- #1
wod58
- 2
- 0
Hi guys,
If I use the definition of the scalar complex field as the combination of two scalar real fields, I can get
\phi (x) = \int \frac{d^3 p}{(2\pi )^3} \frac{1}{\sqrt{2p_0}} [ \hat a _{\vec{p}} e^{-ip.x} + \hat b _{\vec{p}}^{\dagger } e^{ip.x}]
which I can rewrite in terms of (like in Peskin & Schroeder)
\phi (x) = \phi ^{+} (x) + \phi ^{-} (x)
where \langle 0|\phi ^{-} = 0 and \phi ^{+} |0\rangle = 0.
My problem is: when you try to calculate the contraction of the field with itself
\text{\contraction}\{\phi (x)\}\{\phi (y)\} = \begin{cases} [\phi ^{+} (x), \phi ^{-} (y)] , & \text{if } x_0 > y_0 \\ [\phi ^{+} (y), \phi ^{-} (x)] , & \text{if } x_0 < y_0 \end{cases}
which is supposed to be the Feynman Propagator, you obtain it for a scalar real field, but for a scalar complex field as defined above, you obtain terms with \hat a _{\vec{p}} \hat b _{\vec{p \prime}}^{\dagger }. The operators commute, so the vacuum expectation of these terms would be 0.
I guess I'm wrong, but can someone see where? :)
If I use the definition of the scalar complex field as the combination of two scalar real fields, I can get
\phi (x) = \int \frac{d^3 p}{(2\pi )^3} \frac{1}{\sqrt{2p_0}} [ \hat a _{\vec{p}} e^{-ip.x} + \hat b _{\vec{p}}^{\dagger } e^{ip.x}]
which I can rewrite in terms of (like in Peskin & Schroeder)
\phi (x) = \phi ^{+} (x) + \phi ^{-} (x)
where \langle 0|\phi ^{-} = 0 and \phi ^{+} |0\rangle = 0.
My problem is: when you try to calculate the contraction of the field with itself
\text{\contraction}\{\phi (x)\}\{\phi (y)\} = \begin{cases} [\phi ^{+} (x), \phi ^{-} (y)] , & \text{if } x_0 > y_0 \\ [\phi ^{+} (y), \phi ^{-} (x)] , & \text{if } x_0 < y_0 \end{cases}
which is supposed to be the Feynman Propagator, you obtain it for a scalar real field, but for a scalar complex field as defined above, you obtain terms with \hat a _{\vec{p}} \hat b _{\vec{p \prime}}^{\dagger }. The operators commute, so the vacuum expectation of these terms would be 0.
I guess I'm wrong, but can someone see where? :)
