Highlights of Symmetry Groups

The concepts of symmetry and symmetry groups are at the heart of several developments in modern theoretical and mathematical physics. The present paper is devoted to a number of selected topics within this framework: Euclidean and rotation groups; the properties of fullerenes in physical chemistry; Galilei, Lorentz and Poincare groups; conformal transformations and the Laplace equation; quantum groups and Sklyanin algebras. For example, graphite can be vaporized by laser irradiation, producing a remarkably stable cluster consisting of 60 carbon atoms. The corresponding theoretical model considers a truncated icosahedron, i.e. a polygon with 60 vertices and 32 faces, 12 of which are pentagonal and 20 hexagonal. The Carbon 60 molecule obtained when a carbon atom is placed at each vertex of this structure has all valences satisfied by two single bonds and one double bond. In other words, a structure in which a pentagon is completely surrounded by hexagons is stable. Thus, a cage in which all 12 pentagons are completely surrounded by hexagons has optimum stability. On a more formal side, the exactly solvable models of quantum and statistical physics can be studied with the help of the quantum inverse problem method. The problem of enumerating the discrete quantum systems which can be solved by the quantum inverse problem method reduces to the problem of enumerating the operator-valued functions that satisfy an equation involving a fixed solution of the quantum Yang--Baxter equation. Two basic equations exist which provide a systematic procedure for obtaining completely integrable lattice approximations to various continuous completely integrable systems. This analysis leads in turn to the discovery of Sklyanin algebras.Comment: Plain Tex with one figur

Twistors and Spin 3/2 Potentials in Quantum Gravity

Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This paper derives the conditions under which spin-lowering and spin-raising operators preserve these local boundary conditions on a 3-sphere for fields of spin 0,1/2,1,3/2 and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3/2 is applied to the case of a 3-sphere boundary. It is shown that such boundary conditions can only be imposed in a flat Euclidean background, for which the gauge freedom in the choice of the potentials remains. Alternative boundary conditions for supergravity involving the spinor-valued 1-forms for gravitinos and the normal to the boundary are also studied.Comment: 20 pages, plain-tex, recently appearing in: Twistor Theory, edited by Stephen Huggett (Marcel Dekker, New York, 1994). The authors apologize for the delay in circulating the paper, which was due to technical problems now fixe

Classical brackets for dissipative systems

We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.Comment: Latex file, 8 pages, prepared for the Conference Spacetime and Fundamental Interactions: Quantum Aspects, Vietri sul Mare, Italy, 26-31 May 200

Push on a Casimir apparatus in a weak gravitational field

The influence of the gravity acceleration on the regularized energy-momentum tensor of the quantized electromagnetic field between two plane parallel conducting plates is derived. We use Fermi coordinates and work to first order in the constant acceleration parameter. A new simple formula for the trace anomaly is found to first order in the constant acceleration, and a more systematic derivation is therefore obtained of the theoretical prediction according to which the Casimir device in a weak gravitational field will experience a tiny push in the upwards direction.Comment: 14 pages, Plain Tex. Talk given at the 17th SIGRAV Conference on General Relativity and Gravitational Physics, Torino, September 200

Boundary Terms for Massless Fermionic Fields

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal to the boundary and a pair of independent spinor fields. This paper studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. Interestingly, a link is found with the classical boundary-value problem when spectral boundary conditions are imposed on a 3-sphere in the massless case. Moreover, the boundary term in the action functional is derived.Comment: 8 pages, plain-tex, recently appearing in Foundations of Physics Letters, volume 7, pages 303-308, year 199

Ocriplasmin in the treatment of vitreomacular traction in a patient with central retinal vein occlusion. a case report

Aim: To investigate the efficacy of intravitreal injection of ocriplasmin (JETREA®) in the treat-ment of vitreomacular traction (VMT). Materials and Methods: An 81-year-old man with VMT associated with central retinal vein occlusion in his left eye, was treated with a single intravitreal injection of ocriplasmin (25 μg). Best corrected visual acuity (BCVA), ocular fundus, and optical coherence tomography were examined before and after treatment. Results: Complete release of VMT produced a reduction of central macular thickness, ranging from 459 to 141 μm. BCVA remained stable. Discussion and Conclusions: The use of ocriplasmin was effective in the treatment of VMT. Ocriplasmin represents a valid alternative to conventional pars plana vitrec-tomy

New classical brackets for dissipative systems

A set of brackets for classical dissipative systems, subject to external random forces, are derived. The method is inspired to the old procedure found by Peierls, for deriving the canonical brackets of conservative systems, starting from an action principle. It is found that an adaptation of Peierls' method is applicable also to dissipative systems, when the friction term can be described by a linear functional of the coordinates, as is the case in the classical Langevin equation, with an arbitrary memory function. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.Comment: 4 page

Euclidean Maxwell Theory in the Presence of Boundaries. II

Zeta-function regularization is applied to complete a recent analysis of the quantized electromagnetic field in the presence of boundaries. The quantum theory is studied by setting to zero on the boundary the magnetic field, the gauge-averaging functional and hence the Faddeev-Popov ghost field. Electric boundary conditions are also studied. On considering two gauge functionals which involve covariant derivatives of the 4-vector potential, a series of detailed calculations shows that, in the case of flat Euclidean 4-space bounded by two concentric 3-spheres, one-loop quantum amplitudes are gauge independent and their mode-by-mode evaluation agrees with the covariant formulae for such amplitudes and coincides for magnetic or electric boundary conditions. By contrast, if a single 3-sphere boundary is studied, one finds some inconsistencies, i.e. gauge dependence of the amplitudes.Comment: 24 pages, plain-tex, recently appearing in Classical and Quantum Gravity, volume 11, pages 2939-2950, December 1994. The authors apologize for the delay in circulating the file, due to technical problems now fixe

Variations of Casimir energy from a superconducting transition

We consider a five-layer Casimir cavity, including a thin superconducting film. We show that when the cavity is cooled below the critical temperature for the onset of superconductivity, the sharp variation (in the microwave region) of the reflection coefficient of the film produces a variation in the value of the Casimir energy. Even though the relative variation in the Casimir energy is very small, its magnitude can be comparable to the condensation energy of the superconducting film, and thus causes a significant increase in the value of the critical magnetic field, required to destroy the superconductivity of the film. The proposed scheme might also help clarifying the current controversy about the magnitude of the contribution to Casimir free energy from the TE zero mode, as we find that alternative treatments of this mode strongly affect the shift of critical field.Comment: RevTex 4, 15 pages, 8 figures. Revised version with improved presentation, a brief description of the ALADIN experiment, and a few references adde