Identical Particle Systems : Hierarchical Spectral Reconstruction
arXiv:2605.02864
The paper reveals a hierarchical symmetry based on cyclotomic fields and Galois groups that partitions the Hilbert space of identical‑particle quantum systems, enabling a controlled coarse‑graining of many‑body spectra and efficient approximation of the many‑body density of states.
Abstract
We identify a hierarchical symmetry structure underlying the Hilbert space of quantum systems made of identical particles. By studying the linear map between single-body and many-body spectra, we show that the resulting spectral organization admits a natural partition into sectors induced by a symmetry governed by cyclotomic fields and their Galois groups. This approach offers a new perspective on the structure of quantum many-body spectra, defining a controlled coarse-graining of the Hilbert space in which degeneracies and spectral features can be systematically organized through a set of new quantum numbers. The resulting structure induces a hierarchy of spectral resolutions following a renormalization flow, which enables information loss control. We demonstrate that this hierarchical decomposition provides an efficient route to approximate MBDoS calculations for any identical particle system, while preserving physically relevant spectral properties. This approach provides a general symmetry-based framework for organizing and approximating many-body spectra, with potential applications to quantum thermalization, spectral statistics, and large-scale quantum simulations.
10 pages, 4 figures (appendix : 5 pages, 1 figure)