Definitions

$A=⟨\text{out}|S|\text{in}⟩$

where S is the S-matrix or Scattering matrix that relates that relates the initial and final states of a physical system undergoing a scattering process.

$⟨\text{out}|$ and $|\text{in}⟩$ are the states at $t=\pm \infty$. Hence, they are non-interacting states and are the direct products of single-particle states, with definite on-shell momentum and polarisation $|p_i,{ϵ}_i⟩$, where i is the particle label.

On-shell momentum: $p_i^2=+(p_i^0{)}^2-{\vec{p}}_i^2=m_i^2$

Metric convention: ${\eta }_{\mu \nu }=\text{diag}(+1,-1,-1,-1)$

We decompose the S-matrix as $S=1+iT$, so that we can separate out the interaction effects into a new matrix called the T-matrix.

LSZ Reduction Formula:

$⟨\text{out}|iT|\text{in}⟩={\prod }_{i=1}^{p_i+p_f}\sqrt{Z_i}(\text{Sum of all amputated and connected Feynman graphs})$

There are $n_i$ incoming and $n_f$ outgoing legs. Z is the wavefunction renormalisation.

Feynman Diagrams	Amplitudes
- Not gauge invariant	- Gauge invariant
- Have off-shell states in propagators	- On-shell
	- Hidden symmetries