# Perturbation theory question (in Quarks & Leptons)

Hi

I'm referring to the book Quarks and Leptons (Halzen, Martin). On pages 79-82 nonrelativistic perturbation theory is investigated (i.e. by using the Schroedinger equation, which is first order in time). On Page 85, however, the transition amplitude (T_fi) is used that has been derived on pages 79-82. However, in this chapter (chapter 4) the Klein-Gordon equation is investigated (which is relativistic as well as second order in time).
How can this be right?

thx

why should it be wrong?

Well , the derivation of T_fi (on pages 79-82) uses the fact that the Schroedinger equation is first order in time. If a second order equation is substituted, a differential equation for the coefficients (a_f) results that is quite different from the simple differential equation that results from a first order equation.

In addition it is written in the book that these derivation is (only) a recapitulation of non-relativistic perturbation theory. However, then it is used in the next chapter for the (relativistic) Klein Gordon equation (without any mention as to why this is applicable).

thx

Quantum mechanics is always first order in time. In QFT you need to distinguish between the field and the wave-function, which actually turns into a wave-functional. The equations of motion for the field is turned into a Lagrangian, and then into a Hamiltonian, and it is this Hamiltonian that generates the first-order equations for the wave-functional. It is customary, however, to starting hiding the state when talking about QFT, and instead refer only to the operators. In that case, the field is a field of operators, and one can write a 2nd order differential equation for it; states are then referred to by the operators used to construct them from the vacuum state.