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## Main Question or Discussion Point

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

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

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

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

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

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 [itex]\langle 0|\phi^-(x')\phi^+(x)|0\rangle[/itex] is null? How?

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

[itex]\langle 0|\phi^+(x)\phi^+(x')|0\rangle[/itex]

[itex]\langle 0|\phi^-(x)\phi^+(x')|0\rangle[/itex]

[itex]\langle 0|\phi^-(x)\phi^-(x')|0\rangle[/itex]

are all null? How?

Thank you for any help.

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

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

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

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

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 [itex]\langle 0|\phi^-(x')\phi^+(x)|0\rangle[/itex] is null? How?

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

[itex]\langle 0|\phi^+(x)\phi^+(x')|0\rangle[/itex]

[itex]\langle 0|\phi^-(x)\phi^+(x')|0\rangle[/itex]

[itex]\langle 0|\phi^-(x)\phi^-(x')|0\rangle[/itex]

are all null? How?

Thank you for any help.