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arXiv:1311.1720 · International Journal of Geometric Methods in Modern Physics

Frame functions in finite-dimensional quantum mechanics and its Hamiltonian formulation on complex projective spaces

Valter Moretti, Davide Pastorello

Published Nov 1, 2015• 4 citations
arXivHTMLPDF

Abstract

This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by Gleason to prove his celebrated theorem. In particular, the problem of associating quantum states with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical-like observables (i.e. real scalar functions on the projective space) representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical-like objects representing quantum ones result to be frame functions. The requirements of [Formula: see text] covariance and (convex) linearity play a central role in the proof of those theorems. A new characterization of classical-like observables describing quantum observables is presented, together with a geometric description of the [Formula: see text]-algebra structure of the set of quantum observables in terms of classical-like ones.

DOI
10.1142/s0219887816500134

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This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by Gleason to prove his celebrated theorem. In particular, the problem of associating quantum states with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical-like observables (i.e. real scalar functions on the projective space) representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical-like objects representing quantum ones result to be frame functions. The requirements of [Formula: see text] covariance and (convex) linearity play a central role in the proof of those theorems. A new characterization of classical-like observables describing quantum observables is presented, together with a geometric description of the [Formula: see text]-algebra structure of the set of quantum observables in terms of classical-like ones.
This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by Gleason to prove his celebrated theorem. In particular, the problem of associating quantum states with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical-like observables (i.e. real scalar functions on the projective space) representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical-like objects representing quantum ones result to be frame functions. The requirements of [Formula: see text] covariance and (convex) linearity play a central role in the proof of those theorems. A new characterization of classical-like observables describing quantum observables is presented, together with a geometric description of the [Formula: see text]-algebra structure of the set of quantum observables in terms of classical-like ones.
In finite dimension (at least), Quantum Mechanics can be formulated as a proper Hamiltonian theory where a notion of phase space is given by the projective space P(H) constructed on the Hilbert space H of the considered quantum theory. It is well-known P(H) can be equipped with a structure of Kähler manifold, in particular we have a symplectic form and a Poisson structure; Quantum dynamics can be described in terms of a Hamiltonian vector field on P(H). In this paper, exploiting the notion and properties of so-called frame functions, I describe a general prescription for associating quantum observables to real functions on P(H), classical-like observables, and quantum states to probability densities on P(H), Liouville densities, in order to obtain a complete and meaningful Hamiltonian formulation of a finite-dimensional quantum theory.
In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.
My PhD thesis is focused on geometric Hamiltonian formulation of Quanum Mechanics and its interplay with standard formulation. The main result is the construction of a general prescription to set up a quantum theory as a classical-like theory where quantum dynamics is given by a Hamiltonian vector field on a complex projective space with Kähler structure. In such geometric framework quantum states are represented by classical-like Liouville densities. After a complete characterization of classical-like observables in a finite-dimensional quantum theory, the observable C*-algebra is described in geometric Hamiltonian terms. In the final part of the work, the classical-like Hamiltonian formulation is applied to the study of composite quantum systems providing a notion of entanglement measure.
Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.