Path integral formulation of mean-field perturbation theory

Cornell University - arXiv, 2020

A formal expansion for the Green's functions of an interacting quantum field theory in a parameter that somehow encodes its "distance" from the corresponding non-interacting one was introduced more than thirty years ago, and has been recently reconsidered in connection with its possible application to the renormalization of non-hermitian theories. Besides this new and interesting application, this expansion has special properties already when applied to ordinary (i.e. hermitian) theories, and in order to disentangle the peculiarities of the expansion itself from those of non-hermitian theories, it is worth to push further the investigation limiting first the analysis to ordinary theories. In the present work we study some aspects related to the renormalization of a scalar theory within the framework of such an expansion. Due to its peculiar properties, it turns out that at any finite order in the expansion parameter the theory looks as non-interacting. We show that when diagrams of appropriate classes are resummed, this apparent drawback disappears and the theory recovers its interacting character. In particular we have seen that with a certain class of diagrams, the weak-coupling expansion results are recovered, thus establishing a bridge between the two expansions.

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.

Physical Review D, 1996

We show that general cutoff scalar field theories in four dimensions are perturbatively renormalizable through the use of diagrammatic techniques and an adapted BPH renormalization method. Weinberg's convergence theorem is used to show that operators in the Lagrangian with dimension greater than four, which are divided by powers of the cutoff, produce perturbatively only local divergences in the two-, three-, and four-point correlation functions. We also show that the renormalized Green's functions are the same as in ordinary Φ 4 theory up to corrections suppressed by inverse powers of the cutoff. These conclusions are consistent with those of existing proofs based on the renormalization group.

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...

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, 1997

A general discussion of the renormalization of the quantum theory of a scalar field as an effective field theory is presented. The renormalization group equations in a mass-independent renormalization scheme allow us to identify the possibility to go beyond the renormalizable φ 4 theory without losing its predictive power. It is shown that there is a minimal extension with just one additional free parameter (the mass scale of the effective theory expansion) and some of its properties are discussed.

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.

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.

Annals of Physics, 1977

It is proposed that field theories quantized in a curved space-time manifold can be conveniently regularized and renormalized with the aid of Pauli-Villars regulator fields. The method avoids the conceptual difficulties of covariant point-separation approaches, by always starting from a manifestly generally covariant action, and the technical limitations of the dimensional regularization approach, which requires solution of the theory in arbitrary dimension in order to go beyond a weak-field expansion. An action is constructed which renormalizes the weak-field perturbation theory of a massive scalar field in two space-time dimensions, it is shown that the trace anomaly previously found in dimensional regularization and some point-separation calculations also arises in perturbation theory when the theory is Pauli-Villars regulated. We then study a specific solvable two-dimensional model of a massive scalar field in a Robertson-Walker asymptotically flat universe. It is shown that the action previously considered leads, in this model, to a well-defined finite expectation value for the stress-energy tensor. The particle production (<0 in / @(x, t)l 0 in) for t-+ co) is computed explicitly. Finally, the validity of weak-field perturbation theory (in the appropriate range of parameters) is checked directly in the solvable model, and the trace anomaIy computed in the asymptotic regions t + i a independently of any weakfield approximation. The extension of our model to higher dimensions and the renormalization of interacting (scalar) field theories are briefly discussed.

Progress of Theoretical Physics Supplement, 1966