[QUE/QFT-11002] QFT-PROBLEM

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Let \(\Delta_F(x-y)\) be defined in terms of the commutator by \[i \Delta^{(\pm)}(x-y)=\big[\phi^{(\pm)}(x),\phi(y)^{(\mp)}\big]\] where \(\phi(x)\) is quantized free scalar field, Let \(\Delta^\text{ret}(x-y), \Delta^\text{adv}(x-y)\) and be the retarded, advanced Green functions \[\Delta^\text{ret}(x-y)= \Delta^{(+)}(x-y)\theta(x_0-y_0)\] \[\Delta^\text{adv}(x-y)= -\Delta^{(-)}(x-y)\theta(y_0-x_0)\] and \(\Delta_F(x-y)\) be the Feynman propagator defined by \[\Delta_F(x-y) =\matrixelement{0}{T\big(\phi(x)\phi(y)}{0}\big)\]

• Show that \begin{eqnarray} \Delta^{(+)}&=&\matrixelement{0}{\phi(x)\phi(y)}{0}\\ \Delta^{(-)}&=&\matrixelement{0}{\phi(y)\phi(x)}{0} \end{eqnarray} 
• Show that \begin{eqnarray} \Delta_F(x)&=&\Delta^\text{ret}(x) -\Delta^\text{adv}(x-y)\\ \end{eqnarray}
• Show that all the three functions, \(\Delta^\text{ret}, \Delta^\text{adv}\) and \(\Delta_F\) obey the equation \[ (\Box + m^2)G(x-y)=-\delta^{(4)}(x-y)\] for Green function for the Klein Gordon field.