Editing Quantum electrodynamics (section)

===Interaction picture===
This theory can be straightforwardly quantized by treating bosonic and fermionic sectors{{clarify|reason=Definition needed.|date=April 2015}} as free. This permits us to build a set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an [[Hamiltonian (quantum mechanics)|evolution operator]], which for a given initial state <math>|i\rangle</math> will give a final state <math>\langle f|</math> in such a way to have<ref name=Peskin/>{{rp|5}}

<math display="block">M_{fi} = \langle f|U|i\rangle.</math>

This technique is also known as the [[S-matrix]]. The evolution operator is obtained in the [[interaction picture]], where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:<ref name=Peskin/>{{rp|123}}

<math display="block">V = e \int d^3 x\, \bar\psi \gamma^\mu \psi A_\mu,</math>

Which can also be written in terms of an integral over the interaction Hamiltonian density <math>\mathcal{H}_I = e \overline \psi \gamma^\mu \psi A_\mu</math>. Thus, one has<ref name=Peskin/>{{rp|86}}

<math display="block">U = T \exp\left[-\frac{i}{\hbar} \int_{t_0}^t dt'\, V(t')\right],</math>

where ''T'' is the [[Path-ordering|time-ordering]] operator. This evolution operator only has meaning as a series, and what we get here is a [[Perturbation theory (quantum mechanics)|perturbation series]] with the [[fine-structure constant]] as the development parameter. This series expansion of the probability amplitude <math>M_{fi}</math> is called the [[Dyson series]], and is given by:

<math display="block">
M_{fi} = \langle f | U |i\rangle 
 =\left\langle f\left|\sum _{n=0}^{\infty }{\frac {(-i)^{n}}{n!}}\int d^4x_{1} \cdots \int d^4x_{n} T \bigg\{ \mathcal{H}(x_{1})\cdots \mathcal {H}(x_{n}) \bigg \} \right|i\right\rangle
</math>