Topology,unitary representations and charged particles

The quantum mechanics of the charged particles with rigid and local symmetries propagating on the manifoldM is studied. It is shown that the classical rigid symmetries of this model may be anomalous. These anomalies are of local and global type, and they related to topological obstructions to lifting a group action of a groupG onM to a principalU(1) bundleP overM. The charged particles with local symmetries may have additional anomalies and the representation theory of the groupC is used to study these anomalies. Finally, the quantum mechanics of the supersymmetric charged particles with symmetries is examined.     

Some remarks on -invariant Fedosov star products and quantum momentum mappings

In these notes we consider a slightly generalized Fedosov star product * on a symplectic manifold (M,ω), emanating from the fibrewise Weyl product and the triple (,Ω,s) consisting of a symplectic torsion free connection on M, a formal series ΩνZ²_dR(M)[[ν]] of closed two-forms on M, and a certain formal series s of symmetric contravariant tensor fields on M. We prove necessary and sufficient conditions for certain classical symmetries to become symmetries of the star product, only sufficient conditions having been published in special cases when this letter was written (note, however, the different proofs in [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1]). For a given symplectic vector field X on M, it is well known that (= is a sufficient condition for the Lie derivative to be a derivation of *. We prove that these conditions are in fact necessary ones, also providing a very simple proof for their being sufficient. Moreover, we prove a criterion that has first been presented by Gutt [S. Gutt, Star products and group actions, Contribution to the Bayrischzell Workshop, April 26–29, 2002] (see also [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1] for a different proof) and which specifies a necessary and sufficient condition for to be a quasi-inner derivation. The statement that this condition is a sufficient one dates back to Kravchenko [O. Kravchenko, Compos. Math. 123 (2000) 131]. Applying our results, we find necessary and sufficient criteria for a Fedosov star product to be -invariant and to admit a quantum Hamiltonian. Finally, supposing the existence of a quantum Hamiltonian, we present a cohomological condition on Ω that is equivalent to the existence of a quantum momentum mapping. In particular, our results show that the existence of a classical momentum mapping in general does not imply the existence of a quantum momentum mapping and thus give a negative answer to Xu’s question posed in [P. Xu, Commun. Math. Phys. 197 (1998) 167].     

<Emphasis Type="Italic">O</Emphasis>(12) limit and complete classification of symmetry schemes in proton-neutron interacting boson model

It is shown that the proton-neutron interacting boson model (pnIBM) admits new symmetry limits withO(12) algebra which breakF spin but preserves theF _z quantum numberM _F. The generators ofO(12) are derived and the quantum numberU ofO(12) for a given boson numberN is determined by identifying the corresponding quasi-spin algebra. TheO(12) algebra generates two symmetry schemes and for both of them, complete classification of the basis states and typical spectra are given. With theO(12) algebra identified, complete classification of pnIBM symmetry limits with goodM _F is established.     

A remark on the optimal cloning of an N-level quantum system

We study quantum cloning machines (QCM) that act on an unknown N-level quantum state and make M copies. We give a formula for the maximum of the fidelity of cloning and exhibit the unitary transformations that realize this optimal fidelity. We also extend the results to treat the case of M copies from () identical N-level quantum systems. Received 21 September 1999     

Quantum logic and operational quantum mechanics

We characterize the class of the μ-complete F-spaces with unit corresponding to the observables of a quantum logic. We show that, conversely, every μ-complete F-space satisfying Axiom I and Axiom II corresponds to a quantum logic. The latter class of F-spaces generalizes that of “spectral F-spaces” introduced by Alfsen and Shultz and by Edwards.     

