Informational and algebraic renormalization group
Abstract
Renormalization group (RG) is a core concept from statistical mechanics to quantum field theory, yet its schemes differ widely across literature. We formulate Wilsonian RG for generic, including finite-dimensional, quantum systems as a quantum channel combining coarse graining with isometric rescaling, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension without assuming tensor-product factorization: the retained low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. Identifying the channel's Cesàro mean with a resource-destroying map onto fixed states establishes a resource theory of RG and yields a relative-entropy monotone without variational optimization. For thermal states in the zero-temperature limit, the resulting Bogoliubov--Kubo--Mori metric per unit spacetime volume is proportional to the Zamolodchikov metric, directly relating the resource density to c−cIR near a two-dimensional IR fixed point. We also introduce the relative entropy of coarse graining to measure discarded UV information, prove its monotonicity along successive coarse grainings, and evaluate it holographically in global AdS3.
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BibTeX
@misc{mori2026informational,
title = {{Informational and algebraic renormalization group}},
author = {Takato Mori and Teruaki Nagasawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30362},
eprint = {2609.30362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30362},
}
RIS
TY - UNPB
TI - Informational and algebraic renormalization group
AU - Takato Mori
AU - Teruaki Nagasawa
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.30362
UR - https://arxiv.org/abs/2609.30362
AB - Renormalization group (RG) is a core concept from statistical mechanics to quantum field theory, yet its schemes differ widely across literature. We formulate Wilsonian RG for generic, including finite-dimensional, quantum systems as a quantum channel combining coarse graining with isometric rescaling, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension without assuming tensor-product factorization: the retained low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. Identifying the channel's Cesàro mean with a resource-destroying map onto fixed states establishes a resource theory of RG and yields a relative-entropy monotone without variational optimization. For thermal states in the zero-temperature limit, the resulting Bogoliubov--Kubo--Mori metric per unit spacetime volume is proportional to the Zamolodchikov metric, directly relating the resource density to c−c_IR near a two-dimensional IR fixed point. We also introduce the relative entropy of coarse graining to measure discarded UV information, prove its monotonicity along successive coarse grainings, and evaluate it holographically in global AdS_3.
ER -