# S matrix and vanishing fields

• I
• Silviu
In summary, the S matrix assumes that interacting particles are free and that the vacuum is not an important factor in the theory.

#### Silviu

Hello! I am reading about the S matrix, and I see that one of the assumption that the derivations are based on is the fact that interacting particles are free at ##t=\pm \infty## and I am not sure I understand why. One of the given examples is the ##\phi^4## theory which contains an interaction term of the form ##\frac{\lambda}{4!}\phi(x)^4##. Why would this term vanish far away from our experiment. More concretely, if we would have an electron Dirac spinor field (assuming we collide electron and positrons), why would we assume that far from the experiment the electron is free? Isn't it still interacting with the vacuum QED i.e. the vacuum would be ##|\Omega>## and not ##|0>## no matter at which point we are (even at big distances from our collision point). Can someone explain this to me please? Thank you!

The interaction with the vacuum is treated separately as "property of the free electron". It is not part of the collision process.

mfb said:
The interaction with the vacuum is treated separately as "property of the free electron". It is not part of the collision process.
So from the point of view of the S matrix, an interacting particle with momentum p and a non-interacting particle with momentum p are identical at ##\pm \infty##? But the operators used to create a particle with momentum p i.e. ##a_p^\dagger## is different (they evolve differently in time) in the 2 theories as the vacuum is different?

Haag's theorem is irrelevant FAPP. See the excellent book

The conceptual framework of QFT, Oxford University Press

It's precisely filling the gaps Weinberg's three volumes leave.

weirdoguy
I recommend you read section 1 of this (available in the amazon preview) to see how insanely fundamental to relativistic quantum theory this issue is:

Page 3 said:
'the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties of free particles'

## 1. What is an S matrix?

An S matrix, also known as the scattering matrix, is a mathematical tool used in physics to describe the scattering of particles or waves. It relates the incoming and outgoing states of a system and is often used to analyze the interaction between particles or fields.

## 2. How is the S matrix related to vanishing fields?

The S matrix is used to calculate the probability of a particle or wave scattering off a potential. In the case of vanishing fields, the potential is zero, meaning there is no interaction between the incoming and outgoing particles or waves. Therefore, the S matrix for vanishing fields is simply the identity matrix, as there is no scattering or change in the state.

## 3. Can the S matrix be used to study quantum systems?

Yes, the S matrix can be used to study quantum systems, particularly in the field of quantum mechanics. It is a key tool in understanding the behavior of particles at the quantum level and is used to calculate various observables, such as cross-sections and decay rates.

## 4. How is the S matrix calculated?

The S matrix is calculated using the principles of quantum mechanics and relies on the use of mathematical techniques such as perturbation theory and Feynman diagrams. The exact method of calculation depends on the specific system being studied and the level of accuracy required.

## 5. What are some applications of the S matrix?

The S matrix has various applications in different fields of physics, such as nuclear physics, particle physics, and quantum field theory. It is used in the study of scattering processes, particle decays, and the behavior of quantum systems. It also has practical applications in engineering, such as in the design of electronic circuits and antennas.