- #1
rayohauno
- 21
- 0
I need to solve an exercise on Quantum Field Theory that reads as follows:
Supousse that the electron has an anomalous magnetic moment, which makes the QED Lagrangian
(density) to have an additional term:
[tex]
L'_I(x) = \frac{2ie}{m} \bar{\psi}(x) \sigma^{\alpha\beta} \psi(x) F_{\alpha\beta}(x)
[/tex]
where:
[tex]
F_{\alpha\beta} = \partial_{\alpha}A_{\beta}(x) - \partial_{\beta}A_{\alpha}(x)
[/tex]
1. Find the matrix element (Feynman amplitude) for Moller's dispersion, having in account this additional
term.
*****************
I found that
[tex]
\partial_{\alpha}\beta = \sum_k (\mp i k_{\alpha}) A^{\pm}_{\beta}(x)
[/tex]
from where, I found that:
[tex]
2i\sigma^{\alpha\beta}F_{\alpha\beta}(x) = -i2[\check{K},\check{A}^+ - \check{A}^-]
[/tex]
where:
[tex]
\check{K},\check{A}^+,\check{A}^-
[/tex]
denotes slash operators ! (I couldn't find how to draw slash operators here). The problem that I have
as from here is, that I don't know how to calculate Time Contractions to the operator:
[tex]
A^+ - A^-
[/tex]
for example, I don't know how to calculate the Feynman propagator to the Time Contraction:
[tex]
T\{ A(x)[A^+(y) - A^-(y)] \}
=
T\{ [A^+(x) + A^-(x) ][A^+(y) - A^-(y)] \}
[/tex]Does anyone knows how to proceed in this case??
best regards
Rayo
Supousse that the electron has an anomalous magnetic moment, which makes the QED Lagrangian
(density) to have an additional term:
[tex]
L'_I(x) = \frac{2ie}{m} \bar{\psi}(x) \sigma^{\alpha\beta} \psi(x) F_{\alpha\beta}(x)
[/tex]
where:
[tex]
F_{\alpha\beta} = \partial_{\alpha}A_{\beta}(x) - \partial_{\beta}A_{\alpha}(x)
[/tex]
1. Find the matrix element (Feynman amplitude) for Moller's dispersion, having in account this additional
term.
*****************
I found that
[tex]
\partial_{\alpha}\beta = \sum_k (\mp i k_{\alpha}) A^{\pm}_{\beta}(x)
[/tex]
from where, I found that:
[tex]
2i\sigma^{\alpha\beta}F_{\alpha\beta}(x) = -i2[\check{K},\check{A}^+ - \check{A}^-]
[/tex]
where:
[tex]
\check{K},\check{A}^+,\check{A}^-
[/tex]
denotes slash operators ! (I couldn't find how to draw slash operators here). The problem that I have
as from here is, that I don't know how to calculate Time Contractions to the operator:
[tex]
A^+ - A^-
[/tex]
for example, I don't know how to calculate the Feynman propagator to the Time Contraction:
[tex]
T\{ A(x)[A^+(y) - A^-(y)] \}
=
T\{ [A^+(x) + A^-(x) ][A^+(y) - A^-(y)] \}
[/tex]Does anyone knows how to proceed in this case??
best regards
Rayo