What is the explanation for the sequence in the Mandl QFT textbook (p.53)?

[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.

 

[Bill_K]

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?

 

[Bill_K]

I don't have a copy of Mandl to compare, but I'm looking at Weinberg section 6.1 where he says something similar. He's evaluating the S-matrix as a sum of terms, <0| ... |0> where ... is a string of creation and annihilation operators, and he's talking about rearranging the order of the operators. Every time you switch the order of two of them you get a numerical factor.

And on p262 he says: (f) Pairing of a field ψ with a field adjoint ψ^† in H(y) yields a factor -iΔ(x,y). (I'm leaving some subscripts out.) This is close to what you're saying. So I think the context is that Mandl's Eq (1) represents a subexpression that's eventually going to be placed between <0| |0>'s.

 

[Avodyne]

To go from eq.1 to eq.2, sandwich eq.1 between <0| and |0>. Use the fact that the left side of eq.1 is just a c-number (not an operator), and so on the left we just get that function times <0|0>=1.