- #1

Glenn Rowe

Gold Member

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## Main Question or Discussion Point

I'm reading through a couple of books (Lahiri & Pal's "A First Book of Quantum Field Theory" and Greiner & Reinhardt's "Field Quantization" and have come to the derivation of the evolution operator which leads to the S-matrix. In both books, the derivation starts with the Schrodinger equation in the form $$i\frac{d}{dt}\left|\Psi\left(t\right)\right\rangle =\left(H_{0}+H_{I}\right)\left|\Psi\left(t\right)\right\rangle$$ where ##H_0## is the free-field Hamiltonian and ##H_I## is the interaction Hamiltonian. The derivation from this point on is fairly clear in both books, and leads to the differential equation for the evolution operator ##U(t)##

$$

\begin{align}

i\frac{dU\left(t\right)}{dt} & =U_{0}^{\dagger}\left(t\right)H_{I}U_{0}\left(t\right)U\left(t\right)\\

& =H_{I}\left(t\right)U\left(t\right)

\end{align}$$

which leads to the time-ordered product and Wick's theorem.

My question is: is it valid to use the Schrodinger equation (which I always assumed was for nonrelativistic QM) to derive things in a relativistic quantum field theory? Neither book makes any comment on this point. I would have thought you'd need to use the Klein-Gordon or Dirac equation, depending on the particles concerned.

Thanks for any insight.

$$

\begin{align}

i\frac{dU\left(t\right)}{dt} & =U_{0}^{\dagger}\left(t\right)H_{I}U_{0}\left(t\right)U\left(t\right)\\

& =H_{I}\left(t\right)U\left(t\right)

\end{align}$$

which leads to the time-ordered product and Wick's theorem.

My question is: is it valid to use the Schrodinger equation (which I always assumed was for nonrelativistic QM) to derive things in a relativistic quantum field theory? Neither book makes any comment on this point. I would have thought you'd need to use the Klein-Gordon or Dirac equation, depending on the particles concerned.

Thanks for any insight.