Editing S-matrix (section)

== Evolution operator ''U'' ==
Define a time-dependent creation and annihilation operator as follows,
<math display="block">\begin{align}
a^{\dagger}{\left(k,t\right)} &= U^{-1}(t) \, a^{\dagger}_{\rm i}{\left(k\right)} \, U{\left( t \right)} \\[1ex]
          a{\left(k,t\right)} &= U^{-1}(t) \,           a_{\rm i}{\left(k\right)} \, U{\left( t \right)} \, ,
\end{align}</math>
so, for the fields,
<math display="block">\phi_{\rm f}=U^{-1}(\infty)\phi_{\rm i} U(\infty)=S^{-1}\phi_{\rm i} S~,</math>
where
<math display="block">S= e^{i\alpha}\, U(\infty).</math>

We allow for a phase difference, given by
<math display="block">e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1} ~,</math>
because for {{mvar|S}},
<math display="block">S\left|0\right\rangle = \left|0\right\rangle \Longrightarrow \left\langle 0|S|0\right\rangle = \left\langle 0|0\right\rangle =1 ~.</math>

Substituting the explicit expression for {{mvar|U}}, one has
<math display="block">S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau H_{\rm{int}}(\tau)}}~,</math>
where <math> H_{\rm{int}}</math> is the interaction part of the Hamiltonian and <math> \mathcal T </math> is the time ordering.

By inspection, it can be seen that this formula is not explicitly covariant.