Understanding the Bethe-Salpeter Equation: Introduction and Examples

[EuphoGuy]

Hello, I was wondering if anyone could direct me to a good introduction or examples of how the Bethe-Salpeter equation is used. I'm currently looking at the large N section of Sidney Coleman's Aspects of symmetry and find his treatment rather impenetrable.

Thanks!

 

[DrDu]

Hello, I was wondering if anyone could direct me to a good introduction or examples of how the Bethe-Salpeter equation is used. I'm currently looking at the large N section of Sidney Coleman's Aspects of symmetry and find his treatment rather impenetrable.

Thanks!

Good question and I fear I can't help you. But it is curious to note that in nonrelativistic QM, you solve most of the time bound state problems, while in QED you are only shown scattering theory.

 

[Bill_K]

But it is curious to note that in nonrelativistic QM, you solve most of the time bound state problems, while in QED you are only shown scattering theory.

A good place to look for this is Weinberg's QFT Vol I, Chap 14, which devotes 30 pages to bound state calculations.

 

[andrien]

The use of bethe salpeter formalism first arose in the work of hyperfine splitting of positronium by karplus and klein paper however the required derivation for it can be found in itzykson and zuber'quantum field theory'.

 

[EuphoGuy]

Thanks, I'm looking at Itzykson and Zuber right now, and it seems helpful so far. I appreciate it :)

 

[DrDu]

A good place to look for this is Weinberg's QFT Vol I, Chap 14, which devotes 30 pages to bound state calculations.

Yes, but if I remember well, he concludes that the Bethe Salpeter equation is only usefull when the bound state can be described in terms of an effective potential.

 

[DrDu]

I remember that a very good description of the Bethe-Salpeter equation can be found in
D. Lurie, Particles and Fields.
Lurie was using the BS equation in his own research, so it is not a second hand exposition.

 

[andrien]

Yes, but if I remember well, he concludes that the Bethe Salpeter equation is only usefull when the bound state can be described in terms of an effective potential.

In non relativistic calculation of positronium hyperfine splitting ,matrix elements are taken the non relativistic limit and the Fourier transform of those matrix elements(for both annihilation and direct exchange diagram for lowest order) gives the potential which are considered for taking the expectation value for calculating the energy difference.Bethe salpeter formalism is a relativistic version so it should be able to deal with more complexity.

 

[Bill_K]

My impression is that the Bethe-Salpeter Equation is a formulation whose reputation far exceeds its actual usefulness.

I have a copy of B-S's "Quantum Mechanics of One and Two Electron Atoms", in which every aspect of atomic wavefunctions and energy levels is exhaustively discussed, and yet the B-S Equation hardly gets a mention. As a nonrelativistic approximation they say it "gives results equivalent to the Breit Equation", and "little progress has been made in its application to highly relativistic systems."