Why does the propagator has a cut starting from EXACTLY 4m^2?

[ingenue]

Consider the usual phi^4 theory, when we derive the Lehmann-Kallen representation of the propagator, by inserting a complete set we know that the propagator has a branch cut starting from 4m^2, where the m is the physical mass.

My question is: what's the construction of these multiple-particle states? We say the energy of two-particle states starts from 2m because we're using the approximation that these two particles are very far away from each other, thus the total energy starts from 2m. However, this is only an approximation. It's hard to imagine that one uses only "two-particle states" where the two particles are far apart, and yet get a complete set. In my opinion, the complete set should be constructed using truly eigenvectors of the Hamiltonian, and in this case we don't even know what a truly eigenvector is, except the vacuum state and the one-particle states. And the fact that we know the properties of the vacuum and one-particle states, is by virtue of physical arguments. In particular, we don't even have a definition for "multiparticle states".

Please help me understand this. Thanks.

 

[AndyW]

Hello, thank you for your question. The construction of multiple-particle states in the Lehmann-Kallen representation is based on the principles of quantum field theory. In this theory, particles are described as excitations of quantum fields, and interactions between particles are described as exchange of virtual particles. The Hamiltonian operator in this theory is defined as the generator of time translations, and its eigenstates are the energy eigenstates, which correspond to the different possible energy levels of the system. These energy eigenstates can be constructed using the creation and annihilation operators of the quantum fields, which create and destroy particles respectively.

In the case of the phi^4 theory, the Hamiltonian operator includes terms that correspond to the creation and annihilation of particles, and the interaction between them. These terms can be expanded in a series, and the first term corresponds to the one-particle states, the second term corresponds to the two-particle states, and so on. These states are not just an approximation, but are actually part of the complete set of states that can be used to describe the system.

The reason why the energy of two-particle states starts from 2m is due to the conservation of energy and momentum in the system. When two particles are far apart, their interaction is weak and can be neglected, so the energy of the system is just the sum of the energies of the individual particles. However, as the particles get closer together, their interaction becomes stronger and the energy of the system is no longer just the sum of the energies of the individual particles. This is where the branch cut in the propagator comes in, as it takes into account the contribution of these two-particle states to the overall energy of the system.

In summary, the construction of multiple-particle states in the Lehmann-Kallen representation is based on the principles of quantum field theory and the properties of the Hamiltonian operator. These states are not just an approximation, but are an integral part of the complete set of states that describe the system. I hope this helps to clarify the concept for you. Let me know if you have any further questions.