P. Grant, Gauge invariance and relativistic radiative transitions, Journal of Physics B: Atomic and Molecular Physics, vol.7, issue.12, p.1458, 1974.
DOI : 10.1088/0022-3700/7/12/007

C. K. Aharonov and . Au, Gauge invariance and pseudoperturbations, Physical Review A, vol.20, issue.4, p.1553, 1979.
DOI : 10.1103/PhysRevA.20.1553

T. Epstein, On the gauge invariance of transition probabilities, Chemical Physics Letters, vol.65, issue.3, p.417, 1979.
DOI : 10.1016/0009-2614(79)80262-6

E. Grynberg and . Giacobino, Choice of gauge and multiphoton transitions, Journal of Physics B: Atomic and Molecular Physics, vol.12, issue.3, p.93, 1979.
DOI : 10.1088/0022-3700/12/3/005

R. B. Haller and . Sohn, Interaction of Schr??dinger electrons and photons, Physical Review A, vol.20, issue.4, p.1541, 1979.
DOI : 10.1103/PhysRevA.20.1541

I. Olariu, C. B. Popescu, and . Collins, Questions of gauge and basis sets for the representation of electromagnetic interactions in quantum mechanics, Physical Review D, vol.20, issue.12, p.3095, 1979.
DOI : 10.1103/PhysRevD.20.3095

H. Kobe and K. Yang, Gauge-invariant non-relativistic limit of an electron in a time-dependent electromagnetic field, Journal of Physics A: Mathematical and General, vol.13, issue.10, p.3171, 1980.
DOI : 10.1088/0305-4470/13/10/014

P. Leubner and . Zoller, Gauge invariant interpretation of multiphoton transition probabilities, Journal of Physics B: Atomic and Molecular Physics, vol.13, issue.18, p.3613, 1980.
DOI : 10.1088/0022-3700/13/18/013

H. Kobe, E. C. Wen, and K. Yang, Power operator in quantum mechanics, Physical Review D, vol.26, issue.8, p.1927, 1982.
DOI : 10.1103/PhysRevD.26.1927

K. Yang, Gauge-invariant time-dependent perturbation theory. I. Non-degenerate case, Journal of Physics A: Mathematical and General, vol.15, issue.4, p.1201, 1982.
DOI : 10.1088/0305-4470/15/4/023

T. E. Kazes, P. H. Feuchtwang, H. Cutler, and . Grotch, Gauge invariance and gauge independence of the S-matrix in nonrelativistic quantum mechanics and relativistic quantum field theories, Annals of Physics, vol.142, issue.1, p.80, 1982.
DOI : 10.1016/0003-4916(82)90229-9

K. Yang, Gauge-invariant time-dependent perturbation theory. II. degenerate case, Journal of Physics A: Mathematical and General, vol.16, issue.5, p.919, 1983.
DOI : 10.1088/0305-4470/16/5/012

K. Au, On gauge invariance in quantum transition amplitudes, Journal of Physics B: Atomic and Molecular Physics, vol.17, issue.3, p.59, 1984.
DOI : 10.1088/0022-3700/17/3/001

E. Feuchtwang, E. Kazes, and P. H. Cutler, Comment on K.-H. Yang's energy operator and gauge independent transition amplitudes, Journal of Physics A: Mathematical and General, vol.17, issue.5, p.1157, 1984.
DOI : 10.1088/0305-4470/17/5/036

E. Feuchtwang, E. Kazes, P. H. Cutler, and H. Grotch, The physical significance of gauge independence and gauge covariance in quantum mechanics, Journal of Physics A: Mathematical and General, vol.17, issue.1, p.151, 1984.
DOI : 10.1088/0305-4470/17/1/017

H. Kobe and K. Yang, Gauge transformation of the time-evolution operator, Physical Review A, vol.32, issue.2, p.952, 1985.
DOI : 10.1103/PhysRevA.32.952

. Adu-gyamfi, On the gauge invariance of linear response theory, Journal of Physics A: Mathematical and General, vol.19, issue.16, p.3443, 1986.
DOI : 10.1088/0305-4470/19/16/039

K. Yang and D. H. Kobe, Superluminal, advanced and retarded propagation of electromagnetic potentials in quantum mechanics, Annals of Physics, vol.168, issue.1, p.104, 1986.
DOI : 10.1016/0003-4916(86)90112-0

H. Kobe and S. M. Golsham, Gauge invariance of the energy shift of a bound electron in an electromagnetic field, Journal of Physics A: Mathematical and General, vol.20, issue.10, p.2813, 1987.
DOI : 10.1088/0305-4470/20/10/024

W. E. Lamb, R. R. Schlicher, and M. O. Scully, Matter-field interaction in atomic physics and quantum optics, Physical Review A, vol.36, issue.6, p.2763, 1987.
DOI : 10.1103/PhysRevA.36.2763

