Quantum Field Perturbation Theory Revised

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), 2018

Physics Letters B, 2001

We propose a regularization-independent method for studying a renormalizable field theory nonperturbatively through its Dyson-Schwinger equations. Using QED 4 as an example, we show how the coupled equations determining the nonperturbative fermion and photon propagators can be written entirely in terms of renormalized quantities, which renders the equations manifestly finite in a regularization-independent manner. As an illustration of the technique, we apply it to a study of the fermion propagator in quenched QED 4 with the Curtis-Pennington electron-photon vertex. At large momenta the mass function, and hence the anomalous mass dimension γ m (α), is calculated analytically and we find excellent agreement with previous work. Finally, we show that for the CP vertex the perturbation expansion of γ m (α) has a finite radius of convergence.

Communications in Mathematical Physics, 2001

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 A (n) of observables "up to n loops" where 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.

2017

In many domains of physics, methods are needed to deal with non-perturbative aspects. I want here to argue that a good approach is to work on the Borel transforms of the quantities of interest, the singularities of which give non-perturbative contributions. These singularities in many cases can be largely determined by using the alien calculus developed by Jean \'Ecalle. My main example will be the two point function of a massless theory given as a solution of a renormalization group equation.

Il Nuovo Cimento A Series 10, 1969

ysis c'm be applied, with minor forinal modifications, to any renormalizable theory, in particular to quantum eleetrodynamies, a subject of wider physical interest. Section 2 contains a short review of results obtained in previous investigations, in order to introduce the principal tools for our later analysis. One of the main features of the renormalization method by finite-part integration rules is the lack of uniqueness of the regularized theory. In fact as one can see from some specific examples and also as it is quite clear from the axiomatic definition given in Sect. 4 of ref. (~), an infinite number of finitepart integrals does in general exist. With any of these finite-part integrals we can associate a regularized theory, the propagators of which will satisfy branching equations involving the particular finite-part integration rule. Since the physical properties of the system described by the theory do not depend upon the rule selected particularly for this purpose, it is possible to introduce the so-called <~ renormalization group ,~, defined as the group of all transformations among different regularized theories. The physical content of the theory is invariant with respect to this group. The renormalization group, arising from the regularization methods by finite-part integration, appears, in its structure, as an enlargement of the group studied by other authors ('~). This contains the latter as a particular case, when the finite-part integrals are restricted to vary in a special subclass. A first implicit characterization of the group follows easily from results of previous works (~-~) and is given in Secl. 3. On the other hand, tile successes (~) of the analysis of the group according to the interpretation of the above-mentioned authors leads one to believe that very useful information on the functional dependence of the propagators on the field-theoretical parameters could be gained from a deep knowledge of the group structure according' to our interpretation. it should for instance be possible to <(improve ,) a perturbative approximation by considering the fact that the theory in its full content is invariant under the group even though its ~pproximations are not. The present work provides a complete cht~racterization of the transformations among different regularized theories. This marks therefore a first neces-(5) E. C. G. ST[ECKELBFR(~ and A. I'ETERMANN:

Mathematical Physics Studies, 2015

These notes are based on the course given by Klaus Fredenhagen at the Les Houches Winter School in Mathematical Physics (January 29-February 3, 2012) and the course QFT for mathematicians given by Katarzyna Rejzner in Hamburg for the Research Training Group 1670 (February 6-11, 2012). Both courses were meant as an introduction to modern approach to perturbative quantum field theory and are aimed both at mathematicians and physicists.

arXiv (Cornell University), 2023

Perturbative QFT is developed in terms of off-shell fields (that is, functionals on the configuration space not restricted by any field equation), and by quantizing the (underlying) free theory by an-dependent deformation of the classical product (i.e., the pointwise product of functionals). The time-ordered product of local fields is defined axiomatically, and constructed by induction on the number of factors using Stora's version of the Epstein-Glaser construction; in particular, the interaction is adiabatically switched off. The set of solutions of these axioms can be understood as the orbit of the Stückelberg-Petermann renormalization group when acting on a particular solution. Interacting fields are defined in terms of the time-ordered product by Bogoliubov's formula; they satisfy the following, physically desired properties: causality, spacelike commutativity, (off-shell) field equation and existence of the classical limit. Local, algebraic properties of the observables can be obtained without performing the adiabatic limit (i.e., the limit removing the adiabatic switching of the interaction). * This short review is commissioned by the Encyclopedia of Mathematical Physics, edited by M. Bojowald and R.J. Szabo, to be published by Elsevier.

arXiv: Mathematical Physics, 2016

The problem of renormalization in perturbative quantum field theory (pQFT) can be described in a rigorous way through the theory of extension of distributions. In the framework of pQFT a certain type of distribution appears, given by products of Green functions which act by integration with a test function. They present ultraviolet divergences, whenever any pair of arguments coincide on one point of spacetime, and therefore, they are not defined everywhere. In this work we have studied the necessary and sufficient conditions for the extension (or regularization) of this type of distribution. Moreover, we have constructed such extensions explicitly, satisfying a series of physically relevant axioms, such as the axiom of causality.

Physics Letters B, 1989

International Journal of Modern Physics A, 2000

We consider the detailed renormalization of two (1+1)-dimensional gauge theories which are quantized without preserving gauge invariance: the chiral and the "anomalous" Schwinger models. By regularizing the non-perturbative divergences that appear in fermionic Green's functions of both models, we show that the "tree level" photon propagator is illdefined, thus forcing one to use the complete photon propagator in the loop expansion of these functions. We perform the renormalization of these divergences in both models to one loop level, defining it in a consistent and semi-perturbative sense that we propose in this paper.