The invariant charges of the Nambu-Goto theory in WKB-approximation: Renormalization

We discuss the one loop renormalization of the reparametrization invariant non-local conserved charges of the Nambu-Goto string theory. In addition we show the stability of a special well-known state under the corresponding infinitesimal symmetry transformations — at least in the WKB-approximation.Work supported by Deutsche Forschungsgemeinschaft     

The algebra formed by the invariant charges of the Nambu-Goto theory: Identification of a maximal Abelian subalgebra

Continuing the analysis of the algebraic structure of the invariant charges of the Nambu-Goto theory, we identify a complete set of commuting observables for the bosonic strings.     

Algebraic properties of the invariant charges of the Nambu-Goto theory

We analyse the infinite dimensional algebra of observable non-local integrals of motion of the Nambu-Goto string theory.Work supported by Deutsche Forschungsgemeinschaft     

The invariant charges of the Nambu-Goto theory: Their geometric origin and their completeness

We give an alternative construction of the reparametrization invariant non-local conserved charges of the Nambu-Goto theory which elucidates their geometric nature and their completeness property.     

Conformally invariant Yang-Mills theory realizing the Virasoro-Kac-Moody algebra

We will discuss a conformally invariant Yang-Mills theory in two dimensions, which realizes the Virasoro-Kac-Moody algebra as the BRST symmetry. The theory will have the Virasoro anomaly with the central charge −26, but no anomaly for the Kac-Moody algebra.     

“Conserved charges” of the Bondi-Metzner-Sachs algebra in the Brans-Dicke theory

The asymptotic symmetries in the Brans-Dicke theory are analyzed using Penrose's conformal completion method,which is independent of the coordinate system used.These symmetries,indeed,include supertranslations and Lorentz transformations for an asymptotically flat spacetime.With the Wald-Zoupas formalism,“conserved charges”and fluxes of the Bondi-Metzner-Sachs algebra are computed.The scalar degree of freedom contributes only to the Lorentz boost charge,even though it plays a role in various fluxes.The flux-balance laws are further applied to constrain the displacement memory,spin memory,and center-of-mass memory effects.     

Casimir pressure renormalization by the method of efficient surface charges

The problem of regularization and renormalization of the Casimir pressure in a finite area with a smooth boundary is considered using the example of a real scalar field in the two-dimensional case. A procedure for the renormalization of boundary pressure using a certain modified area and a method for approximate calculation of the Green’s function and its derivatives with the use of efficient surface charges are proposed. For problems that allow an exact solution, it is shown that the method has high accuracy and computational efficiency.     

The algebra of non-local charges in non-linear sigma models

We obtain the exact classical algebra obeyed by the conserved non-local charges in bosonic non-linear sigma models. Part of the computation is specialized for a symmetry groupO(N). As it turns out the algebra corresponds to a cubic deformation of the Kac-Moody algebra. We generalize the results for the presence of a Wess-Zumino term. The algebra is very similar to the previous one, now containing a calculable correction of order one unit lower. The relation with Yangians and the role of the results in the context of Lie-Poisson algebras are also discussed.Supported by FAPESP.Supported by CNPq.Supported by CNPq     

On the Casimir operators associated with any Lie algebra

We give a characterization of the Casimir operators of a Lie algebra by polynomial solutions of a system of first-order partial differential equations. Further an upper bound is stated for the number of independent Casimir operators of a nilpotent Lie algebra.     

On the Poisson algebra of a singular map

We construct the Poisson algebra associated to a singular mapping into symplectic space and show that this is an algebra of smooth functions generating solvable implicit Hamiltonian systems.     

Simple evaluation of Casimir invariants in finite-dimensional Poisson systems

We present a procedure for the calculation of the Casimir functions of finite-dimensional Poisson systems which avoids the burden of solving a set of partial differential equations, as is usually suggested in the literature. We show how a simple algebraic manipulation of the structure matrix substantially reduces the difficulty of the problem.     

The Galilean and translationally invariant operator algebra describing nuclei

The deviation of the commutation relations of the Galilean and translationally invariant field operators (given in part I) from the exact Fermi commutation relations is investigated. For this purpose the invariant field operators are integrated with one-particle functions describing a compactn-particle system of nucleons and its low excitations. With this assumption an approximate expression for the (subtractive) deviation of both commutation rules from one another is given. It is carefully estimated that the exact deviation, taken in the operator norm, is of ordern ^?2/3. It is shown that the degree of compactness of nuclear matter in the phase space plays an essential role in the determination of this estimate.     

Some remarks on the integration of the Poisson algebra

We prove the integrability of the Poisson algebra of functions with compact supports of a noncompact manifold. We also determine a Lie subalgebra of vector fields which, weakly, integrate the Poisson algebra of a not necessarily compact manifold covered by an exact symplectic manifold.     

The Galilean and translationally invariant operator algebra describing nuclei

The Galilean and translationally invariant field operator formalism developed in the parts I and II is applied to the shell model with harmonic potential. In spite of the complicate structure of the commutation relations some essential analogies to the non-invariant calculus can be verified. The unperturbed ground state can be described as quasi-vacuum. Further the one-particle density is diagonal in the energy representation. This fact admits the invariant use of the concepts of “occupied” and “unoccupied” states, in a somewhat modified, but well-defined sense.     

On a theory of the Casimir effect

We develop a mathematically precise framework for the Casimir effect. A major role is played by Dietz's idea of identifying the Casimir energy as the regularization-independent Ramanujan sum of an asymptotic series. As an illustration, we treat two cases: parallel plates and the sphere. We finally discuss the open problem of the Casimir force for the cube. We propose an Ansatz for the exterior force and argue why it may provide the exact solution, as well as an explanation of the repulsive sign of the force.     

The Nambu-Goto theory of closed bosonic strings moving in 1 + 3–dimensional Minkowski space: the construction of the quantum algebra of observables up to degree five

The quantum algebra of observables of closed bosonic strings moving in 1 + 3–dimensional Minkowski space is constructed up to degree five. All independent relations of degree four are computed; they involve three as yet undetermined parameters. Definitions and symbols are used as introduced in the above-mentioned article.     

Experiment and theory in the Casimir effect

Casimir effect is the attractive force which acts between two plane parallel, closely spaced, uncharged, metallic plates in vacuum. This phenomenon was predicted theoretically in 1948 and reliably investigated experimentally only in recent years. In fact, the Casimir force is similar to the familiar van der Waals force in the case of relatively large separations when the relativistic effects come into play. We review the most important experiments on measuring the Casimir force by means of torsion pendulum, atomic force microscope and micromechanical torsional oscillator. Special attention is paid to the puzzle of the thermal Casimir force, i.e. to the apparent violation of the third law of thermodynamics when the Lifshitz theory of dispersion forces is applied to real metals. Thereafter we discuss the role of the Casimir force in nanosystems including the stiction phenomenon, actuators, and interaction of hydrogen atoms with carbon nanotubes. The applications of the Casimir effect for constraining predictions of extra-dimensional unification schemes and other physics beyond the standard model are also considered.