Free-field realization of theWKP(N) algebra

TheW _KP ^(N) algebra has been identified with the second Hamiltonian structure in theNth Hamiltonian pair of the KP hierarchy. In this Letter, by constructing the Miura map that decomposes the second Hamiltonian structure in theNth pair of the KP hierarchy, we show thatW _KP ^(N) can also be decomposed toN independent copies ofW _KP ⁽¹⁾ algebras, therefore its free-field realization can be worked out by constructing free fields for each copy ofW _KP ⁽¹⁾ . In this way, the free fields may consist ofN + 2n number of bosons, among them, 2n are in pairs, wheren is an arbitrary integer between 1 andN. We also express the currents ofW _KP ^(N) in terms of the currents ofN —n copies of U(1) andn copies of SL(2,R) _k algebras with levelk = 1. By reductions, we give similar results forW ^(N) andW ₃ ⁽²⁾ algebra.     

Reichenbach's common cause principle and quantum field theory

Reichenbach's principles of a probabilistic common cause of probabilistic correlations is formulated in terms of relativistic quantum field theory, and the problem is raised whether correlations in relativistic quantum field theory between events represented by projections in local observable algebrasA(V₁) andA(V₂) pertaining to spacelike separated spacetime regions V₁ and V₂ can be explained by finding a probabilistic common cause of the correlation in Reichenbach's sense. While this problem remains open, it is shown that if all superluminal correlations predicted by the vacuum state between events inA(V₁) andA(V₂) have a genuinely probabilistic common cause, then the local algebrasA(V₁) andA(V₂) must be statistically independent in the sense of C^*-independence.     

Braid group action on theq-Weyl algebra

We provide a braid group action on theq-deformed Weyl algebraW _q (n). The restriction of this action to the representations ofU _q (A _n–1 ) andU _q (C _n ) inW _q (n) is seen to agree with the braid group action introduced by Lusztig on these quantum algebras.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

On the defining relations of quantum superalgebras

In defining quantum superalgebras, extra relations need to be added to the Serre-like relations. They are obtained for sl _q (m, n) and osp _q (m, 2n) usingq-oscillator representations.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

Formation and quenching mechanisms of excited particles in an oxygen-iodine laser medium

New values of a number of kinetic constants of processes proceeding in oxygen-iodine laser media are presented. The total probabilities of formation of I₂(X, 15 ≤ v ≤ 24) and I₂(X, 25 ≤ v ≤ 47) molecules in the course of quenching of I* atoms by I₂(X) are found to be 0.9 and 0.1, respectively. The quantum yield of singlet oxygen in the reaction O(¹ D) + N₂O → N₂ + O₂(a ¹Δ) is close to 100%. The quenching rate constants of I₂(A’) by O₂, H₂O, CO₂, I₂, and Ar and of I(² P _1/2) by O(³ P), O₃, NO₂, N₂O₄, and N₂O are presented.     

Large-Gap Quantum Spin Hall State in Double-Transition-Metal Homologous Compounds of WSi2N4: A First-Principles Study

Recently, the MoSi₂N₄(MoN)_4n homologous compounds have been synthesized, extending the MA₂Z₄ family materials to the superthick MA₂Z₄(MZ)_n form. Herein, a first-principles study on the double-transition-metal W₂TiSi₂N₆ and related group VI–IV M₂M′Si₂N₆ nanosheets is performed. Robust structural stability is affirmed in this W₂TiSi₂N₆ nanosheet from the energetic, mechanical, dynamical, and thermal perspectives. Unlike the WSi₂N₄ and WSi₂N₄(WN)_n systems, the W₂TiSi₂N₆ nanosheet exhibits a semimetal behavior with the top valence and bottom conduction bands touching each other at the Fermi level. With the inclusion of spin–orbit coupling, a nontrivial band gap is opened in the W₂TiSi₂N₆ nanosheet, which has a sizeable bulk gap of 0.11 eV. The Z₂ invariant is 1 and a pair of topologically protected edge states emerges in the gap region, confirming that W₂TiSi₂N₆ nanosheet is a quantum spin Hall (QSH) insulator. Through in-plane strain engineering, the bulk gap can be further increased to 0.23 eV, which is sufficiently large for the room-temperature QSH effect. Such a large-gap QSH state is also present in other group VI–IV M₂M′Si₂N₆ systems with the combinations of MM′ = WZr, MoZr, and MoHf. This study demonstrates that the double-metal M₂M′Si₂N₆ nanosheets are promising candidates to realize fascinating topological quantum states.     

