Tunable ultra-fast carrier–light field dynamics of quantum dots

A theory of ultra-fast carrier–light field dynamics of quantum dots is presented. The carrier–light field dynamics is described by Maxwell–Bloch equations. A calculation of the dipole matrix elements requires the determination of the electronic wave functions taking into account their dependence on the degeneracy of the carrier states. The ultra-fast carrier–light field dynamics depends strongly on the external applied electric field. PACS 42.55.Px; 42.60.Rn; 74.78.Fk     

Quantum Cosmology in the Bruns–Dicke Theory and its Stability

The Bruns–Dicke theory with a scalar field related to the quantum spinor matter is discussed [1]. The quantum Friedmann cosmology is studied. A solution to the equations of motion describing the quantum Friedmann Universe is examined for stability for the case of a flat model of the Universe. A different exact analytical solution to these equations is derived.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

A question concerning unified field theory

The quantum hydrodynamics of extended particles is advanced by taking into account the gravitational field. A system of equations is obtained for relativistic nonlinear quantum unified field theory.Institute of Terrestrial Magnetism, the Ionosphere and Radiowave Propagation, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 71–76, November, 1993.     

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

Asymptotes of the effective potential in an external gravitational field

Renormalization group equations are derived which permit study of the behavior of the quantum theory effective potential of a field in curved space-time. Within the framework of asymptotically free models the asymptotes of the potential are studied for the limit of a strong gravitational field, the limit of large scalar fields, and the composite limit. The conditions for stability of quantum field theory in an external gravitational field are investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 26–32, October, 1985.The authors are indebted to I. V. Tyutin for evaluating the study.     

Asymptotic behavior of the quasipotential wave function of a two-particle system

The asymptotic behavior of relativistic wave functions of two-particle systems, consisting of spin 0 and 1/2 particles, is found with the help of integral equations derived within the framework of a covariant single-time approach in quantum field theory.Gomel'skii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 102–105, June, 1993.     

Mean-field dynamics of a quantum dot-microcavity system

Mean-field evolution equations for the exciton and photon populations and polarizations (Bloch–Lamb equations) are written and numerically solved in order to describe the dynamics of electronic states in a quantum dot coupled to the photon field of a microcavity. The equations account for phase space filling effects and Coulomb interactions among carriers, and include also (in a phenomenological way) incoherent pumping of the quantum dot, photon losses through the microcavity mirrors, and electron–hole population decay due to spontaneous emission of the dot. When the dot may support more than one electron–hole pair, asymptotic oscillatory states, with periods between 0.5 and 1.5 ps, are found almost for any values of the system parameters.     

Vertex Algebras in Higher Dimensions and Globally Conformal Invariant Quantum Field Theory

We propose an extension of the definition of vertex algebras in arbitrary space–time dimensions together with their basic structure theory. A one–to–one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed.     

Interacting Field Theories in Robertson-Walker Spacetimes: Analytic Approximations

The renormalization of a scalar field theory with a quartic self-coupling via adiabatic regularization in a Robertson-Walker spacetime is discussed. The adiabatic counterterms are presented in a way that is most conducive to numerical computations. A variation of the adiabatic regularization method is presented which leads to analytic approximations for the energy–momentum tensor of the quantum field and the quantum contribution to the effective mass of the mean field. Conservation of the energy–momentum tensor for the field is discussed and it is shown that the part of the energy–momentum tensor which depends only on the mean field is not conserved but the full renormalized energy–momentum tensor is conserved, as expected and required by the semiclassical Einstein's equation. It is also shown that if the analytic approximations are used the resulting approximate energy–momentum tensor is conserved. This allows a self-consistent backreaction calculation to be performed using the analytic approximations. The usefulness of the approximations is discussed.     

Legendre Transformation for Regularizable Lagrangians in Field Theory

Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.     

Quantum Cosmology in CGBD Theory

We apply the theory developed in quantum cosmology to a model of charged generalized Brans–Dicke gravity. This is a quantum model of gravitation interacting with a charged Brans–Dicke type scalar field which is considered in the Pauli frame. The Wheeler–DeWitt equation describing the evolution of the quantum Universe is solved in the semiclassical approximation by applying the WKB approximation. The wave function of the Universe is also obtained by applying both the Vilenkin-like and the Hartle–Hawking-like boundary conditions. We then make predictions from the wave functions and infer that the Vilenkin's boundary condition is more reasonable in the Brans–Dicke gravity models leading a large vacuum energy density at the beginning of the inflation.     

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

The Conformal Metric Associated with the U(1) Gauge of the Stueckelberg–Schrödinger Equation

We review the relativistic classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelbert–Schrödinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields with compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann–Robertson–Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson–Walker scale factor. Using the Einstein equations, one see that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann–Robertson–Walker dynamical model and the configurations in the conformal coordinates.     

Letter: Generation of Solutions of the Einstein Equations by Means of the Kaluza–Klein Formulation

It is shown that starting from a solution of the Einstein–Maxwell equations coupled to a scalar field given by the Kaluza–Klein theory, invariant under a one-parameter group, one can obtain a one-parameter family of solutions of the same equations.     

A modified equation for single-mode superradiance kinetics

A new method is proposed for decoupling the chain of quantum equations, which incorporates the fluctuations in the population difference between levels in the atoms near the maximum in the superradiance intensity. A new system of equations is derived that describes a broader and more unsymmetrical superradiance pulse than does the results of [1–4]. The new solutions improve the agreement between theory and experiment [5].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 27, No. 1, pp. 28–33, January, 1984.     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Effective Field Theory Description of the Higher Dimensional Quantum Hall Liquid

We derive an effective topological field theory model of the four dimensional quantum Hall liquid state recently constructed by Zhang and Hu. Using a generalization of the flux attachment transformation, the effective field theory can be formulated as a U(1) Chern–Simons theory over the total configuration space CP₃, or as a SU(2) Chern–Simons theory over S⁴. The new quantum Hall liquid supports various types of topological excitations, including the 0-brane (particles), the 2-brane (membranes), and the 4-brane. There is a topological phase interaction among the membranes which generalizes the concept of fractional statistics.