Photon emission/absorption by an atom and local gauge invariance I understand that the local gauge invariance leads to a photon emission/absorption when the phase of an electron field is changed while the amplitude being unchanged. 
I'd like to know whether this is true for the case of a photon emission/absorption by an atom, for example, in the Lyman alpha transition 2p-1s in hydrogen. If not, is there any alternative principle for this kind of photon emission/absorption other then the gauge principle? 
 A: The solution of the  Schrodinger equation for the Hydrogen atom (and the Dirac equation yields the same solutions)  is very successful in giving the Lyman series. It is a postulate that the difference in the energy levels leaves as a real photon, and gives the observed spectra. This is first quantization.
When one talks of gauge invariance  leading to emission absorption from electrons one is in the realm of second quantization, field theory.

QED is our first complete example of an interacting Quantum Field Theory.
....
We have written the fields of the photon and the electron in terms of creation and annihilation operators.

This lecture explores the Feynman diagrams needed to treat the electron of the hydrogen atom in QED.
In PDF page 24 of this book it is clear that in the hydrogen atom photons are emitted because of creation and annihilation operators  having a large probability for transition. So for the hydrogen atom your  statement "the local gauge invariance leads to a photon emission/absorption" is irrelevant. Actually you do not give a link so I cannot think how relevant it is even for Compton scattering ( absorption and emission of a photon  ) page 33 and on in the same link.
