# A topological approach to renormalization and its geometrical, dimensional consequences

## TL;DR

This paper presents a topological framework for understanding renormalization in quantum field theories, revealing how dimensional constraints influence the geometry and degrees of freedom, with implications for anomalies and holographic models.

## Contribution

It introduces a topological approach to renormalization, linking invariance and finiteness to geometric and dimensional properties of quantum field theories.

## Key findings

- Renormalization in QED involves specific dimensional restrictions.
- Dimensional conditions reduce degrees of freedom and symmetry.
- Connections to holographic principle models are discussed.

## Abstract

The necessity of renormalization arises from the infinite integrals which are caused by the discrepancy between the orders of differential and integral operators in the four dimensional QFTs. Therefore in view of the fact that finiteness and invariant properties of operators are their topological aspects, essential renormalization tools to extract finite invariant values from those infinities which are comparable with the experimental results, e. g. regularization, perturbation and radiative corrections follow some topological standards. In the second part we consider dimensional and geometrical consequences of topological approach to renormalization for the geometrical structure and degrees of freedom of renormalized theory. We show that regularization and renormalization of QED are performed only by certain restrictive dimensional conditions on QED fields. Further it is shown that in accord with our previous topological approach to renormalization of QED the geometrical evaluation of applied dimensional renormalization conditions and the appearance of anomalies refer to a reduction of number of degrees of freedom according to the reduced symmetry of QED. A conclusion concerning a comparison of our results with holographic principle models is also included.