Uniqueness of the Ground State in Weak Perturbations of Non-Interacting Gapped Quantum Lattice Systems

We consider a general weak perturbation of a non-interacting quantum lattice system with a non-degenerate gapped ground state. We prove that in a finite volume the dependence of the ground state on the boundary condition exponentially decays with the distance to the boundary, which implies in particular that the infinite-volume ground state is unique. Also, equivalent forms of boundary conditions for ground states of general finite quantum systems are discussed.On leave from Institute for Information Transmission Problems, Moscow, Russia.     

The Quantum Entanglement in the Back Ground of Gauss-Bonnet Gravity for a Generically Entangled State

In this work the Hawking-Unruh effect on the quantum entanglement of bosonic field in background of a spherically symmetric black hole of Gauss-Bonnet gravity is investigated beyond the single mode approximation. The entanglement decreases due to Hawking-Unruh effect. However, it has been shown that the dimensions of space time, Gauss-Bonnet term and the parameter β of initial entangled state would be influenced on this degradation. In our investigation, we consider the accelerated observer either near or far from the event horizon and inspect entanglement degradation for them. The mutual information of this bosonic system is also calculated in beyond the single mode approximation and we show that the mutual information will have different behavior when the Hawking temperature increases.     

Logarithmic Quantum Mechanics: Existence of the Ground State

We study nonlinear logarithmic Schrödinger equation in three dimensions. We examine energy levels in this setting, we are especially interested in the ground state. We also show some topological properties of the spectrum. The main contribution of this paper is the first rigorous proof of existence of the ground state in logarithmic quantum mechanics.     

Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity

We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.     

A Quantum Cosmological Model with Static and Dynamic Wormholes

Quantization is performed for a Friedmann-Robertson-Walker universe filled with a conformally invariant scalar field and a perfect fluid with equation of state p = . A well-known discrete set of static quantum wormholes is shown to exist for radiation ( = 1/3), and a novel continuous set is found for cosmic strings ( = –1/3), the latter states having throat radii of any size. In both cases wave-packet solutions to the Wheeler-DeWitt equation are obtained with all the properties of evolving quantum wormholes. In the case of a radiation fluid, a detailed analysis of the quantum dynamics is made in the context of the Bohm-de Broglie interpretation. It is shown that a repulsive quantum force inversely proportional to the cube of the scale factor prevents singularities in the quantum domain. For the states considered, there are no particle horizons either.     

Ground State for Fractional Quantum Hall Effect

We investigate the ground-state wave function for an explicit model of electronsin an external magnetic field with specific interparticle interactions. The excitationstates of this model are also given.     

Lattice Gas with Finite-Range Interaction Under Gravity

Existence of a phase separation is proved for a classical lattice gas with finite-range pair potential under the action of a weak gravitational field.     

Monotonicity of Quantum Ground State Energies: Bosonic Atoms and Stars

The N-dependence of the non-relativistic bosonic ground state energy ? ^B (N) is studied for quantum N-body systems with either Coulomb or Newton interactions. The Coulomb systems are “bosonic atoms,” with their nucleus fixed, and it is shown that $\mathcal {E}_{{C}}^{{B}}(N)/\mathcal {P}_{{C}}(N)$ grows monotonically in N>1, where ? _C (N)=N ²(N?1). The Newton systems are “bosonic stars,” and it is shown that when the Bosons are centrally attracted to a fixed gravitational “grain” of mass M>0, and N>2, then $\mathcal {E}_{{N}}^{{B}}(N;M)/\mathcal {P}_{\!{N}}(N)$ grows monotonically in N, where ? _N (N)=N(N?1)(N?2); in the translation-invariant problem (M=0), it is shown that when N>1 then $\mathcal {E}_{{N}}^{{B}}(N;0)/\mathcal {P}_{{C}}(N)$ grows monotonically in N, with ? _C (N) from the Coulomb problem. Some applications of the new monotonicity results are discussed.     

Stability of Non-asymptotically Flat Thin-Shell Wormholes in Generalized Dilaton-Axion Gravity

We construct a new type of thin-shell wormhole for non-asymptotically flat charged black holes in generalized dilaton-axion gravity inspired by low-energy string theory using cut-and-paste technique. We have shown that this thin shell wormhole is stable. The most striking feature of our model is that the total amount of exotic matter needed to support the wormhole can be reduced as desired with the suitable choice of the value of a parameter. Various other aspects of thin-shell wormhole are also analyzed.     

Quantum Entanglement in Ground State of Extended Hubbard Model

Metal-insulator and CDW-SDW transitions are studied in the one-dimensional Extended Hubbard Model at half-filling by analysing the behaviour of local entanglement in fermionic systems. 1D traditional Hubbard model exhibits metal-insulator transition at critical point U_c = 0, where local entanglement reaches its maximum value. Moreover, a transition between charge- and spin-density- wave (CDW-SDW) occurs in 1D Extended Hubbard Model tUV with long-range interaction at straight line U = 2 V. The analysis of our obtained results shows that CDW-SDW transition has curious properties whose can be used in quantum information processing.

