Mathematical Structure Of Quantum Mechanics Research Articles

Geometric aspects of quantization and relationship to integrability

Abstract It is the case that quantum mechanics has a deep geometric structure and can be presented accordingly. Quantum mechanics is to a certain degree foreshadowed by the geometry inherent in the geometric structure of classical mechanics. The purpose here is to present some new results and proofs which impact the mathematical structure of quantum mechanics. The relationship between integrability and quantum physics is investigated in terms of geometric ideas and structures. Two physical examples are drawn from these mathematical ideas which directly relate to physics. An introduction as to how these ideas can be extended to infinite degrees of freedom is also described.

The Marvel and the Mystery of Quantum Mechanics — Some Reflections

The creation of quantum mechanics is one of the most dramatic developments in the physics of the 20th century. After the period 1900–1924, during which the Law of Black Body Radiation, wave particle duality for light and for matter, the general quantisation of energy and stability of matter and the laws of spectroscopy had begun to be understood, the mathematical structure of quantum mechanics was discovered amazingly rapidly in just under two years, 1925–1927. On the other hand, the physical interpretation and meaning of this structure required an enormous effort, in which the uncertainty and complementarity principles, the Born probability interpretation and rule and the wave function collapse idea, all played important roles. While quantum mechanics has all along been amazingly successful in numerous applications, many puzzling questions about interpretation remain and continue to be pursued till today, though the focus has shifted from wave particle duality to entanglement and its signatures and consequences. This article gives an impressionistic account of these developments, accompanied by comments on the origins of human intuition and the meaning of human understanding of nature [1].

Generic Emergence of Objectivity of Observables in Infinite Dimensions.

Quantum Darwinism posits that information becomes objective whenever multiple observers indirectly probe a quantum system by each measuring a fraction of the environment. It was recently shown that objectivity of observables emerges generically from the mathematical structure of quantum mechanics, whenever the system of interest has finite dimensions and the number of environment fragments is large [F. G. S. L. Brandão, M. Piani, and P. Horodecki, Nat. Commun. 6, 7908 (2015)NCAOBW2041-172310.1038/ncomms8908]. Despite the importance of this result, it necessarily excludes many practical systems of interest that are infinite dimensional, including harmonic oscillators. Extending the study of quantum Darwinism to infinite dimensions is a nontrivial task: we tackle it here by using a modified diamond norm, suitable to quantify the distinguishability of channels in infinite dimensions. We prove two theorems that bound the emergence of objectivity, first for finite mean energy systems, and then for systems that can only be prepared in states with an exponential energy cutoff. We show that the latter class includes any bounded-energy subset of single-mode Gaussian states.

Exploiting the full quantum crystallography

Quantum crystallography (QCr) is a branch of crystallography aimed at obtaining the complete quantum mechanics of a crystal given its X-ray scattering data. The fundamental value of obtaining an electron density matrix that is N-representable is that it ensures consistency with an underlying properly antisymmetrized wavefunction, a requirement of quantum mechanical validity. However, X-ray crystallography has progressed in an impressive way for decades based only upon the electron density obtained from the X-ray scattering data without the imposition of the mathematical structure of quantum mechanics. Therefore, one may perhaps ask regarding N-representability “why bother?” It is the purpose of this article to answer such a question by succinctly describing the advantage that is opened by QCr.

Inverse Bayesian inference as a key of consciousness featuring a macroscopic quantum logical structure

To overcome the dualism between mind and matter and to implement consciousness in science, a physical entity has to be embedded with a measurement process. Although quantum mechanics have been regarded as a candidate for implementing consciousness, nature at its macroscopic level is inconsistent with quantum mechanics. We propose a measurement-oriented inference system comprising Bayesian and inverse Bayesian inferences. While Bayesian inference contracts probability space, the newly defined inverse one relaxes the space. These two inferences allow an agent to make a decision corresponding to an immediate change in their environment. They generate a particular pattern of joint probability for data and hypotheses, comprising multiple diagonal and noisy matrices. This is expressed as a nondistributive orthomodular lattice equivalent to quantum logic. We also show that an orthomodular lattice can reveal information generated by inverse syllogism as well as the solutions to the frame and symbol-grounding problems. Our model is the first to connect macroscopic cognitive processes with the mathematical structure of quantum mechanics with no additional assumptions.

