Equivalence between the Osserman condition and the Rakić duality principle in dimension 4

We show that four-dimensional Riemannian manifolds which satisfy the Raki? duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant). Thus, since it was proved in Raki? (1999) [9] that Osserman manifolds satisfy the Raki? duality principle, both conditions are equivalent.     

Impurity non-preserving 3-point correlators of BMN operators from PP-wave holography II: Fermionic excitations

The holographic principle in the PP-wave limit proposed in our previous works is further confirmed by studying impurity non-preserving processes which contain a fermionic BMN operator with one scalar and one fermion impurities. We show that the previously proposed duality relation between the matrix elements of the three point interaction Hamiltonian in the holographic string field theory and the OPE coefficients in super-Yang–Mills theory holds to the leading order in the large μ limit. Operator mixing is required to obtain the BMN operator of definite conformal dimension which corresponds to the string state with one scalar and one fermion excitations. The mixing term plays a crucial role for our duality relation to be valid. Our results, combined with those in the previous papers, provide a positive support that our duality relation holds for the general process regardless of the kind of impurities and of whether impurities conserve or not.     

Nucleon resonances and anomalous dimensions

It is studied whether duality analogous to that of Bloom and Gilman holds in the case in which the non-canonical scaling proves to be true. The data show that such duality holds well, independently of Q².     

广义量子干涉原理及对偶量子计算机

我们综述最近提出的广义量子干涉原理及其在量子计算中的应用。广义量子干涉原理是对狄拉克单光子干涉原理的具体化和多光子推广,不但对像原子这样的紧致的量子力学体系适用,而且适用于几个独立的光子这样的松散量子体系。利用广义量子干涉原理,许多引起争议的问题都可以得到合理的解释,例如两个以上的单光子的干涉等问题。从广义量子干涉原理来看双光子或者多光子的干涉就是双光子和双光子自身的干涉,多光子和多光子自身的干涉。广义量子干涉原理可以利用多组分量子力学体系的广义Feynman积分表示,可以定量地计算。基于这个原理我们提出了一种新的计算机,波粒二象计算机,又称为对偶计算机。在原理上对偶计算机超越了经典的计算机和现有的量子计算机。在对偶计算机中,计算机的波函数被分成若干个子波并使其通过不同的路径,在这些路径上进行不同的量子计算门操作,而后这些子波重新合并产生干涉从而给出计算结果。除了量子计算机具有的量子平行性外,对偶计算机还具有对偶平行性。形象地说,对偶计算机是一台通过多狭缝的运动着的量子计算机,在不同的狭缝进行不同的量子操作,实现对偶平行性。目前已经建立起严格的对偶量子计算机的数学理论,为今后的进一步发展打下了基础。本文着重从物理的角度去综述广义量子干涉原理和对偶计算机。现在的研究已经证明,一台d狭缝的n比特的对偶计算机等同与一个n比特+一个d比特(qudit)的普通量子计算机,证明了对偶计算机具有比量子计算机更强大的能力。这样,我们可以使用一台具有n+log2d个比特的普通量子计算机去模拟一个d狭缝的n比特对偶计算机,省去了研制运动量子计算机的巨大的技术上的障碍。我们把这种量子计算机的运行模式称为对偶计算模式,或简称为对偶模式。利用这一联系反过来可以帮助我们理解广义量子干涉原理,因为在量子计算机中一切计算都是普通的量子力学所允许的量子操作,因此广义量子干涉原理就是普通的量子力学体系所允许的原理,而这个原理只是是在多体量子力学体系中才会表现出来。对偶计算机是一种新式的计算机,里面有许多问题期待研究和发展,同时也充满了机会。在对偶计算机中,除了幺正操作外,还可以允许非幺正操作,几乎包括我们可以想到的任何操作,我们称之为对偶门操作或者广义量子门操作。目前这已经引起了数学家的注意,并给出了广义量子门操作的一些数学性质。此外,利用量子计算机和对偶计算机的联系,可以将许多经典计算机的算法移植到量子计算机中,经过改造成为量子算法。由于对偶计算机中的演化是非幺正的,对偶量子计算机将可能在开放量子力学的体系的研究中起到重要的作用。     

