What is the Landau pole and how does it relate to the coupling constant in QED?

In summary: So for me, any clarification on perturbation series beyond what I've learned in the past would be helpful.
  • #1
waterfall
381
1
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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
Vanadium 50 said:
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..
 

1. What is the Landau pole in QED?

The Landau pole is a hypothetical singularity in the running coupling constant of Quantum Electrodynamics (QED). It occurs at an energy scale where the coupling constant becomes infinitely large, indicating the breakdown of the theory.

2. How does the Landau pole relate to the coupling constant in QED?

The Landau pole is a critical point in the running coupling constant, where it becomes infinite. This is a mathematical consequence of the renormalization process in QED, where the coupling constant is adjusted to account for the effects of virtual particles.

3. Why is the Landau pole important in QED?

The Landau pole serves as a theoretical limit for the validity of QED. If the energy scale of a physical process reaches the Landau pole, the coupling constant becomes infinitely large, indicating the breakdown of the theory and the need for a more complete description of the system.

4. Can the Landau pole be observed experimentally?

No, the Landau pole is a mathematical singularity and cannot be directly observed in experiments. However, its existence has important implications for the limitations of QED and the need for a more complete theory at high energy scales.

5. How does the Landau pole affect the renormalization of QED?

The Landau pole is a critical point in the renormalization process of QED. It is used to determine the maximum energy scale at which the theory can be valid, and helps to guide the development of new theories that can better describe physical processes at high energies.

Similar threads

  • Quantum Physics
Replies
2
Views
956
Replies
22
Views
9K
Replies
16
Views
1K
Replies
1
Views
1K
Replies
1
Views
724
Replies
1
Views
735
  • Advanced Physics Homework Help
Replies
7
Views
948
Replies
6
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
2K
Back
Top