Derivation and application of quantum Hamilton equations of motion

Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image

Quantum Brownian motion model for the stock market

It is believed by the majority today that the efficient market hypothesis is imperfect because of market irrationality. Using the physical concepts and mathematical structures of quantum mechanics, we construct an econophysical framework for the stock market, based on which we analogously map massive numbers of single stocks into a reservoir consisting of many quantum harmonic oscillators and their stock index into a typical quantum open system—a quantum Brownian particle. In particular, the irrationality of stock transactions is quantitatively considered as the Planck constant within Heisenberg’s uncertainty relationship of quantum mechanics in an analogous manner. We analyze real stock data of Shanghai Stock Exchange of China and investigate fat-tail phenomena and non-Markovian behaviors of the stock index with the assistance of the quantum Brownian motion model, thereby interpreting and studying the limitations of the classical Brownian motion model for the efficient market hypothesis from a new perspective of quantum open system dynamics.

The Hilbert space of conditional clauses

In the absence of a satisfactory interpretation of quantum theory, physical law lacks physical basis. This paper reviews the orthodox, or Dirac–von Neumann interpretation, and makes explicit that Hilbert space describes propositions about measurement results. Kets are defined as conditional clauses referring to measurements in a formal language. It is seen that these clauses are elements of a Hilbert space, such that addition is logical disjunction, the dual space consists of consequent clauses, and the inner product is a set of statements in the subjunctive mood. The probability interpretation gives truth values for corresponding future tense statements when the initial state is actually prepared and the final state is to be measured. The mathematical structure of quantum mechanics is formulated in terms of discrete measurement results at finite level of accuracy and does not depend on an assumption of a substantive, or background, space–time continuum. A continuum of kets, ∣x〉 for x ∈ ℝ3, is constructed from linear combinations of kets in a finite basis. The inner product can be expressed either as a finite sum or as an integral. Discrete position functions are uniquely embedded into smooth wave functions in such a way that differential operators are defined. It is shown that the choice of basis has no effect on underlying physics (quantum covariance). The Dirac delta has a representation as a smooth function. Operators do not in general have an integral form. The Schrödinger equation is shown from the requirements of the probability interpretation. It is remarked that a formal construction of quantum electrodynamics (qed) avoiding divergence problems has been completed using finite dimensional Hilbert space. I conclude that quantum mechanics makes statements about the world with clear physical meaning, such that space is emergent from particle interactions and has no fundamental role.

Quantum mechanics, strong emergence and ontological non-reducibility

We show that a new interpretation of quantum mechanics, in which the notion of event is defined without reference to measurement or observers, allows to construct a quantum general ontology based on systems, states and events. Unlike the Copenhagen interpretation, it does not resort to elements of a classical ontology. The quantum ontology in turn allows us to recognize that a typical behavior of quantum systems exhibits strong emergence and ontological non-reducibility. Such phenomena are not exceptional but natural, and are rooted in the basic mathematical structure of quantum mechanics.

Normal typicality and von Neumann’s quantum ergodic theorem

We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ 0 from an energy shell is ‘normal’: it evolves in such a way that | ψ t 〉〈 ψ t | is, for most t , macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.

From probabilities to mathematical structure of quantum mechanics

A large part of the mathematical formalism of quantum mechanics including the probability density, probability density current, uncertainty relations, wave function, momentum operator, kinetic energy, commutation relations and potentials is related to the probabilistic character of quantum measurements. Equations of motion of quantum mechanics can be obtained from the relativistic invariance of the space–time Fisher information. Some properties of potentials and existence of antiparticles are related to the CPT symmetry. Operators representing parallel or serial physical events in experimental setups can be represented by the sum or product of the corresponding operators.

Contribution to understanding the mathematical structure of quantum mechanics

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, the Born rule, commutation and uncertainty relations, probability density current, momentum operator, and rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, the Schrödinger equation, and the Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of th e δ-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Manyparticle systems and the postulates of quantum mechanics are also discussed.

