Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space
We investigate the classical aspects of quantum theory and under which description quantum theory does appear classical. Although such descriptions or variables are known as “ontological” or “hidden,” they are not hidden at all but are dual classical states (in the sense of the general classical–qua...
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2025-03-01
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| author | Diego J. Cirilo-Lombardo Norma G. Sanchez |
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| description | We investigate the classical aspects of quantum theory and under which description quantum theory does appear classical. Although such descriptions or variables are known as “ontological” or “hidden,” they are not hidden at all but are dual classical states (in the sense of the general classical–quantum duality of nature). We analyze and interpret the dynamical scenario in an inherent quantum structure: (i) We show that the use of the known φ states in the circle [F. London, Z. Phys. 37, 915 (1926) and G. ’t Hooft, “The hidden ontological variable in quantum harmonic oscillators,” arXiv 2407.18153 (2024)] takes a true dimension only when the system is subjected to the minimal group representation action of the metaplectic group Mp(n). The Mp(n) Hermitian structure fully covers the symplectic Sp(n) group and, in certain cases, OSp(n). (ii) We compare the circle φ states and the cylinder ξ states in configuration space with the two sectors of the full Mp(2) Hilbert space corresponding to the even and odd n harmonic oscillators and their total sum. (iii) We compute the projections of the Mp(2) states on the circle φ and cylinder ξ states. The known London circle states are not normalizable. We compute here the general coset coherent states α,φ in the circle, with α being the coherent complex parameter. It allows full normalizability of the complete set of the circle states. (iv) The London states (ontological in ’t Hooft’s description) completely classicalize the inherent quantum structure only under the action of the Mp(n) minimal group representation. (v) For the coherent states in the cylinder (configuration space), all functions are analytic in the disk z=ωe−iφ<1. For the general coset coherent states α,φ in the circle, the complex variable is z′=ze−iα*/2: The analytic function is modified by the complex phase (φ − α*/2). (vi) The analyticity z′=ze−Imα/2<1 occurs when Im α ≠ 0 because of normalizability and Im α > 0 because of the identity condition. The circle topology induced by the α,φ coset coherent state also modifies the ratio of the disk due to the displacement by the coset. (vii) For the coset coherent cylinder states in configuration space, the classicalization is stronger due to screening exponential factors e−2n2, e−(2n+1/2), and e−(2n+1/2)2 for large n arising in the Mp(2) projections on them. The generalized Wigner function shows a bell-shaped distribution and stronger classicalization than the square norm functions. The application of the minimal group representation immediately classicalizes the system, with Mp(2) emerging as the group of the classical–quantum duality symmetry. |
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| spelling | doaj-art-7877cb69d1aa4b02a7007e3e5c6a7b172025-08-20T03:03:49ZengAIP Publishing LLCAPL Quantum2835-01032025-03-0121016104016104-1310.1063/5.0247698Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert spaceDiego J. Cirilo-Lombardo0Norma G. Sanchez1M. V. Keldysh Institute of the Russian Academy of Sciences, Federal Research Center-Institute of Applied Mathematics, Miusskaya sq. 4, 125047 Moscow, Russian FederationThe International School of Astrophysics Daniel Chalonge - Hector de Vega, CNRS, INSU-Institut National des Sciences de l’Univers, Sorbonne University, 75014 Paris, FranceWe investigate the classical aspects of quantum theory and under which description quantum theory does appear classical. Although such descriptions or variables are known as “ontological” or “hidden,” they are not hidden at all but are dual classical states (in the sense of the general classical–quantum duality of nature). We analyze and interpret the dynamical scenario in an inherent quantum structure: (i) We show that the use of the known φ states in the circle [F. London, Z. Phys. 37, 915 (1926) and G. ’t Hooft, “The hidden ontological variable in quantum harmonic oscillators,” arXiv 2407.18153 (2024)] takes a true dimension only when the system is subjected to the minimal group representation action of the metaplectic group Mp(n). The Mp(n) Hermitian structure fully covers the symplectic Sp(n) group and, in certain cases, OSp(n). (ii) We compare the circle φ states and the cylinder ξ states in configuration space with the two sectors of the full Mp(2) Hilbert space corresponding to the even and odd n harmonic oscillators and their total sum. (iii) We compute the projections of the Mp(2) states on the circle φ and cylinder ξ states. The known London circle states are not normalizable. We compute here the general coset coherent states α,φ in the circle, with α being the coherent complex parameter. It allows full normalizability of the complete set of the circle states. (iv) The London states (ontological in ’t Hooft’s description) completely classicalize the inherent quantum structure only under the action of the Mp(n) minimal group representation. (v) For the coherent states in the cylinder (configuration space), all functions are analytic in the disk z=ωe−iφ<1. For the general coset coherent states α,φ in the circle, the complex variable is z′=ze−iα*/2: The analytic function is modified by the complex phase (φ − α*/2). (vi) The analyticity z′=ze−Imα/2<1 occurs when Im α ≠ 0 because of normalizability and Im α > 0 because of the identity condition. The circle topology induced by the α,φ coset coherent state also modifies the ratio of the disk due to the displacement by the coset. (vii) For the coset coherent cylinder states in configuration space, the classicalization is stronger due to screening exponential factors e−2n2, e−(2n+1/2), and e−(2n+1/2)2 for large n arising in the Mp(2) projections on them. The generalized Wigner function shows a bell-shaped distribution and stronger classicalization than the square norm functions. The application of the minimal group representation immediately classicalizes the system, with Mp(2) emerging as the group of the classical–quantum duality symmetry.http://dx.doi.org/10.1063/5.0247698 |
| spellingShingle | Diego J. Cirilo-Lombardo Norma G. Sanchez Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space APL Quantum |
| title | Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space |
| title_full | Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space |
| title_fullStr | Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space |
| title_full_unstemmed | Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space |
| title_short | Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space |
| title_sort | classical ontological dual states in quantum theory and the minimal group representation hilbert space |
| url | http://dx.doi.org/10.1063/5.0247698 |
| work_keys_str_mv | AT diegojcirilolombardo classicalontologicaldualstatesinquantumtheoryandtheminimalgrouprepresentationhilbertspace AT normagsanchez classicalontologicaldualstatesinquantumtheoryandtheminimalgrouprepresentationhilbertspace |