Remarks on a quantum state extension problem

The problem is considered of finding, for a given pair of states on C ^*-algebras A ₁ A ₂ and A ₂ A ₃, a joint extension to A ₁ A ₂ A ₃. The fact that, in contrast to classical probability, such an extension may fail to exist, is related to the fact that different convex decompositions of the same quantum state need not have a common refinement. Improved necessary criteria for extensibility in terms of Bell's inequalities are derived, and are compared to the necessary and sufficient criteria, as well as to entropic bounds in the simplest case.     

Differential calculi on quantum vector spaces with Hecke-type relations

From a vector spaceV equipped with a Yang-Baxter operatorR one may form the r-symmetric algebraS _R V=TV/v w–R(v w), which is a quantum vector space in the sense of Manin, and the associated quantum matrix algebraM _R V=T(End(V))/f g–R(f g)R ^-1. In the case whenR satisfies a Hecke-type identityR ²=(1–q)R+q, we construct a differential calculus _R V forS _R V which agrees with that constructed by Pusz, Woronowicz, Wess, and Zumino whenR is essentially theR-matrix of GL _q (n). Elements of _R V may be regarded as differential forms on the quantum vector spaceS _R V. We show that _R V isM _R V-covariant in the sense that there is a coaction _*: _R V M _R V _R V with _*d=(1 d)_* extending the natural coaction :S _R V M _R V S _R V.     

Analogs of q-Serre relations in the Yang-Baxter algebras

Yang-Baxter bialgebras, as previously introduced by the authors, are shown to arise from a double crossproduct construction applied to the bialgebra R ₁₂ T ₁ T ₂ = T ₂ T ₁ R ₁₂, E ₁ T ₂ = T ₂ E ₁ R ₁₂, (T)=TT, (E)=ET + 1E and its skew dual, with R being a numerical matrix solution of the Yang-Baxter equation. It is further shown that a set of relations generalizing q-Serre ones in the Drinfeld-Jimbo algebras U _q(g) can be naturally imposed on Yang-Baxter algebras from the requirement of non-degeneracy of the pairing.     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

On infinite direct products of Lorentz transformations

An infinite direct product _i =1 U _i (a) of continuous unitary representations of SL(2,C) in Hilbert spaces _i is continuous only on certain incomplete direct product subspaces of _i =1 _i . If no representations of the complementary series occur, then each of these subspaces contains a product vector almost all factors of which are SL(2, C)-invariant.     

Representation for matrix elements of compact Lie groups

A method for evaluation of matrix elements of unitary irreducible representations of compact Lie groups is formulated. matrix elements of some representations of the group SO(p+q) in the SO(p) SO(q) basis and of the group U(p) U(p) in the U(p) basis (diagonal imbedding) are given.     

An application of Bell's inequalities to a quantum state extension problem

Using techniques from the study of quantum violations of Bell's inequalities, we give examples of three C ^*-algebras A, B, C, and states ₁₂ on A B, and ₂₃ on B C, which agree on B, but do not have a common extension to A B C. This situation cannot occur in classical probability, i.e. for commutative algebras.     

High-spin states and band structures in the doubly odd <Superscript>136</Superscript>La nucleus

High-spin states of ¹³⁶La have been investigated with the reaction ¹³⁰Te( ¹¹B, 5n) at a beam energy of 60 MeV. The level scheme with three collective bands has been updated with spin up to 20 . The observed h_11/2 h_11/2 band shows -instability with increasing spin according to the TRS calculations. The band crossing and the signature splitting and inversion have been discussed. Other two collective bands based on 12^-and 16⁺ levels were proposed as oblate deformation with -60^°. They most probably originate from four- and six-quasiparticle configurations, that is, h_11/2 g_7/2h_11/2² and g_7/2 g_7/2²d_5/2h_11/2² respectively.     

