Ultraviolet problems in field theory and multiscale expansions

An introductory survey is given of ultraviolet problems in Euclidean quantum field theory which are heuristically interpreted either with the aid of the classical renormalization theory or with the aid of Wilson's renormalization group strategy. A unification of each of these approaches with the method of multiscale cluster expansions is necessary for strict proofs.Translated from Itogi Nauki i Tekhniki, Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 24, pp. 111–189, 1986.     

Combinatorics,Bethe Ansatz,and representations of the symmetric group

Techniques developed in the realms of the quantum method of the inverse problem are used to analyze combinatorial problems (Young diagrams and rigged configurations).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 50–64, 1986.     

Hamiltonian interpretation of the Volterra model

We consider a differential-difference Volterra system in the new Hamiltonian formulation of L. A. Takhtadzhyan and L. D. Faddeev. The connection with the quantum method of the inverse problem and conformally invariant two-dimensional field theory is discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 17–25, 1986.     

Poisson structure of a periodic classical XYZ -chain

Classical variables ( -variables) are constructed for a periodic classical XYZ -chain (of the discrete Landau-Lifshitz equation). The connection of the quantum method of the inverse problem and the theory of finite-zone integration is discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 154–180, 1986.     

Null-range potentials and M. G. Krein's formula for generalized resolvents

For nonadditive finite-dimensional perturbations of a Hermitian operator, with the aid of M. G. Krein's formula for generalized resolvents, one derives a representation of all scattering suboperators, parametrized by Hermitian matrices. On the basis of this representation, one obtains explicit expressions for a series of new, exactly solvable quantum models with null-range potential. One establishes a connection between the obtained parametrizations with phenomenological S matrices and the corresponding Wigner R functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 7–23, 1986.     

Quantum B-algebras

The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.     

Quantum B-splines

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q (quantum)-blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden’s identity.     

A representation theorem for quantum systems

In this note representations of quantum systems are investigated. We propose a unital bipolar theorem for unital quantum cones, which plays a key role in proving a representation theorem for quantum systems. It turns out that each quantum system is identified with a certain quantum L^∞-system up to a quantum order isomorphism.     

Optimal multiple quantum statistical hypothesis testing

This paper is concerned with the problem of optimal M-alternative determination of quantum statistical states. A review of newest achievement of solving this problem is given. A notion of an effective decision Hilbert space is introduced and necessary and sufficient condkions for optimality of multiple quantum hypothesis testing in this space are formulated. The general solution is found for the case of a two-dimensional decision space. Another problem solved is that of discrimination of quantum pure non-orthogonal states. The result is represented in explicit analytical form for an "equidiagonal" case, which is quite general. In particular, we find explicit solutions of optimal discrimination problem of homogeneous and equiangle sets of pure states. These results are used for the M-ary detection problem in solving for the quantum coherent non-orthogonal signals. It is proved that the simplex signals are optimal elso in quantum case. The optimal estimatesof phaseandamplitude of quantum coherent signals are found. For decision operators a notion of IT-representation is introduced to get a general quasi-classical (optimal in quasi-classical limit) M-ary detection procedure of stochastic fields and particles, which submits to Bose-Einstein statistics. An optimal solution of problem of non-coherent detection of quantum stochastic (including optical) signals are found in the extreme quantum limit (weaknoise and signals with unknown phase).     

Separability for Positive Operators on Tensor Product of Hilbert Spaces

The separability and the entanglement(that is, inseparability) of the composite quantum states play important roles in quantum information theory. Mathematically, a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space. In this paper,in more general frame, the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces. However, not like the quantum state case, there are different kinds of separability for positive operators with different operator topologies. Four types of such separability are discussed; several criteria such as the finite rank entanglement witness criterion,the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established; some methods to construct separable positive operators by operator matrices are provided. These may also make us to understand the separability and entanglement of quantum states better, and may be applied to find new separable quantum states.     

Quantum homogeneous spaces,duality and quantum 2-spheres

For a quantum groupG the notion of quantum homogeneousG-space is defined. Two methods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one is based upon the notion of infinitesimal invariance with respect to certain two-sided coideals in the Hopf algebra dual to the Hopf algebra ofG. These methods are applied to the quantum group SU(2). As two-sided coideals we take the subspaces spanned by twisted primitive elements in the sl(2) quantized universal enveloping algebra. A one-parameter series of mutually non-isomorphic quantum 2-spheres is obtained, together with the spectral decomposition of the corresponding right regular representation of quantum SU(2). The link with the quantum spheres defined by Podle is established.     

