Framework Of Causal Perturbation Theory Research Articles

CONNECTION BETWEEN THE RENORMALIZATION GROUPS OF STÜCKELBERG–PETERMANN AND WILSON

The Stückelberg–Petermann renormalization group is the group of finite renormalizations of the S-matrix in the framework of causal perturbation theory. The renormalization group in the sense of Wilson relies usually on a functional integral formalism, it describes the dependence of the theory on a UV-cutoff Λ; a widespread procedure is to construct the theory by solving Polchinski's flow equation for the effective potential.To clarify the connection between these different approaches we proceed as follows: in the framework of causal perturbation theory we introduce an UV-cutoff Λ, define an effective potential VΛ, prove a pertinent flow equation and compare with the corresponding terms in the functional integral formalism. The flow of VΛ is a version of Wilson's renormalization group. The restriction of these operators to local interactions can be approximated by a subfamily of the Stückelberg–Petermann renormalization group.

Infrared Behavior of Massive Scalar Matter Coupled to Gravity

In the framework of causal perturbation theory we consider a massive scalar field coupled to gravity. In the field theoretic approach to quantum gravity we start with a massless second-rank tensor field. This tensor field is then quantized in a covariant way in Minkowski space. This article deals with the adiabatic limit for graviton radiative corrections in a scattering process of two massive scalar particles. We compute the differential cross section for Bremsstrahlung processes in which one of the outgoing particles emits a graviton of low energy, a so-called soft graviton. Since the emitted graviton will not be detected we must integrate over the energy of all soft gravitons. Furthermore we discuss the infrared behavior of graviton and matter self-energy contributions.

Spin-2 Quantum Gauge Theories and Perturbative Gauge Invariance

In the framework of causal perturbation theory we analyze the gauge structure of a massless self-interacting quantum tensor field. We look at this theory from a pure field theoretical point of view without assuming any geometrical aspect from general relativity. To first order in the perturbation expansion of the S-matrix we derive necessary and sufficient conditions for such a theory to be gauge invariant, by which we mean that the gauge variation of the self-coupling with respect to the gauge charge operator Q is a divergence in the sense of vector analysis. The most general trilinear self-coupling of the graviton field turns out to be the one derived from the Einstein–Hilbert action plus divergences and coboundaries.

Scalar Matter Coupled to Quantum Gravity in the Causal Approach: One-Loop Calculations and Perturbative Gauge Invariance

Quantum gravity coupled to scalar massive matter fields is investigated in the framework of causal perturbation theory using the Epstein–Glaser regularization/renormalization scheme. Detailed one-loop calculations include the matter loop graviton self-energy and the matter self-energy. The condition of perturbative operator gauge invariance to second order implies the usual Slavnov–Ward identities for the graviton two-point connected Green function in the loop graph sector and generates the correct quartic graviton-matter interaction in the tree graph sector. The mass zero case is also discussed.

Epstein-Glaser renormalization and differential renormalization

In the framework of causal perturbation theory by Epstein and Glaser the process of renormalization is precisely equivalent to the extension of time-ordered distributions to coincident points. This is achieved by a modified Taylor subtraction on the corresponding test functions. I show that the pullback of this operation to the distributions yields expressions known from differential renormalization. The subtraction is equivalent to BPHZ subtraction in momentum space. Some examples from Euclidean scalar field theory in flat and curved spacetime will be presented.

A Local (Perturbative) Construction of Observables in Gauge Theories: The Example of QED

Interacting fields can be constructed as formal power series in the framework of causal perturbation theory. The local field algebra $\tilde {\cal F}({\cal O})$ is obtained without performing the adiabatic limit; the (usually bad) infrared behavior plays no role. To construct the observables in gauge theories we use the Kugo-Ojima formalism; we define the BRST-transformation $\tilde s$ as a graded derivation on the algebra of interacting fields and use the implementation of $\tilde s$ by the Kugo-Ojima operator $Q_{\rm int}$. Since our treatment is local, the operator $Q_{\rm int}$ differs from the corresponding operator $Q$ of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED.

Slavnov–Taylor Identities from the Causal Point of View

We continue the investigation of quantized Yang–Mills theories coupled to matter fields in the framework of causal perturbation theory which goes back to Epstein and Glaser. In this approach gauge invariance is expressed by a simple commutator relation for the S matrix and the corresponding gauge transformations are simple transformations of the free fields only. In spite of this simplicity, gauge invariance implies the usual Slavnov–Taylor identities. The main purpose of this paper is to prove the latter statement. Since the Slavnov–Taylor identities are formulated in terms of Green functions, we investigate the agreement of two perturbative definitions of Green functions, namely Epstein and Glaser's definition with the Gell-Mann–Low series.

Non-uniqueness of quantized Yang - Mills theories

We consider quantized Yang - Mills theories in the framework of causal perturbation theory which goes back to Epstein and Glaser. In this approach gauge invariance is expressed by a simple commutator relation for the S-matrix. The most general coupling which is gauge invariant to first order contains a two-parametric ambiguity in the ghost sector: a divergence- and a coboundary-coupling may be added. We prove (not completely) that the higher orders with these two additional couplings are also gauge invariant. Moreover, we show that the ambiguities of the n-point distributions restricted to the physical subspace are only a sum of the divergences (in the sense of vector analysis). It turns out that the theory without divergence- and coboundary-coupling is the simplest one in a quite technical sense. The proofs for the n-point distributions containing coboundary-couplings are given up to third or fourth order only, whereas the statements about the divergence-coupling are proved for all orders.

On gauge invariance of Yang-Mills theories with matter fields

We continue the investigation of quantied Yang-Mills theories coupled to matter fields in the framework of causal perturbation theory. In this approach, which goes back to Epstein and Glaser, one works with free fields throughout, so that all expressions are mathematically well defined. The general proof of theCg-identities (C-number identities expressing gauge invariance) is completed. We attach importance to the correct treatment of the degenerate terms and to theCg-identities with external matter legs. Moreover, the compatibility of allCg-identities withP-, T-, C-invariance and pseudo-unitarity is shown.

Gauge invariance of massless QED

A simple general proof of gauge invariance in QED is given in the framework of causal perturbation theory. It illustrates a method which can also be used in non-abelian gauge theories.

Scalar QED revisited

Scalar QED is investigated in the framework of causal perturbation theory. This approach which is ultraviolet finite differs from the conventional formalism because the inductive construction starts from the simple first-order interaction, without the term quadratic in the electromagnetic potential. The latter appears in the process of distribution splitting as a consequence of gauge invariance. In this way normalizability, gauge invariance and unitarity can be rigorously proven.