Showing papers on "Quantum geometry published in 1989"
TL;DR: The quantum group SU(2)q is discussed in this paper by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose.
Abstract: The quantum group SU(2)q is discussed by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum Such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose, is developed
1,555 citations
TL;DR: An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field of p-adic numbers is considered in this article, where the free particle and the harmonic oscillator are considered.
Abstract: An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.
271 citations
01 Jan 1989
Abstract: In an abstract framework, we present a class of supersymmetric quantum mechanics whose eigenvalue problem is (in part) exactly solvable. In concrete realizations, the class includes supersymmetric quantum mechanical models associated with one-dimensional or radial Schrodinger operators with potentials of a special type, called “shape-invariant potentials” in the physics literature.
129 citations
TL;DR: In this paper, the modal expansion of the wave function in the discontinuity region based on the superposition principle together with a mode-matching technique was used to investigate the transmission characteristics of semiconductor quantum wire structures with discontinuities.
Abstract: We have used the modal expansion of the wave function in the discontinuity region based on the superposition principle together with a mode‐matching technique to investigate the transmission characteristics of semiconductor quantum wire structures with discontinuities. Our calculations compare quite well with published results for the theoretical transmission coefficient and experimental conductance of a T‐stub and split‐gate geometry, respectively. We apply this technique to analyze the effect of right‐angle bends in narrow quantum wires which show strong resonant behavior due to the presence of discontinuities in this geometry.
88 citations
TL;DR: In this paper, the authors present a formalism for treating one-dimensional problems in quaternionic quantum mechanics, and derive an explicit form for the T matrix for scattering from a square barrier, and use this result to calculate the transmission and reflection coefficients.
Abstract: We present a formalism for treating one-dimensional problems in quaternionic quantum mechanics. As an example, we derive an explicit form for the T matrix for scattering from a square (quaternionic) barrier, and use this result to calculate the transmission and reflection coefficients. We show that the qualitative form of these coefficients is the same as in complex quantum mechanics, even when the barrier has a nonzero value for the quaternionic components of the potential.
77 citations
TL;DR: In this article, the authors describe the evolution and implementation of a set of quantum balance equations for examining transport in mescoscopic structures, and describe how they can be applied to the problem of transport analysis.
Abstract: This study describes the evolution and implementation of a set of quantum balance equations for examining transport im mescoscopic structures
64 citations
TL;DR: In this article, the authors consider the quantisation of p-dimensional surfaces moving through spacetime, where p>1, and define the transition amplitude between given surface configurations as a sum over all (p+1)-dimensional surfaces bounded by these configurations.
Abstract: Polyakov's (1981) approach to the quantisation of strings is readily adapted to other types of extended object. The authors consider the quantisation of p-dimensional surfaces moving through spacetime, where p>1. The transition amplitude between given surface configurations is defined as a sum over all (p+1)-dimensional surfaces bounded by these configurations. This definition is made more precise by a careful analysis of boundary conditions. They calculate the 1-loop divergences using the semiclassical approximation and the heat-kernel expansion. Their models incorporate an Einstein-Hilbert term in the action and therefore include the case of (p+1)-dimensional quantum gravity on a manifold with a boundary; it should be possible to deal with more general theories of extended objects using similar techniques. They then consider the quantisation of the spinning membrane. Certain components of the fermion fields must be fixed on the boundary, and it turns out that there is a supersymmetric equivalence relation between different boundary configurations. A 1-loop calculation is performed, and it is found that the spinning membrane is not renormalisable in any number of dimensions.
53 citations
TL;DR: In this paper, the theoretical and physical meaning of dissipation of background fields due to particle creation and statistical effects in interacting quantum field theories and in semiclassical gravitational theories is discussed.
Abstract: We discuss the theoretical and physical meaning of dissipation of background fields due to particle creation and statistical effects in interacting quantum field theories and in semiclassical gravitational theories. We indicate the possible existence of a fluctuation-dissipation relation for non-equilibrium quantum fields as occuring in cosmological particle creation and backreaction processes. We also conjecture that all effective theories, including quantum gravity, could manifest dissipative behavior.
48 citations
TL;DR: This paper discusses the concepts of quantum integrability and nonintegrability based on the concept of a complete set of commuting observables and the Hilbert-space structure of a quantum system and the definitions are given for the quantum-dynamical degrees of freedom and quantum phase space.
