Scale-Space Entanglement and Factorization Caveats

Momentum bands, wavelet scales, and spatial regions define different subsystem questions. A scale-space entropy is meaningful only after a regulator supplies a factorization or an operational algebra of accessible modes. It is generally basis dependent, can conflict with gauge constraints, and should not be identified with spatial entanglement or Wilsonian information loss without an explicit map.

Required background. Modes, regions, and factorization scales supplies the scale decompositions; separating scales distinguishes their limits.

Helpful background. Factorization failure and local algebras explains why continuum spatial subsystems are not literal Hilbert factors.

Fourier and wavelet subsystems

In a finite lattice volume, a canonical Fourier transform can factor the oscillator Hilbert space into modes,

$$\mathcal H_{\rm lat} =\bigotimes_{\mathbf k}\mathcal H_{\mathbf k}.$$

One may trace modes in a band and compute momentum-space entanglement. For a free vacuum in the normal-mode basis, different $\mathbf k$ oscillators factorize, so this entropy can vanish even though spatial entanglement is nonzero. Interactions entangle modes, but the answer depends on the chosen canonical variables and band definition.

Perturbative momentum-space entanglement and its explicit dependence on the mode split are analyzed in Balasubramanian, McDermott, and Van Raamsdonk 2012, §§ II–IV.

A wavelet transform replaces globally supported Fourier modes by functions localized in position and scale. It can make locality and multiresolution structure more transparent, but it is still a choice of basis and filter. Different orthogonal wavelets can assign different entanglement to adjacent scales while reconstructing the same state.

The exact relation between wavelet filters and entanglement-renormalization circuits is shown in Evenbly and White 2016, Eqs. (1)–(8); a continuum real-space field construction is given by Haegeman et al. 2013, Eqs. (2)–(9).

Scale-space partitions require an extra subsystem choice before the chapter’s three branches. A Fourier or wavelet band is not the same object as a spatial region, and tracing it is a channel only in the declared regulated factorization. Schematic and not to scale.

Basis dependence as a diagnostic

Suppose two orthogonal mode bases are related by a unitary $U$ that mixes vectors across the proposed low/high split. The total state is unchanged, but the tensor factors assigned to “low” and “high” change. Their entanglement entropies need not agree. Only transformations block diagonal within each side preserve the bipartition.

This is not a flaw if the basis represents an instrument. A detector with a finite temporal window, aperture, and frequency response selects smeared modes. Its accessible algebra gives the partition operational meaning. Without such a task, “entanglement between scales” is a representation-dependent diagnostic.

Gauge and locality constraints

Gauge-invariant observables are constrained by Gauss law and do not generally factor by links or momentum modes in the naive way. Longitudinal variables, zero modes, and edge fluxes require a center or extension prescription. A Fourier-shell trace performed before imposing constraints can count gauge redundancy as entanglement.

Wavelets also trade exact momentum separation for spatial localization. A compactly supported wavelet has a broad momentum response, so a sharp statement about “integrating out $k>\Lambda$” no longer applies. Conversely, a sharp Fourier shell is nonlocal in position. The two decompositions answer different operational questions.

Reconstruction comparison

For a regulated Gaussian field, compute the covariance matrix $V$ and apply two symplectic basis changes: a Fourier transform and an orthogonal wavelet transform. Check first that both reconstruct the same $V$ and preserve symplectic eigenvalues of the full state. Then split each basis at a nominal scale and compute reduced symplectic spectra.

The resulting entropies may differ even at identical mode counts. The correct conclusion is not that one calculation is wrong, but that the scale algebras differ. Robust statements are those tied to a declared response function, stable under refinement within that response family, or expressed in basis-independent correlators.

Evidence boundary as of 10 August 2026

Momentum-space and multiresolution entanglement remain useful research diagnostics, especially in perturbative QFT and tensor-network constructions. There is no general basis-independent theorem identifying their entropy with a spatial entropic $c$, $F$, or $a$ function, nor a universal physical recovery map for a formal shell integration. Claims should therefore specify regulator, variables, accessible algebra, and task.

Validity map for scale-space claims. Failure to fix a mode algebra, basis family, and operational channel leaves a regulated diagnostic rather than universal RG information loss. Gauge and locality constraints belong in the first two gates. Schematic and not to scale.

Common pitfalls

Calling momentum-shell entropy spatial entropy. The bipartitions are different and can disagree even in a free vacuum.

Treating a basis label as operational. State the detector, smearing, or algebra that makes the selected modes accessible.

Imposing gauge constraints after tracing. Define the physical algebra and center first; otherwise redundant variables can contaminate the entropy.

References

• Balasubramanian, Vijay, Michael B. McDermott, and Mark Van Raamsdonk. “Momentum-Space Entanglement and Renormalization in Quantum Field Theory.” Physical Review D 86 (2012): 045014. DOI.
• Evenbly, Glen, and Steven R. White. “Entanglement Renormalization and Wavelets.” Physical Review Letters 116 (2016): 140403. DOI.
• Haegeman, Jutho, Tobias J. Osborne, Henri Verschelde, and Frank Verstraete. “Entanglement Renormalization for Quantum Fields in Real Space.” Physical Review Letters 110 (2013): 100402. DOI.