under construction

Content

Idea

In perturbation theory, the Dyson series is a perturbative expansion of a unitary time evolution operator, $U(t,t_0)$, primarily employed in quantum field theory. While it is asymptotically divergent, the second-order deviation from experimental data is on the order of ${10}^{-10}$ paradoxically making it the most empirically accurate prediction in all of physics. It is a starting point for the use of Feynman diagram?s.

The Dyson series

The Dyson series is given as

$U(t,t_0)={\sum }_{n=0}^{\mathrm{\infty }}{U}_n(t,t_0)=𝒯e^{-i{\int }_{t_0}^{t}d\tau V(\tau )}$

where $𝒯$ is a time-ordering operator and $V$ is the interaction portion of the Hamiltonian.

Use as an operator

Applying $U(t,t_0)$ to some initial quantum state $\mid \psi (t_0)⟩$, we have

$\mid \psi (t)⟩=U(t,t_0)\mid \psi (t_0)⟩={\sum }_{n=0}^{inty}\frac{(-i{)}^n}{n!}\left({\prod }_{k=1}^n{\int }_{t_0}^t{\mathrm{dt}}_k\right)𝒯\left\{{\prod }_{k=1}^n{e}^{{\mathrm{iH}}_0{t}_k}{\mathrm{Ve}}^{-{\mathrm{iH}}_0{t}_k}\right\}\mid \psi (t_0)⟩$.