The so-called spectral decomposition quantization (SDQ) involving spectral decompositions of the identity operator has been proposed. The quantization yields the standard (sharp) quantum observables in the form of Hermitian operators acting in the Hilbert space of square-integrable functions. Here, the quantizable classical observables are limited as in the case of geometric quantization (GQ). It has been shown that this quantization leads to the same results as those yielded by geometric quantization in the position representation as well as in the momentum representation. The method of quantization also has been formulated in the representation concerning any pair of canonical observables. Then, the quantization has been generalized by replacing the spectral measure appearing in the spectral decompositions of the identity operator with positive-operator-valued measure constructed by using suitable smearing function (e.g. confidence distribution) so that the quantization yields unsharp quantum observables. It has been shown that the unsharp quantum observables are yielded from the quantization of the so-called unsharp quantizable classical observables. Some properties of unsharp classical observables have been investigated. The explicit expressions of unsharp quantum observables have been obtained.
The so-called spectral decomposition quantization (SDQ) involving spectral decompositions of the identity operator has been proposed. The quantization yields the standard (sharp) quantum observables in the form of Hermitian operators acting in the Hilbert space of square-integrable functions. Here, the quantizable classical observables are limited as in the case of geometric quantization (GQ). It has been shown that this quantization leads to the same results as those yielded by geometric quantization in the position representation as well as in the momentum representation. The method of quantization also has been formulated in the representation concerning any pair of canonical observables. Then, the quantization has been generalized by replacing the spectral measure appearing in the spectral decompositions of the identity operator with positive-operator-valued measure constructed by using suitable smearing function (e.g. confidence distribution) so that the quantization yields unsharp quantum observables. It has been shown that the unsharp quantum observables are yielded from the quantization of the so-called unsharp quantizable classical observables. Some properties of unsharp classical observables have been investigated. The explicit expressions of unsharp quantum observables have been obtained.
The so called spectral decomposition method of quantization involving spectral decompositions of the identity operator has been proposed. The quantization yields the standard (sharp) quantum observables in the form of Hermitean operators acting in the Hilbert space of square-integrable functions. Here, the quantizable classical observables are limited as in the case of geometric quantization. It has been shown that this quantization leads to the same results as those yielded by geometric quantization in the position representation as well as in the momentum representation. The method of quantization also has been formulated in the representation concerning any pair of canonical observables. Then, the quantization has been generalized by replacing the spectral measure appearing in the spectral decompositions of the identity operator with positive-operator-valued measure constructed by using suitable smearing function (e.g. confidence distribution) so that the quantization yields unsharp quantum observables. It has been shown that the unsharp quantum observables are yielded from the quantization of the so called unsharp quantizable classical observables. Some properties of unsharp classical observables have been investigated. The explicit expressions of unsharp quantum observables have been obtained.
In the conventional axiomatic formalism of quantum mechanics, a physical measurement of a quantum observable is mathematically represented by the spectral decomposition of the Hermitian operator associated with the observable. This constitutes the scenario of a L\"uders (projective) measurement, which includes a von Neumann (rank-one projective) measurement as a special and prominent instance. In this context, the measurement is called sharp in the sense that each measurement operator is an orthogonal projection (eigenprojection of the observable). In the modern operational formalism, a measurement is represented by a positive-operator valued measure (POVM), which consists of a family of non-negative definite operators (measurement operators, effects) summing to the identity. In this scenario, a measurement is called unsharp (fuzzy) if some measurement operators are not orthogonal projections. A natural question arises as to how to quantify unsharpness of a measurement. In this work, we address this issue in terms of uncertainty. For this purpose, we study a family of observables associated with a measurement and their uncertainty. By exploiting the difference between the (total) measurement uncertainty and the observable uncertainty, we are led to some information-theoretic quantifiers of unsharpness. We reveal their basic properties and illustrate them through some important measurements. In particular, we characterize L\"uders measurements and equiangular POVMs as extreme measurements in terms of unsharpness.
In this chapter we take up systems whose state spaces form a continuum. These are, naturally, infinite dimensional vector spaces (often called Hilbert spaces). Unfortunately, we could not do any better than treating such systems by formal analogy with their finite dimensional counterparts. A rigorous treatment would require a background in functional analysis and is beyond the scope of the present work. Thus, we essentially lay out a dictionary for translating the mathematical ingredients and formulae, from discrete to continuum descriptions, motivated by plausibility arguments. In particular, we argue why states are now described by complex square integrable functions (called wave functions) and observables by Hermitian differential operators. We then move on to show how to quantize quantum systems with classical analogues, which comprise the bulk of real life physical systems. Here, we explicitly show how the most nontrivial component of the quantization, i.e., the description of the representation of the momentum operator (in position space) can be deduced from both, the de Broglie hypothesis, and the canonical commutation relation. The “particle in a box” problem is treated as an illustration. The parity operator, Ehrenfest's theorem and continuity conditions have been relegated to the problem section.
In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasi-hermitian operators. The necessary mathematics to understand rigged Hilbert spaces and their representations is presented. Using the mathematics, we give rigorous meaning to Dirac's kets and bras, using the momentum operator as an example. The mathematical theory culminates with the nuclear spectral theorem(Gelfand-Maurin) which guarantees an eigenket decomposition for the wave functions. Next, we illustrate the theory by applying it to the quantum system defined by a potential barrier. We then introduce non-hermitian operators of interest in Quantum Mechanics, specifically PT-symmetric Hamiltonians. It is known that PT-symmetric Hamiltonians are necessarily pseudo-hermitian. We restrict our attention to a subclass of pseudo-hermitian operators, the quasi-hermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasi-hermitian operator and the metric operator, guarantees an eigenket decomposition for a dense subset of the Hilbert space on which the operator is defined. We discuss how the extension may help put quasi-hermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasi-hermitian operators with respect to an operator-valued measure exists. This is an extension of the spectral decomposition of self-adjoint operators with respect to a projection-valued measure.
We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process. If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory. Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization.
Abstract The paper uses an approach to the formal description of linear quantization as a statistical mixture of operators of the so-called tau quantization. This allows us to consider arbitrary Hermitian linear quantization as a linear integral operator acting on functions from the space of canonical variables and translating them into matrix elements of the quantum operator. In such a scheme, various quantization rules are uniformly studied: Weyl, Born, Jordan, and a number of others. The main result relates to the proof of the uniqueness of Weyl quantization as an operator, which is answered by the inverse operator generating a positive definite probability measure. For any other quantization, the inverse operator (dequantization) generates a measure that is not positive definite.