Functionals and the Quantum Master Equation

The quantum master equation is usually formulated in terms of functionals of the components of mappings (fields in physpeak) from a space–time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the antibracket (odd poisson bracket) of functionals and the other is the Laplacian of a functional. Both of these terms seem to depend on the fact that the mappings on which the functionals act are vector-valued. It turns out that neither the Laplacian nor the antibracket is well-defined for sections of an arbitrary vector bundle. We show that if the functionals are permitted to have their values in an appropriate graded tensor algebra whose factors are the dual of the space of smooth functions on M, then both the antibracket and the Laplace operator can be invariantly defined. This permits one to develop the Batalin–Vilkovisky approach to BRST cohomology for functionals of sections of an arbitrary vector bundle.     

Energies of knot diagrams

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function f and study the simplest nontrivial case f(x) = x ², obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.     

New properties of a class of generalized kinetic equations

We give some properties of a new class of hard-sphere kinetic equations of great generality, introduced earlier by Polewczak. The assumptions used to obtain the general class are very weak, and the equations include not only the standard and revised Enskog equations, but also generalizations thereof that can be expected to yield essentially exact transport coefficients. In particular, there is a natural two-particle realization that is obtained from maximizing the information entropy subject to prescribed two-particle and one-particle probability distribution functions;k-particle analogs fork > 2 also naturally follow. We obtain Liapunov functionals for the whole class of equations under consideration and discuss the question of which of these functionals can be expected to play the role ofH-functions. We also obtain several more special results that include new lower bounds on the potential part of theH-function for the revised Enskog equation. The bounds are instrumental in obtaining global existence theorems and also imply that the necessary condition for invertibility of the nonequilibrium extension of local activity as a functional of local density is satisfied.     

Operator orderings and functional formulations of quantum and stochastic dynamics

We study the connection between operator ordering schemes and thec-number formulations of quantum mechanics, which are based on generating functionals and functional integrals. We show by explicit construction that different operator ordering schemes are related to different functional and functional integral formulations of quantum mechanics. The results of these considerations are applied to classical non-linear stochastic dynamics by using the formal analogy between the Fokker-Planck equation and the Schrödinger equation.     

Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional. Received: 26 November 1998 / Accepted: 2 December 1998     

Benchmark CCSD(T) and DFT study of binding energies in Be7 − 12: in search of reliable DFT functional for beryllium clusters

We present a computational study of the stability of small homonuclear beryllium clusters Be_{7 ? 12} in singlet electronic states. Our predictions are based on highly correlated CCSD(T) coupled cluster calculations. Basis set convergence towards the complete basis set limit as well as the role of the 1s core electron correlation are carefully examined. Our CCSD(T) data for binding energies of Be_{7 ? 12} clusters serve as a benchmark for performance assessment of several density functional theory (DFT) methods frequently used in beryllium cluster chemistry. We observe that, from Be₁₀ clusters on, the deviation from CCSD(T) benchmarks is stable with respect to size, and fluctuating within 0.02 eV error bar for most examined functionals. This opens up the possibility of scaling the DFT binding energies for large Be clusters using CCSD(T) benchmark values for smaller clusters. We also tried to find analogies between the performance of DFT functionals for Be clusters and for the valence-isoelectronic Mg clusters investigated recently in Truhlar's group. We conclude that it is difficult to find DFT functionals that perform reasonably well for both beryllium and magnesium clusters. Out of 12 functionals examined, only the M06-2X functional gives reasonably accurate and balanced binding energies for both Be and Mg clusters.     

Response function analysis of excited-state kinetic energy functional constructed by splitting k-space

Over the past decade, fundamentals of time-independent density functional theory for excited state have been established. However, construction of energy functionals for excited states remains a challenging problem. We have developed a method for obtaining these functionals by splitting k-space according to the occupation of orbitals. In this paper we demonstrate the accuracy of kinetic energy functional thus obtained for excited states with two sets of vacant orbitals. We then perform a response function analysis of the functional proposed and show why our method could be the method of choice for the construction of excited state energy functionals.     

