Today, in my advanced particle physics class, the professor reminded the time-dependent perturbation theory in NRQM and derived the formula:(adsbygoogle = window.adsbygoogle || []).push({});

##\displaystyle \frac{da_m(t)}{dt}=-i \sum_n e^{-i(E_n-E_m)} \int_{\mathbb R^3}d^3 x \phi^*_m (\vec x) V(\vec x,t) \phi_n(\vec x)##.

Then he said that this is true also for Klein-Gordon and Dirac equations which confused me. I asked him how can that be true because we assumed an equation of the form ## (H_0+V)\psi = i \frac{\partial \psi}{\partial t} ## and started from eigenfunctions of ## H_0 ## to do the perturbation but Dirac and Klein-Gordon equations are not of this form. His answer wasn't convincing too. So I investigated a little bit and then suddenly remembered the functional Schrodinger equation. Now I think that using the functional Schrodinger equation, we can derive a similar formula:

##\displaystyle \frac{da_m(t)}{dt}=-i \sum_n a_n(t) \int D \phi \ \psi^*_m [\phi] V[\phi] \psi_n[\phi] e^{-i(E_n-E_m)}##

and interpret the ## a_n ##s as transition amplitudes between different field configurations. Is this true?

Thanks

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# Functional time-dependent perturbation theory

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