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arXiv:2501.11119 · APL Quantum

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

Diego Julio Cirilo-Lombardo, Norma G. Sanchez

Published Jan 9, 2025• 0 citations
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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 oddn 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) completelyclassicalize 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.

DOI
10.1063/5.0247698

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

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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 oddn 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) completelyclassicalize 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.
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.
A precise physical description and understanding of the classical dual content of quantum theory is necessary in many disciplines today: from concepts and interpretation to quantum technologies and computation. In this paper we investigate Quantum Entanglement with the new approach APL Quantum 2, 016104 (2025) on dual Classicalization. Thus, the results of this paper are twofold: Entanglement and Classicalization and the relationship between them. Classicalization truly occurs only under the action of the Metaplectic group Mp(n) (Minimal Representation group, double covering of the Symplectic group). Some of the results of this paper involves the computation and analysis of the entanglement for different types of coherent (coset and non coset) states and topologies: in the circle and the cylinder. We project the entangled wave functions onto the even (+) and odd (-) irreducible Hilbert Mp(n) subspaces, and compute their square norms: Entanglement Probabilities P++, P--, P+-, (eg in the same or in the different subspaces), and the Total sum of them, and more. These theoretical and conceptual results can be of experimental and practical real-world interest.
In this paper, we compute and analyze the geometrical and topological changes in quantum physics within the new framework of our recent work [D. J. Cirilo-Lombardo and N. G. Sanchez, APL Quantum 2 (2025) 016104] on classicalization. Thus, the results of our paper here are twofold: Topology, geometry and classicalization and the relationship between them. With this end, we consider various types of states on the simplest polygon: states with triangle topology as solution of the Schrodinger equation, eigenstates of a lowering operator and coset coherent states. The results of this paper are: (1) From the point of view of the metaplectic group and the classicalization, the application of the Minimal Group Representation (MGR) on the states on the triangle and the circle can be performed only if the ladder operators [Formula: see text] and [Formula: see text] are related to those of the oscillator [Formula: see text] and [Formula: see text]. (2) For the coherent state case, the application of the MGR can be viewed in the same way as (1) but considering an eigenstate of the operator [Formula: see text], or a more general case, as an eigenstate of [Formula: see text]. (3) These two-dimensional states are edge states geometrically dual to the quantum states within the boundary of the triangle: there is interplay between states at the edge and the states inside the triangular boundary. (4) The coherent states in the triangle are generated by applying the group [Formula: see text] to the coset coherent states of the circle as a fiducial vector (e.g. a polygon with infinity sides). (5) The justification for the proposal in item (4) is easy to see by applying the cyclic subgroup [Formula: see text] of [Formula: see text] to the fiducial (coset circle coherent states): as the number of sides tends to infinity, the elements of [Formula: see text] tend to the identity. (6) The application of MGR in the case of [Formula: see text] ([Formula: see text] coherent state, reveals to us the quantum structure underlying each of the sides of the triangle (polygon). (7) The norm (probability) found shows a clear interaction/interference between the sides of the triangle and between the even and odd sectors of the Hibert space. (8) The application of MGR allows topological isolation between the projected sectors (even and odd). It can be seen that the topological effect is of a quantum nature and vice versa. (9) These topological isolation effects do not disappear at the macroscopic level (e.g. they are independent of the space-time parameters of the physical system). In general, the results here show too that quantum states in geometries with angles and vertices classicalize more than in geometries without them. (10) Within this context, a generalization of the Bargmann representation, not previously introduced in the literature, is presented. (11) These theoretical results can have impact and applications in: Classical and quantum information processing, Quantum computation, classical-quantum interaction and interpretation measurements, classical-quantum duality, classical or quantum optimization.
The standard quantum mechanical harmonic oscillator has an exact, dual relationship with a completely classical system: a classical particle running along a circle. Duality here means that there is a one-to-one relation between all observables in one model, and the observables of the other model. Thus the duality we find, appears to be in conflict with the usual assertion that classical theories can never reproduce quantum effects as observed in many quantum models. We suggest that there must be more of such relationships, but we study only this one as a prototype. It reveals how classical hidden variables may work. The classical states can form the basis of Hilbert space that can be adopted in describing the quantum model. Wave functions in the quantum system generate probability distributions in the classical one. One finds that, where the classical system always obeys the rule probability in = probability out, the same probabilities are quantum probabilities in the quantum system. It is shown how the quantum x and p operators in a quantum oscillator can be given a classical meaning. It is explained how an apparent clash with quantum logic can be rationalized.
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase-space of the system under consideration. The ''matter fields'' are local generalizations of states and observables: they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase-space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group theoretical and differential geometrical interpretation. A novel background-quantum split symmetry plays a central role. (orig.)