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arxiv: 2606.10004 · v1 · pith:PMXDSHASnew · submitted 2026-06-08 · 🪐 quant-ph · hep-th

Wave packets from the spectrum

ChunJun Cao , Oliver Friedrich , Marin Girard , Nicolas Loizeau , Ashmeet Singh This is my paper

Pith reviewed 2026-06-27 16:02 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords Fock basislocalitywave packetsHamiltonian spectrumquantum mereologylattice theoriesnon-integrabilityrandom matrix models
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The pith

Any Hamiltonian can be rewritten via Fock basis change as a local 1D lattice theory whose dispersion and integrability depend only on its spectrum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the choice of Fock basis for creation and annihilation operators can always be adjusted so that an arbitrary Hamiltonian generates local dynamics on a one-dimensional lattice. In this representation, particles move as localized wave packets whose propagation speed is set by the energy eigenvalues of the original operator. The same spectrum also fixes whether the resulting lattice model is integrable or chaotic. This reframes apparent non-locality as a feature of the operator basis rather than an intrinsic property of the dynamics, offering a concrete route toward quantum mereology for fields.

Core claim

We show that any Hamiltonian can be made to look like a local 1D lattice theory where particles propagate in localized wave packets, with the lattice dispersion relation and the non-integrability of the theory depending on the spectrum of Ĥ. A highly non-local random-matrix Hamiltonian is explicitly turned into such a local theory by a suitable redefinition of the Fock operators.

What carries the argument

Redefinition of the Fock basis from one set of creation and annihilation operators to another that renders the Hamiltonian local on a 1D lattice.

If this is right

  • The dispersion relation on the effective lattice is completely determined by the eigenvalues of the original Hamiltonian.
  • Whether the lattice model is integrable or non-integrable is likewise fixed by the spectrum.
  • Random-matrix Hamiltonians that appear highly non-local can be rewritten as local lattice theories supporting propagating wave packets.
  • Locality in quantum field theories can be viewed as a consequence of basis choice rather than an absolute requirement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction suggests that many-body localization or thermalization properties might be tunable by basis choice even when the spectrum is held fixed.
  • It opens the possibility of mapping arbitrary quantum systems to lattice simulators by identifying the appropriate operator redefinition.
  • The approach may connect to questions of how classical spacetime emerges from quantum degrees of freedom by selecting a local basis.

Load-bearing premise

A suitable Fock basis redefinition always exists that makes the dynamics of any Hamiltonian local on a 1D lattice, with dispersion and integrability fixed solely by the spectrum.

What would settle it

Exhibit a specific Hamiltonian for which no Fock basis exists that produces local nearest-neighbor dynamics on a 1D lattice, or show that the resulting dispersion cannot be read off from the spectrum alone.

Figures

Figures reproduced from arXiv: 2606.10004 by Ashmeet Singh, ChunJun Cao, Marin Girard, Nicolas Loizeau, Oliver Friedrich.

Figure 1
Figure 1. Figure 1: Demonstrating our procedure for (re-)interpreting a given Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualizing our iterative procedure for factorizing a given Hilbert space into minimum-interaction qubits. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The dispersion relations we obtain when iteratively decomposing either the Pauli field Hamiltonian or a [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Starting from a Gaussian random Hamiltonian in a [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Classical MDS embeddings with compact B-matrix spectra: The top left panel shows the embedding of the spatial Fermionic degrees-of-freedom emerging from a Wigner Hamiltonian in a two-dimensional cartesian space via MDS. The top right panel shows the corresponding embedding for a Pauli field Hamiltonian. We overlay graph edges between embedded Fermionic degrees-of-freedom, with edge color indicating hopping… view at source ↗
Figure 6
Figure 6. Figure 6: We show the squared norm of Hˆ int when fixing the energy gap Eq = Emax −Emin of the qubit Hamiltonian (and thus fixing the norm of Hˆ q). These conditional minima closely follow the asymptotic lower bound we derived in Section 3. The vertical, dotted lines indicate the location of minimum-interaction-peaks that is predicted by Equation 47. In order for a given factorization to satisfy the approximate Equa… view at source ↗
Figure 7
Figure 7. Figure 7: Left panels are showing exact solutions to the decomposition of Equation [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Dispersion relation of the discretized Pauli field when using different schemes to represent the Laplacian [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
read the original abstract

The freedom to change Fock basis seems to ensure a minimum amount of locality in lattice theories in the following sense: If $\lbrace (\hat a_i^\dagger\,,\,\hat a_i)\rbrace$ for $i=1,\dots,n$ is a lattice of creation and annihilation operators and if a given Hamiltonian $\hat H$ induces highly non-local dynamics on that lattice, then it will usually be possible to change to a new set of operators $\lbrace (\hat b_i^\dagger\,,\,\hat b_i)\rbrace$ in terms of which the dynamics appear less non-local. We demonstrate this by turning a highly non-local random matrix model into a local, 1D lattice theory where particles can propagate in localized wave packets. More generally, we show that any Hamiltonian can be made to look like such a theory, with the lattice dispersion relation and the non-integrability of the theory depending on the spectrum of $\hat H$. We argue that our results are a step towards quantum mereology for fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the freedom to redefine the Fock basis ensures a minimum locality in any lattice theory: for an arbitrary Hamiltonian Ĥ, there exists a change to operators {b_i} such that the dynamics appear as those of a local 1D lattice model supporting localized wave packets. The lattice dispersion relation and the degree of non-integrability are asserted to depend only on the spectrum of Ĥ. This is illustrated by an example in which a highly non-local random-matrix Hamiltonian is transformed into a local 1D theory; the authors present the result as a step toward quantum mereology for fields.

