Path Integrals on Riemannian Manifolds with Symmetry and Induced Gauge Structure

We formulate path integrals on any Riemannian manifold which admits the action of a compact Lie group by isometric transformations. We consider a path integral on a Riemannian manifold M on which a Lie group G acts isometrically. Then we show that the path integral on M is reduced to a family of path integrals on a quotient space Q=M/G and that the reduced path integrals are completely classified by irreducible unitary representations of G. It is not necessary to assume that the action of G on M is either free or transitive. Hence our formulation is applicable to a wide class of manifolds, which includes inhomogeneous spaces, and it covers all the inequivalent quantizations. To describe the path integral on inhomogeneous space, stratification geometry, which is a generalization of the concept of principal fiber bundle, is necessarily introduced. Using it we show that the path integral is expressed as a product of three factors; the rotational energy amplitude, the vibrational energy amplitude, and the holonomy factor. When a singular point arises in $Q$, we determine the boundary condition of the path integral kernel for a path which runs through the singularity.Comment: 20 pages, no figur

Classical Exchange Algebra of the Superstring on S^5 with the AdS-time

A classical exchange algebra of the superstring on S^5 with the AdS-time is shown on the light-like plane. To this end we use the geometrical method of which consistency is guaranteed by the classical Yang-Baxter equation. The Dirac method does not work, there being constraints which contain first-class and second-class and one can disentangle with each other keeping the isometry hardly.Comment: 12 pages, v2: argument on alternative representation of S^5 spherical functions added, typos corrected, one reference added, matches journal versio

An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

We solved the Schr{\"o}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / {\Lambda} is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice {\Lambda}. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schr{\"o}dinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.Comment: 32 pages, LaTeX, minor corrections are mad

Magnetic translation groups in an n-dimensional torus

A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG on an n-dimensional torus is isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible unitary representations of the MTG on a three-torus and apply the representation theory to three examples. We shortly describe a representation theory for a general n-torus. The MTG on an n-torus can be regarded as a generalization of the so-called noncommutative torus.Comment: 29 pages, LaTeX2e, title changed, re-organized, to be published in Journal of Mathematical Physic

Tractable circula densities from Fourier series

This article proposes an approach, based on infinite Fourier series, to constructing tractable densities for the bivariate circular analogues of copulas recently coined ‘circulas’. As examples of the general approach, we consider circula densities generated by various patterns of nonzero Fourier coefficients. The shape and sparsity of such arrangements are found to play a key role in determining the properties of the resultant models. The special cases of the circula densities we consider all have simple closed-form expressions involving no computationally demanding normalizing constants and display wide-ranging distributional shapes. A highly successful model identification tool and methods for parameter estimation and goodness-of-fit testing are provided for the circula densities themselves and the bivariate circular densities obtained from them using a marginal specification construction. The modelling capabilities of such bivariate circular densities are compared with those of five existing models in a numerical experiment, and their application illustrated in an analysis of wind directions

The Structure of Screening in QED

The possibility of constructing charged particles in gauge theories has long been the subject of debate. In the context of QED we have shown how to construct operators which have a particle description. In this paper we further support this programme by showing how the screening interactions arise between these charges. Unexpectedly we see that there are two different gauge invariant contributions with opposite signs. Their difference gives the expected result.Comment: 8 pages, LaTe

The Whitham Deformation of the Dijkgraaf-Vafa Theory

We discuss the Whitham deformation of the effective superpotential in the Dijkgraaf-Vafa (DV) theory. It amounts to discussing the Whitham deformation of an underlying (hyper)elliptic curve. Taking the elliptic case for simplicity we derive the Whitham equation for the period, which governs flowings of branch points on the Riemann surface. By studying the hodograph solution to the Whitham equation it is shown that the effective superpotential in the DV theory is realized by many different meromorphic differentials. Depending on which meromorphic differential to take, the effective superpotential undergoes different deformations. This aspect of the DV theory is discussed in detail by taking the N=1^* theory. We give a physical interpretation of the deformation parameters.Comment: 35pages, 1 figure; v2: one section added to give a physical interpretation of the deformation parameters, one reference added, minor corrections; v4: minor correction

