Abstract
The renormalizability of QFT's is a vastly studied issue, and particularly the results concerning a scalar eld theory are well-known through the traditionalrenormalization approach in the literature. However, in this paper we analyze the problem through a less known approach, which justies in a more rigorousand mathematically neat manner, the heuristic arguments of standard treatments of divergencies in QFT's. This paper analyzes the renormalizability ofan arbitrary Scalar Field Theory with interaction Lagrangean L(x) =: 'm(x) : using the method of Epstein-Glaser and techniques of microlocal analysis, inparticular, the concept of scaling degree of a distribution. For a renormalizability proof of perturbative models in the Epstein-Glaser scheme one rst needs to dene an n-fold product of sub-Wick monomials of the interaction Lagrangean. This time ordering is an operator-valued distribution on R4n and the basic issue is its ill-denedness on a null set. The renormalization of a theory in this scheme amounts to the problem of extension of distributions across null sets.
Highlights
The problem of infinities is already present in classical electromagnetism
To the classical case, in Quantum Electrodynamics (QED) there is a problem of divergence that arises when one tries to calculate perturbatively the electron and photon self-energies, for instance
The Renormalization Theory is a process devised initially as a way to eliminate these infinities arising in QED and other QFT’s [5, 7, 10, 9, 8, 13, 11, 12, 15, 14, 16, 17]
Summary
The problem of infinities is already present in classical electromagnetism. It is a known fact that if one tries to calculate the electron self-energy due to its electric field considering the electron as a point-like particle, one is left with an integral which diverges as the radius of the electron goes to zero, i.e. r 0. We prove the renormalizability of an arbitrary scalar field theory with no derivatives and interaction Lagrangean L(x) =: m (x) : through the calculation of the scaling degree of the distribution associated to an arbitrary n -fold time-ordered product of sub-Wick monomials of the interaction lagrangean. The "UV problem" of divergencies consists, in this context, in the extension of the T -product across n This extension is not unique in general, and the choice of possible extensions is restricted by the requirements of covariance and Wick expansion for the T -product, which may be called (re-) normalization conditions. By inspecting equation (2.20) we notice that if m > 4 , the degree of divergence will increase with the number of vertices no matter how large the number of external lines is, which makes the theory non-renormalizable in this case.
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