Quantum Field Notes
by: Victor Ilisie
Hello everybody !!
I am Victor. I am currently a PhD student in theoretical physics at the University of Valencia, Spain.
This is my personal web-site where I shall post some of my personal notes on Quantum Field Theory and Physics in general.
There are based on my course notes, some of my tutor's notes and also on many many books.
Through these notes I seek clarity and transparency in all my calculations, things that, in my opinion, lack in many famous physics books.
Enjoy, and let me know your suggestions !! ( ilisiev@ific.uv.es )
Tensors and Manifolds
Before we discuss some topics on QFT we need to treat a few questions which, in my opinion are very delicate .
In my first years of University, there was mysterious question that was haunting many of us: What are tensors?
They were present almost everywhere in physics, everybody talked about them, but nobody explained them to me formally
until my Differential Geometry class. The problem was that this class was optional. So, I assume that there are still many
people that still have doubts about this. Many physics books assume that the reader is already familiar with tensors,
so they get directly into advanced topics. On the other hand, many mathematical books are somewhat too formal for a physicist
(at least for me). Many notes about tensors on the internet treat them very, very poorly. In my notes I want to explain the basic
notions of Tensors and Manifolds trying to be as explicit as possible.
There is no real mystery here.. Actually, things are quite nice and simple..
Dimensional Regularization
The cornerstone of QFT is renormalization. We shall speak about in in the next chapter. Before, it is necessary to discuss the method.
The best and most simple is, of course, dimensional regularization (doesn't break the symmetries, doesn't violate the Ward Identities...).
When explained clearly, it becomes very simple, clear, obvious...
For the moment we shall only discuss UV divergencies...
QED Renormalization to All Orders
What I try to do here is to present the standard renormalization algorithm to all orders in perturbation theory in a simple transparent manner.
Many authors are not being clear about it, or the renormalization procedure is scattered around in various chapters.
Things are simple and clear. At the end we take a one-loop example. This is the way I see it:
One-loop Higgs Mass Renormalization
In the literature, many authors talk about the hierarchy problem and fine-tuning. In the Standard Model (SM) there is no such thing.
We only encounter these problems when we try to extend the SM up to some higher scale where new heavy particles are supposed to come
into the picture. The quadratic correction proportional to the cut-off lambda (when using the cut-off method) does not contribute to the physical
mass correction. The mass correction can not depend on the cut-off. When applying the renormalization group equations, this infinite correction gets
reabsorbed by the mass redefinition. Therefore unsing a cut-off or dimensional regularization we obtain the exact same result. (A logarithmic corrections
that, in the SM is small)
There is more to come soon..
(Usefull results on Relativistic Kinematics, General formulation of Noether’s Theorem for fields, etc.. )