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this contribution is concerned with the evaluation of atomic energy levels with qed .
such an evaluation yields stringent tests of qed in _ strong electric fields _ , whereas @xmath0-factor experiments and calculations currently probe qed in situations where the magnetic field can be treated perturbatively    the nuclear coulomb field experienced by the inner levels of _ highly - charged ions _ makes the electrons reach _ relativistic _ velocities .
such simple physical systems are thus particularly interesting for testing relativistic effects in quantum systems ( for example , see refs .
@xcite for experimental results with lithiumlike ions ) .
theoretical predictions of energy levels in such systems obviously _ require _ the use of qed .
experiments have reached an accuracy that shows that extremely accurate evaluations of qed effects are also needed in _ helium_. experiments performed during the last ten years in the spectroscopy of this atom have become two orders of magnitude more precise than the current theoretical calculations ( see for instance refs .
@xcite and references therein ) .
several experiments are now focusing on helium and heliumlike ions , and especially their @xmath1 fine structure  @xcite such experiments have implications in metrology , as they could provide a measurement the fine structure constant and provide checks of theoretical higher - order effects .
very precise theoretical calculations of energy levels in heliumlike ions can be also important in the investigation of _ parity violation _
@xcite .
predictions of energy levels are usually more difficult to obtain for states with one or more open shells ( retardation in the interaction and exchange of electrons must be included , and there can be quasidegenerate levels ) .
only a few calculations of excited energy levels in heliumlike and lithiumlike ions have been performed up to now ; the first results have been published quite recently  @xcite . in regards to qed shifts of _ quasidegenerate _ levels , they have only been obtained this year for the first time  @xcite , with the help of the method that we present in this talk .
as is well known , relativistic electrons orbiting a nucleus are well treated with the _ dirac equation _ , in which the nucleus can be considered as point - like or not .
we thus treat the binding to the nucleus _ non - perturbatively _ by using `` bound - state qed ''  @xcite ( the coupling constant of the nucleus - electron interaction is @xmath2 , which is not small for highly - charged ions ) . in this formalism , however , qed effects are taken into account by treating the electron - electron interaction _ perturbatively _ ( with coupling constant @xmath3 ) , and both the electron and photon fields are _ quantum _ fields ( i.e. , in _ second_-quantized form ) ; the only difference with the free - field case used in high - energy physics is that electronic creation and annihilation operators create and destroy atomic states instead of free particles .
a few methods allow one to extract energy levels from the bound - state qed hamiltonian : the two - time green s function method  @xcite , the method being developed by lindgren ( based on relativistic many - body perturbation theory merged with qed )  @xcite , the adiabatic @xmath4-matrix formalism of gell - mann , low and sucher  @xcite , and the evolution operator method  @xcite .
some other methods yield atomic energy levels , but they include qed effects only _ partly _ or approximately ( such as the multiconfiguration dirac - fock method  @xcite , configuration interaction calculations  @xcite and relativistic many - body perturbation theory  @xcite ) .    however ,
only _ two _ methods can in principle be employed in order to calculate energy levels of _ quasidegenerate _ atomic states [ e.g. , the @xmath5 and the @xmath6 levels in heliumlike ions , which are experimentally important ] : the two - time green s function method and the method being elaborated by lindgren .
we present in this talk a non - technical introduction to the first method .
the two - time green s function method has also the advantage of yielding a _ simpler renormalization procedure _ than the gell - mann  low  sucher method in the case of _ degenerate _ levels  @xcite .
all the methods that extract atomic energy levels from the bound - state qed hamiltonian study the _ propagation _ of electrons between two different times .
the methods differ in the number of _ infinite _ times used :    ( a )  in the gell - mann  low  sucher method , the atomic state under consideration evolves from time @xmath7 to time @xmath8 with an adiabatic switching of the interaction ; ( b )  in lindgren s formalism  @xcite , the evolution is from time @xmath7 to time @xmath9 , which avoids problems associated with the two infinite times in the @xmath4-matrix approach of gell - mann  low  sucher ; ( c )  in the two - time green s function method , that we present here , the adiabatic switching is completely avoided by studying the propagation of electrons between _ two finite times_. we note that adiabatic switching of the interactions is physically motivated in the study of collisions between particles that start very far from each other , but this switching is not so easily related to the physical description of the orbiting electrons of an atom .
