- #1
Diracobama2181
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- TL;DR Summary
- I am attempting to find the off-forward quark-quark amplitude in momentum space.
I am having difficulty writing out
##\bra{p',\lambda}\psi^{\dagger}(-\frac{z^-}{2})\gamma^0\gamma^+\psi\frac{z^-}{2})\ket{p,\lambda}## in momentum space.
Here, I am working in light-cone coordinates, where I am defining ##z^-=z^0-z^3##, ##r'=r=(0,z^{-},z^1,z^2)##.
My attempt at this would be
$$\bra{p',\lambda}\psi^{\dagger}(-\frac{z^{-}}{2})\gamma^0\gamma^{+}\psi\frac{z^{-}}{2})\ket{p,\lambda}=\Sigma_{r,r'}\bra{p',\lambda}(\ket{r'}\bra{r'})\psi^{\dagger}(-\frac{z^{-}}{2})\gamma^0\gamma^{+}\psi\frac{z^{-}}{2}(\ket{r}\bra{r})\ket{p,\lambda}\\\\
=\int d^3r exp[i(p'-p)\cdot z^{-}]\psi^{\dagger}(-\frac{z^-}{2})\gamma^0\gamma^+\psi(\frac{z^-}{2})$$.
From here, I can substitute in
##\psi(t,\vec{r})=\int\frac{d^3\vec{k}}{(2\pi)^3}exp[-i(k^0t-\vec{k}\cdot \vec{r})]\phi(\vec{k})##
Is this attempt correct so far, or am I overlooking something? Any comments are appreciated.
##\bra{p',\lambda}\psi^{\dagger}(-\frac{z^-}{2})\gamma^0\gamma^+\psi\frac{z^-}{2})\ket{p,\lambda}## in momentum space.
Here, I am working in light-cone coordinates, where I am defining ##z^-=z^0-z^3##, ##r'=r=(0,z^{-},z^1,z^2)##.
My attempt at this would be
$$\bra{p',\lambda}\psi^{\dagger}(-\frac{z^{-}}{2})\gamma^0\gamma^{+}\psi\frac{z^{-}}{2})\ket{p,\lambda}=\Sigma_{r,r'}\bra{p',\lambda}(\ket{r'}\bra{r'})\psi^{\dagger}(-\frac{z^{-}}{2})\gamma^0\gamma^{+}\psi\frac{z^{-}}{2}(\ket{r}\bra{r})\ket{p,\lambda}\\\\
=\int d^3r exp[i(p'-p)\cdot z^{-}]\psi^{\dagger}(-\frac{z^-}{2})\gamma^0\gamma^+\psi(\frac{z^-}{2})$$.
From here, I can substitute in
##\psi(t,\vec{r})=\int\frac{d^3\vec{k}}{(2\pi)^3}exp[-i(k^0t-\vec{k}\cdot \vec{r})]\phi(\vec{k})##
Is this attempt correct so far, or am I overlooking something? Any comments are appreciated.