Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants. Received: 26 May 1999 / Accepted: 9 November 1999     

Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the Stückelberg–Bogoliubov–Epstein–Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, K?hler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space-times, we construct time-ordered operator-valued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of time-ordered products to all points of space-time. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space-times. Finite renormalizations are deferred to a subsequent paper. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surface. Received: 31 March 1999 / Accepted: 10 June 1999     

Scattering Theory for Quantum Fields¶with Indefinite Metric

In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag–Ruelle theory do not carry over to the case of indefinite metric [4], we propose an axiomatic framework for the construction of in- and out-states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out-fields, called the “form factor functional”, which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework. Received: 13 September 1999/ Accepted: 1 August 2000     

Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M ₂(ℂ)=ℂℤ₂·ℂℤ₂. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct q-monopoles on all the Podleś quantum spheres S ² _q,s . Received: 25 September 1998 / Accepted: 23 February 2000     

Quantum Teleportation with Entangled States¶Given by Beam Splittings

Quantum teleportation is rigorously demonstrated with coherent entangled states given by beam splittings. The mathematical scheme of beam splitting has been used to study quantum communication [2] and quantum stochastic [8]. We discuss the teleportation process by means of coherent states in this scheme for the following two cases: (1) Delete the vacuum part from coherent states, whose compensation provides us a perfect teleportation from Alice to Bob. (2) Use fully realistic (physical) coherent states, which gives a non-perfect teleportation but shows that it is exact when the average energy (density) of the coherent vectors goes to infinity. We show that our quantum teleportation scheme with coherent entangled state is more stable than that with the EPR pairs which was previously discussed. Received: 21 January 2000 / Accepted: 23 April 2001     

Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional. Received: 26 November 1998 / Accepted: 2 December 1998     

Free Massive Fermions¶Inside the Quantum Discrete sine-Gordon Model

We extend the notion of space shifts introduced in [FV3] for certain quantum light cone lattice equations of sine-Gordon type at root of unity (e.g. [FV1,FV2,BKP,BBR]). As a result, we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation, which is obtained by exponentiating these roots, is exactly the evolution equation for the?“classical background” as described in [BBR]. As an application for the introduced constructions, we derive a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice, as a special case of the lattice Thirring model constructed in [DV]. Received: 2 December 1996 / Accepted: 19 January 1999     

A Spin-Statistics Theorem for Quantum Fields¶on Curved Spacetime Manifolds¶in a Generally Covariant Framework

A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial. Received: 1 March 2001 / Accepted: 28 May 2001     

Evolution of a Model Quantum System¶ Under Time Periodic Forcing:¶Conditions for Complete Ionization

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as t→∞. This is irrespective of the magnitude or frequency (resonant or not) of η(t). There are however exceptional, very non-generic η(t), that do not lead to full ionization, which include rather simple explicit periodic functions. For these η(t) the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator. Received: 1 November 2000 / Accepted: 5 February 2001     

Quantum Vertex Representations via Finite Groups¶and the McKay Correspondence

We establish a q-analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ×ℂ^× and the symmetric groups. An important special case is obtained when Γ is a finite subgroup of SU ₂, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of ADE type. Received: 17 August 1999 / Accepted: 2 December 1999     

Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra

We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone. Received: 28 October 1998 / Accepted: 7 March 1999     

Twisted Index Theory on Good Orbifolds, II:¶Fractional Quantum Numbers

This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes–Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. The new phenomenon that we observe in our case is that the Connes–Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers. Received: 4 November 1999 / Accepted: 22 September 2000     

The Split Property and the Symmetry Breaking¶of the Quantum Spin Chain

We consider the relationship between the symmetry breaking and the split property of pure states of quantum spin chains. We obtain a representation theoretic condition implying that the half-sided uniform mixing condition leads to symmetry breaking of translationally invariant pure states. This is a mathematical generalization of Dichotomy previously found by I. Affleck and E. Lieb and M. Aizenman and B. Nachtergaele for ground states of a special class of Hamiltonians. Received: 1 February 1999 / Accepted: 5 December 2000     

A Change of Coordinates on the Large Phase Space¶of Quantum Cohomology

The Gromov–Witten invariants of a smooth, projective variety V, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin–Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes. Received: 2 August 1999 / Accepted: 30 September 2000     

Local Wick Polynomials and Time Ordered Products¶of Quantum Fields in Curved Spacetime

In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and K?hler, and by Brunetti and Fredenhagen, but they did not impose any “locality” or “covariance” condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an “extended Wick polynomial algebra”– large enough to contain the Wick polynomials and their time ordered products – by generalizing a construction of Dütsch and Fredenhagen to curved spacetime. We then define the notion of a local, covariant quantum field, and seek a definition of local Wick polynomials and their time ordered products as local, covariant quantum fields. We introduce a new notion of the scaling behavior of a local, covariant quantum field, and impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements – such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation. Received: 27 March 2001 / Accepted: 6 June 2001     

Multiparty Quantum Secret Sharing of Quantum States with Quantum Registers

A quantum secret sharing scheme is proposed by making use of quantum registers. In the proposed scheme, secret message state is encoded into multipartite entangled states. Several identical multi-particle entanglement states are generated and each particle of the entanglement state is filled in different quantum registers which act as shares of the secret message. Two modes, i.e. the detecting mode and the message mode, are employed so that the eavesdropping can be detected easily and the secret message may be recovered. The security analysis shows that the proposed scheme is secure against eavesdropping of eavesdropper and cheating of participants.     

Quantum Central Processing Unit and Quantum Algorithm

Based on a scalable and universal quantum network,quantum central processing unit,proposed in our previous paper [Chin.Phys.Lett.18(2001) 166],the whole quantum network for the known quantum algorithms,including quantum Fouries transformation,Shor‘s algorithm and Grover‘s algorithm,is obtained in a unified way.     

Quantum Gravity on a Quantum Computer?

EPR-type measurements on spatially separated entangled spin qubits allow one, in principle, to detect curvature. Also the entanglement of the vacuum state is affected by curvature. Here, we ask if the curvature of spacetime can be expressed entirely in terms of the spatial entanglement structure of the vacuum. This would open up the prospect that quantum gravity could be simulated on a quantum computer and that quantum information techniques could be fully employed in the study of quantum gravity.     

Quantum Key Distribution Using High-Dimensional Quantum Error-Avoiding Code

Using high-dimensional quantum error-avoiding code, we present two new quantum key distribution protocols over a collective noisy channel, i.e. six-photon and five-photon quantum error-avoiding codes. Compared with the previous protocols using four-photon and three-photon quantum error-avoiding code, the qubit efficiencies of the new protocols have increases of 16.67% and 5% respectively. In addition, the security of these protocols is analysed with a conclusion that the new protocols are much more secure than the four-photon and three-photon ones.