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A Minimal Bicomplex Extension of the Complex Scalar Algebra of Quantum Mechanics with an Ideal-Valued Sector — arXiv:2610.09747

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.09747
License (as archived): CC0-1.0
Archived: 2026-10-08T12:59:58.497Z
SHA-256 of the archived text: 55641e5cd01b4e3d3d88123428e59a8b1d59b73a9eb313f6c58337a24306bc80
Stable URL of this page: https://metraipolis.com/documento/dc_79fd5ce707ee

How to cite

A Minimal Bicomplex Extension of the Complex Scalar Algebra of Quantum Mechanics with an Ideal-Valued Sector — arXiv:2610.09747 — arXiv (en), https://arxiv.org/abs/2610.09747 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_79fd5ce707ee (Metraipolis archive dc_79fd5ce707ee, sha256 55641e5cd01b4e3d3d88123428e59a8b1d59b73a9eb313f6c58337a24306bc80)

BibTeX

@misc{dc79fd5ce707ee,
  title = {A Minimal Bicomplex Extension of the Complex Scalar Algebra of Quantum Mechanics with an Ideal-Valued Sector — arXiv:2610.09747},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.09747},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_79fd5ce707ee; sha256 55641e5cd01b4e3d3d88123428e59a8b1d59b73a9eb313f6c58337a24306bc80},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.09747. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Standard quantum mechanics takes the complex numbers as its scalar algebra. We ask whether this complex structure is algebraically fundamental or may instead form a distinguished embedded sector of a larger commutative and associative scalar algebra. Requiring the ordinary complex sector to remain intact while admitting an additional stable complex-like analytic sector turns the latter into a proper ideal, makes zero divisors unavoidable in finite dimension, and implies a lower bound of four real dimensions. The bicomplex algebra realizes this minimum, and under an additional compatibility assumption the four-dimensional realization is unique up to isomorphism. The resulting algebra exhibits two distinct decompositions: the canonical decomposition into orthogonal idempotent ideals and a decomposition adapted to the embedded complex sector and the additional ideal-valued sector. The ordinary complex scalars therefore lie diagonally across the canonical idempotent decomposition. The ideal sector supports Taylor series, oscillatory exponentials, Fourier-type kernels, and Schrödinger-type expressions, and carries a formal ideal-valued eigenvalue structure. Restricting the Born rule to the embedded complex sector recovers its standard form. Whether this dual scalar architecture has physical significance beyond the algebraic framework remains an open question.

Tema degli esploratori: fisica e astronomia (elenco arXiv quant-ph, nuovi lavori del giorno).
Categorie arXiv: Quantum Physics (quant-ph).
Pagina del lavoro: https://arxiv.org/abs/2610.09747 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T12:21:30.337Z dalla pagina https://arxiv.org/list/quant-ph/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).