- #1

Markus Kahn

- 112

- 14

- TL;DR Summary
- I don't understand why in a particular situation the expectation value of normal ordered fields and partially contracted fields is either irrelevant or zero.

I have trouble understanding the solution to a homework problem.

Consider the interaction Lagragian ##\mathcal{L}_{\rm int} = -iqA_\mu \bar{\psi}\gamma^\mu \psi##, i.e. photon-electron/positron interaction. We want to focus on the Compton scattering

$$e^-(\vec p_1, \alpha) + \gamma(\vec p_2, \sigma)\quad \longrightarrow\quad e^-(\vec q_1, \beta) + \gamma(\vec q_2, \rho),$$

where the ##\vec p_i, \vec q_i## describe the momentum of the particles and the greek letters the spin/polarization. The original question of the homework was to find the matrix elements in ##\mathcal{O}(q^2)##, without the use of Feynman rules and diagrams.

To do this we set up final and initial state

$$\vert i\rangle := a_\alpha^\dagger (\vec p_1)c_\sigma^\dagger (\vec p_2)\vert 0\rangle \qquad\langle f\vert := \langle 0\vert c_\rho (\vec q_2) a_\beta (\vec q_1),$$

where ##a,c## are annihilation operators of electrons and photons respectively,

and expand the time evolution operator ##U_{\rm int} = T[ e^{-iS_{\rm int}}]## up to second order. We can then calculate the matrix element:

$$

\begin{align*}

F^{(2)}

&= -\frac{i}{2}q^2 \int d^4 x d^4 y \langle f\vert T[A_\mu(x) \bar{\psi}(x)\gamma^\mu \psi(x)A_\nu(y) \bar{\psi}(y)\gamma^\mu \psi(y)] \vert i \rangle\\

&= -\frac{i}{2}q^2 \int d^4 x d^4 y \langle 0\vert c_\rho (\vec q_2) a_\alpha (\vec q_1) T[A_\mu(x) \bar{\psi}(x)\gamma^\mu \psi(x)A_\nu(y) \bar{\psi}(y)\gamma^\mu \psi(y)] a_\alpha^\dagger (\vec p_1)c_\sigma^\dagger (\vec p_2)\vert 0\rangle

\end{align*}

$$

To evaluate this we need to replace the time ordering operator ##T## with the normal ordering operator ##N## plus all

Here is where the problem starts:

Consider the interaction Lagragian ##\mathcal{L}_{\rm int} = -iqA_\mu \bar{\psi}\gamma^\mu \psi##, i.e. photon-electron/positron interaction. We want to focus on the Compton scattering

$$e^-(\vec p_1, \alpha) + \gamma(\vec p_2, \sigma)\quad \longrightarrow\quad e^-(\vec q_1, \beta) + \gamma(\vec q_2, \rho),$$

where the ##\vec p_i, \vec q_i## describe the momentum of the particles and the greek letters the spin/polarization. The original question of the homework was to find the matrix elements in ##\mathcal{O}(q^2)##, without the use of Feynman rules and diagrams.

To do this we set up final and initial state

$$\vert i\rangle := a_\alpha^\dagger (\vec p_1)c_\sigma^\dagger (\vec p_2)\vert 0\rangle \qquad\langle f\vert := \langle 0\vert c_\rho (\vec q_2) a_\beta (\vec q_1),$$

where ##a,c## are annihilation operators of electrons and photons respectively,

and expand the time evolution operator ##U_{\rm int} = T[ e^{-iS_{\rm int}}]## up to second order. We can then calculate the matrix element:

$$

\begin{align*}

F^{(2)}

&= -\frac{i}{2}q^2 \int d^4 x d^4 y \langle f\vert T[A_\mu(x) \bar{\psi}(x)\gamma^\mu \psi(x)A_\nu(y) \bar{\psi}(y)\gamma^\mu \psi(y)] \vert i \rangle\\

&= -\frac{i}{2}q^2 \int d^4 x d^4 y \langle 0\vert c_\rho (\vec q_2) a_\alpha (\vec q_1) T[A_\mu(x) \bar{\psi}(x)\gamma^\mu \psi(x)A_\nu(y) \bar{\psi}(y)\gamma^\mu \psi(y)] a_\alpha^\dagger (\vec p_1)c_\sigma^\dagger (\vec p_2)\vert 0\rangle

\end{align*}

$$

To evaluate this we need to replace the time ordering operator ##T## with the normal ordering operator ##N## plus all

**partial and full**normal ordered contractions.Here is where the problem starts:

- The solutions to the exercise now claim that we don't need to consider the partially contracted terms, but only the fully contracted ones. I don't see why this is supposed to be true. If we were looking at something of the form ##\langle 0\vert T[\dots]\vert 0\rangle##, then I would agree, since any field inside the time ordering operator that gets not contracted will annihilate the vacuum. But since we have ##\langle f \vert T[\dots]\vert i\rangle##, we can't be sure that the vacuum will directly get annihiliated. Can someone explain the reasoning here?
- The second claim was that we can ignore the uncontracted normal ordered fields, i.e. If ##T[\dots] = N[\dots] + \text{contractions}##, then ##\langle f\vert N[\dots] \vert i\rangle## is irrelevant. I don't see why. Chances are that if will not be zero, so does this generate only loop diagrams or tadpoles? How can I see this?