Abstract
The third method we shall consider for numerical QED calculation on bound states is the covariant-evolution-operator (CEO) method, developed during the last decade by the Gothenburg group [9]. This procedure is based upon the nonrelativistic time-evolution operator, discussed in Chap. 8, but it is made covariant in order to be applicable in relativistic calculations. Later, we shall demonstrate that this procedure forms a convenient basis for a covariant relativistic many-body perturbation procedure, including QED as well as correlational effects, which for two-electron systems is fully compatible with the Bethe–Salpeter equation. This question will be the main topic of the rest of the book.
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Notes
- 1.
See footnote in the Introduction.
- 2.
An “n-body operator” is an operator with n pairs of creation/absorption operators (for particles), while an “m-particle” function or operator is an object of m particles outside our vacuum. In principle, n can take any value n ≤ m, although we shall normally assume that n = m.
- 3.
It should be observed that a Goldstone diagram is generally distinct from a “time-ordered Feynman diagram,” as is further analyzed in Appendix I.
- 4.
Concerning the definition of the concepts “reducible” and “irreducible,” see Sect. 7.6.
- 5.
Also, the Fock space is a form of Hilbert space, and therefore we shall refer to the Hilbert space with a constant number of photons as the restricted (Hilbert) space and the space with a variable number of photons as the (extended) photonic Fock space (see Appendix A.2).
- 6.
This equation is not completely covariant, because it has a single time, in accordance with the established quantum-mechanical picture. This is the equal-time approximation, mentioned above and further discussed later. In addition, a complete covariant treatment would require that also the interaction between the electrons and the nucleus is treated in a covariant way by means of the exchange of virtual photons (see, for instance, [14]).
- 7.
In the following, we shall leave out the subscript “Rel.”
- 8.
The Green’s operator is closely related – but not quite identical – to the reduced covariant evolution operator, previously introduced by the Gothenburg group [9].
- 9.
This can be compared with the situation in the MBPT Bloch equation (2.56), where – using the heavy dot – the folded term could be expressed \(\Omega \cdot P{V }_{\mathrm{eff}}P\), indicating that the energy parameters of the wave operator depend on the intermediate model-space state.
- 10.$$\begin{array}{rcl} \frac{\delta \mathcal{G}} {\delta \mathcal{E}}& =& \frac{{\mathcal{G}}_{\mathcal{E}}-{\mathcal{G}}_{\mathcal{E}\prime }} {\mathcal{E}-\mathcal{E}\prime } ; \quad \frac{\delta } {\delta \mathcal{E}}\Big{(}\frac{\delta \mathcal{G}} {\delta \mathcal{E}}V \Big{)} = \frac{\big{(}\frac{\delta \mathcal{G}} {\delta \mathcal{E}}{\big{)}}_{\mathcal{E}}{V }_{\mathcal{E}}-\big{(}\frac{\delta \mathcal{G}} {\delta \mathcal{E}}{\big{)}}_{\mathcal{E}\prime }{V }_{\mathcal{E}\prime }} {\mathcal{E}-\mathcal{E}\prime } \\ & =& \frac{\big{(}\frac{\delta \mathcal{G}} {\delta \mathcal{E}}{\big{)}}_{\mathcal{E}}{V }_{\mathcal{E}}-\big{(}\frac{\delta \mathcal{G}} {\delta \mathcal{E}}{\big{)}}_{\mathcal{E}\prime }{V }_{\mathcal{E}} + \big{(}\frac{\delta \mathcal{G}} {\delta \mathcal{E}}{\big{)}}_{\mathcal{E}\prime }{V }_{\mathcal{E}}-\big{(}\frac{\delta \mathcal{G}} {\delta \mathcal{E}}{\big{)}}_{\mathcal{E}\prime }{V }_{\mathcal{E}\prime }} {\mathcal{E}-\mathcal{E}\prime } = \frac{{\delta }^{2}\mathcal{G}} {\delta {\mathcal{E}}^{2}} \,V + \frac{\delta \mathcal{G}} {\delta \mathcal{E}}\frac{\delta V } {\delta \mathcal{E}} \\ \frac{\delta } {\delta \mathcal{E}}\;{V }^{2}& =& \frac{\delta } {\delta \mathcal{E}}{V }_{\mathcal{E}\prime \prime }{V }_{\mathcal{E}} = {V }_{\mathcal{E}\prime \prime }\frac{{V }_{\mathcal{E}}- {V }_{\mathcal{E}\prime }} {\mathcal{E}-\mathcal{E}\prime } = V \frac{\delta V } {\delta \mathcal{E}} \end{array}$$
This can be generalized to
$$\frac{{\delta }^{n}(AB)} {\delta {\mathcal{E}}^{n}} ={ \sum \nolimits }_{m=0}^{n}\frac{{\delta }^{m}A} {\delta {\mathcal{E}}^{m}} \,\frac{{\delta }^{n-m}B} {\delta {\mathcal{E}}^{n-m}}$$(see further [11, Appendix B]).
