Non-linear and weak-coupling expansion in Quantum Field Theory

Physical Review D, 1979

We derive a simple and general diagrammatic procedure for obtaining the strong-coupling expansion of a ddimensional quantum field theory, starting from its Euclidean path-integral representation. At intermediate stages we are required to evaluate diagrams on a lattice; the lattice spacing provides a cutofF for the theory. We formulate a simple Pade-type prescription for extrapolating to zero lattice spacing and thereby obtain a series of approximants to the true strong-coupling expansion of the theory. No infinite quantities appear at any stage of the calculation. Moreover, all diagrams are simple to evaluate (unlike the diagrams of the ordinary weak-coupling expansion) because nothing more than algebra is required, and no diagram, no matter how complex, generates any transcendental quantities. We explain our approach in the context of a g$ field theory and calculate the two-point and four-point Green's functions. Then we specialize to d = 1 (the anharmonic oscillator) and compare the locations of the poles of the Green's functions with the tabulated numerical values of the energy levels. The agreement is excellent, Finally, we discuss the application of these techniques to other models such as g$'", g(PQ)', and quantum electrodynamics.

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.

Journal of Nonlinear Mathematical Physics, 2004

We first review regularization methods based on matrix geometry which provide an ultraviolet cutoff for scalar fields respecting the symmetries. Sections of bundles over the sphere can be quantized, too. This procedure even allows to regularize supersymmetry without violating it. Recently, this work was extended to include quantum group covariant regularizations. In a second part recent attempts to renormalize four-dimensional deformed quantum field theory models are reviewed. For scalar models the well-known UV/IR-mixing does not allow to use standard techniques. The same applies to the Yang-Mills model in four dimensions. Only additional symmetry, as it occurs in the Wess-Zumino model, may allow to avoid this problem. There was some hope that circumventing UV/IR-mixing by application of the Seiberg-Witten map one can achieve renormalizability of noncommutative field theories. Although the photon self-energy is renormalizable to all orders, θ-expanded noncommutative QED was shown to be not renormalizable, thus ruling out this approach as well.

2011

We discuss a special Euclidean φ 4 -quantum field theory over quantized space-time as an example of a renormalizable field theory. Using a Ward identity, it was possible to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger-Dyson equation a nonlinear integral equation for the renormalised two-point function alone. The non-trivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalisation from this almost solvable model.

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:

Physical Review D, 1981

%'e study two continuum methods of regulating the formal strong-coupling expansion of the Green's functions, obtained by expanding the path integral in powers of the kinetic energy {inverse free propagator). Our continuum regulations amount to introducing either a hard (0 function) or soft (Gaussian) cutoff A in momentum space. The cutoA' takes the place of the usual spatial cutoff, the lattice spacing, which arises when the path integral is defined as the continuum limit of,ordinary integrals on a Euclidean space-time lattice. We find, by investigating free field theory and g$4 field theory in one dimension, that the 8-function regulation is more accurate than the Gaussian and, unlike the Gaussian, preserves certain continuum Green s-function identities. The extension to field theories with fermions is trivial and we give the strong-coupling graphical rules for an arbitrary field theory with fermions and bosons in d dimensions.

2020

As applied to quantum theories, the program of renormalization is successful for ‘renormalizable models’ but fails for ‘nonrenormalizable models’. After some conceptual discussion and analysis, an enhanced program of renormalization is proposed that is designed to bring the ‘nonrenormalizable models’ under control as well. The new principles are developed by studying several, carefully chosen, soluble examples, and include a recognition of a ‘hard-core’ behavior of the interaction and, in special cases, an extremely elementary procedure to remove the source of all divergences. Our discussion provides the background for a recent proposal for a nontrivial quantization of nonrenormalizable scalar quantum field models, which is briefly summarized as well. Dedication: It is a pleasure to dedicate this article to the memory of Prof. Alladi Ramakrishnan who, besides his own important contributions to science, played a crucial role in the development of modern scientific research and educat...

Physical Review D, 2011

We show how expansions in powers of Planck's constant = h/2π can give new insights into perturbative and nonperturbative properties of quantum field theories. Since is a fundamental parameter, exact Lorentz invariance and gauge invariance are maintained at each order of the expansion. The physics of the expansion depends on the scheme; i.e., different expansions are obtained depending on which quantities (momenta, couplings and masses) are assumed to be independent of. We show that if the coupling and mass parameters appearing in the Lagrangian density are taken to be independent of , then each loop in perturbation theory brings a factor of. In the case of quantum electrodynamics, this scheme implies that the classical charge e, as well as the fine structure constant are linear in. The connection between the number of loops and factors of is more subtle for bound states since the binding energies and bound-state momenta themselves scale with. The expansion allows one to identify equal-time relativistic bound states in QED and QCD which are of lowest order in and transform dynamically under Lorentz boosts. The possibility to use retarded propagators at the Born level gives valence-like wave-functions which implicitly describe the sea constituents of the bound states normally present in its Fock state representation.

International Journal of Theoretical Physics, 2009

It was shown that quantum metric fluctuations smear out the singularities of Green's functions on the light cone [1], but it does not remove other ultraviolet divergences of quantum field theory. We have proved that the quantum field theory in Krein space, i.e. indefinite metric quantization, removes all divergences of quantum field theory with exception of the light cone singularity [2, 3]. In this paper, it is discussed that the combination of quantum field theory in Krein space together with consideration of quantum metric fluctuations, results in quantum field theory without any divergences.

International Journal of Modern Physics, 2012

We discuss a special Euclidean φ 4 4-quantum field theory over quantized space-time as an example of a renormalizable field theory. Using a Ward identity, it was possible to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger-Dyson equation a nonlinear integral equation for the renormalized two-point function alone. The nontrivial renormalized four-point function fulfills a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalization from this almost solvable model.