Testing two versions of lattice gauge theory: Creutz ratios inU(1)3

In our simplicial version of lattice gauge theory, we approximate Euclidean path integrals by tiling space-time with simplexes and by linearly interpolating the fields throughout each simplex from their values at the vertices. We compare this method with Wilson's lattice gauge theory forU(1) in three dimensions. As a standard of comparison, we compute the exact values of Creutz ratios of Wilson loops in the continuum theory. Monte Carlo computations using the simplicial method give Creutz ratios within a few percent of the exact values for reasonably sized loop atβ=1, 2, and 10. Similar computations using Wilson's method give ratios that typically differ from the exact values by factors of 2 or more for 1⩽β⩽3.5 and that have the wrongβ dependence. The better accuracy of the simplicial method is due to its use of the action and domain of integration of the exact theory, unaltered apart from the granularity of the simplicial lattice. We also present data on the action density and the mass gap. Research supported by the U.S. epartment of Energy under grant DE-FG04-84ER40166.     

Discrete quantum gravity and anomalies

A new version of quantum gravity on discrete spaces (simplicial complexes) is proposed. A theory of gravitation interacting with Dirac field is considered. This theory is shown to be free of reparametrization anomaly. The problem of axial gauge anomaly and the associated problem of the doubling of fermion states on a lattice are discussed.     

Equivalence of the Regge and Einstein equations for almost flat simplicial space-times

The theory of distributions is applied to almost flat simplicial space-times. Explicit expressions are given for the first-order defects. It is shown explicitly that the Riemann tensor for an almost flat simplicial space-time contains delta-functions on the bones and derivatives of delta-functions on the 3-dimensional faces of the boundary of the space-time. The latter terms have not previously been seen in the Regge calculus. It is shown that the Regge and Hilbert actions have equal values on almost fiat simplicial space-times and that the Einstein equations lead directly to the Regge field equations.     

Comparison of the simplicial method with Wilson's lattice gauge theory for U(1)3

In the simplicial version of lattice gauge theory, euclidean path integrals are approximated by tiling spacetime with simplexes and by linearly interpolating the fields throughout each simplex from their values at the vertices. This method is compared with Wilson's lattice gauge theory for U(1) in three dimensions. As a standard of comparison, the exact values of Creutz ratios of Wilson loops in the continuum theory are computed. Monte Carlo computations using the simplicial method give Creutz ratios within a few percent of the exact values for reasonably sized loops at β = 1,2,and10. Similar computations using Wilson's method give ratios that typically differ from the exact values by factors of two or more for 1 ⩽ β ⩽ 3.5 and that have the wrong β dependence. The better accuracy of the simplicial method is due to its use of the action and domain of integration of the exact theory, unaltered apart from the granularity of the simplicial lattice. Data on the action density and the mass gap are also presented.     

Simplicial vs. continuum string theory and loop equations

We derive loop equations in a scalar matrix field theory. We discuss their solutions in terms of simplicial string theory—the theory describing embeddings of two-dimensional simplicial complexes into the spacetime of the matrix field theory. This relation between the loop equations and the simplicial string theory gives further arguments that favor one of the statements of the paper hep-th/0407018. The statement is that there is an equivalence between the partition function of the simplicial string theory and the functional integral in a continuum string theory—the theory describing embeddings of smooth two-dimensional world-sheets into the spacetime of the matrix field theory in question.     

Intramolecular bonding in a statistical associating fluid theory of ring aggregates

The first-order thermodynamic perturbation theory of Wertheim (TPT1) is extended to treat ring aggregates, formed by inter- and intramolecular association. The expression for the residual association contribution to the Helmholtz free energy for ring aggregates, incorporating the appropriate terms in Wertheim's fundamental graph sum of the TPT1 density expansion, is derived to calculate the distribution of the molecular bonding states. This requires the introduction of two new parameters to characterise each possible ring type: the ring size τ, which is equal to one in the case of intramolecular association, and a parameter W that captures the likelihood of two ring-forming sites bonding. The resulting framework can be incorporated in equations of state that account for the residual association contribution to the free energy, such as the statistical associating fluid theory (SAFT) family, or the cubic plus association (CPA) equation of state. This extends the applicability of these equations of state to mixtures with an arbitrary number of association sites capable of hydrogen bonding to form intramolecular and intermolecular rings. The formalism is implemented within SAFT-VR Mie to calculate the fluid-phase equilibria of model chain-like molecules containing two associating sites A and B, allowing for the formation of open-chain aggregates and intramolecular bonds. The effect of adding a second component that competes for the association sites that mediate intramolecular association in the chain is also examined. Accounting for intramolecular bonding is shown to have a significant impact on the phase equilibria of such systems.     

Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes–Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes–Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.     

金属Nb中H的量子行为研究

采用基于密度泛函原理的赝势平面波方法,计算了H在Nb晶体中的势场.通过求解H在该势场中的Schrdinger方程,得到了H的振动状态.计算结果表明,H的基态和第一激发态是局域的,第二激发态是非局域的,可以在T点之间迁移以及做4T/6T环运动. 关键词： 第一性原理 金属氢化物 量子振动     

State sum models and simplicial cohomology

We study a class of subdivision invariant lattice models based on the gauge groupZ _p , with particular emphasis on the four dimensional example. This model is based upon the assignment of field variables to both the 1- and 2-dimensional simplices of the simplicial complex. The property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-p flatness condition. By explicit computation of the partition function for the manifoldRP ³×S ¹, we establish that the theory has a quantum Hilbert space which differs from the classical one.Supported by Stichting voor Fundamenteel Onderzoek der Materie (FOM)     

Nuclear Algebraic Geometry and SO(10)

In this contribution nuclear representations of the Dirac ring, developed over many years, are shown to be a particular case of a theorem in algebraic geometry which at the same time associates them with a Hodge decomposition of a Kaehler manifold. This yields a shape that in some cases is independent of any appeal to a symmetry group. However, because the nuclear representations are in the infinitesimal ring of SO(4) and the internal space of each representation is in a Kaehler (even Calabi-Yau) manifold K; the group SO(10) = SO(4) × K can give additional information. This paper develops the very fruitful symbiosis between algebra and irreducible representations of SO(10) and covers some aspects of string theory.     