- 11.
Distinguishing the various interactions, we can write
$$\begin{array}{rcl} {\mathcal{G}}_{0}& =& {\mathcal{G}}^{(0)}\big{(}1 + {\Gamma }_{ Q}{V }_{1} + {\Gamma }_{Q}{V }_{1}{\Gamma }_{Q}{V }_{2} + \cdots \big{)} \\ {\Delta }_{1}& =& \Big{[}\frac{\delta {\mathcal{G}}_{0}} {\delta \mathcal{E}} - {\Gamma }_{Q}{V }_{1}\,\frac{\delta {\mathcal{G}}_{0}} {\delta \mathcal{E}} \Big{]}{W}_{0} = \Big{[}\frac{\delta {\mathcal{G}}^{(0)}} {\delta \mathcal{E}} + {\mathcal{G}}_{0}\frac{\delta ({\Gamma }_{Q}{V }_{1})} {\delta \mathcal{E}} \,\big{(}1 + {\Gamma }_{Q}{V }_{2} + \cdots \big{)}\Big{]}{W}_{0} \\ & =:& \frac{{\delta }^{{_\ast}}{\mathcal{G}}_{1}} {\delta \mathcal{E}} \Big{[}\frac{{\delta }^{2}{\mathcal{G}}_{0}} {\delta {\mathcal{E}}^{2}} - {\Gamma }_{Q}{V }_{1}\,\frac{{\delta }^{2}{\mathcal{G}}_{0}} {\delta {\mathcal{E}}^{2}} \Big{]}{W}_{0} \\ & =& \Bigg{[}\frac{{\delta }^{2}{\mathcal{G}}^{(0)}} {\delta {\mathcal{E}}^{2}} + \frac{\delta {\mathcal{G}}^{(0)}} {\delta \mathcal{E}} \frac{\delta ({\Gamma }_{Q}{V }_{1})} {\delta \mathcal{E}} \,\big{(}1 + {\Gamma }_{Q}{V }_{2} + \cdots \big{)} + {\mathcal{G}}^{(0)}\frac{{\delta }^{2}({\Gamma }_{Q}{V }_{1})} {\delta {\mathcal{E}}^{2}} \,\big{(}1 + {\Gamma }_{Q}{V }_{2} + \cdots \big{)} \\ & & \quad +\, {\mathcal{G}}^{(0)}\frac{\delta ({\Gamma }_{Q}{V }_{1})} {\delta \mathcal{E}} \,\frac{\delta ({\Gamma }_{Q}{V }_{2})} {\delta \mathcal{E}} \,\big{(}1 + {\Gamma }_{Q}{V }_{3} + \cdots \big{)} + \cdots \Bigg{]}{W}_{0} =: \frac{{\delta }^{{_\ast}}{\mathcal{G}}_{1}} {\delta \mathcal{E}} \end{array}$$ - 12.
We observe here that also the zeroth-order term has changed its time dependence, which is a consequence of the fact that the zeroth-order Green’s operator, \({\mathcal{G}}^{(0)}\), is being modified by the expansion (6.96).
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Lindgren, I. (2011). Covariant Evolution Operator and Green’s Operator. In: Relativistic Many-Body Theory. Springer Series on Atomic, Optical, and Plasma Physics, vol 63. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8309-1_6
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