# Cluster decomposition of the S-matrix

• acipsen
In summary, the conversation discusses a confusion about the signs in the cluster decomposition of the S-matrix, specifically in equation 4.3.2. It explores the possibility of permuting αi and βi, and how this affects the number of fermion interchanges. The speaker also mentions that Weinberg may have left out an obvious explanation.
acipsen
Hi,

I am reading through Weinberg's "Quantum Theory of Fields" (vol. 1)
and I am somewhat confused about the signs in the cluster decomposition
of the S-matrix. Specifically, referring to eq. 4.3.2, let's say the term
coming from the partition
$\alpha \to \alpha_1\alpha_2$,$$\beta \to \beta_1\beta_2$$
would be
$$+S^C_{\beta_1,\alpha_1}S^C_{\beta_2,\alpha_2}$$.
Lets now assume that permuting $$\alpha_1,\alpha_2$$ gives a sign, while permuting $$\beta_1,\beta_2$$ doesn't,
then the equivalent partition
$\alpha \to \alpha_2\alpha_1$,$$\beta \to \beta_2\beta_1$$
would give rise to the term
$$-S^C_{\beta_2,\alpha_2}S^C_{\beta_1,\alpha_1}$$,
i.e. minus what I had before. This clearly cannot be right, but I'm not sure where
the flaw in my reasoning is. Does it have something to do with conservation
of $$(-1)^F$$, where $$F$$ is the number of Fermions?

I would be grateful for any hints. And no, this is not homework , the course I'm
following is using Peskin & Schroeder.

Regards,
Asger

Welcome to PF!

Hi Asger! Welcome to PF!

(have an alpha: α and a beta: β and try using the X2 tag just above the Reply box )

Since αi and βi together form a separate experiment, for each i, the number of fermions in αi will be odd only if the number in βi is.

So swapping two experiments will give the same number of fermion interchanges for α as for β.

Hi Tim!

Thanks for the answer. I'm sure Weinberg thought that it too obvious to mention.

Regards,
Asger

acipsen said:
Hi Tim!

Thanks for the answer. I'm sure Weinberg thought that it too obvious to mention.

Regards,
Asger

That is Weinberg's style :) If it is too simple, he does not mention it ;)

## 1. What is the S-matrix in cluster decomposition?

The S-matrix, or scattering matrix, is a mathematical tool used in quantum field theory to calculate the probability amplitudes for particles to interact and scatter with each other. It describes the overall dynamics of a system in terms of the incoming and outgoing particles' states.

## 2. What does cluster decomposition mean in relation to the S-matrix?

Cluster decomposition is a property of the S-matrix that states that the scattering amplitude for a multi-particle process can be factorized into products of amplitudes for individual two-particle processes. This allows for simpler calculations and better understanding of the overall scattering process.

## 3. How does cluster decomposition relate to quantum field theory?

Cluster decomposition is a fundamental property of quantum field theory, which describes the interactions between particles at the most fundamental level. It is a result of the mathematical structure of the theory and is crucial for understanding and calculating scattering processes.

## 4. What are the implications of cluster decomposition for particle physics?

The property of cluster decomposition has important implications for particle physics, as it allows for easier calculations and better understanding of particle interactions. It also helps to explain the behavior of particles in high-energy collisions and plays a crucial role in the development of theories such as the Standard Model.

## 5. Are there any exceptions to cluster decomposition in the S-matrix?

While cluster decomposition is a general property of the S-matrix, there are some exceptions in certain theories or situations. For example, it may not hold in theories with non-local interactions or in the presence of bound states. However, cluster decomposition remains a powerful tool in understanding particle interactions and is applicable in most cases.

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