Coupling constant is turned off adiabatically? To me, adiabatic processes are idealisation. What do people mean with statements such as: "turning off the coupling constant (in QED say) adiabatically"?
 A: In general it means varying (or turning off an interaction in your particular question) a parameter on a time scale that is much larger  than the smallest energy separation of your Hamiltonian. More explicitly : Suppose you have a Hamiltonian $H$ with energy levels $E_n$ and suppose that $\left|E_a-E_b\right|$ ($a\neq b$) is your smallest level splitting (assume no degeneracy for simplicity). Then suppose you turn on some knob to include a magnetic field in your system : $H_{\text{total}}=H+H_B(t)$, where $H_B(t)$ is the time dependent part of your Hamiltonian that couples to the magnetic field $B(t)$. The adiabatic limit is when the time scale, $\tau$ over which you turn on $H_B$ satisfies $\tau\gg\hbar/\left|E_a-E_b\right|$.
Of course this is all discussed in more details for instance on the wikipedia page.
A: This means that you assume the energy of the initial state is approximately conserved and the initial state goes over smoothly into the state from which it originates when first adding the perturbation to your theory.
Adiabatic here means "without changing the energy".
