Approximate Synthetic Spectra of Quantum Systems Associated with Compact Lie Groups
Abstract
How can noncommuting quantum observables retain the geometry of a classical phase space? Motivated by Mumford's question on approximately macroscopically unique (AMU) states and the work of Lin and Wang, we study synthetic spectra of position--momentum and angular-momentum systems. Under the hypotheses of their AMU theorems, nonempty synthetic spectra yield AMU states. For sufficiently small ℏ, the position--momentum spectrum fills the plane, while the angular-momentum spectrum contains any prescribed ball. The latter calculation leads to compact Lie groups: we prove that highest-weight coadjoint orbits lie in regional synthetic spectra under an explicit variance bound. Function-space representations of SU(n), SO(n), and Sp(n) provide further examples.
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BibTeX
@misc{liu2026approximate,
title = {{Approximate Synthetic Spectra of Quantum Systems Associated with Compact Lie Groups}},
author = {Junyang Liu and Hang Wang and Houjin Wang and Haoran Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04306},
eprint = {2611.04306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04306},
}
RIS
TY - UNPB
TI - Approximate Synthetic Spectra of Quantum Systems Associated with Compact Lie Groups
AU - Junyang Liu
AU - Hang Wang
AU - Houjin Wang
AU - Haoran Xu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04306
UR - https://arxiv.org/abs/2611.04306
AB - How can noncommuting quantum observables retain the geometry of a classical phase space? Motivated by Mumford's question on approximately macroscopically unique (AMU) states and the work of Lin and Wang, we study synthetic spectra of position--momentum and angular-momentum systems. Under the hypotheses of their AMU theorems, nonempty synthetic spectra yield AMU states. For sufficiently small ℏ, the position--momentum spectrum fills the plane, while the angular-momentum spectrum contains any prescribed ball. The latter calculation leads to compact Lie groups: we prove that highest-weight coadjoint orbits lie in regional synthetic spectra under an explicit variance bound. Function-space representations of SU(n), SO(n), and Sp(n) provide further examples.
ER -