     

Ground State of a Quantum Particle in a Potential Field

A solution of the Schrödinger equation for the ground state of a particle in a potential field is analyzed. Since the wavefunctions of the ground state are nodeless, potentials of various kinds can be unambiguously determined. It turns out that the ground state corresponds to zero energy for a wide class of model potentials. Moreover, the zero level can be a single one at the boundary of the continuous spectrum. Crater-like potentials monotonically dependent on coordinates in one-, two-, and three-dimensional cases are studied. Instanton-type potentials with two local minima are of interest in the one-dimensional case. For the Coulomb potential, the energy of the ground state is stable with respect to both long- and short-range screening of this potential. Two-soliton solutions of the nonlinear Schrödinger equation are found. It is demonstrated that the proposed version of the inverse scattering transform is efficient in the analysis of solutions of differential equations.

     

A Dynamics for Discrete Quantum Gravity

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the “completed” universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space H on the set of paths. The quantum dynamics is governed by a sequence of positive operators ρ _n on H that satisfy normalization and consistency conditions. The pair (H,{ρ _n }) is called a quantum sequential growth process (QSGP). We next discuss a concrete realization of a QSGP in terms of a natural quantum action. This gives an amplitude process related to the “sum over histories” approach to quantum mechanics. Finally, we briefly discuss a discrete form of Einstein’s field equation and speculate how this may be employed to compare the present framework with classical general relativity theory.     

Quantum Space-Time: A Review

The review presents systematically the results of studies which develop an idea of quantum properties of space-time in the microworld or near exotic objects (black holes, magnetic monopoles and others). On the basis of this idea motion equations of nonrelativistic and relativistic particles are studied. It is shown that introducing concept of quantum space-time at small distances (or near superdense matter) leads to an additional force giving rise to appearance of spiral-like behaviour of a particle along its classical trajectory. Given method is generalized to nonrelativistic quantum mechanics and to motion of a particle in gravitational force. In the latter case, there appears to be an antigravitational effect in the motion of a particle leading to different value of free-fall time (at least for gravitational force of exotic objects) for particles with different masses. Gravitational consequences of quantum space-time and tensor structures of physical quantities are inveatigated in detail. From experimental data on testing relativity and anisotropy of inertia estimation L ≦ 10⁻²² cm on the value of the fundamental length is obtained.     

Quantum Linear Gravity in de Sitter Universe on Gupta-Bleuler Vacuum State

Application of Krein space quantization to the linear gravity in de Sitter space-time have constructed on Gupta-Bleuler vacuum state, resulting in removal of infrared divergence and preserving de Sitter covariant. By pursuing this path, the non uniqueness of vacuum expectation value of the product of field operators in curved space-time disappears as well. Then the vacuum expectation value of the product of field operators can be defined properly and uniquely.     

Small Quantum Structures with Small State Spaces

We summarize and extend results about “small” quantum structures with small dimensions of state spaces. These constructions have contributed to the theory of orthomodular lattices. More general quantum structures (orthomodular posets, orthoalgebras, and effect algebras) admit sometimes simplifications, but there are problems where no progress has been achieved.     

A Possible Origin of Quantum Correlations

We intend to eliminate the known conflict between relativity and quantum mechanics. We believe the “instant” correlation between entangled distant quantum particles can be explained by the fact that in a laboratory reference frame the photon traveling duration is positive and finite while its proper (in vacuum) traveling duration is equal to zero. In the latter case, any two events that are separated (in a laboratory reference frame) by an arbitrary finite distance can be considered as simultaneous ones. So, the photon nonlocal correlation turns out to be a relative property and may be explained like known twins paradox in relativity. In such a situation, any standard causal interaction between the correlated particles is absent in a laboratory reference frame; however, some specific mutual couple appears between them; this couple is strictly oscillating without some oriented energy or/and information transferring. We also motivate the basic hypothesis extension on quantum particles having nonzero masses.     

Ground State Energy of the One-Component Charged Bose Gas

The model considered here is the “jellium” model in which there is a uniform, fixed background with charge density −eρ in a large volume V and in which N=ρV particles of electric charge +e and mass m move – the whole system being neutral. In 1961 Foldy used Bogolubov's 1947 method to investigate the ground state energy of this system for bosonic particles in the large ρ limit. He found that the energy per particle is −0.402 in this limit, where . Here we prove that this formula is correct, thereby validating, for the first time, at least one aspect of Bogolubov's pairing theory of the Bose gas. Received: 23 August 2000 / Accepted: 5 October 2000     

Ground State Energy of the Two-Component Charged Bose Gas

We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for N particles is at least as negative as –CN^7/5 for large N and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant C was given by a mean-field minimization problem that used, as input, Foldys calculation (using Bogolubovs 1947 formalism) for the one-component gas. Earlier we showed that Foldys calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dysons conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state.Dedicated to Freeman J. Dyson on the occasion of his 80th birthday©2003 by the authors. This article may be reproduced in its entirety for non-commercial purposes.Work partially supported by NSF grant DMS-0111298, by EU grant HPRN-CT-2002-00277, by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation, and by grants from the Danish research council.Work partially supported by U.S. National Science Foundation grant PHY01 39984-A01.     

Glafka 2004: Generalizing Quantum Mechanics for Quantum Gravity

Familiar quantum mechanics assumes a fixed spacetime geometry. Quantummechanics must therefore be generalized for quantum gravity where spacetime geometry is not fixed but rather a quantum variable. This extended abstract sketches a fully fourdimensional generalized quantum mechnics of cosmological spacetime geometries that is one such generalization.This contribution to the proceedings of the Glafka Conference is an extended abstract of the author's talk there. More details can be found in the references cited at the end of the abstract expecially (Hartle, 1995).