Three questions on Montonen-Olive duality

[metroplex021]

Hi folks -- I have three questions on Montonen-Olive duality. This duality switches the electric and magnetic couplings, where these couplings are related by the Dirac quantization condition

eg = 2$\hbar\pi$n

Here e is the electric charge and g the magnetic (monopole) charge.

1. First of all, do these couplings both 'run' in the theory (thus have their own, non-zero beta function)? If so, does that mean that there are energy regimes in which e is small and g large, and other regimes where g is small and e large?

2. It is said that, in one way of taking the classical limit, magnetically charged particles arise as composite particles while the electrically charged particles are fundamental, and yet in another way of taking the limit it's the electrically charged particles that are composite and the magnetic monopoles that are now elementary. In the first case, it's when e is taken to go to zero while g is held fixed as h tends to zero, and in the other case it is g that goes to zero while e is held fixed in the classical limit. Do any principles govern which of these limits is the right one to take? That is, do any facts about the physics of the situation inform you that it's g (or e) that should be held fixed in the h-> 0 limit?

3. Finally, why do we look at particle content in the classical limit anyway? Isn't the spectrum of a quantum field theory an intrinsic property of it, and moreover one that can be assigned in regimes of finite h?

Any help on any of these questions would be very much appreciated!

metroplex

 

[AndyW]

, thank you for your questions on Montonen-Olive duality. This duality is a very interesting topic in particle physics and has been studied extensively by many scientists.

To answer your first question, yes, both the electric and magnetic couplings have their own beta functions in the theory. This means that they can both change as the energy scale of the system changes. In some regimes, the electric coupling may be small while the magnetic coupling is large, and vice versa. This is due to the fact that the two couplings are related by the Dirac quantization condition, and therefore their values can change in different energy regimes.

As for your second question, the choice of which limit to take (e or g held fixed in the classical limit) is not determined by any specific principles. It is a matter of convenience and depends on the specific situation being studied. In some cases, it may be more useful to hold g fixed while taking the classical limit, while in others it may be more useful to hold e fixed. This choice is usually made based on the physics being studied and the desired outcome.

Finally, regarding your third question, it is true that the spectrum of a quantum field theory is an intrinsic property that can be assigned in regimes of finite h. However, studying the particle content in the classical limit can give us important insights into the behavior of the theory at high energies. It can also help us understand the relationship between different particles and their interactions. Therefore, studying the classical limit can provide valuable information about the quantum field theory as a whole.

I hope this helps answer your questions about Montonen-Olive duality. Please let me know if you have any further inquiries.