- #1

- 118

- 7

## Summary:

- Why can the perturbation matrix element be replaced with the transition part of the S-matrix?

## Main Question or Discussion Point

Hey there,

This question was asked elsewhere, but I wasn't really satisfied with the answer.

When I learned about Fermi's golden rule, ##{ \Gamma }_{ if }=2\pi { \left| \left< { f }|{ \delta V }|{ i } \right> \right| }^{ 2 }\rho \left( { E }_{ f } \right)##, it was derived from first order perturbation theory in the context of quantum mechanics.

In the context of QFT, the perturbation was replaced by the transition part ##\hat { T }## of the S-matrix, ##\hat { S } ≔\hat { I } +i\hat { T }##. However, ##\hat { T }## is not necessarily given only up to first order, so why can we just make this replacement in general?

This question was asked elsewhere, but I wasn't really satisfied with the answer.

When I learned about Fermi's golden rule, ##{ \Gamma }_{ if }=2\pi { \left| \left< { f }|{ \delta V }|{ i } \right> \right| }^{ 2 }\rho \left( { E }_{ f } \right)##, it was derived from first order perturbation theory in the context of quantum mechanics.

In the context of QFT, the perturbation was replaced by the transition part ##\hat { T }## of the S-matrix, ##\hat { S } ≔\hat { I } +i\hat { T }##. However, ##\hat { T }## is not necessarily given only up to first order, so why can we just make this replacement in general?