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     

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     

A Discrete Density Matrix Theory for Atoms¶in Strong Magnetic Fields

This paper concerns the asymptotic ground state properties of heavy atoms in strong, homogeneous magnetic fields. In the limit when the nuclear charge Z tends to ∞ with the magnetic field B satisfying B>> Z ^4/3 all the electrons are confined to the lowest Landau band. We consider here an energy functional, whose variable is a sequence of one-dimensional density matrices corresponding to different angular momentum functions in the lowest Landau band. We study this functional in detail and derive various interesting properties, which are compared with the density matrix (DM) theory introduced by Lieb, Solovej and Yngvason. In contrast to the DM theory the variable perpendicular to the field is replaced by the discrete angular momentum quantum numbers. Hence we call the new functional a discrete density matrix (DDM) functional. We relate this DDM theory to the lowest Landau band quantum mechanics and show that it reproduces correctly the ground state energy apart from errors due to the indirect part of the Coulomb interaction energy. Received: 20 October 2000 / Accepted: 3 November 2000     

Møller’s Energy-Momentum Complex for a Spacetime Geometry on a Noncommutative Curved D3-Brane

Møller’s energy-momentum complex is employed in order to determine the energy and momentum distributions for a spacetime described by a “generalized Schwarzschild” geometry in (3+1)-dimensions on a noncommutative curved D3-brane in an effective, open bosonic string theory. The geometry considered is obtained by an effective theory of gravity coupled with a nonlinear electromagnetic field and depends only on the generalized (effective) mass and charge which incorporate corrections of first order in the noncommutativity parameter.     

Quantum Inequality Bounds for Free Rarita-Schwinger Field in Flat Spacetime

Although quantum field theory allows the local energy density negative, it also places severe restrictions on the negative energy. One of the restrictions is the quantum energy inequality （QEI）, in which the energy density is averaged over time, or space, or over space and time. By now temporal QEIs have been established for various quantum fields, but less work has been done for the spacetime quantum energy inequality. In this paper we deal with the free Rarita-Schwinger field and present a quantum inequality bound on the energy density averaged over space and time. Comparison with the QEI for the Rarita-Schwinger field shows that the lower bound is the same with the QEI. At the same time, we find the quantum inequality for the Rarita-Schwinger field is weaker than those for the scalar and Dirac fields. This fact gives further support to the conjecture that the more freedom the field has, the more easily the field displays negative energy density and the weaker the quantum inequality becomes.     

A Riemann–Roch Theorem¶for One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C ₀(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L ^∞(Σ)⋊Γ. Received: 1 February 2000 / Accepted: 3 December 2000     

A Perron–Frobenius Type of Theorem for Quantum Operations

We define a special class of quantum operations we call Markovian and show that it has the same spectral properties as a corresponding Markov chain. We then consider a convex combination of a quantum operation and a Markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the Perron–Frobenius theorem if its Markovian part does. Moreover, under a compatibility condition of the two operations, we show that its limiting distribution is the same as the corresponding Markov chain. We apply our general results to partially decoherent quantum random walks with decoherence strength \(0 \le p \le 1\). We obtain a quantum ergodic theorem for partially decoherent processes. We show that for \(0 < p \le 1\), the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk.     

The Reeh–Schlieder Property for Quantum Fields on Stationary Spacetimes

We show that as soon as a linear quantum field on a stationary spacetime satisfies a certain type of hyperbolic equation, the (quasifree) ground- and KMS-states with respect to the canonical time flow have the Reeh–Schlieder property. We also obtain an analog of Borchers' timelike tube theorem. The class of fields we consider contains the Dirac field, the Klein–Gordon field and the Proca field. Received: 1 March 2000 / Accepted: 30 May 2000     

A Geometric Interpretation of the χy Genus¶on Hyper-Kähler Manifolds

The group SL(2) acts on the space of cohomology groups of any hyper-K?hler manifold X. The χ _y genus of a hyper-K?hler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the χ _y genus. Received: 3 December 1999 / Accepted: 30 January 2000     

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     

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.     

A Quantum Theory of Electron Emission from a Metal–Dielectric Structure in High Electric Fields

Technical Physics - A generalized formula for electron emission current as a function of temperature, field, and electron work function in a metal–dielectric system has been derived with...     

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     

A Gibbons–Penrose Inequality for Surfaces in Schwarzschild Spacetime

We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.     

Continuous Quantum Nondemolition Measurements of a Particle in Electromagnetic and Gravitational Fields

The detection of a particle in electromagnetic plus gravitational fields is investigated. We obtain a set of quantum nondemolition variables. The continuous measurements of these nondemolition parameters are analyzed in the framework of restricted path integral formalism. We manipulate the corresponding propagators, and deduce the probabilities associated with the possible measurement outputs.     

Quantum Inequality for Negative Energy Density States of Massive Dirac Field in Four-Dimensional Spacetime

Negative energy density and the quantum inequality are examined for the Dirac field. A proof is given of the quantum inequality for negative energy densities in the massive Dirac field produced by the superposition of two single particle electron states.     

Egorov’s Theorem for Transversally Elliptic Operators on Foliated Manifolds and Noncommutative Geodesic Flow

The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations. Mathematics Subject Classifications (2000) 58J40, 58J42, 58B34.     

Effects of Crossed Electric and Magnetic Fields on Shallow Donor Impurity Binding Energy in a Parabolic Quantum Well

We have calculated variationally the ground state binding energy of a hydrogenic donor impurity in a parabolic quantum well in the presence of crossed electric and magnetic fields. These homogeneous crossed fields are such that the magnetic field is parallel to the heterostructure layers and the electric field is applied perpendicular to the magnetic field. The dependence of the donor impurity binding energy to the well width and the strength of the electric and magnetic fields are discussed. We hope that the obtained results will provide important improvements in device applications, especially for a suitable choice of both fields in the narrow well widths.     

Off—Center D—Centers in a Quantum Dot in the Presence of a Perpendicular Magnetic Fields

We investigate the effect of the position of the donor in quantum dots on the energy spectrum in the presence of a perpendicular magnetic field by using the method of few-body physics,As a function of the magnetic field,we find,when D^- centers are placed sufficiently off-center,discontinuous ground-state transitions which are similar to those found in many-electron parabolic quantum dots.Series of magic numbers of angular momentum which minimize the ground-state electron-electron interaction energy have been discovered.The dependence of the binding energy of the gound-state of the D^- center on the dot radius for a few values of the magnetic field strength is obtained and compared with other results.