Consider QFT of scalar field

• We know that most of Physics is contained in \({G(x_{1},x_{2},x_{3},....x_{n})\braket{\omega }{T(\pi_{i=1} ^ {N} \hat{\phi(x)}) \ket{ \omega }}\) where,
  • \({x=(\vec{x},t)}\)
  • \(\ket{omega}\) = Interactive ground state
  • \(T(\phi (t_{1})\phi (t_{2}))=\Theta(t_{1}-t_{2})\phi(t_{1})\phi(t_{2}) + \Theta(t_{2}-t_{1})\phi(t_{2})\phi(t_{1})\) i.e. Largest time on left.

The reason greens function is important is that it contains:

1. Information about masses of particles for a given theory; which are given by poles of 2 point functions in momentum space.
2. It contains info about S-matrix elements via LSZ reduction formalism.

1. First goal is to develop new ways of calculating greens function. example: Path integral formalism and study the properties of Green's function; we use path integral formalism because
   1. Canonical formalism breaks manifest Lorentz invariance (i.e we give special role to time in Hamiltonian formalism(Hamiltonian generates time translations not space translations)) but path integral preserves Lorentz invariance.
   2. For case of Hamiltonian formalism; if the interaction term in Lagrangian has time derivative then Hamiltonian might become complicated. Consider 1-d QM: \[ \begin{align*} L &=\frac{1}{2} \dot{q}^2+ \frac{\lambda}{2} \dot{q}^2 q \\ p &= \frac{\partial \mathcal{L}}{\partial q} = \dot{q} + \lambda \dot{q} q \\ & \implies \dot{q}=\frac{p}{1+\lambda q} \\ H &=\dot{q}p-L \\ .\end{align*} \]