Non-trivial extension of the (1+2)-Poincaré algebra and conformal invariance on the boundary of {\mathrm{AdS}}_3

Using recent results on strings on AdS$_3\times N^d$, where N is a d dimensional compact manifold, we re-examine the derivation of the non-trivial extension of the (1+2)-dimensional-Poincaré algebra obtained by Rausch de Traubenberg and Slupinsky. We show by explicit computation that this new extension is a special kind of fractional supersymmetric algebra which may be derived from the deformation of the conformal structure living on the boundary of AdS. The two so(1,2) Lorentz modules of spin used in building of the generalization of the (1+2) Poincaré algebra are re-interpreted in our analysis as highest weight representations of the left and right Virasoro symmetries on the boundary of AdS. We also complete known results on 2d-fractional supersymmetry by using spectral flow of affine Kac–Moody and superconformal symmetries. Finally we make preliminary comments on the trick of introducing Fth roots of g-modules to generalize the so(1,2) result to higher rank Lie algebras g. Received: 20 July 2000 / Published online: 19 September 2001     

Symmetries in quantum logics

A symmetry in the quantum logic (L, M) is defined as a pair of bijections :L L andv :M M such that the probabilities are preserved. Some properties of the symmetries are investigated.     

ZN-Graded Noncommutative Differential Calculus

We consider the algebra M _N (C) ofN × N matrices as a cyclic quantum plane.We also analyze the coaction of the quantum group F and the action of its dualquantum algebra H on it. Then we study the decomposition ofM _N (C) in termsof the quantum algebra representations. Finally, we develop the differential algebraof the cyclic group Z _N with d ^N = 0, where Z _N is viewed as the the subalgebraof diagonal N × N complex matrices, and treat the particularcase N = 3.     

Macroscopic retrocausation

Referring to Stapp's and Schmidt's recent papers: a Feynman transition amplitude |I><M><F| used retrodictively with post-selected outgoing bras<F| formalizes psychokinesis—Jaynes' qualification of the mind-induced-quantum-collapse concept. Time extendedness of matter, final cause, information-negentropy reversibility, are features inherent in relativistic quantum mechanics.     

Density functional calculations on the geometric structure and properties of the 3d transition metal atom doped endohedral fullerene M@C20F20（M=Sc-Ni）

This paper uses the generalised gradient approximation based on density functional theory to analyse the geometric structure and properties of the 3d transition metal atom doped endohedral fullerene M@C20F20(M=Sc-Ni).The geometric optimization shows that the cage centre is the most stable position for M,forming the structure named as M@C 20 F 20-4.The inclusion energy,zero-point energy,and energy gap calculations tell us that Ni@C 20 F 20-4 should be thermodynamically and kinetically stablest.M@C 20 F 20-4(M = Sc-Co) possesses high magnetic moments varied from 1 to 6 μ B,while Ni@C 20 F 20-4 is nonmagnetic.The Ni-C bond in Ni@C 20 F 20-4 contains both the covalent and ionic characters.     

Dimensional Reduction Over the Quantum Sphere and Non-Abelian q-Vortices

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a K?hler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU _q (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-K?hler quotient construction.     

Vector Bundle Moduli

We give the general presciption for calculating the number of moduli of irreducible, stable U(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi–Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B = F _r. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.     

MC20F20（M=Li，Na，Be和Mg）几何结构和电子性质的密度泛函计算研究

采用密度泛函理论（density functional theory, DFT）中的广义梯度近似（generalized gradient approximation, GGA）对MC₂₀F₂₀（M=Li,Na,Be和Mg）的几何结构和电子性质进行了计算研究.几何结构研发现：随着内掺原子序数的增加,金属原子M对C₂₀F₂₀中的C—C键的影响越来越大,而对C—F键的影响甚微.掺杂能计算表明：MC₂₀F₂₀的掺杂能均为负值,需要在一定的实验条件下才能被合成.内掺碱金属和碱土金属分别产生了两类截然不同的能隙和磁性.其中,内掺碱金属的能隙非常小,且带有1μ_B的净磁矩,表现出磁性;而内掺碱土金属的能隙比C₆₀的能隙还大,净自旋为0,表现出非磁性. 关键词： 富勒烯 几何结构 电子结构 密度泛函     

Automorphisms and symmetries of quantum logics

Given an amalgam of groups then every quantum logicQ ₀ = (L ₀,M ₀) (L ₀ is aσ-orthomodular poset,M ₀ is a full set of states on it) satisfying some reasonable conditions can be embedded in a quantum logicQ = (L, M), in which (1) all the automorphisms ofL form a group ∼-G ₁, (2) all the automorphisms ofM form a group ∼-G ₂, and (3) all the symmetries ofQ form a group ∼-G ₀. The quantum logic of all closed subspaces of a Hilbert spaceH and all its measures satisfies the conditions required fromQ ₀; hence, enlarging it, one can obtain “anything.”     

