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Anticommuting Displacement Operators, Exact Pauli Algebras,and the Singlet State from the Oscillator Vacuum

  • S. P. Sorella
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Abstract

We give an explicit construction of a pair of anticommuting Weyl displacement operators for a harmonic oscillator and use bounded functional calculus to obtain exact dichotomic representatives of the Pauli algebra. Two independent oscillators provide commuting Pauli algebras for Alice and Bob. The standard singlet-sector projector, written entirely in terms of these bounded oscillator observables, is then applied to the two-oscillator vacuum. We show explicitly that the projected vacuum has nonzero norm, so its normalization is well defined, has the exact singlet correlations, and reaches the Tsirelson value 2√2 for the usual Bell--CHSH settings. We also clarify that the construction generally realizes a singlet sector with multiplicity, rather than selecting a unique vector in the full oscillator Hilbert space, and we compare the construction with the older pseudospin approach to continuous-variable Bell tests.

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Cite

BibTeXRIS
BibTeX
@misc{sorella2026anticommuting,
  title = {{Anticommuting Displacement Operators, Exact Pauli Algebras,and the Singlet State from the Oscillator Vacuum}},
  author = {S. P. Sorella},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23264},
  eprint = {2610.23264},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23264},
}
RIS
TY  - UNPB
TI  - Anticommuting Displacement Operators, Exact Pauli Algebras,and the Singlet State from the Oscillator Vacuum
AU  - S. P. Sorella
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.23264
UR  - https://arxiv.org/abs/2610.23264
AB  - We give an explicit construction of a pair of anticommuting Weyl displacement operators for a harmonic oscillator and use bounded functional calculus to obtain exact dichotomic representatives of the Pauli algebra. Two independent oscillators provide commuting Pauli algebras for Alice and Bob. The standard singlet-sector projector, written entirely in terms of these bounded oscillator observables, is then applied to the two-oscillator vacuum. We show explicitly that the projected vacuum has nonzero norm, so its normalization is well defined, has the exact singlet correlations, and reaches the Tsirelson value 2√2 for the usual Bell--CHSH settings. We also clarify that the construction generally realizes a singlet sector with multiplicity, rather than selecting a unique vector in the full oscillator Hilbert space, and we compare the construction with the older pseudospin approach to continuous-variable Bell tests.
ER  -