Extension Of Quantum Mechanics Research Articles

Fundamentals of fuzzy probability theory

The canonical classical extension of quantum mechanics studied recently by E. G. Beltrametti and S. Bugajski opens a new way toward generalizing the standard probability theory. The emerging fuzzy probability theory is able to give a full account of both classical and quantal probabilities, and—like the standard probability theory—could be of universal use, far outside the borders of physics. A specific feature of this hypothetical theory of probability is its mixed, classical-quanta character: classical as well as quantal random variables are described on an equal footing in a unified framework. Some new features of the fuzzy probability theory are shown on simple examples.

Quantum Field Theory: Renormalization and the renormalization group

AbstractIn the last twenty years experiments at accelerators have established with increasing accuracy the Standard Model of the weak, electromagnetic and strong interactions, a (non‐abelian) generalization of quantum electrodynamics (QED). This model is based on a mathematical structure, quantum field theory (QFT) which is a relativistic extension of quantum mechanics. An essential ingredient in the construction of the Standard Model has been that the corresponding local field theory be renormalizable, i.e. that a recipe called renormalization can be applied which allows physical results to be extracted from a theory which is affected by unavoidable short‐distance or large‐momentum divergences.

Exactly solvable quantum macroscopic motions and decoherence mechanism based on many quantum-mechanical space theory

Exactly solvable macroscopic systems composed of harmonic oscillators are investigated on the basis of non-standard analysis. Wave functions for the motions of the centre-of-mass coordinate of the systems, which are described as ultra-eigenfunctions representing a classical ensemble of the macroscopic objects in a non-standard extension of quantum mechanics, are explicitly obtained. Many quantum-mechanical spaces are naturally introduced as the physical background of the ultra-eigenfunctions. The so-called wave function collapse described by the transition from pure states to mixed states, that is decoherence among different ultra-eigenfunctions, is derived from integrations with respect to the freedom of the many quantum-mechanical spaces contained in the ultra-eigenfunctions. It is also shown that this decoherence mechanism is quite different from that of coherent states reproducing classical trajectories. The relation to the coarse-grained operators by van Kampen will also be discussed.

SECOND ORDER DERIVATIVE SUPERSYMMETRY, q DEFORMATIONS AND THE SCATTERING PROBLEM

In a search for pairs of quantum systems linked by dynamical symmetries, we give a systematic analysis of novel extensions of standard one-dimensional supersymmetric quantum mechanics. The most general supercharges involving higher order derivatives are introduced, leading to an algebra which incorporates a higher order polynomial of the Hamiltonian. We investigate the condition for irreducibility of such a higher order generator to a product of standard first derivative Darboux transformations. As a new example of application of this approach we study the quantum-mechanical radial problem including the scattering amplitudes. We also investigate the links between this higher derivative SUSY and a q-deformed supersymmetric quantum mechanics and introduce the notion of self-similarity in momentum space. An explicit model for the scattering amplitude is constructed in terms of a hypergeometric function which corresponds to a reflectionless potential with infinitely many bound states.

A classical extension of quantum mechanics

A physically natural generalization of the notion of observable that encompasses both the classical and the quantum ones is derived. Based on it, the idea of the classical extension of a theory is developed; the states of the extended theory being the probability measures on the pure states of the original one. It is shown that quantum theory admits such a classical extension and that the qualifying features of quantum observables are preserved in the extended model.

Anosov actions on noncommutative algebras

An axiomatic framework for a quantum mechanical extension to the theory of Anosov systems is constructed, and it is shown that this retains some of the characteristic features of its classical counterpart, e.g., nonvanishing Lyapunov exponents, a vectorial K-property, and exponential clustering. The effects of quantization are investigated on two prototype examples of Anosov systems, namely, the iterations of an automorphism of the torus (the ‘‘Arnold Cat’’ model) and the free dynamics of a particle on a surface of negative curvature. It emerges that the Anosov property survives quantization in the case of the former model, but not of the latter one. Finally, we show that the modular dynamics of a relativistic quantum field on the Rindler wedge of Minkowski space is that of an Anosov system.