Enhancement of dye regeneration kinetics in dichromophoric porphyrin-carbazole triphenylamine dyes influenced by more exposed radical cation orbitals

Reduction kinetics of oxidized dyes absorbed on semiconductor surfaces and immersed in redox active electrolytes has been mainly modeled based on the free energy difference between the oxidation potential of the dye and the redox potential of the electrolyte. Only a few mechanisms have been demonstrated to enhance the kinetics by other means. In this work, the rate constant of the reduction of oxidized porphyrin dye is enhanced by attaching non-conjugated carbazole triphenylamine moiety using iodine/triiodide and tris(2,2′-bispyridinium)cobalt II/III electrolytes. These results are obtained using transient absorption spectroscopy by selectively probing the regeneration kinetics at the porphyrin radical cation and the carbazole triphenylamine radical cation absorption wavelengths. The enhancement in the reduction kinetics is not attributed to changes in the driving force, but to the more exposed dye cation radical orbitals of the dichromophoric dye. The results are important for the development of high efficiency photo-electrochemical devices with minimalized energy loss at electron transfer interfaces

Experiments quantifying elemental and isotopic fractionations during evaporation of CAI-like melts in low-pressure hydrogen and in vacuum : Constraints on thermal processing of CAI in the protoplanetary disk

This work was supported by NASA grant NNX17AE84G (to R.M.). Magnesium isotopic measurements were supported by NSF grant EAR-17407706 (to F.-Z. T.). P.S. and the Si isotope measurements made at the St Andrews Isotope Group (STAiG) at the University of St Andrews were supported by NERC grant NE/R002134/1 a Carnegie Trust Research Incentive Grant. Evaporation experiments at Hokkaido University were supported by the Ministry of Education, Sports, Science, and Technology KAKENHI Grant (to S.T.).It is widely believed that the precursors of coarse-grained CAIs in chondrites are solar nebula condensates that were later reheated and melted to a high degree. Such melting under low-pressure conditions is expected to result in evaporation of moderately volatile magnesium and silicon and their mass-dependent isotopic fractionation. The evaporation of silicate melts has been extensively studied in vacuum laboratory experiments and a large experimental database on chemical and isotopic fractionations now exists. Nevertheless, it remains unclear if vacuum evaporation of CAI-like melts adequately describes the evaporation in the hydrogen-rich gas of the solar nebula. Here we report the results of a detailed experimental study on evaporation of a such melt at 1600°C in both vacuum and low-pressure hydrogen gas, using 1.5- and 2.5-mm diameter samples. The experiments show that although at 2×10−4 bar H2 magnesium and silicon evaporate ∼2.8 times faster than at 2×10−5 bar H2 and ∼45 times faster than in vacuum, their relative evaporation rates and isotopic fractionation factors remain the same. This means that the chemical and isotopic evolutions of all evaporation residues plot along a single evaporation trajectory regardless of experimental conditions (vacuum or low-PH2) and sample size. The independence of chemical and isotopic evaporation trajectories on PH2 of the surrounding gas imply that the existing extensive experimental database on vacuum evaporation of CAI-like materials can be safely used to model the evaporation under solar nebula conditions, taking into account the dependence of evaporation kinetics on PH2. The experimental data suggest that it would take less than 25 minutes at 1600°C to evaporate 15–50% of magnesium and 5–20% of silicon from a 2.5-mm diameter sample in a solar nebula with PH2∼2×10−4 bar and to enrich the residual melt in heavy magnesium and silicon isotopes up to δ25Mg ∼ 5–10‰ and δ29Si ∼ 2–4‰. The expected chemical and isotopic features are compatible to those typically observed in coarse-grained Type A and B CAIs. Evaporation for ∼1 hour will produce δ25Mg ∼30–35‰ and δ29Si ∼10–15‰, close to the values in highly fractionated Type F and FUN CAIs. These very short timescales suggest melting and evaporation of CAI precursors in very short dynamic heating events. The experimental results reported here provide a stringent test of proposed astrophysical models for the origin and evolution of CAIs.PostprintPeer reviewe