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Recently I was confronted with some difficulties in understanding the perturbative expansion of S matrix .

The conventional treatment is expansing it in the interaction picture,which have to first transform Lagrangian to Hamiltonian and then replace the original field operator by new operator under the interaction picture. If the interactive part of Lagrangian lacks derivative, especially the time derivative, things sitll can be well handled by few steps of calculation and just simplily "treat" the interactive Lagrangian as interactive Hamiltonian( they share a same form).

But if the interactive part of Lagrangian contains time derivative,the interactive Hamiltonian would be much more different with the form of interactive Lagrangian.Also the expansion at each order break the Lorentz invariance.

I know there is another way of calculating S matrix by applying LSZ formula and path integration. Such kind of expansion guarantees the Lorentz invariance at each order and seems to be more direct and beautiful.

I really can't work out the connections between both of the two different expansions.Could anyone tell me some things about it.