We shall now describe the CEO method and see how it can be used to combine QED and electron correlation. We start from a general second-quantized Hamiltonian[1, Eq. 6.41], where the unperturbed part is a sum of Dirac operators, like the nonrelativistic counterpart (4) and the perturbation is the Coulomb interaction together with the transverse part of interaction between the electrons and the radiation field is given by
The single-particle field-theoretical Green's function can in the single-reference case be defined
where T is the Wick time-ordering operator and are the electron-field operators in the Heisenberg representation. The state is the vacuum in this representation, that is, the state with no particles or holes. In a single-reference case, the model state is identical to the vacuum state. Both the numerator and the denominator of the definition (17) can be singular, but the ratio is always regular.
The standard, nonrelativistic evolution operator represents the time evolution of the nonrelativistic wave function or state vector
where N is a normalization constant. This is illustrated for single-photon exchange in the first diagram of Figure 3. Here, only particle states (positive energy) are involved and time flows only in the positive direction. Therefore, this operator is not relativistically covariant. The second diagram represents the corresponding Green's function, where the free lines are replaced by electron propagators, allowing positive- and negative-energy states, making the concept covariant. The last diagram represents the corresponding CEO, where in addition electron-field operators are attached to the free ends, making CEO an operator, while the Green's function is a function.
Figure 3. Comparison between the standard evolution operator, the Green's function and the CEO for single-photon exchange. (Reproduced with permission from I. Lindgren, Relativistic Many-Body Theory: A New Field-Theoretical Approach, 2011,
© Springer-Verlag: New York
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Generally, the single-particle (CEO) can be defined
using the same vacuum expectation as in the definition of the Green's function. In the interaction picture this leads to the expansion
coupled to a one-body operator. Note that a contraction of TWO interactions of is needed to form a single-photon exchange. Generally, the evolution operator falls in the photonic Fock space, where the number of photons is not constant.
The CEO represents the time evolution of the relativistic state vector
We consider now the CEO for a two-particle ladder (Fig. 4), analogous to the pair function in Figure 2 but with a general energy-dependent interaction [c.f. Eq. (8)]
Intermediate model-space states will make the CEO (quasi-)singular. We shall refer to the regular part of CEO as the Green's operator, , defined by
where the heavy dot indicates that the Green's operator operates on the intermediate model-space state.
From the relations (21) and (23) we have
We shall in the following always assume that the initial time is , which implies that we start from an unperturbed state (infinitesimal damping is applied). We shall leave out the initial time and instead include the energy parameter. The definition (23) can then be expressed
The last term is referred to as the counterterm that eliminates the singularities of the evolution operator. Note that the parameter of the Green's operator of the counterterm is the energy of the intermediate model-space state ( ).
For time t = 0 the Green's operator becomes equivalent to the standard MBPT wave operator (2)
The covariant effective interaction is found to be
The Green's operator satisfies the Bloch-type equation
The similarity between the Bloch equations for the Green's operator and the standard MBPT wave operator demonstrates that the perturbation expansion based upon the CEO or Green's function is completely compatible with the standard procedure. Therefore, it can serve as a convenient basis for a unified procedure, where QED and Coulomb interactions can be mixed arbitrarily.
QED effects into the expansion
The energy-dependent interaction V(E), used above, can contain all kinds of QED effects, such as retardation, virtual pairs, electron self energy, electron polarization, and so on. Some low-order effects are indicated in Figure 1. The third diagram is an example of a diagram with a negative-energy orbital. The diagrams are time-ordered, and negative-energy states are represented by lines directed downward to indicate that then can in some interpretation be considered as result of time going backward.
We assume here that the Coulomb gauge is applied, which allows all developments of standard MBPT to be utilized. To evaluate the QED effects in this gauge is more complicated than in the Feynman gauge that is commonly used, but this problem has recently been solved in our group,[34, 35] based upon works of Adkins.[36, 37]
The QED effects can be combined with an arbitrary number of Coulomb (and instantaneous Breit) interactions, which might also cross the retarded interactions. By iteration, diagrams that are reducible, that is, can be separated into simpler diagrams by a horizontal cut, can be generated, as illustrated in Figure 5.
In principle, also general irreducible multiphoton effects can be included in the potential (see Fig. 6). They are expected to have only a minor effect (beyond the two-photon exchange that can be evaluated by standard methods) and much more tedious to evaluate. They are not included in the procedure we perform at present. It should be noted that the potential should include only irreducible interactions, since the reducible combinations can be generated by means of interaction (Fig. 5).
An important advantage of using the Coulomb gauge is that the instantaneous Breit interaction [Eq. (15)] can be used in the same fashion as the Coulomb interaction, which will in the single-photon approximation yield important effects that otherwise would require irreducible multiphoton effects.