LSZ Representation: A Simple Explanation

[Itzhak the cat]

I am looking for simple explanation for the idea of LSZ representation in quantum fields theory.
thanks in advance

 

[dextercioby]

I don't know what you mean by "representation",but I'm sure there's no simple explanation of this issue.Heavy mathematics is needed.

I infer you to chapters 13 & 14 from Bogolubov's book [1] for a serious treatment.

Daniel.

[1]N.N.Bogolubov et al.,"Introduction to Axiomatic Quantum Field Theory",Benjamin/Cummings,1975.

 

[selfAdjoint]

I don't know what you mean by "representation",but I'm sure there's no simple explanation of this issue.Heavy mathematics is needed.

I infer you to chapters 13 & 14 from Bogolubov's book [1] for a serious treatment.

Daniel.

[1]N.N.Bogolubov et al.,"Introduction to Axiomatic Quantum Field Theory",Benjamin/Cummings,1975.

Let me try this coarse overview. Corrections always welcome.

First of all interactive quantum field theory is not well-defined. It has as many Hilbert spaces as there are points on a line, which is not physically meaningful. This is Haag's theorem. This problem shows up as a shaky vacuum state condition for interactive QFT; field states that should be over and gone continue to affect the vacuum. The LSZ formalism intends to work around this by going into the remote past and seeking "asymptotically localized" solutions of the theory. These will than be used as the incoming particles to the intreractions defined, e.g., by Feynman diagrams. As Daniel said, the math of doing this is very non-trivial, and Bogoliubov's book is a good intro to it. See also R. Haag, Local Quantum Physics, p. 81 ff.

 

[Itzhak the cat]

Finding the spectrum.

Let me try this coarse overview. Corrections always welcome.

First of all interactive quantum field theory is not well-defined. It has as many Hilbert spaces as there are points on a line, which is not physically meaningful. This is Haag's theorem. This problem shows up as a shaky vacuum state condition for interactive QFT; field states that should be over and gone continue to affect the vacuum. The LSZ formalism intends to work around this by going into the remote past and seeking "asymptotically localized" solutions of the theory. These will than be used as the incoming particles to the intreractions defined, e.g., by Feynman diagrams. As Daniel said, the math of doing this is very non-trivial, and Bogoliubov's book is a good intro to it. See also R. Haag, Local Quantum Physics, p. 81 ff.

This is the reduction formalism. I am more interested in the spectrum- why the masses are the poles of the propagator, and why the spin is the number of degrees of freedom of the propagator?
I more or less understand the technique, using Poincare eigenstates, but sometimes (many times ) I can follow the math but miss the basic ideas.

 

[vanesch]

This is the reduction formalism. I am more interested in the spectrum- why the masses are the poles of the propagator

The masses are the poles of the propagators because in free field theory, this is the case. We somehow want to consider our particles "free" in the far past and future. Of course they aren't, really, and there are serious problems in doing so. But even if interacting field theory is not free field theory, "it plays free field theory on TV, early in the morning, and late in the evening" And so you try to identify what aspects of the interacting theory mimick a free field theory: the pole of the propagator (the full, interacting one) looks a lot like the pole of a free field theory propagator ; so that pole is then identified as the mass. At least that's how I understand this correspondence.

cheers,
Patrick.

 

[reilly]

The relationship between poles and bound states and resonances goes back to basic scattering theory, as in the Lippman Schwinger Eq.. In fact, scattering theory is usually expressed with Green's Function based solutions. (For all practical purposes, the LSZ formalism is like a covariant form of scattering theory. Horrible things that can happen to make the asymptotic limits of the S-Matrix suspect are all over the place -- unbounded operators, and stuff like that. This is great grist for those that work in formal field theory. Then there are the highly practical ones, who can show that the use of wave packets rather than plane waves cures many evils.)

Suppose that |W> = |W0> + (1/(E-H0))*V*|W>, where H0 is the unperturbed hamiltonian, V is the interaction so, that H = H0 + V. |W0> is an eigenstate of H0. This eq. is not much help with bound states, so we use a formal solution,

|W> = (1 - (1/((E-H))*V) |W0>. Thus, there are poles at the bound states, E<0, of H, and a branch point at E = 0 (How's that again?) Weinberg's Vol 1 does LSZ, an older book by Goldberger and Watson, Scattering Theory does wave packets, Lippman Schwinger and formal scattering theory, and poles and branch cuts (dispersion relations), and you name it -- it's a superb book.) (Greens functions are sometimes called resolvants by mathematicians. There's a huge literature on this subject usually under the general field of Functional Analysis.
Regards,
Reilly Atkinson