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

TL;DR

The paper addresses how classical behavior emerges from quantum theory by invoking dual classical (ontological) states and the principle of minimal group representation, with the metaplectic group $Mp(2)$ as the key symmetry. It develops a formal framework where London ontological states classicalize only under $Mp(n)$, introduces coset coherent states on the circle, and demonstrates stronger classicalization for cylinder configurations via analytic projections and weight factors like $e^{-2n^2}$. The main contributions include a detailed metaplectic reduction, the construction of normalizable coset coherent states on the circle, a generalized analytic Wigner function, and explicit group-action formulas that connect circle phase-space dynamics to a discrete quantum structure. Taken together, these results illuminate a deep, symmetry-driven route to quantum-classical duality with potential implications for quantum gravity and semiclassical phase-space descriptions.

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). The application of the Minimal Group Representation immediately classicalizes the system, Mp(2) emerging as the group of the classical-quantum duality symmetry. (Abridged)

Figures (4)

• Figure 1: Graphical representation of the the norm of the projection of the total Megaplectic $Mp(2)$ states onto the circle $\left\vert \varphi \right\rangle$ states: The function $\left\vert \left\langle \varphi\right\vert \left\vert \Psi\left( \omega\right) \right\rangle \right\vert ^{2}$. As shown, analyticity is evident since it clearly respects $\left\vert z = \omega e^{i\varphi}\right\vert = \left\vert \omega\right\vert < 1.$
• Figure 2: Three-dimensional representation of the norm of the projection of the total Megaplectic $Mp(2)$ states on the circle $\left\vert \varphi \right\rangle$ states: The function $\left\vert \,\left\langle\, \varphi\right\vert \left\vert \Psi\left( \omega\right)\, \right\rangle \, \right\vert^{2}$, showing clearly the analytical character due to $\left\vert z = \omega\, e^{i\varphi} \right\vert =\left\vert \omega\right\vert < 1.$
• Figure 3: Graphical representation of the generalized Wigner function for the approximate $W_{mm}$ case: the shape of this distribution appears more bell-shaped than the function in Figure 1 (square norm).
• Figure 4: In the Figure we see graphically represented the square norm of the projections under the application of the Minimal Group Representation, against the variable $z = \omega \, e^{\,i\, \varphi}$: The huge curve $\left\vert \,\left\langle\, \varphi_{+}\right\vert \left\vert \,\Psi\left( \omega\right)\, \right\rangle\, \right\vert ^{2}$ corresponds to the $s = 1/4$, even $n$ sector in the Hilbert space of the analytical functions, and the small curve $\left\vert\, \left\langle \varphi_{-}\right\vert \left\vert \Psi\left( \omega\right)\, \right\rangle\, \right\vert ^{2}$ corresponds to the $s = 3/2$, odd $n$ sector in the Hilbert space of the analytical functions ).