Abstract: In this paper we discuss the concepts of quantum integrability and nonintegrability. Based on the concept of a complete set of commuting observables and the Hilbert-space structure of a quantum system, the definitions are given for the quantum-dynamical degrees of freedom and quantum phase space from which the quantum integrability is defined. A criterion for quantum integrability then emerges; the system is integrable if it possesses dynamical symmetry. Breaking of dynamical symmetry is connected with the nonintegrability of systems and thus is the inherent mechanism of chaotic motion. A number of examples are discussed.
46 citations
TL;DR: In this paper, it was shown that boundary contributions of fermion fields must vanish for bosonic and spinning strings in arbitrary spacetimes of the critical dimensions in order to obtain the quantum transition amplitudes.
29 citations
01 Jan 1989
01 Jan 1989
TL;DR: The main source of motivation for quantum groups was the Quantum Inverse Scattering Method (QISM), and quantum Lie groups appeared afterwards as abstraction of concrete algebraic constructions constituting the mathematical formalism of QISM.
Abstract: Publisher Summary This chapter focuses on the quantum groups and integrable models. The main source of motivation for quantum groups was the Quantum Inverse Scattering Method (QISM). Quantum Lie groups and quantum Lie algebras appeared afterwards as abstraction of concrete algebraic constructions constituting the mathematical formalism of QISM. One can also define the quantum exterior algebras of the quantum Euclidean and symplectic spaces and introduce the algebras of functions on the quantum homogeneous spaces.
01 Jan 1989
TL;DR: The problem is reviewed, arguments for taking a particular version of this proposal as the appropriate way to resolve the problem are presented, and a quantum ergodic theory of macrosystems is presented.
Abstract: Recent work by Machida and Namiki [9, 10] on the measurement problem formulates a new and detailed version of the proposal that the problem can be resolved by exploiting the macroscopic nature of the measuring instrument — an idea which has been developed in the literature before in various ways (e.g. by Daneri, Loinger, and Prosperi [6] in terms of a quantum ergodic theory of macrosystems, and by Hepp [7] who, like Machida and Namiki, treats the measuring instrument as a quantum system with an infinite number of degrees of freedom). I review the problem here, and present arguments for taking a particular version of this proposal as the appropriate way to resolve the problem.
TL;DR: In this article, it was shown that the torus universe in the (2+1)-dimensional quantum gravity is a quantum chaos in a rigorous sense, and that it is possible to construct a (2 + 1)-dimensional (1 + 2)-quantum chaos model from a toy model.
Abstract: We consider the (2+1)-dimensional quantum gravity as a toy model. It is shown that the torus universe in the (2+1)-dimensional quantum gravity is a quantum chaos in a rigorous sense.
TL;DR: In this article, non-composable reducible time representations are used in non-Abelian gauge theories and touched upon in some naive regularization schemes of quantum field theories.
Abstract: Nondecomposable reducible time representations are used in non-Abelian gauge theories and touched upon in some naive regularization schemes of quantum field theories. The explicit forms of such representations are given together with their implementation in quantum algebras. The associated probability forms are constructed which legitimize them for quantum theories. Their embedding into Minkowski time-space allows a framework for locally interacting quantum fields without divergencies.
01 Mar 1989
TL;DR: In this article, a good way to search with high sensitivity for such nonlinearities is to look for violations of predictions of symmetry principles, like the Wigner-Eckart theorem, that depend not only on the assumed symmetry, but also on the linear superposition principle of quantum mechanics.
Abstract: After commenting on the historical importance of Wigner's 1939 paper, some general considerations are presented regarding the introduction of nonlinearities in the structure of quantum mechanics. It appears that a good way to search with high sensitivity for such nonlinearities is to look for violations of predictions of symmetry principles, like the Wigner-Eckart theorem, that depend not only on the assumed symmetry, but also on the linear superposition principle of quantum mechanics.
24 Jul 1989
TL;DR: A phenomenological non-classical model to describe highly complex systems in developed since quantum mechanical observables cannot be applied directly to complex systems, a categorical analysis of those systems is necessary.
Abstract: Many of the early founders of quantum theory expected to find “quantum-like” features also within the domain of living systems. But these ideas have not found much support in biology or psychology until now. However, modern system theory dealing with complex self-referential systems might cast a new light upon this question. Since quantum mechanical observables cannot be applied directly to complex systems, a categorical analysis of those systems is necessary. After discussing some general features of quantum theory a phenomenological non-classical model to describe highly complex systems in developed.
TL;DR: In this paper, the role of time in the interpretation of quantum mechanics and quantum gravity is analyzed, and changes to the form of quantum gravity to make it interpretable are suggested.
