An Indefinite Metric Model for Interacting Quantum Fields on Globally Hyperbolic Space-Times

We consider a relativistic ansatz for the vacuum expectation values (VEVs) of a quantum field on a globally hyperbolic space-time which is motivated by certain Euclidean field theories. The Yang-Feldman asymptotic condition w.r.t. an in-field in a quasi-free representation of the canonic commutation relations (CCR) leads to a solution of this ansatz for the VEVs. A GNS-like construction on a non-degenerate inner product space then gives local, covariant quantum fields with indefinite metric on a globally hyperbolic space-time. The non-trivial scattering behavior of quantum fields is analyzed by construction of the out-fields and calculation of the scattering matrix. A new combined effect of non-trivial quantum scattering and non-stationary gravitational forces is described for this model, as quasi-free in- fields are scattered to out-fields which form a non-quasi-free representations of the CCR. The asymptotic condition, on which the construction is based, is verified for the concrete example of de Sitter space-time. Communicated by Klaus Fredenhagen submitted 03/05/02, accepted: 18/07/03     

Tree Quantum Field Theory

We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e., it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define non-perturbatively differential renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse–Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a non-perturbative definition of field theory in non-integer dimension.     

Quantum Field Theory on Affine Bundles

We develop a general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes. All concepts and results are formulated using the language of category theory, which allows us to prove that these models satisfy the principle of general local covariance. Our analysis is a preparatory step towards a full-fledged quantization scheme for the Maxwell field, which emphasises the affine bundle structure of the bundle of principal U(1)-connections. As a by-product, our construction provides a new class of exactly tractable locally covariant quantum field theories, which are a mild generalization of the linear ones. We also show the existence of a functorial assignment of linear quantum field theories to affine ones. The identification of suitable algebra homomorphisms enables us to induce whole families of physical states (satisfying the microlocal spectrum condition) for affine quantum field theories by pulling back quasi-free Hadamard states of the underlying linear theories.     

Conformal Quantum Field Theory and Subfactors

We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.     

Schwinger Functions in Noncommutative Quantum Field Theory

It is shown that the n-point functions of scalar massive free fields on the noncommutative Minkowski space are distributions which are boundary values of analytic functions. Contrary to what one might expect, this construction does not provide a connection to the popular traditional Euclidean approach to noncommutative field theory (unless the time variable is assumed to commute). Instead, one finds Schwinger functions with twistings involving only momenta that are on the mass-shell. This explains why renormalization in the traditional Euclidean noncommutative framework crudely differs from renormalization in the Minkowskian regime.     

拓扑量子场理论与相交指标

该文构造了一新的上同调型拓扑量子场理论并证明了其配分函数是相交指标 (crossingindex)     

An Indefinite Kahler Metric on the Space of Oriented Lines

The total space of the tangent bundle of a Kähler manifoldadmits a canonical Kähler structure. Parallel translationidentifies the space T of oriented affine lines in R³ with thetangent bundle of S². Thus the round metric on S² induces aKähler structure on T which turns out to have a metricof neutral signature. It is shown that the identity componentof the isometry group of this metric is isomorphic to the identitycomponent of the isometry group of the Euclidean metric on R³. The geodesics of this metric are either planes or helicoidsin R³. The signature of the metric induced on a surface inT is determined by the degree of twisting of the associatedline congruence in R³, and it is shown that, for Lagrangian,the metric is either Lorentz or totally null. For such surfacesit is proved that the Keller–Maslov index counts the numberof isolated complex points of J inside a closed curve on .     

Casimir Effect and Quantum Field Theory in Dielectrics

We apply the path-integral method in the coordinate space to the Casimir effect. We consider several examples: the Casimir energy of a dilute dielectric ball with dispersion, the Casimir energy of a polarized particle near a dielectric ball, and the Casimir energy of a polarized particle inside a perfectly conducting wedge-shaped cavity. The renormalization group equation for the ⁴ model is obtained in the coordinate space by a new method that emphasizes the relation between the background field method and the Casimir energy.     

New Perturbation Theory for Quantum Field Theory: Convergent Series Instead of Asymptotic Expansions

We present a new approach to perturbation theory for quantum field theory based on convergent series instead of asymptotic expansions. This approach could be considered as the next step after traditional perturbation theory calculations, which allows more comprehensive use of previously obtained information in finding numerical values with greater accuracy.     

Massless Relativistic Wave Equations and Quantum Field Theory

We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincaré group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fiber degrees of freedom of the covariant representation. They are characterized by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between (the two-fold cover of the 2-dimensional Euclidean group) and We conclude with a brief comparison between the free nets constructed in Section 3 and a recent alternative construction that uses the notion of modular localization. Communicated by Klaus FredenhagenSubmitted 16/03/03, accepted 11/12/03     

Dimensional Regularization and Renormalization of Non-Commutative Quantum Field Theory

Using the recently introduced parametric representation of noncommutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized model on the Moyal space. Submitted: June 8, 2007. Accepted: December 11, 2007.     

About a Quantum Field Theory for 3D Gravity

We outline some aspects of a dilogarithmic quantum field theory (DQFT) for a 2+1 bordism category based on 3-manifolds equipped with principal flat bundles. We give some geometric evidence that it could be pertinent to 3D gravity, and we describe the building blocks of DQFT: the matrix dilogarithms.Lecture held in the Seminario Matematico e Fisico on November 24, 2003Received: April, 2004     

A Note on Quantum Field Theory and P-brans innDimensions

Euclidean n−1 dimensional spheres are postulated as P-brans sweeping an n dimensional world sheet. It is conjectured that the resulting space, S ^(∞) , is a hierarchical one with an effective topological dimension equal to theexpectationvalue of n, where 0⩽n⩽∞. Numerical estimation shows that 〈n〉=〈dim_T S ^(∞) 〉 is very close to the topological dimensionof actualspacetime.     

The One-dimensional Fractional Supersymmetric Quantum Mechanical Operator of Momentum

In the case of the quantum generalization of stable Lévy processes, expressions for the Hermitian operator of momentum and its eigenfunctions are proposed. The normalization constant has been determined and its relation to the translation operator is shown. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer dimensionality α. The simplest nonlocal superalgebra is introduced.      

格上度量理论

通过对格上两种度量理论,即M．A．Erceg意义下的度量理论与史福贵意义下的度量理论的介绍与评述,展现出格上拓扑学中有关度量理论发展的概况,从这里可以看到,这一般拓扑学听度量理论比较,格上度量理论的研究有基本思想和研究方法上都有其独到之处。     

L-拓扑中的度量理论

对L-拓扑中已有度量理论的研究进展作一个简要的介绍和评述,力图使读者了解L-拓扑中度量理论的发展概况和发展方向。     

Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories

We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.     

Quantum Field Theory in Static External Potentials and Hadamard States

We prove that the ground state for the Dirac equation on Minkowski space in static, smooth external potentials satisfies the Hadamard condition. We show that it follows from a condition on the support of the Fourier transform of the corresponding positive frequency solution. Using a Klein space formalism, we establish an analogous result in the Klein–Gordon case for a wide class of smooth potentials. Finally, we investigate overcritical potentials, i.e. which admit no ground states. It turns out, that numerous Hadamard states can be constructed by mimicking the construction of ground states, but this leads to a naturally distinguished one only under more restrictive assumptions on the potentials.