# Understand Mandl & Shaw's Derivation of Moller Scattering

• jdstokes
In summary, Mandl and Shaw have derived the part of the S-operator which affects Moller scattering using Wick's theorem and the fact that two electrons must be annihilated and then created. Four contributions to the Moller transition are then claimed due to each initial (final) electron (positron) being able to be annihilated (created) by either of the \psi^+ (\psi^-) operators. However, I am having trouble understanding why we would get four terms out of this. After a lot of anguish, I managed to get Mandl and Shaw's equation (7.18) to within an overall phase factor of negative one. Could this be an error in the text?
jdstokes
I'm trying to understand Mandl and Shaw's derivation of the part of the S-operator which affects Moller scattering.

Starting with the second term in the Dyson series expansion for the S-matrix, they extract using Wick's theorem the term with a contracted electromagnetic field and four uncontracted fermion fields.

Then using the fact that two electrons must be annihilated and then created they conclude that the relevant term is

$S(2e^-\to 2e^-) = -\frac{e^2}{2!} \iint d^4x_1 d^4x_2 : (\bar{\psi}^-\gamma^\alpha\psi^+)_{x_1}(\bar{\psi}^-\gamma^\beta \psi^+)_{x_2}:\mathrm{i} D_{\mathrm{F}\alpha\beta}(x_1 -x_2)$

where D_F is the Feynman propagator representing transmission of a virtual photon between the spacetime points x_1 and x_2.

Now here comes the tricky bit. They claim that the above expression gives four contributions to the Moller transition since each initial (final) electron (positron) can be annihilated (created) by either of the $\psi^+$ ($\psi^-$) operators.

I'm having trouble seeing why we should get four terms out of this. If we sandwich $S(2e^-\to 2e^-)$ between initial and final Fock states, how can this result in a sum of four terms mathematically?

I'm slowly working my way towards a resolution of this problem.

I am pretty sure it is to do with the fact that the initial and final Fock states consist of linear superpositions because of the fact that the fermions are identical e.g.

$|initial\rangle = \frac{1}{\sqrt{2}}(|p_1\rangle p_2\rangle - |p_2 \rangle p_1\rangle)$.

So far I have not been able to account for the minus sign prefactor in the first term on the RHS of equation (7.18).

After a lot of anguish, I managed to get Mandl and Shaw's (7.18) to within an overall phase factor of negative one. Could this be an error in the text?

## 1. What is Moller scattering and why is it important?

Moller scattering is a type of scattering that occurs when two electrons collide and interact with each other. It is important because it helps us understand the behavior of particles at the subatomic level and is used in various scientific fields such as particle physics and nuclear physics.

## 2. Who are Mandl and Shaw and what is their derivation of Moller scattering?

Mandl and Shaw are two scientists who developed a mathematical formula to describe the process of Moller scattering. Their derivation is a theoretical framework that allows us to calculate the probability of electron-electron interactions and the resulting energy and momentum transfers.

## 3. How does the derivation of Moller scattering work?

The derivation of Moller scattering uses principles from quantum mechanics and relativity to calculate the probability of electron-electron interactions. It involves solving a set of equations that describe the initial and final states of the electrons and the interaction between them.

## 4. What are the assumptions made in Mandl & Shaw's derivation of Moller scattering?

The derivation assumes that the electrons are point particles with no internal structure, and that the interaction between them is described by the electromagnetic force. It also assumes that the collision is elastic, meaning that there is no energy loss during the interaction.

## 5. How is the derivation of Moller scattering applied in scientific research?

The derivation of Moller scattering is used in various scientific research, such as in particle accelerators to study the behavior of subatomic particles, and in nuclear physics to understand the structure of the nucleus. It is also used in theoretical calculations to test the validity of different theories and models in particle physics.

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