For any given algebra of local observables in relativistic quantum field theory there exists an associated scaling algebra which permits one to introduce renormalization group transformations and to construct the scaling (short distance) limit of the theory. On the basis of this result it is discussed how the phase space properties of a theory determine the structure of its scaling limit. Bounds on the number of local degrees of freedom appearing in the scaling limit are given which allow one to distinguish between theories with classical and quantum scaling limits. The results can also be used to establish physically significant algebraic properties of the scaling limit theories, such as the split property.For any given algebra of local observables in relativistic quantum field theory there exists an associated scaling algebra which permits one to introduce renormalization group transformations and to construct the scaling (short distance) limit of the theory. On the basis of this result it is discussed how the phase space properties of a theory determine the structure of its scaling limit. Bounds on the number of local degrees of freedom appearing in the scaling limit are given which allow one to distinguish between theories with classical and quantum scaling limits. The results can also be used to establish physically significant algebraic properties of the scaling limit theories, such as the split property.

arXiv:hep-th/9505130May 22, 199517 citations

An Active Notion of Localization

Christian Beck

Fundamental theories of physicsJan 1, 20210 citations

Phenomenological Aspects of Localizability

Lawrence Paul Horwitz

Springer eBooksJan 1, 19740 citations

Local Scale Invariance in Quantum Theory: A Non-Hermitian Pilot-Wave Formulation

Indrajit Sen, Matthew Leifer

We show that Weyl's abandoned idea of local scale invariance has a natural realization at the quantum level in pilot-wave (de Broglie-Bohm) theory. We obtain the Weyl covariant derivative by complexifying the electromagnetic gauge coupling parameter. The resultant non-hermiticity has a natural interpretation in terms of local scale invariance in pilot-wave theory. The conserved current density is modified from $|ψ|^2$ to the local scale invariant, trajectory-dependent ratio $|ψ|^2/ \mathbf 1^2[\mathcal C]$ , where $\mathbf 1[\mathcal C]$ is a scale factor that depends on the pilot-wave trajectory $\mathcal C$ in configuration space. All physical predictions are local scale invariant, even in the presence of mass terms. Our approach is general, and we implement it for the Schrödinger and Pauli equations, and for the Dirac equation in curved spacetime, each coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique. Our results provide a pivotal understanding of local scale invariance in quantum theory.We show that Weyl's abandoned idea of local scale invariance has a natural realization at the quantum level in pilot-wave (de Broglie-Bohm) theory. We obtain the Weyl covariant derivative by complexifying the electromagnetic gauge coupling parameter. The resultant non-hermiticity has a natural interpretation in terms of local scale invariance in pilot-wave theory. The conserved current density is modified from $|ψ|^2$ to the local scale invariant, trajectory-dependent ratio $|ψ|^2/ \mathbf 1^2[\mathcal C]$, where $\mathbf 1[\mathcal C]$ is a scale factor that depends on the pilot-wave trajectory $\mathcal C$ in configuration space. All physical predictions are local scale invariant, even in the presence of mass terms. Our approach is general, and we implement it for the Schrödinger and Pauli equations, and for the Dirac equation in curved spacetime, each coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique. Our results provide a pivotal understanding of local scale invariance in quantum theory.

quant-pharXiv:2601.03567Jan 8, 20260 citations

A reinterpretation of the notion of localization

Andrés J. Kálnay, B. P. Toledo

Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fieldsApr 1, 196743 citations

Spectral multifractality and emergent energy scales across the many-body localization transition

Sthitadhi Roy

We present a scaling theory of the many-body localization transition in terms of emergent, characteristic energy scales. The analysis is based on the decomposition of the eigenstates in the basis of trivially localized states, resolved in the energies of the latter, which we refer to as the spectral decomposition of the eigenstates. The characteristic energy scales emerge when the multifractal properties, or lack thereof, of the spectral decomposition are studied at different scales. These characteristic scales correspond to the ones, above which the spectral decompositions exhibit their global behavior, namely full ergodicity in the ergodic phase and multifractality in the many-body localized phase. On the other hand, at scales below the characteristic ones, the decomposition in the ergodic phase shows finer (multi)fractal structures whereas in the localized phase, the decomposition picks out well-separated, localized resonant peaks. The scaling of these characteristic energy scales across the many-body localization transition admits a scaling theory consistent with a Kosterlitz-Thouless-type scenario and bears striking resemblances to that of inverse participation ratios of eigenstates.We present a scaling theory of the many-body localization transition in terms of emergent, characteristic energy scales. The analysis is based on the decomposition of the eigenstates in the basis of trivially localized states, resolved in the energies of the latter, which we refer to as the spectral decomposition of the eigenstates. The characteristic energy scales emerge when the multifractal properties, or lack thereof, of the spectral decomposition are studied at different scales. These characteristic scales correspond to the ones, above which the spectral decompositions exhibit their global behavior, namely full ergodicity in the ergodic phase and multifractality in the many-body localized phase. On the other hand, at scales below the characteristic ones, the decomposition in the ergodic phase shows finer (multi)fractal structures whereas in the localized phase, the decomposition picks out well-separated, localized resonant peaks. The scaling of these characteristic energy scales across the many-body localization transition admits a scaling theory consistent with a Kosterlitz-Thouless-type scenario and bears striking resemblances to that of inverse participation ratios of eigenstates.

arXiv:2404.07975Physical review. B./Physical review. BMay 8, 20252 citations

|ψ|^2

to the local scale invariant, trajectory-dependent ratio

|ψ|^2/ \mathbf 1^2[\mathcal C]

, where

\mathbf 1[\mathcal C]

is a scale factor that depends on the pilot-wave trajectory

\mathcal C

in configuration space. All physical predictions are local scale invariant, even in the presence of mass terms. Our approach is general, and we implement it for the Schrödinger and Pauli equations, and for the Dirac equation in curved spacetime, each coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique. Our results provide a pivotal understanding of local scale invariance in quantum theory.