The Multi-form Generalised Gleason Theorem

Arising from investigations of decoherence functionals in the “histories” approach to quantum mechanics, the following result is given by the methods of [14]. Let A ₁ and A ₂ be von Neumann algebras without Type I ₂ direct summands and let P(E _j ), (j=1,2) be their lattices of projections. Let m\colon; P(A ₁)×(P(A ₂)→ℂ be a bounded quantum bi-measure. Then there is a unique bounded bilinear functional M on A ₁×A ₂ which extends m. In this note we use a different approach to establish a generalisation of this result to k-fold, vector valued, quantum multi-measures. This tool is needed for further investigations of decoherence functionals in quantum theory. Received: 10 April 1998 / Accepted: 17 April 1998     

Normal ordering and antinormal ordering of the operator （fQ＋gP）n and some of their applications

In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator (f Q + gP )n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.     

Factorized Weight Functions vs. Factorized Scattering

Starting from an extensive class of factorized weight functions W(p) on the N-dimensional torus ?, we construct an orthonormal base of symmetric N-variable polynomials for L ² _s (?,W(p)dp) via lexicographic ordering of the monomial symmetric functions (free boson states) and the Gram-Schmidt procedure. We show that the dominant asymptotics of these polynomials is factorized. As a corollary, we obtain a large class of quantum integrable soliton systems on the symmetric subspace of l ²(ℤ ^N ). The class of weight functions contains in particular the weight function yielding Macdonald polynomials. For that special case, the quantum soliton system can be viewed as the dual relativistic Calogero–Sutherland system. Received: 4 September 2001 / Accepted: 4 January 2002     

The quantum group structure associated with non-linearly extended Virasoro algebras

Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universalR-matrix ofSL(2) _q , has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebrasA _N,N>1, is considered along the same line. Contrary to the previous case (A ₁) the standardR-matrix ofSL(N+1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

Gleason's Theorem for Rectangular JBW-Triples

A JBW^*-triple B is said to be rectangular if there exists a W^*-algebra A and a pair (p,q) of centrally equivalent elements of the complete orthomodular lattice P(A)\mathcal{P}(A) of projections in A such that B is isomorphic to the JBW^*-triple pAq. Any weak^*-closed injective operator space provides an example of a rectangular JBW^*-triple. The principal order ideal CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} of the complete ^*-lattice CP(A)\mathcal{C}\mathcal{P}(A) of centrally equivalent pairs of projections in a W^*-algebra A, generated by (p,q), forms a complete lattice that is order isomorphic to the complete latticeI(B)\mathcal{I}(B) of weak^*-closed inner ideals in B and to the complete lattice S(B)\mathcal{S}(B) of structural projections on B. Although not itself, in general, orthomodular, CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} possesses a complementation that allows for definitions of orthogonality, centre, and central orthogonality to be given. A less familiar notion in lattice theory, that is well-known in the theory of Jordan algebras and Jordan triple systems, is that of rigid collinearity of a pair (e₂,f²) and (e₂,f²) of elements of CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)}. This is defined and characterized in terms of properties of P(A)\mathcal{P}(A). A W^*-algebra A is sometimes thought of as providing a model for a statistical physical system. In this case B, or, equivalently, pAq, may be thought of as providing a model for a fixed sub-system of that represented by A. Therefore, CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} may be considered to represent the set consisting of a particular kind of sub-system of that represented by pAq. Central orthogonality and rigid collinearity of pairs of elements of CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} may be regarded as representing two different types of disjointness, the former, classical disjointness, and the latter, decoherence, of the two sub-systems. It is therefore natural to consider bounded measures m on CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} that are additive on centrally orthogonal and rigidly collinear pairs of elements. Using results of J.D.M. Wright, it is shown that, provided that neither of the two hereditary sub-W^*-algebras pAp and qAq of A has a weak^*-closed ideal of Type I², such measures are precisely those that are the restrictions of bounded sesquilinear functionals {_m on pAp 2 qAq with the property that the action of the centroid Z(B) of B commutes with the adjoint operation. When B is a complex Hilbert space of dimension greater than two, this result reduces to Gleason's Theorem.     

