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| CATEGORIES ABOUT DYSON SERIES | |
| scattering theory | |
| quantum field theory | |
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THE DYSON OPERATOR We suppose we have a Hamiltonian H which for some reason or other, we split into a "free" part ''H0'' and an "interacting" part ''V'' i.e. ''H=H0+V''. We will work in the Interaction Picture here. In the Interaction Picture , the ''evolution operator U'' defined by the equation: : is called Dyson operator. We have : and then ( Tomonaga-Schwinger Equation ) : Thus: : DERIVATION OF THE DYSON SERIES This leads to the following Neumann Series : : If we assume that we can say that the fields are Time Ordered , and so it is useful to introduce an operator called '' Time-ordering operator''. Defining: : We can now try to make this integration simpler. in fact, in the following example: : If K is symmetric in its arguments, we can define (look at integration limits): : And so it is true that: : Returning to our previous integral, it holds the identity: : Summing up all the terms we obtain the Dyson series: : THE DYSON SERIES FOR WAVEFUNCTIONS Then, going back to the wavefunction for t>t0, | ||
| :<math>\langle\psi Ft F\psi It I Angle | \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}\langle\psi_ft_fe^{-iH_0(t_f-t_1)}Ve^{-iH_0(t_1-t_2)}\cdots Ve^{-iH_0(t_n-t_i)}\psi_it_i
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