Theoretical investigations of the structural,electronic and optical properties of Hg1−xCdxTe alloys

The structural, electronic and optical properties of mercury cadmium telluride (Hg_1?xCd_xTe; x = 0.0, 0.25, 0.5, 0.75) alloys are studied using density functional theory within full-potential linearized augmented plane wave method. We used the local density approximation (LDA), generalized gradient approximation (GGA), hybrid potentials, the modified Becke–Johnson (LDA/GGA)-mjb and Hubbard-corrected functionals (GGA/LDA + U), for the exchange-correlation potential (E_ex). We found that LDA functional predicts better lattice constants than GGA functional, whereas, both functionals fail to predict the correct electronic structure. However, the hybrid functionals were more successful. For the case of HgTe binary alloy, the GGA + U functional predicted a semi-metallic behaviour with an inverted band gap of ?0.539 eV, which is closest to the experimental value (?0.30 eV). Ternary alloys, however, are found to be semiconductors with direct band gaps. For the x = 0.25 and 0.50, the best band gaps are found to be 0.39 and 0.81 eV using LDA-mbj functional, whereas, the GGA-mbj functional predicted the best band gap of 1.09 eV for Hg_0.25Cd_0.75Te alloy, which is in a very good agreement with the experimental value (1.061 eV). The optical properties of the alloys are obtained by calculating the dielectric function ?(ω). The peaks of the optical dielectric functions are consistent with the electronic gap energies of the alloys.     

Duality for Coloured Quantum Groups

Duality between the coloured quantum group and the coloured quantum algebra corresponding to GL(2) is established. The coloured L ^± functionals are constructed and the dual algebra is derived explicitly. These functionals are then employed to give a coloured generalisation of the differential calculus on quantum GL(2) within the framework of the R-matrix approach.     

La structure algébrique de la représentation et le problème de la symétrie pour une classe de fonctionelles de Wightman

The special setQ ofWightman's functionals is considered. All the pure functionals that belong toQ have the strong spectral property and the single vacuum. All other functionals fromQ may be represented by integrals over (weak) compact sets of pure functionals fromQ. The Borchers structure theorem is proved for this set. The conditions of the unitary equivalence of the representations and their similarity are given. These results are applied for the analysis of the symmetry problem. In particular we formulate the necessary and sufficient condition for the given group of automorphisms to be unitarily implemented.     

Faddeev–Jackiw canonical path integral quantization for a general scenario,its proper vertices and generating functionals

We generalize the Faddeev–Jackiw canonical path integral quantization for the scenario of a Jacobian with J=1 to that for the general scenario of non-unit Jacobian, give the representation of the quantum transition amplitude with symplectic variables and obtain the generating functionals of the Green function and connected Green function. We deduce the unified expression of the symplectic field variable functions in terms of the Green function or the connected Green function with external sources. Furthermore, we generally get generating functionals of the general proper vertices of any n-points cases under the conditions of considering and not considering Grassmann variables, respectively; they are regular and are the simplest forms relative to the usual field theory.     

Gamow state vectors as functionals over subspaces of the nuclear space

Exponentially decaying ‘Gamow state’ vectors are obtained from S-matrix poles in the lower half of the second sheet and are defined as functionals over a subspace of the nuclear space Φ. Exponentially growing ‘Gamow state’ vectors are obtained from S-matrix poles in the upper half of the second sheet and are defined as functionals over another subspace of Φ. On functionals over these two subspaces the dynamical group of time development splits into two semigroups.     

Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity

Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I₂ direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give: Let d be a countably additive decoherence functional defined on all pairs of projections in L(H). If H is infinite dimensional then d must be bounded. By contrast, when H is finite dimensional, unbounded (countably additive) decoherence functionals always exist for L(A). Received: 6 December 1996 / Accepted: 18 May 1997     

Chern-Simons states at genus one

We present a rigorous analysis of the Schrödinger picture quantization for theSU(2) Chern-Simons theory on 3-manifold torusxline, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals ofsu(2)-connections on the torus, are expressed by degree 2k theta-functions satisfying additional conditions. The conditions are obtained by splitting the space of semistablesu(2)-connections into nine submanifolds and by analyzing the behavior of states at four codimension 1 strata. We construct the Knizhnik-Zamolodchikov-Bernard connection allowing to compare the states for different complex structures of the torus and different positions of the Wilson lines. By letting two Wilson lines come together, we prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins>1/2 level implies the Verlinde dimension formula.     