Abstract: The role of time in the interpretation of quantum mechanics and quantum gravity are analyzed, and changes to the form of quantum gravity to make it interpretable are suggested
TL;DR: The formalism of non-equilibrium quantum statistical mechanics developed by R. Balescu provides a convenient framework for deriving kinetic equations as mentioned in this paper, which requires the definition of concepts such as vacuum of correlation and free motion operator.
Abstract: The formalism of non-equilibrium quantum statistical mechanics developed by R. Balescu provides a convenient framework for deriving kinetic equations. It requires the definition of concepts such as vacuum of correlation and free motion operator. We propose a realization of such concepts for bound states in the concrete case of quantum gases starting from a description in terms of electrons and nuclei. Polarized hydrogen and unpolarized helium gases are considered.
01 Jan 1989
TL;DR: In this paper, a quantum dynamical system is modeled by associating a complete set of orthogonal eigenfunctions labeled by position q (pixels), defined only at lattice points, plus other labelling indices (colors).
Abstract: A quantum dynamical system is modeled by associating a complete set of orthogonal eigenfunctions labeled by position q (“pixels”), defined only at lattice points, plus other labelling indices (“colors”), and a complete set of orthogonal eigenfunctions labelled by momentum p, defined only inside the Brillouin zone, plus other corresponding labelling indices. Measurement of position q and momentum p are represented by operators \(\hat Q\) and \(\hat P\) multiplying the position (q) and momentum (p) eigenfunctions, respectively. The uncertainty relations for \(\hat Q\) and \(\hat P\) hold. We demonstrate that this model enable us to formulate quantum mechanics in a lattice, representing a reduced number of degrees of freedom.
TL;DR: In this paper, the choice of suitable basis sets in ab initio calculations of the electronic structure and properties of molecules is briefly discussed, with particular attention paid to a relatively new basis set of Hermite-Gaussian functions.
Abstract: Importance of the choice of suitable basis sets in ab initio calculations of the electronic structure and properties of molecules is briefly discussed. Particular attention is payed to a relatively new basis set of Hermite-Gaussian (HG) functions. Their properties and merits are considered in some detail and a review of recent results is given.
01 Jan 1989
TL;DR: The interpretative framework of quantum mechanics loosely subsumed under the name "Copenhagen interpretation" contains two central assumptions which seem incompatible with a quantum cosmology built on a covariant quantum theory of spacetime as mentioned in this paper.
Abstract: The interpretative framework of quantum mechanics loosely subsumed under the name “Copenhagen interpretation” contains two central assumptions which seem incompatible with a quantum cosmology built on a covariant quantum theory of spacetime. The first is a distinguished class of classical systems. The second is a distinguished time variable and its associated notion of causality. The first assumption is incompatible with the uniform application of quantum mechanics to the universe as a whole. The second is incompatible with the general covariance of gravitational theory. This paper explores the possibility that both of these distinguished features of our world arise, not as special features of the formalism of quantum mechanics, but rather as consequences of specific initial conditions for cosmology.
TL;DR: In this paper, a theory of quantum gravity with interactions introduced by topological principle was proposed, and the fundamental property of such a theory is that its energy-momentum tensor is a BRST anticommutator.
Abstract: We propose the theory of quantum gravity with interactions introduced by topological principle. The fundamental property of such a theory is that its energy-momentum tensor is a BRST anticommutator. Physical states are elements of the BRST cohomology group. The model with only topological excitations, introduced recently by Witten, is discussed from the point of view of the induced gravity program. We find that the mass scale is induced dynamically by gravitational instantons. The low-energy effective theory has gravitons, which occur as the collective excitations of geometry, when the metric becomes dynamical. Applications of cobordism theory to quantum gravity are discussed.
TL;DR: In this paper, the authors considered the non-archimedean quantum mechanics with wave packet evolution and showed that the norms of nonarchimmedean coordinates and time correspond to our usual notion of space and time.
TL;DR: A generalized Floreanini-Jackiw Lagrangian density is obtained for the Majorana-Weyl fermions in two dimensions with chiral bosons on a circle, and the corresponding gravitational anomalies are found as expected.
Abstract: We extend the geometric treatment done for the Majorana-Weyl fermions in two dimensions by Sanielevici and Semenoff to chiral bosons on a circle. For this case we obtain a generalized Floreanini-Jackiw Lagrangian density, and the corresponding gravitational (or Virasoro) anomalies are found as expected.