Publicação
De rerum infinitesimarum et quantorum natura: from leibnizian algebraic infinitesimals to the semigroup language of quantum mechanics
| Resumo: | This work develops a non-formal Leibnizian algebraic framework whose internal quasigroup mechanism produces, after external projection, an Archimedean algebraic infinitesimal hierarchy encoded by the geometric weights of the distorted operation. These infinitesimal scales are ordinary real quantities, distinct both from analytic infinitesimals arising from asymptotic limits and from the non-Archimedean infinitesimals present in Euclidean number fields. Externally, the M-Algebraic hierarchy realises the analytic deformation Ds = ∂ + s. We prove (Theorem 4.1) that this operator coincides on a core with the shifted generator in Pazy’s theory of C0-semigroups, thereby revealing a direct bridge between algebraic iteration and the analytic structures underlying propagators, resolvents and causal prescriptions in quantum mechanics. A parallel comparison with the Euclidean non-Archimedean framework of ultrafunctions, including the model case of the delta potential (Theorem 5.1), shows that the three infinitesimal mechanisms— Archimedean algebraic, analytic, and non-Archimedean algebraic— nevertheless converge to the same external physical predictions. Taken together, these results indicate that the semigroup– resolvent picture offered by Ds captures core analytic features of quantum mechanics within a single coherent formalism |
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| Autores principais: | Pinho-da-Cruz, J. |
| Assunto: | Algebraic infinitesimals Analytic deformation Non-Archimedean structures Euclidean numbers Ultrafunctions C0 semigroups Resolvents Abel–Laplace regularisation Delta potential Quantum mechanics |
| Ano: | 2025 |
| País: | Portugal |
| Tipo de documento: | preprint |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade de Aveiro |
| Idioma: | inglês |
| Origem: | RIA - Repositório Institucional da Universidade de Aveiro |
| Resumo: | This work develops a non-formal Leibnizian algebraic framework whose internal quasigroup mechanism produces, after external projection, an Archimedean algebraic infinitesimal hierarchy encoded by the geometric weights of the distorted operation. These infinitesimal scales are ordinary real quantities, distinct both from analytic infinitesimals arising from asymptotic limits and from the non-Archimedean infinitesimals present in Euclidean number fields. Externally, the M-Algebraic hierarchy realises the analytic deformation Ds = ∂ + s. We prove (Theorem 4.1) that this operator coincides on a core with the shifted generator in Pazy’s theory of C0-semigroups, thereby revealing a direct bridge between algebraic iteration and the analytic structures underlying propagators, resolvents and causal prescriptions in quantum mechanics. A parallel comparison with the Euclidean non-Archimedean framework of ultrafunctions, including the model case of the delta potential (Theorem 5.1), shows that the three infinitesimal mechanisms— Archimedean algebraic, analytic, and non-Archimedean algebraic— nevertheless converge to the same external physical predictions. Taken together, these results indicate that the semigroup– resolvent picture offered by Ds captures core analytic features of quantum mechanics within a single coherent formalism |
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