Duality theorems in BRST cohomology

In classical mechanics, there is no duality theorem relating the BRST cohomologies at positive and negative ghost numbers since these generically fail to be isomorphic. It is shown in this paper, however, that a duality theorem for the BRST operator cohomology can be established in quantum mechanics. Furthermore, when the hermicity properties of the quantum BRST formalism—which are in general just formal—turn out to be actually well defined, this duality theorem also holds for the state cohomology, as a consequence of the non degenerate pairing between subspaces at positive and negative ghost numbers defined by the BRST scalar product. In the case of gauge systems quantized in the Schrödinger representation with compact gauge orbits, the duality theorem contains ordinary Poincaré duality for a compact manifold. In the Fock representation, the duality theorem sheds a new light on existing decoupling theorems. The comparison with the classical situation is also briefly discussed.     

Regge behaviour for inclusive annihilation and decay processes

By exploiting annihilation processes it is shown that inclusive experiments are also relevant at low energies, if duality holds.     

Punctured Haag Duality in Locally Covariant Quantum Field Theories

We investigate a new property of nets of local algebras over 4-dimensional globally hyperbolic spacetimes, called punctured Haag duality. This property consists in the usual Haag duality for the restriction of the net to the causal complement of a point p of the spacetime. Punctured Haag duality implies Haag duality and local definiteness. Our main result is that, if we deal with a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, then also the converse holds. The free Klein-Gordon field provides an example in which this property is verified.     

K-Theoretic Duality for Shifts of Finite Type

We will study the stable and unstable Ruelle algebras associated to a hyperbolic homeomorphism of a compact space. To do this, we will describe a notion of K-theoretic duality for -algebras which generalizes Spanier-Whitehead duality in topology. A criterion for checking that it holds is presented. As an application it is shown that the Ruelle algebras which are associated to subshifts of finite type, (and agree with Cuntz-Krieger algebras in this case) satisfy this criterion and hence are dual. Received: 23 February 1996 / Accepted: 10 December     

Mathematical Theory of Generalized Duality Quantum Computers Acting on Vector-States

Following the idea of duality quantum computation, a generalized duality quantum computer (GDQC) acting on vector-states is defined as a tuple consisting of a generalized quantum wave divider (GQWD) and a finite number of unitary operators as well as a generalized quantum wave combiner (GQWC). It is proved that the GQWD and GQWC of a GDQC are an isometry and a co-isometry, respectively, and mutually dual. It is also proved that every GDQC gives a contraction, called a generalized duality quantum gate (GDQG). A classification of GDQCs is given and the properties of GDQGs are discussed. Some applications are obtained, including two orthogonal duality quantum computer algorithms for unsorted database search and an understanding of the Mach-Zehnder interferometer.     

Universal curvature identities II

We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in the Riemannian setting, they are given solely in terms of curvature (and not in terms of covariant derivatives of curvature) even in the pseudo-Riemannian setting.     

黎曼面上的温度对偶性与亏格数g=1和2的弦宇宙学解

考察了一个在引力场g_μv和dilaton场背景下的有限温度玻色弦模型,导出了高亏格黎曼面上能量动量张量满足的对偶关系式;同时,还在四维Robertson–Walker(R–W)度规下证明了弦气体物质作用量的温度对偶不变性,获得了亏格数g=1和2的弦宇宙学解,并研究了运动方程的温度变换性质.     

