We are focused on the idea that observables in quantum physics are a bit more than just hermitian operators and that this is, in general, a "tricky business". The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.
We consider deformations of unbounded operators by using the novel construction tool of warped convolutions. By using the Kato-Rellich theorem we show that unbounded self-adjoint deformed operators are self-adjoint if they satisfy a certain condition. This condition proves itself to be necessary for the oscillatory integral to be well-defined. Moreover, different proofs are given for self-adjointness of deformed unbounded operators in the context of quantum mechanics and quantum field theory.
Observables in quantum mechanics are represented by self-adjoint operators on Hilbert space. Such ubiquitous, well-known, and very foundational fact, however, is traditionally subtle to be explained in typical first classes in quantum mechanics, as well as to senior physicists who have grown up with the lesson that self-adjointness is "just technical". The usual difficulties are to clarify the connection between the demand for certain physical features in the theory and the corresponding mathematical requirement of self-adjointness, and to distinguish between self-adjoint and hermitian operato…
We show that allowing magnetic fields to be complex-valued leads to an improvement in the magnetic Hardy-type inequality due to Laptev and Weidl. The proof is based on the study of momenta on the circle with complex magnetic fields, which is of independent interest in the context of PT-symmetric and quasi-Hermitian quantum mechanics. We study basis properties of the non-self-adjoint momenta and derive closed formulae for the similarity transforms relating them to self-adjoint operators.
We consider the descendants of self-adjointly extended Hamiltonians in supersymmetric quantum mechanics on a half-line, on an interval, and on a punctured line or interval. While there is a 4-parameter family of self-adjointly extended Hamiltonians on a punctured line, only a 3-parameter sub-family has supersymmetric descendants that are themselves self-adjoint. We also address the self-adjointness of an operator related to the supercharge, and point out that only a sub-class of its most general self-adjoint extensions is physical. Besides a general characterization of self-adjoint extensions …
This paper is devoted to the construction of what we will call {\em exactly solvable models}, i.e. of quantum mechanical systems described by an Hamiltonian $H$ whose eigenvalues and eigenvectors can be explicitly constructed out of some {\em minimal ingredients}. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in …
We analize the Berry-Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provide a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann-von …
We construct a non-Grassmann spinning-particle model which, by analogy with quantum mechanics, does not admit the notion of a trajectory within the position space. The pseudo-classical character of the model allows us to avoid the inconsistencies arising in the quantum-mechanical interpretation of a one-particle sector of the Dirac equation.
We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the supports of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state independent uncertainty relations. Our res…
The kinematical foundations of Schwinger's algebra of selective measurements were discussed in a previous paper (arXiv:1905.12274) and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analysed. For that, the algebra generated by the observables, as well as the notion of state, are dicussed, and the structure of the transition functions, that plays an instrumental role in Schwinger's picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers …
Opposite-subsystem twin events and twin observables, studied previously in the context of distant correlations, are first generalized to pure states of not-necessarily-composite systems, and afterwards they are further generalized to delayed twins that are due to unitary evolution of the quantum system. The versatile aspects of delayed twin observables are studied in terms of necessary and sufficient conditions to make possible various applications. Three of these are sketched: Preparation of some quantum experiments, easy solution of a puzzle in an important Scully et al. real experiment, and…
During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are manifestly non-Hermitian. The mathematical consistency of the resulting models of stable quantum systems requires a reconstruction of an alternative, amended, physical inner product of states. We point out the less known fact that for more than one observable the task is not always feasible. The difficulty is re-analyzed and its elementary linear-algebraic interpretation and treatment are outlined.
