LSZ reduction formula derivation following Peskin's book.

[andreluzoro]

Question: In page number 225 of Introduction to quantum field theory of peskin and schroeder, looking at the equation that follows to Eq. (7.41), I cannot understand how the general state $|\lambda_K>$ (one eigenstate of the full Hamiltonian Lorentz shifted so its momentum be K) is annihilated by two operators constrained to lie in distant wavepackets. This assumption allows to decompose this state as the product state of two excitations of the vacuum, and therefore is essencial to the proof of LSZ reduction formula. What is difficult for me to understand is that the fields that appear in the formula are the Heisenberg picture field for the interacting theory and as the vacuum of the interacting theory is also different to the free theory, I think that one cannot do any analogy with the free theory, so how must I think?
Thanks!

 

[eljose79]

Thank you for your question about the equation in Introduction to quantum field theory of peskin and schroeder. I can understand your confusion about the general state $|\lambda_K>$ being annihilated by two operators constrained to lie in distant wavepackets. This assumption is crucial for the proof of the LSZ reduction formula, but it may not be immediately clear how it relates to the Heisenberg picture fields for the interacting theory.

First, it is important to note that the LSZ reduction formula is a mathematical tool used to calculate scattering amplitudes in quantum field theory. It allows us to relate the asymptotic states (particles in the far past and far future) to the initial and final states of the interacting theory. In order to do this, we need to use the Heisenberg picture fields for the interacting theory, as these are the operators that create and destroy particles in the interacting theory.

Now, you are correct in saying that the vacuum state for the interacting theory is different from the vacuum state for the free theory. However, this does not mean that we cannot make an analogy with the free theory. In fact, the LSZ reduction formula works precisely because we are able to use the free theory as a starting point, and then incorporate the interactions through the Heisenberg picture fields.

To understand how the general state $|\lambda_K>$ is decomposed into a product state of two excitations of the vacuum, it may be helpful to think about it in terms of Feynman diagrams. The LSZ reduction formula essentially takes into account all possible interactions between the initial and final states, and this is represented by the Feynman diagrams. The two operators constrained to lie in distant wavepackets represent the initial and final states, while the general state $|\lambda_K>$ represents the intermediate state with all possible interactions.

In summary, the LSZ reduction formula is a powerful tool for calculating scattering amplitudes in quantum field theory. While it may seem confusing at first, it is based on the Heisenberg picture fields for the interacting theory and makes use of the analogy with the free theory. I hope this helps to clarify your understanding. If you have any further questions, please don't hesitate to ask.