From probabilities to quantum and classical mechanics

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that basic mathematical structures of quantum mechanics like the probability amplitudes, Born rule, commutation and uncertainty relations, momentum operator, probability density current, rules for including the scalar and vector potentials and antiparticles can be obtained from the definition of the mean values of powers of the space coordinates and time. Equations of motion of quantum mechanics, the Klein–Gordon equation, Schrödinger equation and Dirac equation, are obtained from the requirement of relativistic invariance of the theory. Limit case of localized probability densities yields the Hamilton–Jacobi equation of classical mechanics. Many particle systems are also discussed.

Quantum Mechanics Needs No Interpretation

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.

The quantum measurement problem

The measurement problem in quantum mechanics still appears to be an unresolved issue. Here we present a new quantum theory of measurement that overcomes many of the difficulties previously found. It is based on a consistent use of the linear superposition principle and distinguishes two aspects: recording and observation. A recording elicits the full interaction of the object quantum system with the quantum measuring apparatus. No wavefunction collapse is introduced. Statistics may appear at the observation of the recording only and depends on filtering processes. The theory presented here uses the existing mathematical structure of quantum mechanics but requires no ad hoc measurement postulates. Well-known paradoxical aspects in standard quantum mechanics, for instance, wave-particle duality, Schrodinger's cat, and Zeno effects do not appear in the current formulation. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 97: 670 - 678, 2004

Max Born

Max Born was one of the small group of theoretical physicists who established the mathematical structure of quantum mechanics in the 1920s. Born himself recognized that Heisenberg had the crucial idea that the variables in quantum mechanics should be observable quantities, not classical coordinates that were not observable. Heisenberg proposed to take instead quantities that were related to the transitions between states of an atom. The photograph shows Max Born (Copyright © Anne Meitner)

The contextuality of the possessed values

The mathematical structure of quantum mechanics is determined by two algorithms: quantization algorithm (QUAN) which says that an observed value of an observable Q is one of the eigenvalues q of the corresponding physical operator , and the statistical algorithm (STAT) which says that the probability for the observed physical quantity vm(Q) to be q is | q||2. From these two algorithms we can derive the principle of statistical function composition (SFC), Prob(vm(f(Q)) = r|) = Prob(vm(Q) = f-1(r)|). With the assumption of the principle of function composition (FC) that the algebraic relations between observables of a given physical system are identical with the algebraic relations between the values possessed by the system before the observation is made, it could be explained why the result of measurement obeys SFC. According to Bell-KS theorem, FC is not compatible with quantum theory. We analyse, in this paper, the incompatibility of FC with quantum theory in detail, focusing on the analysis of Redhead (1989 Incompleteness, Nonlocality and Realism (Oxford: Oxford University Press)). Redhead assumes the faithful measurement principle (FM) in order to construct FC. However, we show, that FC can be derived without the help of FM.

Conditional probabilities in quantum mechanics

We consider the generalized probability space which arises from the mathematical structure of quantum mechanics. In this space we characterize a conditional probability that is additive with respect to orthogonal decompositions of the conditioning event. The physical meaning of this notion in quantum mechanics is examined. We consider also the relation between this conditional probability and the conditional expectation on a von Neumann algebra.

Conditional probabilities in quantum mechanics. I.—Conditioning with respect to a single event

The mathematical structure of quantum mechanics suggests the definition of a generalized probability space, consisting of the lattice of projections of a von Neumann algebra and a probability measure defined on it. The conditional probability with respect to a single event is defined in this space and the physical relevance of this notion for quantum mechanics is pointed out.

Description of many separated physical entities without the paradoxes encountered in quantum mechanics

We show that it is impossible in quantum mechanics to describe two separated physical systems. This is due to the mathematical structure of quantum mechanics. It is possible to give a description of two separated systems in a theory which is a generalization of quantum mechanics and of classical mechanics, in the sense that this theory contains both theories as special cases. We identify the axioms of quantum mechanics that make it impossible to describe separated systems. One of these axioms is equivalent to the superposition principle. We show how these findings throw a different light on the paradox of Einstein, Podolsky, and Rosen.