Quantum Monte Carlo assessment of density functionals for π-electron molecules: ethylene and bifuran

ABSTRACT

We perform all-electron, pure-sampling quantum Monte Carlo (QMC) calculations on ethylene and bifuran molecules. The orbitals used for importance sampling with a single Slater determinant are generated from Hartree-Fock and density functional theory (DFT). Their fixed-node energy provides an upper bound to the exact energy. The best performing density functionals for ethylene are BP86 and M06, which account for 99% of the electron correlation energy. Sampling from the π-electron distribution with these orbitals yields a quadrupole moment comparable to coupled cluster CCSD(T) calculations. However, these, and all other density functionals, fail to agree with CCSD(T) while sampling from electron density in the plane of the molecule. For bifuran, as well as ethylene, a correlation is seen between the fixed-node energy and deviance of the QMC quadrupole moment estimates from those calculated by DFT. This suggests that proximity of DFT and QMC densities correlates with the quality of the exchange nodes of the DFT wave function for both systems.     

FeP(Im)−AB bonding energies evaluated with a large number of density functionals (P = porphine,Im = imidazole,AB = CO,NO, and O2)

Sixty-four density functionals, ranging from GGA, meta-GGA, hybrid GGA to hybrid meta-GGA, were tested to evaluate the FeP(Im)–AB bonding energies (E _bond) in the heme model complexes FeP(Im)(AB) (P?=?porphine, Im?=?imidazole, AB?=?CO, NO, and O₂). The results indicate that an accurate prediction of E _bond for the various ligands to heme is difficult with the DFT methods; usually, a functional successful for one system does not perform equally well for other system(s). Relatively satisfactory results for the various FeP(Im)–AB bonding energies are obtained with the meta-GGA functionals BLAP3 and Bmτ1; they yield E _bond values of ca. 1.1, 1.2, and 0.4?eV for AB?=?CO, NO, and O₂, respectively, which are in reasonable agreement with experimental data (0.78–0.85?eV for CO, 0.99?eV for NO, and 0.44???0.53?eV for O₂). The other functionals show more or less deficiency for one or two of the systems. The performances of the various functionals in describing the spin-state energetics of the five-coordinate FeP(Im) complex were also examined.     

Banach spaces of weights on quasimanuals

Section 1 is a brief introduction. Section 2 contains the basic definitions of quasimanuals, weights, and operational logics. The linear spaceW of all weights on a quasimanualA is introduced and given a norm.W with this norm is seen to be a Banach space. The subspaceV ofW generated by the positive cone ofW is given the base norm and is also shown to be an Archimedian ordered Banach space with an additive norm. In Section 3 normal linear functionals onV ^* are defined in analogy with normal linear functionals onw ^* algebras. The spaceV is shown to be the set of normal functionals onV ^* and we showV to be the unique partially ordered Banach space with a closed generating cone which is predual toV ^*. Next, weakly compact subsets ofW are characterized in terms of eventwise convergence. This is the Hahn-Vitali-Saks theorem of classical measure theory in this noncommutative setting; several weak compactness results are drawn from this and compared with their classical counterparts. Section 4 introduces the ultraweak topology forV ^* in analogy with the same for the trace class operators on Hubert space. Here the condition for a compact base for the cone ofV is examined and shown to be a poor and unnecessary hypothesis in many circumstances. Many connections with the existent literature are made and throughout the paper there are many examples and open questions.     

On GNS Representations¶on Inner Product Spaces

A generalization of the GNS construction to hermitian linear functionals W defined on a unital *-algebra is considered. Along these lines, a continuity condition (H) upon W is introduced such that (H) proves to be necessary and sufficient for the existence of a J-representation on a Krein space . The property whether or not the Gram operator J leaves the (common and invariant) domain of the representation invariant is characterized as well by properties of the functional W as by those of . Furthermore, the interesting class of positively dominated functionals is introduced and investigated. Some applications to tensor algebras are finally discussed. Received: 21 May 1996 / Accepted: 12 May 1997     

Unbounded functionals on a group algebra and applications to quantum theory

A certain class of positive functionals on a group algebra is examined that is pertinent to the induced representations of Frobenius and Mackey. Though these functionals are not bounded in theL ¹ norm, continuity still persists to an extent that secures the existence of a continuous group representation obtained from Gelfand's construction. The theory thus developed provides a new aspect of both the improper states in quantum theory and the induced representations of groups. The method is applied to the Poincaré group and it is shown that the representations, in which particles can be accommodated, are determined up to unitary equivalence by unbounded functionals of a simple structure. It is stressed that representations describing an infinitely degenerate vacuum emerge from mass nonzero representations as the mass tends to zero.