Zero-mass limit in perturbative quantum field theory

A newR-operation which satisfies Bogolubov-Parasiuk and Hepp recurrence and which is infrared and ultra violet convergent graph by graph, is defined in perturbative quantum field theory. This new subtraction scheme is used to achieve the zero-mass limit of a massive field theory.Chargé de recherche CNRS. — Work supported in part by Alexander Von Humboldt fellowship.Work supported in part by Deutsche Forschungsgemeinschaft. — Institut für theoretische Physik, Freie Universität Berlin, Berlin, Federal Republic of Germany, and present address: Radiological Research Laboratory, Columbia University, 630 West 168 St. New York, New York 10032, USA.     

Large-mass and high-temperature behaviour in perturbative quantum field theory

Bounds for large-mass behaviour in renormalized perturbation expansions at zero temperature, which were previously obtained by Manoukian and Caswell-Kennedy in momentum space, are rederived in the parametric representation. A very simple unified proof of the BPHZ theorem and the decoupling theorem is also given. A new technique for asymptotic analysis, based on a generalized Kontorovich-Lebedev integral transform, is introduced. This method is applied to find the leading high-temperature behaviour of perturbative field theories in the imaginary-time formalism. We prove that diagrams containing nonstatic modes, which at high temperature behave like particles with a large mass, are suppressed relative to purely static diagrams. This rigorously proves a limited form of dimensional reduction at infinite temperature.     

Estimating perturbative coefficients in quantum field theory and statistical physics

We present a method for estimating perturbative coefficients in quantum field theory and statistical physics. We are able to obtain reliable error bars for each estimate. The results are in excellent agreement with known exact calculation.     

Kolmogorov exponents for near-incompressible turbulence from perturbative quantum field theory

I derive the Kolmogorov exponents for the energy spectrum of freely-decaying, fully-developed, near-incompressible turbulence, using the methods of perturbative quantum field theory. In contrast to the approach involving Gaussian random forces, the leading-order result is determined uniquely through selfconsistency. At the first order in , I find a unique and nontrivial, IR (infrared) stable fixed-line. I show that the upper critical dimension of this system is 6, and E(k)k^–2 in 3 dimensions and E(k)k^–3 in 2 dimensions along this nontrivial fixed-line (at the one-loop level).     

Finite element approximation in quantum field theory

Lattices are conventionally used to regulate quantum field theory. However, lattices have a number of drawbacks, one of the most noteworthy being the awkwardness which arises in treating the first difference operator associated with the fermion kinetic energy. We propose here an alternative to the conventional lattice formulation of quantum field theories. This alternative is based on the method of finite elements, which has been a powerful tool in continuum mechanics and fluid dynamics for many years. We believe that this method may be advantageous in quantum field theory because it treats bosons and fermions on a nearly equal footing. This is possible because in the finite element method one does not introduce a finite difference approximation for derivatives. In this paper, we adapt the finite element technique to quantum field theory, and apply it to several elementary field theory models.     

A new scheme for strong-coupling calculations in supersymmetric field theory

We propose a resummation scheme for strong-coupling diagrams in two-dimensional lattice supersymmetric theories originally derived by Bender et al. To any given order, this reduces the number of diagrams drastically. We calculate the ground state energy density up to eighth order and obtain a modified strong-coupling series in terms of a new expansion parameter which seems to have a smoother continuum limit.     

Massive supersymmetric quantum gauge theory

We continue the study of the supersymmetric vector multiplet in a purely quantum framework. We obtain some new results which make the connection with the standard literature. First we construct the one‐particle physical Hilbert space taking into account the (quantum) gauge structure of the model. Then we impose the condition of positivity for the scalar product only on the physical Hilbert space. Finally we obtain a full supersymmetric coupling which is gauge invariant in the supersymmetric sense in the first order of perturbation theory. By integrating out the Grassmann variables we get an interacting Lagrangian for a massive Yang‐Mills theory related to ordinary gauge theory; however the number of ghost fields is doubled so we do not obtain the same ghost couplings as in the standard model Lagrangian.     