Significance. If rigorously established, the result would offer a basis-dependent perspective on locality and wave-packet propagation, potentially clarifying when apparent non-locality is an artifact of operator choice rather than an intrinsic feature. The explicit dependence of dispersion and integrability on the spectrum alone would be a strong, falsifiable statement with implications for many-body localization and quantum simulation.

major comments (2)
  1. [Abstract] Abstract and introductory claim: the assertion that 'any Hamiltonian can be made to look like such a theory' with locality fixed solely by the spectrum is not accompanied by an explicit construction, unitary matrix, or derivation. The random-matrix example is referenced but no supporting equations, basis transformation, or verification that the resulting operator is strictly local (finite-range on a 1D chain) are supplied.
  2. [Abstract / general claim] General result: the claim that a U(n) redefinition of the single-particle modes suffices to map an arbitrary Ĥ onto the O(n)-dimensional manifold of local 1D lattice Hamiltonians is dimensionally inconsistent. The space of Hermitian operators on the Fock space has dimension exponential in n, while a single-particle unitary supplies only n² parameters; hence a generic Ĥ cannot lie in the orbit of any local model under such a transformation. This directly undermines the universality statement.
minor comments (1)
  1. [Abstract] The phrase 'usually be possible' in the abstract is left undefined; a precise statement of the measure or conditions under which a basis change reduces non-locality would clarify the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising these substantive issues. We respond point by point to the major comments and indicate the revisions that will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introductory claim: the assertion that 'any Hamiltonian can be made to look like such a theory' with locality fixed solely by the spectrum is not accompanied by an explicit construction, unitary matrix, or derivation. The random-matrix example is referenced but no supporting equations, basis transformation, or verification that the resulting operator is strictly local (finite-range on a 1D chain) are supplied.

    Authors: We agree that the abstract and introduction would benefit from greater explicitness. The random-matrix example was intended to illustrate the transformation of a dense single-particle Hamiltonian into tridiagonal form, but the supporting details were omitted. In the revised manuscript we will add an explicit construction (via, for example, Householder reflections or the Lanczos algorithm), the resulting unitary matrix elements or their defining properties, the transformed local Hamiltonian, and a direct verification that all couplings are nearest-neighbor, confirming finite range on the 1D chain. This will also show how the dispersion relation is fixed by the eigenvalues. revision: yes

  2. Referee: [Abstract / general claim] General result: the claim that a U(n) redefinition of the single-particle modes suffices to map an arbitrary Ĥ onto the O(n)-dimensional manifold of local 1D lattice Hamiltonians is dimensionally inconsistent. The space of Hermitian operators on the Fock space has dimension exponential in n, while a single-particle unitary supplies only n² parameters; hence a generic Ĥ cannot lie in the orbit of any local model under such a transformation. This directly undermines the universality statement.

    Authors: The dimensional observation is correct for a fully general many-body Hamiltonian. Our construction applies specifically to quadratic Hamiltonians, whose single-particle matrices lie in an n²-dimensional space; any such matrix can be unitarily transformed to tridiagonal (local 1D) form, with the spectrum determining the eigenvalues and hence the effective dispersion. We will revise the abstract, introduction, and general claim to restrict the scope explicitly to quadratic Hamiltonians (or to cases where the Hamiltonian remains quadratic under the basis change) and to remove the unqualified universality statement for arbitrary many-body operators. The dependence of dispersion and integrability properties on the spectrum will be retained and clarified within this restricted setting. revision: yes

Circularity Check

1 steps flagged

Any Hamiltonian made local on 1D lattice by Fock basis change reduces to redefinition by construction

specific steps
  1. self definitional [Abstract]
    "More generally, we show that any Hamiltonian can be made to look like such a theory, with the lattice dispersion relation and the non-integrability of the theory depending on the spectrum of Ĥ."

    The phrase 'such a theory' refers to a local 1D lattice model obtained precisely by the Fock-basis change described in the preceding sentence. Locality is therefore defined into existence by the choice of {b_i}, with no external criterion supplied that would make the statement non-tautological. Spectrum dependence is likewise automatic because eigenvalues are invariant under the unitary redefinition.

full rationale

The paper's strongest claim—that any Ĥ can be rewritten as a local 1D lattice theory with dispersion and integrability fixed solely by spec(Ĥ)—follows immediately from the stated freedom to perform an arbitrary unitary redefinition of the single-particle operators. No independent constraint on the form of the new operators or on the resulting interaction range is imposed beyond the redefinition itself, so the locality is achieved tautologically. The skeptic dimension-counting argument (U(n) orbit vs. O(n)-dimensional local Hamiltonians) is not addressed in the provided text, confirming the reduction is definitional rather than derived.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mathematical freedom to perform arbitrary unitary changes of Fock basis to induce locality, with no free parameters, invented entities, or additional axioms specified beyond standard quantum mechanics.

axioms (1)
  • domain assumption A unitary transformation always exists that maps any set of Fock operators to a local 1D lattice set with localized wave packets.
    This is the core premise enabling the general claim for any Hamiltonian.

pith-pipeline@v0.9.1-grok · 5707 in / 1269 out tokens · 38276 ms · 2026-06-27T16:02:26.007001+00:00 · methodology

discussion (0)

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Reference graph

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