[QUE/QFT-04001] QFT-PROBLEM

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Consider free Schrodinger equation as a quantized field.

• Show that \begin{equation} G(x -x{'}, t -t{'} ) = \matrixelement{0}{T ψ\psi(x, t)\psi^\dagger (x{'}, t{'}))}{0} \end{equation} obeys the equation for Green function of the free particle Schrodinger equation.
• Use expansion of the field operators in terms of free particle wave function \(N \exp(ikx - iE_k t)\), where \(E_k = \frac{\hbar^2k^2}{2m}\). Obtain an explicit expression for this time ordered product as a function of \(x, t, x{'} , t{'}\).
• Have you seen this object before? Where?