G. P. Durante, C. Arrighini, and . Guidotti, A simple model of atomic collision in the presence of an intense radiation field: A comparison test for gauge invariant vs. conventional quantum transition probabilities, Zeitschrift f???r Physik D Atoms, Molecules and Clusters, vol.79, issue.1, p.63, 1988.
DOI : 10.1007/BF01384525

G. Woolley, Charged particles, gauge invariance, and molecular electrodynamics, International Journal of Quantum Chemistry, vol.94, issue.5, p.531, 1999.
DOI : 10.1002/(SICI)1097-461X(1999)74:5<531::AID-QUA9>3.0.CO;2-H

M. Savukov and W. R. Johnson, Equality of length-form and velocity-form transition amplitudes in relativistic many-body perturbation theory, Physical Review A, vol.62, issue.5, p.52506, 2000.
DOI : 10.1103/PhysRevA.62.052506

G. Woolley, Advanced Problems and Complex Systems, New Trends in Quantum Systems in Chemistry and Physics. of Progress in Theoretical Chemistry and Physics, 1999.

R. W. Rzazewski and . Boyd, Equivalence of interaction hamiltonians in the electric dipole approximation, Journal of Modern Optics, vol.61, issue.8, p.1137, 2004.
DOI : 10.1103/PhysRev.163.147

A. Stokes, On the gauge of the natural lineshape, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.46, issue.14, p.145505, 2013.
DOI : 10.1088/0953-4075/46/14/145505

D. Bandrauk, F. Fillion-gourdeau, and E. Lorin, Atoms and molecules in intense laser fields: gauge invariance of theory and models, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.46, issue.15, p.153001, 2013.
DOI : 10.1088/0953-4075/46/15/153001

K. L. Mandal and . Hunt, Gauge-invariant expectation values of the energy of a molecule in an electromagnetic field, The Journal of Chemical Physics, vol.144, issue.4, p.44109, 2016.
DOI : 10.1088/0953-4075/46/17/175601

F. W. Blagojevi´cblagojevi´c and . Hehl, Gauge Theories of Gravitation, 2013.

E. Feuchtwang and E. Kazes, Comment on K-H Yang's energy operator and gauge independent transition amplitudes, Journal of Physics A: Mathematical and General, vol.18, issue.14, p.2859, 1985.
DOI : 10.1088/0305-4470/18/14/035

T. Goldman, Gauge invariance, time-dependent Foldy-Wouthuysen transformations, and the Pauli Hamiltonian, Physical Review D, vol.15, issue.4, p.1063, 1977.
DOI : 10.1103/PhysRevD.15.1063

T. E. Kazes, H. Feuchtwang, P. H. Grotch, and . Cutler, Gauge-independent minimal electromagnetic coupling, Physical Review D, vol.27, issue.6, p.1388, 1983.
DOI : 10.1103/PhysRevD.27.1388

. Weinberg, The Quantum Theory of Fields. I Foundations, 1995.

. Strocchi, An Introduction to Non-Perturbative Foundations of Quantum Field Theory
DOI : 10.1093/acprof:oso/9780199671571.001.0001

. Bialynicki-birula, Renormalization, Diagrams, and Gauge Invariance, Physical Review D, vol.2, issue.12, p.2877, 1970.
DOI : 10.1103/PhysRevD.2.2877

B. Manoukian, Gauge invariance properties of transition amplitudes in gauge theories. I, International Journal of Theoretical Physics, vol.1, issue.7, p.787, 1988.
DOI : 10.1007/BF00669292

A. Batalin, P. M. Lavrov, and I. V. Tyutin, A systematic study of finite BRST-BV transformations in field???antifield formalism, International Journal of Modern Physics A, vol.17, issue.29, p.1450166, 2014.
DOI : 10.1063/1.529145

Y. Moshin and A. A. Reshetnyak, Field-dependent BRST???anti-BRST Lagrangian transformations, International Journal of Modern Physics A, vol.17, issue.04n05, p.1550021, 2015.
DOI : 10.1142/S0217732395002817

H. W. Lenz, K. Naus, M. Ohta, and . Thies, Quantum Mechanics of Gauge Fixing, Annals of Physics, vol.233, issue.1, p.17, 1994.
DOI : 10.1006/aphy.1994.1059

. Grigore, Gauge invariance of quantum electrodynamics in the causal approach to renormalization theory, Annalen der Physik, vol.28, issue.6-7, p.439, 2001.
DOI : 10.1002/1521-3889(200106)10:6/7<439::AID-ANDP439>3.0.CO;2-N

K. Das, R. R. Francisco, and J. Frenkel, Gauge independence of the fermion pole mass, Physical Review D, vol.88, issue.8, p.85012, 2013.
DOI : 10.1103/PhysRevD.88.085012

L. F. Haller and . Landovitz, Subsidiary Condition in Quantum Electrodynamics, Physical Review, vol.171, issue.5, p.1749, 1968.
DOI : 10.1103/PhysRev.171.1749

