Perturbation theory question (in Quarks & Leptons)

[gop]

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

 

[ansgar]

why should it be wrong?

 

[gop]

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

 

[genneth]

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.