# Meson propagator development

intervoxel
Could anyone please explain the sequence below taken from Mandl QFT textbook (p.53)?

1. $i\hbar c\Delta^+(x-x')=[\phi^+(x),\phi^-(x')]$

2. $i\hbar c\Delta^+(x-x')=\langle 0|[\phi^+(x),\phi^-(x')]|0\rangle$

3. $i\hbar c\Delta^+(x-x')=\langle 0|\phi^+(x)\phi^-(x')|0\rangle$

4. $i\hbar c\Delta^+(x-x')=\langle 0|\phi(x)\phi(x')|0\rangle$

From 1. to 2. does it mean that the vacuum expected value of the commutator is the commutator itself? How?

From 2. to 3. does it mean that the term $\langle 0|\phi^-(x')\phi^+(x)|0\rangle$ is null? How?

From 3. to 4. does it mean that the terms

$\langle 0|\phi^+(x)\phi^+(x')|0\rangle$

$\langle 0|\phi^-(x)\phi^+(x')|0\rangle$

$\langle 0|\phi^-(x)\phi^-(x')|0\rangle$

are all null? How?

Thank you for any help.

Yes to 2, 3 and 4. φ+ is an absorption operator, so φ+|0> = 0 since there is nothing to absorb. Likewise <0|φ- = 0.

intervoxel
Thank you. What about the first transformation?