Reggeon quantum mechanics: A critical discussion

The quantum-mechanical problem of reggeon field theory in zero transverse dimensions is re-examined in order to set up a precise mathematical framework for the case μ = α(0) ? 1 > 0. We establish a Hamiltonian formulation in a Hilbert space for (ifμ > 0) and we prove the equivalence of the related eigenvalue problem with a “radial” Schrödinger-type equation in an L²(0, ∞) space. We prove that the S-matrix and the pomeron Green functions, at fixed rapidity Y and triple-pomeron coupling λ ≠ 0, have a spectral decomposition and are analytic in μ for ?∞ < μ < + ∞. For μ > 0, we confirm most of the qualitative results found by previous authors, and in particular the tunnelling shift [～ exp(?μ²/2λ²)] setting the scale for the asymptotic behaviour in Y.In the classical limit of λ/μ small we find that the action, for μ > 0, develops a singularity in Y at some value Y_c. We give arguments to show that for Y ? Y_c the perturbative result is reached, while for $\text{Y ? Y}{}_{\mathrm{<mtext>c</mtext>}}$ perturbation theory breaks down. Most of these results are shown to be stable against the addition of a small quartic coupling of the simplest type [$\text{λ′(}\text{Ψ}\text{Ψ)}{}^2$] up to the “magic” vvalue λ′ = λ²/μ. The existence of a level crossing at this value is confirmed by an analytic continuation in λ′.     

Electro-absorption and refraction in Fabry-Perot quantum well modulators: a general discussion

Fabry-Perot resonators have long been advocated to improve the limited contrast ratio of multiple quantum well optical modulators used in photonic switches based on self electrooptic effect devices (SEEDs) and in other array based optical interconnection schemes. Using data on field dependent GaAs/AlGaAs quantum well absorption and refraction, we have modelled the reflectivity, modulation depth and contrast ratio of resonant modulators. Our results are generally valid for any quantum well modulator and demonstrate 23the important role played by electro-refraction even in regions of strong absorption. Resonators give large contrast ratios but there are trade-offs in the maximum reflectivity change achievable with Fabry-Perot resonators compared to simple modulators. The model gives the optimum number of quantum wells and reflectivity values required to make a resonator at any wavelength for a given quantum well structure. Understanding the limits of Fabry-Perot quantum well modulator performance is important for their application in symmetric self electrooptic effectiveness for photonic switching where modulation and detection properties are both used and for optical interconnection systems.     

Crystal lattice quantum computer

31 P nucleus can be used to represent a quantum bit (‘qubit’) with a relatively long relaxation time. In a CeP crystal lattice, ³¹P nuclei are periodically situated in three dimensions at distances of about 6 Å. The application of a static magnetic field gradient in one direction causes differences in the Zeemanfrequencies of separate nuclei. This allows thousands of distinct qubits to be individually addressed. Initializations of the qubits can be done efficiently by the Pound–Overhauser double resonance effect on the nuclear spins and the antiferromagnetically ordered 4f electron spins of cerium ions. Logic operations can be performed by simple pulse sequences, and computational results after logic operations can be measured by the nuclear magnetic resonance of neighboring nuclei, or the electron resonance of neighboring 4f electrons of cerium ions. Received: 26 October 1998/Accepted: 29 October 1998     

Classical and quantum general relativity: A new paradigm

We argue that recent developments in discretizations of classical and quantum gravity imply a new paradigm for doing research in these areas. The paradigm consists in discretizing the theory in such a way that the resulting discrete theory has no constraints. This solves many of the hard conceptual problems of quantum gravity. It also appears as a useful tool in some numerical simulations of interest in classical relativity. We outline some of the salient aspects and results of this new framework. Fifth Award in the 2005 Essay Competition of the Gravity Research Foundation. - Ed.     

From quantum cellular automata to quantum lattice gases

A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one-parameter family of evolution rules which are best interpreted as those for a one-particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second, of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.     

A numerical investigation about quantum measure in lattice gravity

We explore the effect of a whole class of quantum measures in euclidean lattice gravity. It turns out that the critical behaviour of the theory stays unchanged: only first-order transitions are detected.     

The quantum mechanical toda lattice

The aim of this paper is to establish the exact quantization conditions for the three-body Toda lattice. The Hamiltonian consists of the kinetic energy for three particles in one dimension, and of the potential energy which couples each particle to its two companions through an exponential spring. After eliminating the center of mass motion, one is left with a system of two degrees of freedom and two constants of motion, the total energy E and a third integral A which commute. Nevertheless, no transformation has been found to separate the classical equations of motion or Schrödinger's equation. The wave function is written as a double Laurent series. Its coefficients have to satisfy two sets of recursion relations on a triangular grid where each set insures that we have a simultaneous eigenfunction of E and A. The condition for the convergence of this series can be expressed as the vanishing of a tridiagonal infinite determinant with 1 in the diagonal and the inverse of a third-order polynomial in the first off-diagonals. The coefficients in this polynomial are E and A, and the variable corresponds to a component of the wave vector associated with the wave function. This determinant can be treated exactly as Hill's, and yields the 3 components. The condition for the square integrability of the wave function requires the phase angle of the principal minors to be equal to 0, $\text{π}\text{3}$, or $\text{2π}\text{3}$ according as the representation of the cyclic groups, for each component of the wave vector. But the third condition follows from the two others. The analogy with the corresponding two-body problem is pointed out.     

