[QUE/QFT-11001] QFT-PROBLEM

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For a quantized real scalar field \(\phi(x)\), define functions \(i\Delta^{(\pm)}(x-y)\) defined by \begin{eqnarray} i\Delta^{(+)}(x-y) &=& \matrixelement{0}{\phi(x)\,\phi(y)}{0} \\ i\Delta^{(-)}(x-y) &=& - \matrixelement{0}{\phi(y)\,\phi(x)}{0}. \end{eqnarray} Use \(\phi^{(\pm)}(x)\) to denote positive and negative frequency parts of the scalar field \(\phi(x)\). Show that

• \(\Delta^{(\pm)}(x-y)\) are real.
• \(\Delta^{(+)}(x-y) = \matrixelement{0}{\big[\phi^{(+)}(x)\,,\,\phi^{(-)}(y)\big]}{0}\) \\ \(\Delta^{(-)}(x-y) = - \matrixelement{0}{\big[\phi^{(-)}(x)\,,\,\phi^{(+)}(y)\big]}{0}\).
• The Feynman propagator \(\Delta_F\) is defined as the vacuum expectation value of time ordered product of fields: \[ \Delta_F(x-y) = \matrixelement{0}{T\big(\phi(x)\phi(y)\big)}{0}.\] Show that the Feynman propagator satisfies the equation \[ (\Box + M^2) \Delta_F(x-y)= - \delta^{(4)}(x-y).\]

Hint Write \(\Delta_F(x-y)\) as \[\Delta_F(x-y)= \Delta^{(+)}(x-y)\theta(x_0-y_0) - \Delta^{(-)}(x-y)\theta(y_0-x_0).\]