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arXiv:quant-ph/0208047 · Journal of Mathematical Physics

Hilbert space structure in classical mechanics. II

E. Deotto, E. Gozzi, D. Mauro

Published Nov 22, 2003• 13 citations
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Abstract

In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper I, it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space.

DOI
10.1063/1.1623334

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In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper I, it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space.
In this paper we study the Hilbert space structure underlying the Koopman-von Neumann operatorial formulation of classical mechanics. While the Hilbert space of zero-forms can be endowed with a positive definite scalar product and the evolution turns out to be unitary, we prove that this is not the case when we include higher forms. In this last case we explore all possible scalar products and prove that for those which are positive definite the evolution is not unitary and vice versa. This feature is due to the Grassmannian nature of the forms and it appears only in classical mechanics. It is known in fact that in a similar structure, which is supersymmetric quantum mechanics, this does not happen.
In this paper we study the Hilbert space structure underlying the Koopman–von Neumann (KvN) operatorial formulation of classical mechanics. KvN limited themselves to study the Hilbert space of zero-forms that are the square integrable functions on phase space. They proved that in this Hilbert space the evolution is unitary for every system. In this paper we extend the KvN Hilbert space to higher forms which are basically functions of the phase space points and the differentials on phase space. We prove that if we equip this space with a positive definite scalar product the evolution can turn out to be nonunitary for some systems. Vice versa, if we insist in having a unitary evolution for every system then the scalar product cannot be positive definite. Identifying the one-forms with the Jacobi fields we provide a physical explanation of these phenomena. We also prove that the unitary/nonunitary character of the evolution is invariant under canonical transformations.
I describe, in the simplified context of finite groups and their representations, a mathematical model for a physical system that contains both its quantum and classical aspects. The physically observable system is associated with the space containing elements fxf for f an element in the regular representation of a given finite group G. The Hermitian portion of fxf is the Wigner distribution of f whose convolution with a test function leads to a mathematical description of the quantum measurement process. Starting with the Jacobi group that is formed from the semidirect product of the Heisenberg group with its automorphism group SL(2,F{N}) for N an odd prime number I show that the classical phase space is the first order term in a series of subspaces of the Hermitian portion of fxf that are stable under SL(2,F{N}). I define a derivative that is analogous to a pseudodifferential operator to enable a treatment that parallels the continuum case. I give a new derivation of the Schrodinger-Weil representation of the Jacobi group. Keywords: quantum mechanics, finite group, metaplectic. PACS: 03.65.Fd; 02.10.De; 03.65.Ta.
Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space mathbbRmathrm2N\\mathbb{R}^{\\mathrm{2N}}mathbbRmathrm2N. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Bohr quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality. There is also an alternative and a priori distinct notion of effective Hamiltonian used in the context of a WKB approximation. These two notions are compared for the example of loop quantum cosmology.
Constructing a classical mechanical system associated with a given quantum-mechanical one entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum-mechanical system naturally has the structure of an infinite-dimensional symplectic manifold ('quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straightforwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and one recovers the linear classical phase space . We report on how the procedure also allows extraction of nonlinear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin- j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.
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.