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QED and the Landau Pole

by waterfall
Tags: landau, pole
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waterfall
#1
Feb21-12, 06:01 PM
P: 381
According to Wiki:

"If a beta function is positive, the corresponding coupling increases with increasing energy. An example is quantum electrodynamics (QED), where one finds by using perturbation theory that the beta function is positive. In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127.

Moreover, the perturbative beta function tells us that the coupling continues to increase, and QED becomes strongly coupled at high energy. In fact the coupling apparently becomes infinite at some finite energy. This phenomenon was first noted by Lev Landau, and is called the Landau pole."

I understood the above since years ago that probing with high energy causes the coupling constant to be larger. But BHobba was claiming that "As the cutoff is made larger and larger the coupling constant gets larger and larger until in the limit it is infinite". He was saying that even at low energy, α ≈ 1/137 would become 1/127 if you increase the terms of the perturbation series. To put in mathematical form.

[tex]
\sum_{n=0}^\infty c_n g^n
[/tex] (where g is the coupling constant).

with one term
[tex]
\sum_{n=0}^\infty c_n (1/137)^n .
[/tex]
with two terms or three terms
[tex]
\sum_{n=0}^\infty c_n (1/15)^n .
[/tex]

with 1000 terms

[tex]
\sum_{n=0}^\infty c_n (1/0)^n .
[/tex]

The above is true even at low energy (before I thought it's only when the probing is high energy). Can anyone science advisor please confirm if this is true and the context of this? Note I'm talking about normal perturbation and let's not complicate things by including the Renormalization Group and the trick of regulator and stuff. Thanks.
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strangerep
#2
Feb21-12, 06:47 PM
Sci Advisor
P: 1,881
Perturbation techniques can be notoriously tricky. There are other cases where the first few terms of the perturbation series give extremely close approximations to the exact solution, but then start to diverge wildly after that.
Vanadium 50
#3
Feb21-12, 08:38 PM
Mentor
Vanadium 50's Avatar
P: 16,101
You may have missed my message on another thread:

If you are just now studying Power Series, you are by my count about 23 courses prior to where renormalization will be discussed. I think you're going to have to accept that the answers you get will be kind of hand-wavy.
If you picked up a murder mystery, read a few pages, and then jumped to the end, you shouldn't be surprised if the killer's identity seems to make no sense.

waterfall
#4
Feb21-12, 08:49 PM
P: 381
QED and the Landau Pole

Quote Quote by Vanadium 50 View Post
You may have missed my message on another thread:



If you picked up a murder mystery, read a few pages, and then jumped to the end, you shouldn't be surprised if the killer's identity seems to make no sense.
I have spent years reading about feynman diagrams and virtual particles and higher order virtual contributions (or more terms in the power series) so I'm familiar already with the landscape and a little tie-up or updates would make me see the wider vistas.

Right now. I'm taking a crash course in relativistic quantum field theory with aims at applications in quantum gravity and beyond..


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