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I am studying Mandl and Shaw's book on QFT and I am trying to understand the different definitions of the propagator functions, or $\Delta$-functions. One $\Delta$-function is defined (and derived) in one section as

$$ i\hbar c \Delta^+(x-y) = [\phi^+(x), \phi^-(y)] = \frac{ic}{2(2\pi)^3} \int \frac{d^3\textbf{k}}{\omega_{\textbf{k}}} e^{-ik(x-y)}, $$

where $\phi^{\pm}(x)$ are the positive and negative frequency parts of $\phi(x)$ which is a complex scalar field. This derivation is somewhat straightforward and I think I understand it. The problem for me is when the same function is later presented as

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

This seems to imply that

$$ \langle 0 | [\phi^+(x), \phi^-(y)] | 0 \rangle = [\phi^+(x), \phi^-(y)], $$

which looks a bit strange. This definition with the vacuum expectation value seems very important for further chapters but is barely motivated at all. Am I missing something obvious here?

Qmechanic
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Tjommen
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1 Answers1

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It's simply because that commutator is proportional to identity operator, as your result imply. Considering a normalized vacuum state then $$ \left[ \hat{\phi}^+(x), \hat{\phi}^-(y) \right] = \text{const} \, \hat{I} \\ \langle 0 | \left[ \hat{\phi}^+(x), \hat{\phi}^-(y) \right] | 0 \rangle = \text{const}$$

Rob Tan
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