dyson series

  Dyson operator

In quantum mechanics, suppose we have a Hamiltonian H which for some reason or other, we split into a "free" part H₀ and an "interacting" part H_int i.e. H=H₀+H_int. We will work in the interaction picture here. Then, for t>t₀,

|\psi(t)>=\sum_{n=0}^\infty {(-i)^n\over n!}\left(\prod_{k=1}^n \int_{t_0}^t dt_k\right) \mathcal{T}\left\{\prod_{k=1}^n e^{iH_0 t}H_{int}e^{-iH_0 t}\right \}|\psi(t_0)>

where

\mathcal{T}

is the time-ordering operator. This is a perturbative series which typically diverges asymptotically. This is called the Dyson series. Returning to the Schrödinger picture, for t_f > t_i,

<\psi_f;t_f|\psi_i;t_i>=\sum_{n=0}^\infty (-i)^n\begin{matrix}\underbrace{\int dt_1 \cdots dt_n}\\t_f\ge t_1\ge \dots\ge t_n\ge t_i\end{matrix}<\psi_f;t_f|e^{-iH_0(t_f-t_1)}H_{int}e^{-iH_0(t_1-t_2)}\cdots H_{int}e^{-iH_0(t_n-t_i)}|\psi_i;t_i>

This is the old-fashioned way of deriving the Feynman diagrams.

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