Crossed modules,quantum braided groups,and ribbon structures

Previous results about crossed modules over a braided Hopf algebra are applied to the study of quantum groups in braided categories. Cross products for braided Hopf algebras and quantum braided groups are constructed. Criteria for when a braided Hopf algebra or a quantum group is a cross product are obtained. A generalization of Majid's transmutation procedure for quantum braided groups is considered. A ribbon structure on a quantum braided group and its compatibility with cross product and transmutation are studied.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 368–387, June, 1995.     

Pairings and actions for dynamical quantum groups

Dynamical quantum groups constructed from a FRST-construction using a solution of the quantum dynamical Yang-Baxter equation are equipped with a natural pairing. The interplay of the pairing with *-structures, corepresentations and dynamical representations is studied, and natural left and right actions are introduced. Explicit details for the elliptic U(2) dynamical quantum group are given, and the pairing is calculated explicitly in terms of elliptic hypergeometric functions. Dynamical analogues of spherical and singular vectors for corepresentations are introduced.     

Relative entropy between quantum ensembles

Relative entropy between two quantum states, which quantifies to what extent the quantum states can be distinguished via whatever methods allowed by quantum mechanics, is a central and fundamental quantity in quantum information theory. However, in both theoretical analysis (such as selective measurements) and practical situations (such as random experiments), one is often encountered with quantum ensembles, which are families of quantum states with certain prior probability distributions. How can we quantify the quantumness and distinguishability of quantum ensembles? In this paper, by use of a probabilistic coupling technique, we propose a notion of relative entropy between quantum ensembles, which is a natural generalization of the relative entropy between quantum states. This generalization enjoys most of the basic and important properties of the original relative entropy. As an application, we use the notion of relative entropy between quantum ensembles to define a measure for quantumness of quantum ensembles. This quantity may be useful in quantum cryptography since in certain circumstances it is desirable to encode messages in quantum ensembles which are the most quantum, thus the most sensitive to eavesdropping. By use of this measure of quantumness, we demonstrate that a set consisting of two pure states is the most quantum when the states are 45° apart.     

Quantum stochastic convolution cocycles III

Every Markov-regular quantum Lévy process on a multiplier C ^*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C ^*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Lévy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C ^*-bialgebra, to locally compact quantum groups and multiplier C ^*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.     

Quantum Optimal Control Problems with a Sparsity Cost Functional

In this article, the investigation of a class of quantum optimal control problems with L¹ sparsity cost functionals is presented. The focus is on quantum systems modeled by Schrödinger-type equations with a bilinear control structure as it appears in many applications in nuclear magnetic resonance spectroscopy, quantum imaging, quantum computing, and in chemical and photochemical processes. In these problems, the choice of L¹ control spaces promotes sparse optimal control functions that are conveniently produced by laboratory pulse shapers. The characterization of L¹ quantum optimal controls and an efficient numerical semi-smooth Newton solution procedure are discussed.     

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

Algebraic quantum hypergroups

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman.We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.     

Centers,cocenters and simple quantum groups

We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of Banica and Vergnioux. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very “far from classical”, are thus seen to be simple. Examples include quotients of free unitary groups by their centers, recovering previous work, as well as quotients of quantum reflection groups by their centers.     

(Co)bordism groups in quantum super PDE's. III: Quantum super Yang–Mills equations

In this third part of a series of three papers devoted to the study of geometry of quantum super PDE's [A. Prástaro, (Co)bordism groups in quantum super PDE's. I: quantum supermanifolds, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.007; (Co)bordism groups in quantum super PDE's. II: quantum super PDE's, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.008], we apply our theory, developed in the first two parts, to quantum super Yang–Mills equations and quantum supergravity equations. For such equations we determine their integral bordism groups, and by using some surgery techniques, we obtain theorems of existence of global solutions, also with nontrivial topology, for Cauchy problems and boundary value problems. Quantum tunnelling effects are described in this context. Furthermore, for quantum supergravity equations we prove existence of solutions of the type quantum black holes evaporation processes just by using an extension to quantum super PDEs of our theory of integral (co)bordism groups. Our proof is constructive, i.e., we give geometric methods to build such solutions. In particular a criterion to recognize quantum global (smooth) solutions with mass-gap, for the quantum super Yang–Mills equation, is given. Finally it is proved that quantum super PDE's contain also solutions that come from Dirac quantization of their superclassical counterparts. This proves that quantum super PDE's are (nonlinear) generalizations of Dirac quantized superclassical PDE's. Applications of this result to free quantum super Yang–Mills equations are given.