: Even though the starting point was pure randomness, their algorithm found a new way to describe it where particles could form "wave packets." These are little bundles of energy that stay together and move smoothly across the system, rather than instantly exploding everywhere.
• They found that these particles move at speeds determined by a "dispersion relation." Think of this as a rulebook that says, "If you have this much energy, you will travel at this specific speed."

Why This Matters (According to the Paper)
The authors suggest this is a step toward "Quantum Mereology." This is a fancy term for asking: "How do we figure out what the fundamental building blocks of the universe are, just by looking at the math of how energy moves?"

Usually, we assume the universe is made of fields and particles from the start. This paper suggests that maybe the universe is just a giant, abstract quantum system, and "particles" and "space" are just the most convenient ways for us to describe it. If we use the right mathematical "lens" (the one they invented), even a completely random, chaotic system can look like a world with local rules, traveling particles, and a clear sense of space.

In Summary:
The paper shows that you can take a quantum system that looks like total chaos and, by mathematically reorganizing how you view its parts, reveal a hidden, orderly world where particles travel in waves. They proved that the "locality" (the idea that things only affect their neighbors) we see in our universe might not be a fundamental rule, but rather a feature that emerges when we choose the right perspective to look at the quantum data.

Technical Summary

Technical Summary: Wave Packets from the Spectrum

Problem Statement
The paper addresses a central challenge in quantum mereology: the task of identifying the "correct" degrees of freedom (dofs) within an abstract quantum theory defined solely by a Hamiltonian $\hat{H}$ acting on a Hilbert space $\mathcal{H}$. While the physics of the universe is described by unitary dynamics, the decomposition of $\mathcal{H}$ into tensor factors (e.g., lattice sites or field modes) is not fundamental but must be chosen to satisfy operational purposes. Specifically, the authors aim to determine if an arbitrary Hamiltonian—potentially one that induces highly non-local dynamics in a given basis—can be re-interpreted as a local lattice theory where particles propagate as localized wave packets. This falls under the category of field mereology, which seeks to identify dynamical building blocks behaving like field theory modes rather than semi-classical objects.

Methodology
The authors propose an iterative algorithm to decompose a Hilbert space of dimension $d=2^n$ into a set of $n$ qubits (interpreted as Fermionic modes) that minimize interaction.

1. Minimum Interaction Factorization: The core procedure involves splitting the Hilbert space $\mathcal{H} \simeq \mathcal{H}_q \otimes \mathcal{H}_r$ (where $\mathcal{H}_q$ is a qubit) such that the interaction Hamiltonian $\hat{H}_{int}$ is minimized. The authors minimize the Hilbert-Schmidt norm loss function $L = \text{Tr}(\hat{H}_{int}^2)$.
   • Theorem 1 establishes that at local minima of this loss function, the interaction takes the form $\hat{H}_{int} \simeq \sigma_z \otimes \hat{X}_{int}$, where $\hat{X}_{int}$ commutes with the rest Hamiltonian $\hat{H}_r$. This ensures that the interaction terms contain only Pauli strings with $\sigma_z$ or identity on the qubit factor.
2. Iterative Retrieval: The algorithm iteratively extracts these low-interaction qubits. At each step $i$, the remaining Hilbert space is further factorized. The authors select the factorization that minimizes the ratio $|\hat{H}_{int}|^2 / |\hat{H}_q|^2$ to ensure robustness (high self-energy).
3. Reconstruction of Lattice Theory:
   • The extracted qubits are treated as Fourier modes of an emergent lattice.
   • The self-energies $E_i$ of these qubits are mapped to a momentum grid $k$ to construct a dispersion relation $E(k)$.
   • A Jordan-Wigner transformation converts the Bosonic operators of the qubits into Fermionic operators $\hat{c}_k$.
   • A discrete Fourier transform yields real-space operators $\hat{a}_x$, expressing the Hamiltonian in a form resembling a local lattice theory: $\hat{H} \approx \sum_{x,y} M_{xy} (\hat{a}_x^\dagger \hat{a}_y - \frac{1}{2}\delta_{xy}) + \hat{H}_{int}$.
4. Spectral Analysis (GUE): To understand the structure of low-interaction peaks found in random Hamiltonians, the authors analyze the asymptotic eigenvalue density of Gaussian Unitary Ensemble (GUE) matrices. They derive that the interaction lower bound exhibits discrete dips (peaks of low interaction) corresponding to qubit energies $E_q$ where the Bessel function $J_1$ in the spectral convolution cancels poles of the integrand.

Key Results

• Recovery of Known Physics: When applied to a discretized Pauli field (an integrable theory), the algorithm successfully recovers the exact Fourier mode energies and the corresponding dispersion relation, validating the method against a known local theory.
• Emergence of Locality from Randomness: When applied to a random Hamiltonian drawn from the GUE (which is highly non-local in a standard basis), the algorithm identifies a new set of operators $\{\hat{b}_i^\dagger, \hat{b}_i\}$ (the emergent lattice).
  • In this new basis, the dynamics allow for localized wave packets that propagate over time.
  • The propagation speed is governed by an effective dispersion relation $E_{eff}(k)$, which is the bare dispersion relation renormalized by the residual interaction $\hat{H}_{int}$.
  • Corollary 3 proves that for any Hamiltonian processed this way, particle number is conserved in the emergent lattice, and the one-particle subspace is diagonal in the momentum basis.
• Spectral Origin of Structure: The authors demonstrate that the discrete sequence of low-interaction qubit energies found in GUE models corresponds analytically to the zeros of the Bessel function $J_1$, derived from the asymptotic spectral density of the ensemble.
• Geometry: Using Multidimensional Scaling (MDS) on the hopping matrix derived from the dispersion relation, the emergent lattice is shown to embed naturally into a 1D circular geometry (periodic boundary conditions), consistent with the 1D nature of the constructed theory.

Significance and Claims
The paper claims to be a step towards quantum mereology for fields. It argues that the freedom to change the Fock basis ensures a minimum amount of locality in lattice theories. Specifically:

• Universality of Local Interpretation: Any Hamiltonian can be made to look like a local lattice theory with propagating particles, provided one chooses the appropriate basis. The "locality" of the dynamics is not an intrinsic property of the Hamiltonian's spectrum alone but depends on the choice of basis, which the algorithm optimizes.
• Spectrum-Driven Emergence: The structure of the emergent theory (specifically the dispersion relation and the existence of wave packets) is determined by the spectrum of $\hat{H}$.
• Limitations and Open Questions: The authors remain modest about the scope. They note that while the one-particle sector behaves classically (propagating wave packets), the interaction structure in higher $N$-particle sectors is not fully characterized. They identify several open tasks, including:
  • Defining a global loss function rather than a step-wise one.
  • Addressing whether real-space modes or harmonic modes are the true minimum-interaction dofs for strongly interacting theories.
  • Extending the framework to Bosonic fields and higher-dimensional lattices.
  • Investigating "overlapping degrees of freedom," as the iterative algorithm forces an exact tensor product factorization, potentially missing compatible but overlapping structures that might better approximate holographic bounds.
  • Incorporating state-dependent information (quantum information structure) rather than relying solely on the Hamiltonian spectrum, which is crucial for applications in quantum gravity.

In summary, the work demonstrates that even chaotic, random matrix models can be re-interpreted as theories of propagating particles with localized wave packets, suggesting that "classical" field-like behavior can emerge from generic quantum dynamics through an appropriate choice of degrees of freedom.