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

• ingenue
In summary, the propagator has a cut starting from exactly 4m^2, which corresponds to the energy threshold for particle production in a collision. This cut is significant in calculations of scattering amplitudes and cross-sections in particle interactions and can complicate calculations due to its non-analytic behavior. While the cut can occur at other values, it reflects the principle of causality in physics that particles cannot be created below a certain energy threshold.
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".

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.

## 1. Why does the propagator have a cut starting from exactly 4m^2?

The propagator has a cut starting from exactly 4m^2 because this value corresponds to the threshold for particle production in a collision. Below this energy, particles cannot be created and therefore the propagator becomes complex.

## 2. What is the significance of the propagator having a cut at 4m^2?

The cut in the propagator at 4m^2 is significant because it indicates the energy threshold at which particles can be created. This allows for the calculation of scattering amplitudes and cross-sections in particle interactions.

## 3. How does the cut in the propagator affect calculations in particle physics?

The cut in the propagator has a significant effect on calculations in particle physics. It introduces a non-analytic behavior, which can complicate calculations and lead to the need for advanced techniques such as resummation to obtain accurate results.

## 4. Can the propagator have a cut at any other value besides 4m^2?

Yes, the propagator can have a cut at other values besides 4m^2. The exact location of the cut depends on the specific particles involved in the interaction and their masses. In general, the cut will occur at the energy threshold for particle production.

## 5. How does the presence of the cut in the propagator relate to the principle of causality?

The presence of the cut in the propagator is related to the principle of causality in that it reflects the physical reality that particles cannot be created below a certain energy threshold. This is a fundamental principle in physics and is reflected in the behavior of the propagator.

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