Expansion in Feynman graphs as simplicial string theory

We show that the series expansion of quantum field theory in Feynman diagrams can be explicitly mapped on the partition function of simplicial string theory—the theory describing embeddings of two-dimensional (2D) simplicial complexes into the spacetime of the field theory. The summation over 2D geometries in this theory is obtained from the summation over the Feynman diagrams and the integration over the Schwinger parameters of the propagators. We discuss the meaning of the obtained relation and derive the one-dimensional analog of the simplicial theory using the example of a free relativistic particle.     

Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics

An optical ring resonator with asymmetric three-layered dielectrics is shown to have an anomalously highQ-factor compared with the conventional symmetric three-layered ring resonator by several orders of magnitude for suitably chosen parameters. A new leaky mode which can propagate on the asymmetric structure is found having an extremely small attenuation loss near the cut-off frequency of the guided mode, but which does not exist in the symmetric structure. Optimum values of the normalized frequency for a single-mode operation to suppress the undesired leaky modes are discussed. It is also shown that the size of the ring resonator using the asymmetric structure can be reduced considerably from that of the ordinary symmetric resonator.     

Remarks on the entropy of 3-manifolds

We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show that all closed simplicial manifolds that can be constructed in this manner are homeomorphic to S3. We discuss the problem of proving that all 3-dimensional simplicial spheres can be obtained by this construction and give an example of a simplicial 3-ball whose boundary triangles can be identified pairwise such that no triangle is identified with any of its neighbours and the resulting 3-dimensional simplicial complex is a simply connected 3-manifold.     

Quantum Deformation of Lattice Gauge Theory

A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a modular Hopf algebra. In the topological (weak-coupling) limit, the gauge theory partition function gives a 3-fold invariant, coinciding in the simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge theory on Riemann surfaces and find a connection with the algebraic Alekseev-Grosse-Schomerus approach. Received: 29 April 1996 / Accepted: 24 September 1996     

Probing the structure and Franck–Condon region of protochlorophyllide a through analysis of the Raman and resonance Raman spectra

The structure and Franck–Condon region of protochlorophyllide a, a precursor in the biosynthesis of chlorophyll and substrate of the light‐regulated enzyme protochlorophyllide oxidoreductase (POR), were investigated by Raman and resonance Raman (RR) spectroscopy. The spectroscopic results are compared to the spectra of the structurally closely related porphyrin model compound magnesium octaethylporphyrin (MgOEP), and interpreted on the basis of density functional theory (DFT) calculations. It is shown that the electronic properties of the two porphyrin macrocycles are affected by different vibrational coupling modes, resulting in a higher absorption cross section of protochlorophyllide a in the visible spectral region. Furthermore, a comparison of the Fourier transform (FT)‐Raman and RR spectra of protochlorophyllide a indicates the modes that are resonantly enhanced upon excitation. Based on vibrational normal mode calculations, these modes include C C ring‐breathing and CC stretching vibrations of the porphyrin macrocycle. In particular, the strong band at 1703 cm⁻¹ can be attributed to the CO carbonyl vibration of the cyclopentanone ring, which is attached in conjugation to the π‐electron path of the porphyrin ring system. The enhancement of that mode upon electronically resonant excitation is discussed in the light of the reaction model suggested for the photoreduction of protochlorophyllide a in the POR. Copyright © 2009 John Wiley & Sons, Ltd.     

Graph Hypersurfaces and a Dichotomy in the Grothendieck Ring

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of \mathbbZ{\mathbb{Z}} generated by the class of a point. This clarifies the extent to which the graph hypersurfaces ‘generate the Grothendieck ring of varieties’: while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of \mathbbZ[\mathbbL]{\mathbb{Z}[\mathbb{L}]} in which \mathbbL{\mathbb{L}} becomes invertible, the span of the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the Lefschetz motive \mathbbL{\mathbb{L}} to zero. In particular, this shows that the graph hypersurfaces do not generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne–Hodge polynomials. These observations are certainly not surprising for the expert reader, but are somewhat hidden in the literature. The treatment in this note is straightforward and self-contained.     

Discrete Torsion Phases as Topological Actions

In this paper, we show that discrete torsion phases in string orbifold partition functions, and membrane discrete torsion phases, are topological actions on the simplicial manifolds associated to orbifold group actions. For this purpose, we introduce an integration theory of smooth Deligne cohomology on a general simplicial manifold, and prove that the integration induces a well-defined paring between the smooth Deligne cohomology and the singular cycles.     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.     

The quantum pentacle

Quantum set theory permits the formulation of a quantum simplicial topology suitable for a quantum theory of time space and gravity without prior time space structure. The quantum simplex differs strikingly from the classical: It is isotropic (points in all directions) and all quantum simplexes of the same signature are congruent. Quantum simplexes and complexes are described byS numbers, elements of the Clifford algebra of quantum sets. The isotropy groups of noncontiguous simplexes commute, like local invariance groups in a gaugeinvariant theory.