the _ effective hamiltonian _ derived from qed by the two - time green s function method has matrix elements between the various degenerate and/or quasidegenerate states under study ; the eigenvalues of this hamiltonian are the atomic energy levels predicted by qed ( to a given order ) .
this effective hamiltonian is however _ not _ associated to a schr " odinger equation of motion ; our hamiltonian is equivalent to the submatrix used in the perturbation theory of degenerate and quasidegenerate states ; in this respect , the approach of the two - time green s function method differs from the spirit of the bethe - salpeter equation .
the qed hamiltonian of the method is defined with the help of a green s function that represents the propagation of @xmath10 electrons between two different ( finite ) times ( @xmath10 is the number of electrons of the atom or ion that we want to study ) ; this propagation is represented in fig .
[ fig : green ] .
the energy levels of an @xmath10-electron ion or atom can be recovered by studying the _ energy _
representation @xmath11 of the green s function , i.e. , by doing a fourier transform : this function _ has ( simple ) poles at the atomic energy levels
_  @xcite .
such a result is similar to the k " all ' en - lehmann representation  @xcite . as an example , fig .
[ fig : greenpoles ] depicts the poles of the two - particle green s function .
the two - time green s function method provides a way of _ mathematically _ extracting from the green s function the positions of the poles , i.e. , the atomic energy levels  @xcite ; the procedure handles degenerate and quasidegenerate atomic levels without any special difficulty  @xcite .
one of the basic ideas behind the pole extraction is found in the following mathematical device , which uses _ any _ contour @xmath12 that encloses the pole in order to find its exact position : if the function @xmath13 has a simple pole at @xmath14 , then we have from complex analysis @xmath15 the contour @xmath12 is only required to encircle the pole and to be positively oriented , as shown in fig .
[ fig : contourint ] . since the green s function has simple poles at the atomic energy levels  @xcite , eq .
( [ eq : polerecover ] ) is a way of obtaining them .
when qed shifts of degenerate of quasidegenerate levels are calculated , the _ scalar _ green s function @xmath0 of eq .
( [ eq : polerecover ] ) is simply replaced by a finite - size _ matrix _ defined on the space of levels under consideration  @xcite .
obviously , _ analytic _ properties of the green s function  @xcite are important in the evaluation of eq .
( [ eq : polerecover ] ) .
we have developed a set of _ graphical _ techniques that allow one to obtain the laurent series of the green s function @xmath11 by a _
systematic _ procedure .
the idea behind these techniques consists in displaying the analytic structure of the green s function step by step ; each step explicitly extracts _
one _ singularity , and we proceed until we have exhausted all the singularities of the green s function ; at this point , contour integrals such as eq .
( [ eq : polerecover ] ) can be calculated quite simply .
it is impossible to give here a full account of the method we use for deriving the effective , finite - size qed hamiltonian .
however , we can mention a particular feature of our calculational strategy : a very special `` particle '' appears in our algorithm ; this particle is quite simple since it `` disintegrates '' immediately ( zero life time ) and can not move ( zero probability for going from one position to a different one ) . in mathematical terms , the coordinate - space propagator of this particle is a four - dimensional delta function @xmath16 $ ] that we represent by a special line in feynman diagrams .
the experimental accuracy on transition energies is so high that second - order ( i.e. , two - photon ) effects must be taken into account in order to compare experiments with theory .
we thus have very recently calculated the contribution of the self - energy screening  @xcite to the qed effective hamiltonian ; this contribution corresponds to the following physical processes : @xmath17 our result is part of the current theoretical effort developed with the aim of matching experimental precisions .
we present an introduction to the principles behind atomic energy level calculations with quantum electrodynamics ( qed ) and the two - time green s function method ; this method allows one to calculate an effective hamiltonian that contains all qed effects and that can be used to predict qed lamb shifts of degenerate , quasidegenerate and isolated atomic levels .