1. Need of QFT:
   1. QM is not compatible with spatial theory of relativity as Schrodinger equation is first order in time and second order in space. An attempt was made to use relativistic Hamiltonian of particle (\(H=\sqrt{p^2+m^2}\) and using \(\hat{p}=-i\partial_{x},\hat{H}=i\partial_{t}\)) which resulted in Klein-Gordon equation (\((\partial^2+m^2)\psi=0\)) for wave function \(\psi\). This solution has the following problems.
      1. This attempt results in negative energy eigenvalues (using \(\psi=e^{ik.x}\) implies \(E=\pm\sqrt{p^2+m^2}\)), we can not ignore these negative energy states since once system is in contact with heat bath it is going to do transition in all possible states.
      2. For free particles, what does it mean to have negative energy?
      3. \(\psi\) have interpretation of probability density, it is not constant and is time dependent due to KG equation being second order in time. Dirac came up with first order equation in space and time (only for spinors) but still had the problem of negative energy solutions, which he tried explaining by stating that all negative energy states are pre occupied and there is no more state left for other spinors(spin 1/2) to go back to negative energy states. This solution offered explanation of positrons (if negative energy states jumps to positive energy states (electron), then the absence of such state is an positron). Dirac solution could only explain spin 1/2, but what about other kind of particles?
      4. Here x(space),t(time) are still not on equal footing, since 'x' is an operator (\(\hat{x}\)) whereas time 't' is just a label.
      5. It only describes spinless particles what about spin 1/2, spin 1?
   2. Schrodinger equation describes fixed number of particles, it does not explain particle creation and destruction. At very small scale (\(\Delta x \le \frac{1}{4m}\)) Heisenberg uncertainty principle implies that new particle anti-particle can be created. So we can not use QM at such small scales. You can not explain the process of creation of so many photons from the sun, tube light and all light sources without use of QFT.
   3. It also does not explain how identical particles are "identical" in all respect with no error in their formation.