Topology lattice as quantum logic

We discuss the relations between the lattice of topologies for the simplest case of a three-point set and quantum logic. A hypothetical topologymeter is considered as a measuring apparatus, and it is shown that it necessarily possesses some quantum features, such as complementarity.     

The lattice quantum Sine-Gordon model

The integrable statistical physics model on the rectangular two-dimensional lattice which we call the L-model is constructed. This model generates the integrable quantum sine-Gordon model on the one-dimensional lattice in the same way as the ice model generates the XXZ model.     

A discussion on the origin of quantum probabilities

We study the origin of quantum probabilities as arising from non-Boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorovian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).     

A general framework for cosmological quantum theory

A self-contained structure for interpreting quantum theory in a cosmological context is presented which is free from the “unique predecessor” restriction previously imposed, and which allows mixed states involving only a finite part of the universe.     

Equivalence of ensembles in quantum lattice systems: States

The general analysis of the equivalence of ensembles in quantum lattice systems, which was undertaken in paper I of this series, is continued.The properties of equilibrium states are considered in a variational sense. It is then shown that there exists a canonical as well as a microcanonical variational formulation of equilibrium both of which are equivalent to the grandcanonical formulation.Equilibrium states are constructed both in the canonical and in the microcanonical formalism by means of suitable limiting procedures.It is shown, in particular, that the invariant equilibrium states for a given energy and density are those for which the maximum of the mean entropy is reached. The mean entropy thus obtained coincides with the microcanonical entropy.     

A general method for non-perturbative renormalization of lattice operators

We propose a non-perturbative method for computing the renormalization constants of generic composite operators. This method is intended to reduce some systematic errors, which are present when one tries to obtain physical predictions from the matrix elements of lattice operators. We also present the results of a calculation of the renormalization constants of several two-fermion operators, obtained, with our method, by numerical simulation of QCD, on a 16³ x 32 lattice, at β = 6.0. The results of this simulation are encouraging, and further applications to four-fermion operators and to the heavy quark effective theory are proposed.     

A general method for constructing vector integrable lattice systems

A method for generating vector-value integrable analogies of integrable lattice systems or integrable differential-difference equations is presented. The basic ingredient of the method is to insert permutation matrices. We formulate the zero-curvature representations and Hamiltonian structures of the resulting vector lattice systems. The effectiveness of the method is illustrated using some examples such as the Volterra lattice, the Belov–Chaltikian lattice, the Ablowitz–Ladik lattice and the Heisenberg ferromagnet lattice.     

A one-dimensional lattice model for a quantum mechanical free particle

Two types of particles, A and B with their corresponding antiparticles, are defined in a onedimensional cyclic lattice with an odd number of sites. In each step of time evolution, each particle acts as a source for the polarization field of the other type of particle with nonlocal action but with an effect decreasing with the distance: . It is shown that the combined distribution of these particles obeys the time evolution of a free particle as given by quantum mechanics.     

A survey of lattice results on finite temperature quantum chromodynamics

The talk summarizes some new results of lattice investigations of QCD at finite temperature. The topics discussed cover the flavor dependence of the critical temperature and the equation-of-state as well as hadronic correlation functions.     

A quantum field theory of lattice dynamics at finite temperature

A quantum field theoretical treatment of three-dimensional cubic crystals at finite temperature is presented from the view-point of the spontaneous breakdown of the spatial translational invariance using thermo field theory. The effective interaction Hamiltonian is constructed by taking into account the dynamical map of the molecular density operator which is obtained from the Ward-Takahashi relations. The acoustic phonons are expected to be the excitation of particle-hole pairs. The conventional secular equation for the lattice vibrations is obtained by neglecting some quantum effects in the Bethe-Salpeter equation for the molecular density fluctuations. The phonon spectra, the phonon propagators and the dynamical map of the molecular density operator are calculated at finite temperature.     

Decoherence in crystal lattice quantum computation

Nuclear magnetic resonance (NMR) quantum computation in a crystal lattice holds more promise for scalability than its solution NMR counterpart, but dephasing is a severe concern. Pulse sequence refocusing can help bring the qubit dephasing time closer to the limit of the intrinsic decoherence time, but the intrinsic transverse relaxation time (T₂) and the longitudinal relaxation time (T₁) of the crystal must be sufficiently long for a successful implementation. We discuss these time scales and their relation to parameters relevant to quantum computation for several crystal types, discussing in detail the examples of CaF₂, MnF₂, and CeP. Included in the calculation of coherence times for CeP is the development of spin-wave spectra in a type-1 antiferromagnetic FCC lattice.