The effect of supersymmetry applied at two-body level on few-body systems

We show that the removal of Pauli forbidden s-states in the local α-α potential of Buck et al., by means of successive supersymmetric transformations, leads to a simultaneous removal of groups of “unphysical” states of the 3 α- and 4 α-systems, but that the remaining states are largely unaffected. Potentials of different strength but similar shape have the same property of spectrum conservation. Counter examples show that it is not valid for all types of potential however. Its main significance lies in the extension of nonrelativistic supersymmetric quantum mechanics form two to many bodies.

Time in quantum gravity: An hypothesis.

A solution to the issue of time in quantum gravity is proposed. The hypothesis that time is not defined at the fundamental level (at the Planck scale) is considered. A natural extension of canonical Heisenberg-picture quantum mechanics is defined. It is shown that this extension is well defined and can be used to describe the "non-Schr\"odinger regime," in which a fundamental time variable is not defined. This conclusion rests on a detailed analysis of which quantities are the physical observables of the theory; a main technical result of the paper is the identification of a class of gauge-invariant observables that can describe the (observable) evolution in the absence of a fundamental definition of time. The choice of the scalar product and the interpretation of the wave function are carefully discussed. The physical interpretation of the extreme "no time" quantum gravitational physics is considered.

Anomaly cancellation in the path integral quantisation of the N=1 chiral spinning particle

The quantum mechanics of the N=1 supersymmetric chiral algebra of first class constraints is studied. It is shown that the model suffers from anomalies and their cancellation requires the quantum mechanical extension of N=1 local supersymmetry to N=2. The amplitudes associated with this algebra are also calculated using path integral methods.

NON-LINEAR QUANTUM MECHANICS: TWO POSSIBILITIES

A comparative discussion of two possible non-linear extensions of quantum mechanics is presented. It is argued that the two extensions are mutually consistent but applicable to completely different phenomena and that, as a consequence, quantum mechanics will be demonstrated to be truly linear only if both extensions are precluded by experiment.

Five-dimensional quantum mechanics

In this paper we consider an extension of quantum mechanics to a five-dimensional space-time. By taking probability densities as being independent of the fifth dimension and that the effect of a coordinate transformation is just a traslation, we get a functional form for the state vector. So, the wave function results multiplied by a phase factor that could modify the outcoming of an interference experiment. By requiring the invariance of the interference pattern we get that the phase of the multiplying factor is quantized. To show a sample application, we apply the argument to the electric charge and consider the hypothesis that a pointlike object in ordinary four-dimensional space-time could have an extension in the fifth dimension.

Quantisation of Hamiltonians from high-order lagrangians

There is yet to be a complete quantum mechanical version of the Fokker-Feynman model for many-particle systems with interactions of electromagnetic origin, because the corresponding Lagrangian is of infinite order. It is still an open question as to how the resultant Hamiltonian should be quantised. It is shown, using a simple illustrative example, that a straightforward means of quantising such systems exists, leading to a natural extension of standard quantum mechanics.

Supersymmetry of the photon

The supersymmetric massless states of a relativistic extension of Witten's supersymmetric quantum mechanics are shown to correspond to Abelian gauge fields realised by antisymmetric tensors. The massive states, which are only partially supersymmetric, include spin-1 and spin-0 particles. In particular the Maxwell and Proca fields are obtained from N=2 supersymmetric relativistic mechanics by first quantisation in much the same way as the Dirac theory arises in the N=1 case. The particles may be coupled supersymmetrically to external scalar and complex Hermitian tensor fields. The latter reduce in special cases to the Riemannian metric of external gravitation and the Kahler metric implied by a vector field coupling. All the couplings exhibit a quadrupole characteristic of the particles. In particular supersymmetry requires their electric charge and magnetic dipole moment to vanish, and there is no coupling to torsion in Riemann-Cartan spacetime. Quantisation in external fields yields covariant generalisations of the classical tensor field equations. Finally classical equations of translational and spin motion involving only real quantities are obtained from the Heisenberg equations.

Valence bond theory for conjugated hydrocarbons

Abstract

Retrodiction of the transferred momentum in measurement and nonlocal determinism in quantum theory

According to quantum mechanics, the change in a measured variable, following a measurement of it, cannot be obtained. In the present paper, a thought-experiment is developed in which the change can be obtained by means of an extension of quantum mechanics. It is, then, claimed that quantum mechanics permits nonlocal determinism through state reduction, in terms of its extension.