Symmetry entanglement in polarization biphoton optics

We study kinematic and dynamic ways of forming entangled states of quantum light fields due to their local and global polarization SU(2) symmetries. The kinematic entanglement is shown to be associated with particular polarization bases in the spaces of quantum states of multi-mode radiation, which are generated by the global SU(2) — symmetry. Dynamic entanglement is due to SU(2) symmetries of the Hamiltonians of the matter-radiation interaction. We also define some entanglement measures, which are related to characteristics of light depolarization. Applications of results obtained in biphoton optics are briefly discussed.     

The ultimate sensitivity of a high-temperature microwave SQUID magnetometer operating in the hysteresis regime

A method for the determination of the noise spectral density in a high-temperature microwave SQUID operating in the hysteresis regime is developed. Under these conditions, the reflection coefficient serves as an output signal. It is shown that if a directional coupler used for extracting the reflected wave is placed as close to the SQUID loop as possible, the magnetometer can be designed as a microwave integrated circuit with a noise flux spectral density S_F ^1/2 < 10 ^{- 5} F₀ /\textHz^{\text0\text.5} ,\textwhere F_\text0 S_\Phi ^{1/2} < 10^{ - 5} \Phi _0 /{\text{Hz}}^{{\text{0}}{\text{.5}}} ,{\text{where }}\Phi _{\text{0}} , is the magnetic flux quantum.     

Coupling and dissociation in artificial molecules

We show that the spin-and-space unrestricted Hartree-Fock method, in conjunction with the companion step of the restoration of spin and space symmetries via Projection Techniques (when such symmetries are broken), is able to describe the full range of couplings in two-dimensional double quantum dots, from the strong-coupling regime exhibiting delocalized molecular orbitals to the weak-coupling and dissociation regimes associated with a Generalized Valence Bond combination of atomic-type orbitals localized on the individual dots. The weak-coupling regime is always accompanied by an antiferromagnetic ordering of the spins of the individual dots. The cases of dihydrogen (H₂, 2e) and dilithium (Li₂, 6e) quantum dot molecules are discussed in detail. Received 19 December 2000     

Multiple-Quantum J-Resolved NMR Spectroscopy (MQ-JRES): Measurement of Multiple-Quantum Relaxation Rates and Relative Signs of Spin Coupling Constants

The technique of multiple-quantum J-resolved NMR spectroscopy (MQ-JRES) is introduced and applied to the spin system SI₃–M (such as in the example given here, the ¹³CH₃–¹²CH in alanine). The SI₃ spin system was excited to its highest quantum state (8S_yI_xI_yI_y), which consists of four coherences: quadruple quantum of (3I + S), double quantum of (3I − S), double quantum of (I + S), and zero quantum of (I − S). In the MQ spectrum generated from the projection onto the F₁ dimension, the resonances of the different multiple-quantum coherences are resolved by their coupling constants to the remote spin (M). The absorptive lineshapes in both F₁ and F₂ dimensions enable accurate measurements of transverse relaxation rates, and both amplitude and relative signs of the long-range coupling constants are to be derived from either frequency or time domain data. The selective detection of MQ-JRES spectra of the individual MQ coherences using either phase cycling or pulsed field gradients is presented.     

Spectra of Sol-Manifolds: Arithmetic and Quantum Monodromy

The spectral problem of three-dimensional manifolds M³_A admitting Sol-geometry in Thurston's sense is investigated. Topologically M³_A are torus bundles over a circle with a unimodular hyperbolic gluing map A. The eigenfunctions of the corresponding Laplace-Beltrami operators are described in terms of modified Mathieu functions. It is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form. As a result the spectral statistics is shown to disagree with the Berry-Tabor conjecture. The topological nature of the monodromy for both classical and quantum systems on Sol-manifolds is demonstrated.