Classical (“ontological”) dual states in quantum theory and the minimal group representation Hilbert space

Abstract

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 thedisk∣z = ω exp(−iφ)∣ < 1.For the general coset coherent states ∣α,φ⟩in the circle,the complex variable isz′ = z exp( −i α∗/2):Theanalyticfunction is modified by the complex phase (φ−α∗/2). (vi) The analyticity ∣z′∣ = ∣z∣e(−Imα/2) < 1 occurs when Imα ≠ 0 because of normalizability and Imα >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 exp(−2n^2) , e−(2n+1/2), and exp(−(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.