Renormalization


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My central claim is that the effective approach to renormalization provides a more physically perspicuous, conceptually coherent and widely applicable framework to construct perturbative QFTs than the continuum approach. I also show how a careful comparison between the two approaches: i helps to dispel the mystery surrounding the success of the renormalization procedure; ii clarifies the various notions of renormalizability; and iii gives reasons to temper Butterfield and Bouatta's claim that some continuum QFTs are ripe for metaphysical inquiry Butterfield and Bouatta, More information and software credits.

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The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics

Figure 6 a First-principles static dielectric function of n -doped BP in different direction. Sign up to receive regular email alerts from Physical Review B. Journal: Phys. X Rev. A Phys. B Phys.

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C Phys. D Phys. In the cases where perturbation theory can be constructed using a regularization which preserves the symmetries - such as the renowned dimensional regularization used in non-abelian Yang-Mills gauge theories without axial couplings 't Hooft G and Veltman M, , with minimal subtractions - the counterterms generated by the radiative corrections are automatically symmetric, their coefficients corresponding to a redefinition of the physical parameters, a process called multiplicative renormalization.

Quantum Field Theory and Quantum Gravity

One should also include in this category renormalization schemes such as BPHZL and Epstein-Glaser see below , which do not use any regularization. Lorentz invariance, for instance, is explicitly preserved by these latter schemes.


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There are however various cases where no invariant regularization is known, and others where it is indeed known that no such regularization exists. A well known example of the former is provided by the supersymmetric field theories Piguet O and Sibold K, , whereas obvious examples of the latter are given by the theories which have been shown to be anomalous, such as the Gauge theories with axial couplings.


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The main tools of algebraic renormalization are:. By the dimension of an action - or of any other integral of a local field polynomial - we mean the maximal dimension of the various terms of its integrand in mass units, within a system of units where Planck constant and light velocity are equal to 1.

If the functional is a sum of terms, its dimension is defined as that of the highest dimension term. Brennecke and M. These statements are general results of perturbative renormalization theory, which were proved in the momentum space subtraction scheme of Bogoliubov, Parasiuk, Hepp, Zimmermann and Lowenstein BPHZL , in the minimal dimensional regularization scheme P.

Breitenlohner and D. A more complex situation was encountered in Piguet O, , where some of the possible breaking terms could not be compensated by counterterms, and they had to be proved to vanish using a recursive consistency procedure. It is based on the two principles stated above. For a power-counting renormalizable model, at the level of the radiative corrections, it reduces the proof of the preservation of the symmetries or the determination of all possible anomalies to the solution of the cohomology of its symmetry group: trivial elements co-boundaries correspond to breakings which can be compensated by counterterms, whereas the non-trivial elements are the possible anomalies.

In short, the aim of the algebraic renormalization procedure is to prove that the full perturbative quantum theory has the same symmetry properties as the classical theory, i. This will be explained with some more details in the rest of this note. In physics, one always formulates basic equations for the fundamental quantities. The basic equation is the quantum action principle, a generalization of the classical action principle taking into account the quantum fluctuations , which tells us the response of the theory to infinitesimal variations of the external conditions: variations of parameters or external fields.

This happens when the algebra of the infinitesimal symmetry generators closes only on shell, i. In theoretical physics the Noether theorem plays a fundamental role in expressing the equivalence between two basic concepts of physics, namely that of symmetry and that of conservation law. In the classical theory, a symmetry - some of these symmetries may be supersymmetries - is expressed by the covariance of the field equations or, equivalently, by the invariance of the action under a group of rigid global transformations of the dynamical variables.

renormalization

On the other hand local or gauge symmetries are needed, in the so-called gauge theories, for their physical interpretation in terms of a well defined quantum theory. In both cases, local or rigid, the symmetry is expressed by a set of relations between the Green functions of the theory, known as Ward identities. The first equality may be interpreted as the infinitesimal symmetry transformations, the ghosts playing the role of the infinitesimal parameters. This action is invariant under a discrete U 1 symmetry corresponding to ghost number conservation.

At the quantum level, these identities imply relations between the divergences of Green functions and thus between the counterterms which render the theory finite. From these relations one derives the generic form of the counterterms. Such analysis is based on a loop-wise expansion of perturbation theory. However, before entering this discussion, a study of the stability of the classical action should be performed.

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