We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity - some of which new - in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We cl…
We present a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governe…
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 A.M. Gleason to prove his celebrated theorem. In particular, the problem of associating quantum state 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 observables representing quantum one…
Maximal Beable Subalgebras of Quantum-Mechanical Observables
给定一个有界量子力学观测量的代数（C*-代数的自伴部分）上的一个状态，我们研究那些关于给定状态对子代数的限制是无色散状态的混合的性质是最大的子代数--我们称之为最大“可”子代数（借用J. S. Bell）。我们还将我们的研究扩展到无界可观测量的代数理论（由R。Kadison），并展示了我们的结果如何阐明了一个坚实的数学基础的核心原则的正统哥本哈根解释的量子理论（如联合不确定性的正则共轭观测，和玻尔的辩护的完整性的量子理论的论点爱因斯坦，波多尔斯基和罗森）。
Recently there has been much interest in deriving the quantum formalism and the set of quantum correlations from simple axioms. In this paper, we provide a step-by-step derivation of the quantum formalism that tackles both these problems and helps us to understand why this formalism is as it is. We begin with a structureless system that only includes real-valued observables, states and a (not specified) state update, and we gradually identify theory-independent conditions that make the algebraic structure of quantum mechanics be assimilated by it. In the first part of the paper, we derive esse…
To simulate a quantum system with continuous degrees of freedom on a quantum computer based on quantum digits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to discrete observables. We consider this problem based on expanding quantum observables in series in powers of two and three analogous to the binary and ternary representations of real numbers. The coefficients of the series ("digits") are, therefore, Hermitian operators. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary and ternary …
This paper presents some of the basic properties of conditioned observables in finite-dimensional quantum mechanics. We begin by defining the sequential product of quantum effects and use this to define the sequential product of two observables. The sequential product is then employed to construct the conditioned observable relative to another observable. We then show that conditioning preserves mixtures and post-process of observables. We consider conditioning among three observables and a complement of an observable. Corresponding to an observable, we define an observable operator in a natur…
The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from…
We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quantized version of a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $. Using the formalism of rigged Hilbert spaces, we define eigenstates for all the observables. Then we work out the explicit form of the corresponding transformation of these eigenstates. A few examples are included at the end of the paper.
Quantum information-theoretic approach has been identified as a way to understand the foundations of quantum mechanics as early as 1950 due to Shannon. However there hasn't been enough advancement or rigorous development of the subject. In the following paper we try to find relationship between a general quantum mechanical observable and it von Neumann entropy. We find that the expectation values and the uncertainties of the observables have bounds which depend on the entropy. The results also show that von Neumann entropy is not just the uncertainty of the state but also encompasses the infor…
In quantum mechanical experiments one distinguishes between the state of an experimental system and an observable measured in it. Heuristically, the distinction between states and observables is also suggested in scattering theory or when one expresses causality. We explain how this distinction can be made also mathematically. The result is a theory with asymmetric time evolution and for which decaying states are exactly unified with resonances. A consequence of the asymmetric time evolution is a beginning of time. The meaning of this beginning of time can be understood by identifying it in da…
The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert sp…
The problem of reconstructing information on a physical system from data acquired in long sequences of direct (projective) measurements of some simple physical quantities - histories - is analyzed within quantum mechanics; that is, the quantum theory of indirect measurements, and, in particular, of non-demolition measurements is studied. It is shown that indirect measurements of time-independent features of physical systems can be described in terms of quantum-mechanical operators belonging to an algebra of asymptotic observables. Our proof involves associating a natural measure space with cer…
Algebra of quantum mechanics via classical phonons. I: The Schrodinger equation as the Newtonian equation of motion and quantum observables as classical averages
We discuss the classical statistics of isolated subsystems. Only a small part of the information contained in the classical probability distribution for the subsystem and its environment is available for the description of the isolated subsystem. The "coarse graining of the information" to micro-states implies probabilistic observables. For two-level probabilistic observables only a probability for finding the values one or minus one can be given for any micro-state, while such observables can be realized as classical observables with sharp values on a substate level. For a continuous family o…
Phenomenological quantum mechanics: I. Phenomenology of quantum observables
Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院斐兹克量子中心）；The International Centre for Theory of Quantum Technologies, University of Gdańsk（格但斯克大学量子技术理论国际中心）；Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University（尼古拉·哥白尼大学物理、天文和信息学学院物理研究所）
The notion coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of…