Realizations of causal manifolds by quantum fields

Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e., by the homogenous spacesD(n)=GL(C _R ⁿ )/U(n) withn=1 for mechanics andn=2 for relativistic fields. The rankn gives the number of both the discrete and continuous invariants used in the harmonic analysis, i.e., two characteristic masses in the relativistic case. ‘Canonical’ field theories with the familiar divergencies are inappropriate realizations of the real 4-dimensional causal manifoldD(2). Faithful timespace realizations do not lead to divergencies. In general they are reducible, but nondecomposable—in addition to representations with eigenvectors (states, particle), they incorporate principal vectors without a particle (eigenvector) basis as exemplified by the Coulomb field. In theorthogonal andunitary groupsO(N ₊,N ₋), respectively, thepositive orthogonal and unitary ones areO(N) andU(N), respectively.     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Let M be a smooth compact manifold of dimension at least 2 and Diff ^r (M) be the space of C ^r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diff ^r (M) the sequence P _n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diff ^r (M) such for a Baire generic diffeomorphism f∈N the number of periodic points P _n f grows with a period n faster than any following sequence of numbers {a _n } _{n ∈ Z +} along a subsequence, i.e. P _n (f)>a _ni for some n _i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented. Received: 27 January 1999 / Accepted: 23 November 1999     

Passenger's fluctuation and chaos on ferryboats

We study the fluctuation of shipping passengers on a few ferryboats which shuttle between an origin and a destination repeatedly. We present the dynamical model for the ferryboats. The model is described in terms of nonlinear maps defined from the vectors T_i(n) and W_i(n), i=1, 2, …, N for N ferryboats where T_i(n) is the arrival time of ferryboat i at the origin on trip n and W_i(n) the number of passengers waiting at the origin on trip n. We clarify the fluctuations of shipping passengers and tour time for the ferry schedule. It is found that the dynamical transitions among the regular, periodic, and chaotic motions occur by varying the ferry's capacity F_max, headway T_min, and loading parameter ΓΠ. Even if the second ferryboat follows the leader (first ferryboat) keeping the constant headway, the passengers shipping on the second fluctuate highly when the parameters take specific values.     

厄米多项式算符的新恒等式及其在量子压缩中的应用

通过发现有关厄米多项式算符H_n(X)的恒等式, 并结合有序算符内的积分技术, 得到了一些关于量子压缩的算符新恒等式, 这对于研究压缩粒子数态波函数十分有用.     

The W N Minimal Model Classification

We first rigourously establish, for any N ≥ 2, that the toroidal modular invariant partition functions for the (not necessarily unitary) W _N (p, q) minimal models biject onto a well-defined subset of those of the SU(N) × SU(N) Wess-Zumino-Witten theories at level (p − N, q − N). This permits considerable simplifications to the proof of the Cappelli-Itzykson-Zuber classification of Virasoro minimal models. More important, we obtain from this the complete classification of all modular invariants for the W ₃(p, q) minimal models. All should be realised by rational conformal field theories. Previously, only those for the unitary models, i.e. W ₃(p, p + 1), were classified. For all N our correspondence yields for free an extensive list of W _N (p, q) modular invariants. The W ₃ modular invariants, like the Virasoro minimal models, all factorise into SU(3) modular invariants, but this fails in general for larger N. We also classify the SU(3) × SU(3) modular invariants, and find there a new infinite series of exceptionals.     

三明治结构Tan(B3N3H6)n+1 团簇的磁性和量子输运性质

利用密度泛函理论和非平衡格林函数方法, 本文对小尺寸团簇Ta_n(B₃N₃H₆)_n+1 (n ≤ 4)的磁性和量子输运性质进行了系统的研究. 计算结果表明, 此类体系采用三明治结构作为其基态并且具有较高的稳定性. 体系的磁矩随团簇尺寸的增大而线性增大. 当把Ta_n(B₃N₃H₆)_n+1团簇耦合到Au电极上时, 形成的Au-Ta_n(B₃N₃H₆)_n+1-Au体系在有限偏压下展示出了较强的自旋过滤能力, 因而可以被看做是一类新型的低维自旋过滤器.     

The quantum Fokker-Planck equation for certain boson systems

Quantum relations between a class of boson Langevin equations and the associated Fokker-Planck equations are derived. The Fokker-Planck equations for the Wigner distribution Φ_sym related with symmetric ordering of the boson operators, the distribution Φ_A related with antinormal ordering, and the distribution Φ_N related with normal ordering (P-representation) are given.