General Quantum Interference Principle and Duality Computer

In this article, we propose a general principle of quantum interference for quantum system, and based on this we propose a new type of computing machine, the duality computer, that may outperform in principle both classical computer and the quantum computer. According to the general principle of quantum interference, the very essence of quantum interference is the interference of the sub-waves of the quantum system itself. A quantum system considered here can be any quantum system: a single microscopic particle, a composite quantum system such as an atom or a molecule, or a loose collection of a few quantum objects such as two independent photons. In the duality computer, the wave of the duality computer is split into several sub-waves and they pass through different routes, where different computing gate operations are performed. These sub-waves are then re-combined to interfere to give the computational results. The quantum computer, however, has only used the particle nature of quantum object. In a duality computer, it may be possible to find a marked item from an unsorted database using only a single query, and all NP-complete problems may have polynomial algorithms. Two proof-of-the-principle designs of the duality computer are presented: the giant molecule scheme and the nonlinear quantum optics scheme. We also propose thought experiment to check the related fundamental issues, the measurement efficiency of a partial wave function.     

On the uniqueness of the Invariant Equilibrium State and surface tension

Symmetric Equilibrium States and their properties under duality transformation are investigated. Necessary and sufficient conditions are derived for equilibrium states to be transformed into equilibrium states by duality. It is shown that ferromagnetic systems satisfying those conditions have correlation functions bounded by those corresponding to the (+) and free boundary conditions. It is then proved than any Invariant Equilibrium State of a ferromagnetic system is transformed into an equilibrium state by duality and is thus unique if the states defined by the (+), and free boundary conditions coincide on the symmetric algebra. The existence of surface tension between two pure phases is established.     

The duality principle and inversion of Laplace–Stielties transforms

The fundamental relation between the Laplace transform, the Stielties transform, and the generalized integral equation of refraction is revealed, and a duality principle is formulated for the solution of inverse problems of radio physics. New formulas of the Laplace-transform inversion satisfying the duality principle are obtained. There is no necessity of contour integration in a complex plane for the relations found, which considerably simplifies the reconstruction of originals and makes it possible to control systematic errors in the experimental data.     

SYZ duality for parabolic Higgs moduli spaces

We prove the SYZ (Strominger–Yau–Zaslow) duality for the moduli space of full flag parabolic Higgs bundles over a compact Riemann surface. In Hausel and Thaddeus (2003) [12], the SYZ duality was proved for moduli spaces of Higgs vector bundles over a compact Riemann surface.     

Generalized holomorphic structures

We investigate generalized holomorphic structures in generalized complex geometry. We find that a generalized holomorphic vector bundle carries a generalized complex structure on its total space if some additional conditions hold. We prove that generalized holomorphicity is equivalent to the integrability of a distribution on the total space, and a family of linear Dirac structures associated with this distribution is a generalized complex structure if a further condition holds. Under the same condition, we also prove that local generalized holomorphic frames exist around a regular point.     

Path-integral derivation of quantum duality in nonlinear sigma-models

We show that a previously derived shift in the dilaton field, which necessarily augments the classical effects of duality transformation on the geometry of a nonlinear sigma-model if conformal invariance is to be preserved at the one-loop level, can be extended without change to the case of sigma-models with Wess-Zumino-Witten term (torsion) before and after duality. We also construct a path-integral implementation of the duality transformation, and discover the origin of the dilaton shift in a functional determinant resulting from the elimination of the first-order field. The path-integral formulation in principle allows a derivation of “quantum” duality transformations which preserve conformal invariance to all orders in α', the string tension parameter.     

Spacelike parallels and evolutes in Minkowski pseudo-spheres

We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper [S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Preprint] we have shown a basic Legendrian duality theorem between pseudo-spheres. We define the spacelike parallels by using the basic Legendrian duality theorem. This definition unifies the notions of parallels of spacelike hypersurfaces in pseudo-spheres. We also define the evolute as the locus of singularities of the spacelike parallels. These notions are investigated as applications of Lagrangian or Legendrian singularity theory. We consider geometric properties of non-singular spacelike hypersurfaces corresponding to singularities of spacelike parallels or evolutes.     

On wave/particle duality

The interference of electrons is considered in the light of a variational principle given earlier. It is shown that the phenomenon can be explained on the basis of the information available when satisfying the variational principle. The concept of wave/particle duality is not required.