Renormalization is a method to extract finite results comparable with the experimental values from the infinite results of QFTs. Such as the theoretical desription of Lamb shift and anomalous magnetic moment of electron. Nevertheless the method is incomprehensible physically as beyond the poineers like Heisenberg and Dirac also S. Weinberg mentioned in a footenote of his book "Qunatum theory of fields": (by analogy) "No two expert of renormalization agree"!
Theoretically renormalization is needed in view of the discrepancy between the order of differential operators (~ propagators) in QFTs which are up to two and the order of integral operators in 4D QFTs which is four.
In its QED version where renormalization the is worked out needs beyond the rgulariozation which menas geometrically a compactification of integral ranges, several dimensional conditions like Ward identity (L^2 = 0), constant magnetic fields (1/L^2 = cte.)and plane waves (2 degrees of freedom). These althogether means in my geometrical view a reduction of 4D QED to its 2D version. Then 2D QFT are by definition renormalized in view of the fact that in this case the mentioned orders of differential and integral operators are two. Also in the case of more incomprehensible renormalization of non-abelian QFTs there are similar dimensional conditions where a. o. some 1/L^2 dimensional (B-field of order B^2) is neglected (set to be zero) and the 1/L^2 dimensional Landau gauge ∂μBaμ=0 and the gauge fixing for Faddaey-Poppov ghost field ∂μCμ=0!
Considering all geometrical and dimesnional conditions on QFTs such as compactification of integration manifold by regularization/cut off and vanishing or constancy of several dimensional quantities as above renormalization menas geometrically as mentioned above the rduction of 4D QFTs to their 2D versions.
I started a geometrical program to prove this general statement by the following work:
"Topological approach to renormalization" added here.
The renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. For example, in quantum electrodynamics (QED), an electron appears to be composed of electrons, positrons (anti-electrons) and photons, as one views it at higher resolution, at very short distances. The electron at such short distances has a slightly different electric charge than does the dressed electron seen at large distances, and this change, or running, in the value of the electric charge is determined by the renormalization group equation.
The simple electron/photon interaction that determines the electron's charge at one renormalization point is revealed to consist of more complicated interactions at another.
My investigations of "Topology of renormalization" includes all these local aspects like electron, photon cloud and coupled particles to electron, etc.
The "Topology of renormalization" explains for the first time why and what it means to consider electron as "to be composed of electrons, positrons and photons"!
Then electron or any elementary particle is a finite topological object and not as it considered in differential equation of QFT including RG an infinitesimal local ("point") particle or wave!
The illness of four dimensional QFT (4DQFT) including why it must be renormalized is that it is designed, under the hierarchy of differential equations in physics, as a local theory and not by a topological action functional.
The "Topology of renormalization" gives according to the invariant aspects of topological investigations for the first time a topological invariant meaning for all renormalization tools and method. It explains what perturbation or radiative corrections, etc. are and why one needs them in any four dimensional QFT. It explains why mathematical tools like Hopf algebra appear in renormalization. Otherwise one has a lot of different methods in a model without knowing why one needs them.
The standard renormalization including RG is a local (differential) superficial method without any explanation of the background of renormalization problem. Its Objects are functions, coordinates and coupling constants which are rescaled. Nevertheless the renormalization is not understood yet in view of its different purely local aspects ( S. Weinberg, The quantum theory of fields, II). Furthermore despite of hierarchy of local/ differential methods in physics the renormalization is like any basic method in science a topological matter. I show for the first time how renormalization works and what it means from invariant topological/ differential topological stand point.
The main illness of 4DQFT is that they are designed traditionally in 4D! Although there is no proof for the four dimensionality of physics which is announced even as the millennium problem of physics ("String 2000", Ann Arbor).
Then any 4DQFT must be renormalized in retrospect which is in my view a correction of their local 4-dimensionality. This aspect, i. e. that renormalization of 4DQFT is defacto its reduction to a topological 2DQFT should be explained more, although I mentioned it in my "Topology of renormalization".
So the background of renormalization is much deeper than those local games in RG. RG did not explains any thing. It is just a calculation in retrospect without naming the foundation of problem where it is applied and why it is applied.
One has to know that a renormalized theory is a 2D theory and no more as RG game insists a 4D theory! Only by topological investigation of renormalization as I deed in "Topology of renormalization" it is possible to look deeper than local superficial RG games and to understand the basic 2D structure of topological physics.
In other words electron, positron, photon are finite topological objects and not as they are considered in physical differential equations like RG local objects.
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