[QUE/QFT-04005] QFT-PROBLEM

Taking the case of free Schrodinger field answer the following questions.

• Find Heisenberg equations of motion for the operators \(a(k,t)\) and \(a^\dagger(k,t)\)
• Solve the equations of motion and expre
• \(a(k,t)\) and \(a^\dagger(k,t)\) as functions of time. Calculate the unequal time commutators \[\big[a(k,t), a(k{'}, t{'})\big],\quad \big[a^\dagger(k,t), a^\dagger(k{'}, t{'})\big],\quad \big[a(k,t), a^\dagger(k{'}, t{'})\big].\]
• Use your answers and work out the unequal time commutator \[\big[\psi(x,t),\psi^\dagger(x{'},t{'})\big].\]
• Use your result for unequal time commutator and express \(\psi(x_1,t_1) \psi^\dagger(x_2,t_2)\) in a normal ordered form.