Average Fermion Current: Understanding the Relation to Background Fermions

In summary, the conversation discusses a problem with averaging a 4-fermion interaction over a background consisting of fermions. The left-handed current is approximated by its average value and a relation is provided for the averaged value produced by the interaction. The remaining terms in the relation are explained in terms of an exchange term and a Fierz transformation. A reference is requested for the averaging procedure and it is clarified that it is not a Fierz reshuffling.
  • #1
parton
83
1
Hi!

I have a little problem.

Consider a 4-fermion interaction (neglecting constant factors) of the form [itex] \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} [/itex] .
I want to average this interaction over a background consisting of fermions (so it corresponds to the situation where fermions propagate in a background consisting of fermions).

To this purpose, the left-handed current [itex] \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} [/itex] is approximated by the average value [itex] \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \rangle [/itex]

There is the following relation for the averaged value produced by this interaction:
[tex]
\begin{align}
\begin{split}
\overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} \to & \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \rangle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} + \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} \rangle
\\
& \quad + \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{d \mathrm{L}} \rangle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}} + \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{d \mathrm{L}} \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}} \rangle
\\
& \quad - \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{b \mathrm{L}} \rangle \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} \rangle - \langle \overline{\psi_{a \mathrm{L}}} \gamma^{\lambda} \psi_{d \mathrm{L}} \rangle \langle \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}} \rangle.
\end{split}
\end{align}
[/tex]

But I don't really understand how to obtain this relation. The first two terms look reasonable, but I don't understand the remaining ones.

Ok, the 3rd and 4th terms might correspond to an "exchange term", where fermion c and d are interchanged, because in the case where c is equal to d, we cannot distinguish between the propagating and the background fermion (on the other hand, if c is not equal to d, this exchange term should be 0). But why is there no additional minus sign, because of Fermi-Dirac statistics?

Maybe it has something to do with a Fierz transformation, where the minus sign cancels out, i.e.,
[tex] \overline{\psi}_{a \mathrm{L}} \gamma^{\lambda} \psi_{b \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{d \mathrm{L}} = \overline{\psi}_{a \mathrm{L}} \gamma^{\lambda} \psi_{d \mathrm{L}} \overline{\psi_{c \mathrm{L}}} \gamma_{\lambda} \psi_{b \mathrm{L}}.[/tex]

And how could the last two terms be interpreted?

I hope somebody could help me understanding the relation above.
 
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  • #2
can you provide a reference for that averaging procedure.How is it defined.Also it is not Fierz reshuffling otherwise you will also get pseudoscalar and pseudovector part(apart from identity)
 

FAQ: Average Fermion Current: Understanding the Relation to Background Fermions

What is the definition of average fermion current?

The average fermion current is a measure of the flow of fermions, which are particles with half-integer spin, through a given space or system. It is a key quantity in quantum field theory and is often used to study the behavior of elementary particles and their interactions.

How is average fermion current calculated?

The average fermion current is typically calculated using a mathematical formula that involves the expectation value of the fermion field operator. This operator describes the behavior of fermions in a given system and can be used to determine the average number of fermions passing through a specific point in space over a certain period of time.

What is the significance of average fermion current in physics?

The study of average fermion current is important in many areas of physics, including particle physics, condensed matter physics, and cosmology. It provides insights into the properties and behavior of elementary particles and can help researchers better understand the fundamental forces of nature.

How does average fermion current differ from average boson current?

The main difference between average fermion current and average boson current lies in the spin of the particles being studied. While fermions have half-integer spin, bosons have integer spin. This difference in spin leads to distinct behaviors and characteristics, which are reflected in their respective average currents.

What are some applications of average fermion current?

Average fermion current has many practical applications, such as in the development of new technologies, such as quantum computing and nanotechnology. It also plays a crucial role in understanding the behavior of matter in extreme conditions, such as in high-energy particle accelerators and in the early universe.

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