This article is devoted to the perturbative renormalization of the abelian Higgs-Kibble model, within the class of renormalizable gauges which are odd under charge conjugation. The Bogoliubov Parasiuk Hepp-Zimmermann renormalization scheme is used throughout, including the renormalized action principle proved by Lowenstein and Lam. The whole study is based on the fulfillment to all orders of perturbation theory of the Slavnov identities which express the invariance of the Lagrangian under a supergauge type family of non-linear transformations involving the Faddeev-Popov ghosts. Direct combinatorial proofs are given of the gauge independence and unitarity of the physicalS operator. Their simplicity relies both on a systematic use of the Slavnov identities as well as suitable normalization conditions which allow to perform all mass renormalizations, including those pertaining to the ghosts, so that the theory can be given a setting within a fixed Fock space. Some simple gauge independent local operators are constructed.

/pdf/renormalization-of-the-abelian-higgs-kibble-model-14fr0vp7oj.pdf

Renormalization of the Abelian Higgs-Kibble Model

We propose additional conditions (beyond those considered in our previous papers) that should be imposed on Wick products and time-ordered products of a free quantum scalar field in curved spacetime. These conditions arise from a simple "Principle of Perturbative Agreement": for interaction Lagrangians L1 that are such that the interacting field theory can be constructed exactly — as occurs when L1 is a "pure divergence" or when L1 is at most quadratic in the field and contains no more than two derivatives — then time-ordered products must be defined so that the perturbative solution for interacting fields obtained from the Bogoliubov formula agrees with the exact solution. The conditions derived from this principle include a version of the Leibniz rule (or "action Ward identity") and a condition on time-ordered products that contain a factor of the free field φ or the free stress-energy tensor Tab. The main results of our paper are (1) a proof that in spacetime dimensions greater than 2, our new conditions can be consistently imposed in addition to our previously considered conditions and (2) a proof that, if they are imposed, then for any polynomial interaction Lagrangian L1 (with no restriction on the number of derivatives appearing in L1), the stress-energy tensor Θab of the interacting theory will be conserved. Our work thereby establishes (in the context of perturbation theory) the conservation of stress-energy for an arbitrary interacting scalar field in curved spacetimes of dimension greater than 2. Our approach requires us to view time-ordered products as maps taking classical field expressions into the quantum field algebra rather than as maps taking Wick polynomials of the quantum field into the quantum field algebra.

Conservation of the Stress Tensor in Perturbative Interacting Quantum Field Theory in Curved Spacetimes

On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for classical field theory presented in our previous publication, we construct in this paper the Batalin-Vilkovisky complex in perturbatively renormalized quantum field theory. The crucial technical ingredient is an extended notion of the renormalized time-ordered product as a binary product equivalent to the pointwise product of classical field theory. Originally, in causal perturbation theory, the time-ordered product is understood merely as a sequence of multilinear maps on the space of local functionals. Our extended notion of the renormalized time-ordered product (denoted by ${\cdot_{{}^{\mathcal{T}_{\rm r}}}}$) is consistent with the old one and we found a subspace of the quantum algebra which is closed with respect to ${\cdot_{{}^{\mathcal{T}_{\rm r}}}}$ . On this space the renormalized Batalin-Vilkovisky algebra is then the classical algebra but written in terms of the time-ordered product, together with an operator which replaces the ill defined graded Laplacian of the unrenormalized theory. We identify it with the anomaly term of the anomalous Master Ward Identity of Brennecke and Dutsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.

pdf/batalin-vilkovisky-formalism-in-perturbative-algebraic-2munizo9my.pdf

Batalin-Vilkovisky Formalism in Perturbative Algebraic Quantum Field Theory

A new formalism for the perturbative construction of algebraic quantum field theory is developed. The formalism allows the treatment of low-dimensional theories and of non-polynomial interactions. We discuss the connection between the Stuckelberg–Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski’s Flow Equation to the Epstein–Glaser method. We also show that the renormalization group in the sense of Gell–Mann–Low (which characterizes the behaviour of the theory under the change of all scales) is a one-parametric subfamily of the Stuckelberg–Petermann group and that this subfamily is in general only a cocycle. Since the algebraic structure of the Stuckelberg–Petermann group does not depend on global quantities, this group can be formulated in the (algebraic) adiabatic limit without meeting any infrared divergencies. In particular we derive an algebraic version of the Callan–Symanzik equation and define the β-function in a state independent way.

/pdf/perturbative-algebraic-quantum-field-theory-and-the-4uutnh3v2c.pdf

Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ${\cal A}^{(n)}$ of observables ``up to $n$ loops'' where ${\cal A}^{(0)}$ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions.

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

In the framework of perturbative algebraic quantum field theory a local construction of interacting fields in terms of retarded products is performed, based on earlier work of Steinmann [42]. In our formalism the entries of the retarded products are local functionals of the off-shell classical fields, and we prove that the interacting fields depend only on the action and not on terms in the Lagrangian which are total derivatives, thus providing a proof of Stora's "Action Ward Identity" [45]. The theory depends on free parameters which flow under the renormalization group. This flow can be derived in our local framework independently of the infrared behavior, as was first established by Hollands and Wald [32]. We explicitly compute non-trivial examples for the renormalization of the interaction and the field.

Causal perturbation theory in terms of retarded products, and a proof of the action ward identity

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.

Slavnov-Taylor identities from the causal point of view

A fresh look at the renormalization group (in the sense of Stuckelberg-Petermann) from the point of view of algebraic quantum field theory is given, and it is shown that a consistent definition of local algebras of observables and of interacting fields in renormalized perturbative quantum field theory can be given in terms of retarded products The dependence on the Lagrangian enters this construction only through the classical action This amounts to the commutativity of retarded products with derivatives, a property named Action Ward Identity by Stora

Action Ward Identity and the Stückelberg-Petermann Renormalization Group

In perturbative quantum field theory the maintenance of classical symmetries is quite often investigated by means of algebraic renormalization, which is based on the Quantum Action Principle. We formulate and prove this principle in a new framework, in causal perturbation theory with localized interactions. Throughout this work a universal formulation of symmetries is used: the Master Ward Identity.

The Quantum Action Principle in the Framework of Causal Perturbation Theory

It is proved that classical BRS-invariance of the Lagrangian implies perturbative gauge invariance for tree diagrams to all orders. The proof applies in particular to the Einstein Hilbert Lagrangian of gravity.

Proof of perturbative gauge invariance for tree diagrams to all orders

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