# Can I add a classical potential to S-matrix

I'm a junior learner of QFT, and I wonder if I can add a classical approximated potential (like Coulomb potential) to a total interaction $$$$V=V_{\mathrm{Coulomb}}+V_{\mathrm{internal}}$$$$ Where $$V_{\mathrm{internal}}$$ is calculated by $$V_{\mathrm{internal}}=\int{H(x)d^3x}$$ Therefore the $S$ matrix is $$S=T \exp \left [-i\int_{-\infty}^{\infty}{dt} {V_{\mathrm{internal}}(t)}+{V_{\mathrm{external}}(-t)} \right]$$ $V_{\mathrm{external}}(-t)$ is to suppose the external field doesn't change by time so I reversed the time evolution picture. The $S$ matrix can still be expanded by Dyson series. So during a collison reaction, like electron postrion annihilation in solid, can I add the external potential accroding to the DFT to the Dyson series to make a correction of $S$-matrix? Indeed it's a violation of unitrary of $S$ matrix, but still, can it be applied as a perturbation?

• Please some one help me – Mmmao Oct 1 at 10:53
• Please someone help me – Mmmao Oct 3 at 5:00
• Why won’t someone help me – Mmmao Oct 3 at 6:44