Energy-structure correlation in metalloporphyrins and the control of oxygen binding by hemoglobin.

The contribution of the porphyrin skeleton to the potential energy surface metalloporphyrins is calculated by the semiempirical method of quantum mechanical extension of the consistent force field to eta electron molecules. This calculation makes it possible to correlate the observed structure of metalloporphyrins with the strain energy of the porphyrin skeleton. It is found that the out-of-plane metal displacement in pentacoordinate heme systems is due to both the restricted size of the porphyrin hole and the "1-3" steric interaction between the axial ligand and the heme nitrogens. The main components of the active site of hemoglobin are simulated by a histidine-heme-oxygen system. The energy surface of this system provides a quantitative explanation for the control of ligand binding by hemoglobin. It is shown that the heme acts as a diaphragm, designed to provide simultaneous binding to the histidine and the sixth ligand under the steric requirements of the 1-3 interactions. The dependence of the hemoglobin potential surface on the distance between the proximal histidine and the heme plane is evaluated for the R and T states, using the calculated heme potential and the observed energy of heme-heme interaction.

Thermodynamic Limit in Quantum Stochastic Process

Asymptotic evaluation of quantum mechanical master equation of specific form is presented. It is shown in the thermodynamic limit that time evolution of intensive observa­ bles and that of fluctuation observables are given by a set of ordinary differential equations and a master equation of the type of damped harmonic oscillators, respectively. This can be regarded as a quantum mechanical extension of van Kampen's system size expansion method. Two applications are shown: 1) super-radiance from two level atoms and from Josephson oscillators, and 2) laser oscillation caused by two-photon process.

Limitation on the quantum-mechanical extension of Baierlein’s entropy theorem

Limitation on the quantum-mechanical extension of Baierlein’s entropy theorem

The quantum mechanical extension of the Boltzmann transport equation and optical absorption in semiconductors

Free carrier absorption at frequencies just below the fundamental absorption edge in the semiconducting compounds InAs, InP, GaAs, and CdTe, requires a quantum mechanical treatment for its correct description. This frequency region, which lies in the near infrared, is characterized by a theoretically predicted and experimentally observed wavelength dependence of the absorption coefficient, which is close to λ 3. At sufficiently low frequencies, however, the quantum treatment reduces to the quasiclassical Boltzmann one, giving a λ 2 dependence of the absorption coefficient, which is tentatively indicated by experimental measurements on the same materials. This paper examines the intermediate and low frequency region of free carrier absorption, and the connection of the quantum theory with the quasiclassical Boltzmann equation in the low frequency limit, by deriving the quantum result for free carrier absorption with inelastic scattering from the generalized Boltzmann equation obtained by a density matrix approach. Use of the density matrix results in a generalized Boltzmann equation, which has been shown to reduce to the quasiclassical Boltzmann equation in the limit of elastic scattering and photon energies small compared with characteristic energies of the system. It is shown here that the generalized Boltzmann equation gives the results obtained by a quantum mechanical treatment based on time-dependent perturbation theory, and hence, gives the absorption coefficient, or optical conductivity, over the entire frequency range below the fundamental absorption edge. The rate equation for the photon occupation number, for direct interband and for indirect free carrier absorption, is shown to follow from the equation of motion of the quantum density matrix and use of the generalized Boltzmann equation.

Quantum theory of correlation functions for liquid metals and normal liquids in the zero sound wave range

The correlation functions of liquid metals and normal liquids in the zero sound wave range are formulated using the quantum-mechanical two-time Green function method, and the classical formula of Bogolyubov-Sadovnikov and Nelkin-Ranganathan for normal liquids is extended to the quantum-mechanical case. The formula obtained clarifies the relation between the coherent-correlation function and the self-correlation function of a free gas, and becomes the quantum-mechanical extension of Kerr's classical formula for the relation. The effective potential obtained classically by Percus and Zwanzig from the direct correlation function is extended to the quantum-mechanical case, and the relation between the effective potential and S( k, E) is obtained. The effective potential between ions of liquid metals is formulated by taking into account the contribution from the conduction electrons. And the formula obtained by March is improved by taking into account the electron correlation effect together with the exchange interaction.