Collective structures of the <Superscript>131</Superscript>Cs nucleus

The collective structures of ¹³¹Cs have been investigated by in-beam -ray spectroscopic techniques following the ¹²⁴Sn (^11B , 4n) reaction at a beam energy E_lab = 57MeV. The previously established rotational bands, built on g_7/2, d_5/2 and the unique-parity h_11/2 orbitals, have been extended and evolve into new bands involving rotationally aligned (h_11/2)² and (h_11/2)² quasiparticles. In addition, a new multiquasiparticle band based on the g_7/2 g_7/2 h_11/2 configuration has also been observed. Theoretical interpretations for the assigned configurations are discussed in the framework of Total Routhian Surface (TRS) and Tilted Axis Cranking (TAC) model calculations. TAC model calculations predict a decrease in the B(M1) values with increasing rotational frequency for the g_7/2/d_5/2 (h_11/2)² and h_11/2 (h_11/2)² bands, thus indicating a magnetic rotation character for these bands.     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

Low-energy elementary surface excitations of symmetric films of liquid4He atT=0 K

It is known that low-energy elementary excitations of symmetric films of liquid⁴He atT=0 K are characterized by a momentum q parallel to the surface and may be described by bound states. We have evaluated wave functions and energies of these states for both best short-ranged and optimal long-ranged correlations. Quantities of physical interest may be expressed in terms of these eigenstates and, in particular, for very small momenta (q<0.2 Å^–1) they are mainly determined by the contribution due to the lowest-lying one. We propose analytic expressions for the lowest-lying excitations and fluctuations in the long-wavelength limit. It is proved that in this limiting case, the excitation energy _LW(q) and the averaged static structure functionS _LW(q) should go linearly to zero asq0, whereas the averaged direct correlationX _LW ^Dg (q) should diverge at the origin as 1/q. It is shown that numerical solutions exhibit the expected long-wavelength behavior provided that optimal correlations are used. All these results are displayed in a series of figures and are discussed in detail.     

Solutions of the braid equation related to a Hopf algebra

Two solutionsT andT of the braid equation acting onA A (whereA is a Hopf algebra) are described. IfA is a cocommutative, thenT=. IfA is commutative, thenT= ( denotes the flip: (a b) =b a for anya,b A).Supported by a grant of the Ministry of Education of Poland.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Mass operator in theSU(3)-symmetric strong coupling theory with pseudoscalar mesons

In this paper theSU(3)-symmetric model of a static baryon octet source interacting with pseudoscalar meson octet fields by the coupling of Yukawa type is considered in the strong coupling limit. Using the result derived earlier that the isobaric states form the basis of the unitary irreducible representation of the dynamical group G=T ₂₄ [SU(3) SU(2)], the mass operator is specified by the kinetic part of the Hamiltonian as a particular element of the universal enveloping algebra of the symmetry groupSU(3) SU(2) acting in the space of isobaric states.     

Representations of central extensions of differentiably simple lie superalgebras

LetS be a simple Lie algebra of characteristic 0 and (n) be the Grassmann superalgebra inn indeterminates. We can form the Lie superalgebraS (n). The purpose of this paper is to classify all finite dimensional irreducible representations of all central extensions ofS (n). We will also give a character formula for these representations.     

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

The quark aspect of partons

It is shown in this article that the baryon nonet as well as the meson nonet can be represented as sets of three different triplets. The values of Q, Y, and 2T₃ are the same as the corresponding values of the three integral quark triplets of Nambu. It is proposed to describe the Nambu quarks not in terms of the usualSU(3) SU(3), but rather in terms ofSU (3) SU (3) SU (3) Like the baryons, such quarks will then be composites, made up of other quarks; the latter will also be made up of other quarks, and in this way we obtain a whole hierarchy of quarks. It is proposed to identify the totality of all of these states as partons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 8, pp. 105–109, August, 1975.     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

Produits tensoriels continus d'espaces et d'algèbres de Banach

The notion of tensor product of a family (A _i ) _i I of Banach algebras is generalized to the case whenI is a topological space; in this case A _i is generated by some elements x _i , the family (x _i ) being subjected to certain conditions: for instance the functioni x _i must be continuous. This notion is applied to Quantum Field Theory in the following sense: certain algebras of observables can be considered as continuous tensor products of simpler ones, namely of algebras of observables with one degree of freedom.