In quantum field theory, a crucial role is played by renormalization. Let us now study this phenomenon in a very simplified manner.
-
We want to show how mathematical difficulties arise if nonlinear equations are linearized in the incorrect place.
-
Furthermore, we will discuss how to overcome these difficulties by using the methods of bifurcation theory.
The main trick is to replace the original problem by an equivalent one by introducing so-called regularizing terms. We have to distinguish between
-
the non-resonance case (N) (or regular case), and
-
the resonance case (R) (or singular case).
In celestial mechanics, it is well-known that resonance may cause highly complicated motions of asteroids.
In rough terms, the complexity of phenomena in quantum field theory is caused by resonances.
In Sect. 7.16, the non-resonance case and the resonance case were studied for linear operator equations. We now want to generalize this to nonlinear problems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Zeidler, E. (2006). Rigorous Finite-Dimensional Perturbation Theory. In: Quantum Field Theory I: Basics in Mathematics and Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34764-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-34764-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34762-0
Online ISBN: 978-3-540-34764-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)