L. F. Haller and . Landovitz, Renormalization Constants, Wave Functions, and Energy Shifts in the Coulomb and Lorentz Gauges, Physical Review D, vol.2, issue.8, p.1498, 1970.
DOI : 10.1103/PhysRevD.2.1498

L. F. Haller and . Landovitz, Catastrophic Events and Gauge Invariance, Physical Review Letters, vol.22, issue.6, p.245, 1969.
DOI : 10.1103/PhysRevLett.22.245

Y. Ohtaka and . Tanabe, Theory of the soft-x-ray edge problem in simple metals: historical survey and recent developments, Reviews of Modern Physics, vol.62, issue.4, p.929, 1990.
DOI : 10.1103/RevModPhys.62.929

E. Klevak, J. J. Kas, and J. J. Rehr, Charge transfer satellites in x-ray spectra of transition metal oxides, Physical Review B, vol.89, issue.8, p.85123, 2014.
DOI : 10.1103/PhysRevB.89.085123

P. Grant, Relativistic Quantum Theory of Atoms and Molecules, 2007.
DOI : 10.1007/978-0-387-35069-1

G. Heitler, G. Panati, and . Stoltz, The Quantum Theory of Radiation, Phys. Rev. Lett, vol.103, p.230401, 1984.

W. Milonni, R. Loudon, P. R. Berman, and S. M. Barnett, Linear polarizabilities of two- and three-level atoms, Physical Review A, vol.77, issue.4, p.43835, 2008.
DOI : 10.1103/PhysRevA.77.043835

S. Mukamel, Causal versus noncausal description of nonlinear wave mixing: Resolving the damping-sign controversy, Physical Review A, vol.76, issue.2, p.21803, 2007.
DOI : 10.1103/PhysRevA.76.021803

. Hatfield, Quantum Field Theory of Point Particles and Strings, 1992.

. Strange, Relativistic Quantum Mechanics, 1998.
DOI : 10.1017/CBO9780511622755

D. Baye, Extension of the Siegert theorem for photon emission, Physical Review C, vol.86, issue.3, p.34306, 2012.
DOI : 10.1103/PhysRevC.86.034306

S. L. Foldy and . Wouthuysen, On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit, Physical Review, vol.78, issue.1, p.29, 1950.
DOI : 10.1103/PhysRev.78.29

E. Eriksen, Foldy-Wouthuysen Transformation. Exact Solution with Generalization to the Two-Particle Problem, Physical Review, vol.111, issue.3, p.1011, 1958.
DOI : 10.1103/PhysRev.111.1011

W. Weiglhofer, Delta???function identities and electromagnetic field singularities, American Journal of Physics, vol.57, issue.5, p.455, 1989.
DOI : 10.1119/1.16001

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms, 1977.

D. Morrison, R. Moss, . Molec, and . Phys, 111 A. L. Fetter and J. D. Walecka, Quantum Theory of Many- Particle Systems, p.491, 1971.

J. D. Walecka, Advanced Modern Physics, Theoretical Foundations (World Scientific, 2010.
DOI : 10.1142/7555

C. Brouder, M. Alouani, and K. H. Bennemann, edge, Physical Review B, vol.54, issue.10, p.7334, 1996.
DOI : 10.1103/PhysRevB.54.7334

URL : https://hal.archives-ouvertes.fr/hal-00421973

. Gougoussis, ExcitationsélectroniquesExcitationsélectroniques et magnétisme des matériaux: calcul ab initio de l'absorption X et du dichro¨?smedichro¨?sme circulaire magnétique au seuil K (Université Pierre et Marie Curie, Elements of Modern X-Ray Physics, 2000.

. M. Nieto, Ambiguity, Physical Review Letters, vol.38, issue.19, p.1042, 1977.
DOI : 10.1103/PhysRevLett.38.1042

. L. Foldy, The Electromagnetic Properties of Dirac Particles, Physical Review, vol.87, issue.5, p.688, 1952.
DOI : 10.1103/PhysRev.87.688

N. Takahashi and . Hiraoka, Theory of magnetic circular dichroism of nonresonant x-ray Raman scattering, Physical Review B, vol.92, issue.9, p.94441, 2015.
DOI : 10.1103/PhysRevB.92.094441

S. D. Matteo, private communication. 130 R. Loudon, The Quantum Theory of Light, 2000.

M. Brown, The Compton effect as one path to QED, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, vol.33, issue.2, p.211, 2002.
DOI : 10.1016/S1355-2198(02)00005-9

. Baum, Spin-Strukturen und Dirac-OperatorenüberOperatoren¨Operatorenüber pseudoriemannschen Mannigfaltigkeiten, 1981.

M. Pourahmadi, Foundations of Time Series Analysis and Prediction Theory, 2001.

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, 1991.