Random phase approximation and extensions applied to a bosonic field theory

An application of a self-consistent version of RPA to quantum field theory with broken symmetry is presented. Although our approach can be applied to any bosonic field theory, we specifically study the ϕ⁴ theory in 1 + 1 dimensions. We show that the standard RPA approach leads to an instability which can be removed when going to a superior version, i.e. the renormalized RPA. We present a method based on the so-called charging formula of the many-electron problem to calculate the correlation energy and the RPA effective potential. Received: 18 February 2002 / Accepted: 8 May 2002     

Dirac operators in Boson-Fermion Fock spaces and supersymmetric quantum field theory

Infinite dimensional analysis is developed on an abstract Boson-Fermion Fock space. A general class of Dirac operators acting there is introduced and properties of them are investigated. An index theorem for the Dirac operators is established in terms of a path integral on a loop space. It is shown that the abstract formalism presented here gives a mathematical unification for some models of supersymmetric quantum field theory.     

Theorems on estimating perturbative coefficients in quantum field theory and statistical physics

We present rigorous proofs for several theorems on using Padé approximants to estimate coefficients in perturbative quantum field theory and statistical physics. As a result, we find new trigonometric and other identities where the estimates based on this approach are exact. We discuss hypergeometric functions, as well as series from both perturbative quantum field theory and statistical physics.     

A new lattice approximation forthe Høegh-Krohn quantum field model

The lattice approximation for the exponential interaction model (Høbegh-Krohn quantum field model) with different lattice cutoffs a, a′ in the free and interacting parts is discussed. It is shown that the continuum limit exists under certain conditions on the dependence a′(a).     

Central charges and Lorentz and internal symmetries in massive supersymmetric quantum field theory

The properties of central charges in the framework of the massive supersymmetric quantum field theory related to internal symmetries, Lorentz covariance and locality of the fields are investigated. It is shown that in the presence of z central charges the largest semisimple part of the internal symmetry algebra is a direct sum of z compact symplectic group algebras and possibly an additional term representing the unimodular unitary group algebra. Next it is shown that 4j ? N + K, where j is the highest spin value of the underlying fields, N is the number of spinorial charges and K the number of these spinorial charges which are not linked to other spinorial charges by a central charge. It is further demonstrated that, in general, the central charge can not be redefined in such a way that it is at the same time real and preserves the locality principle. The discussion of the obtained results concludes the paper.     

A vanishing theorem for supersymmetric quantum field theory and finite size effects in multiphase cluster expansions

We apply cluster expansion methods to to theN=2 Wess-Zumino models in finite volume, in two space-time dimensions. We show that in the region of convergence of the cluster expansion, a vanishing theorem holds for the supercharge of the theory; that is, the dimension of the kernel of the Hamiltonian is equal to the index of the supercharge.Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 90-07206Supported in part by National Science Foundation Mathematical Sciences Postodoctoral Research Fellowship DmS 88-07291     

Non-local charges: A new concept in quantum field theory

Non-local charges are defined as a natural generalization of standard charges in relativistic quantum field theory. The general form of a non-local charge in terms of asymptotic fields is given and preliminary results on the restrictions imposed on non-local charges in interacting theories are reported [8].On leave of absence from the II Institute of Theoretical Physics, University of Hamburg, F.R.G.     

Stochastic quantization of supersymmetric field theory

The Parisi-Wu stochastic quantization method is applied to supersymmetric field theory. The Langevin equation, which reproduces the Green functions of euclidean field theory, is written in terms of superfields. Supersymmetric U(1) theory under gauge fixing and the large N reduction in chiral SU(N) theory are discussed. Regularization based on the stochastic method is studied also.     

A perturbative approach to fuzzifying field theories

We propose a procedure for computing noncommutative corrections to the metric tensor, and apply it to scalar field theory written on coordinate patches of smooth manifolds. The procedure involves finding maps to the noncommutative plane where differentiation and integration are easily defined, and introducing a star product. There are star product independent, as well as dependent, corrections. Applying the procedure for two different star products, we find the lowest order fuzzy corrections to scalar field theory on a sphere which is stereographically projected to the plane.