1 3
Theor Chem Acc (2015) 134:123
DOI 10.1007/s00214-015-1707-6
REGULAR ARTICLE
Combination of many-body perturbation theory and quantum
electrodynamics
Ingvar Lindgren
1
· Johan Holmberg
2 · Sten Salomonson
1
Received: 22 May 2015 / Accepted: 25 July 2015 / Published online: 6 October 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
The procedures for many-body perturbation calculations
(MBPT) for atomic and molecular systems are nowadays
very well developed, and the dominating electrostatic as
well as magnetic perturbations can be taken to essentially
all orders of perturbation theory (see, for instance, [ 1 ]).
Less pronounced, but in many cases still quite signifi -
cant, are the quantum electrodynamical (QED) perturbations—retardation, virtual pairs, electron self-energy,
vacuum polarization and vertex correction. Sophisticated
procedures for their evaluation have also been developed,
but for practical reasons such calculations are prohibitive beyond second order (two-photon exchange). Pure
QED effects beyond that level can be expected to be very
small, but the combination of QED and electrostatic perturbations (electron correlation) can be signifi cant. However, none of the previously existing methods for MBPT
or QED calculations is suited for this type of calculation.
We have during the past decade developed an energydependent perturbation procedure, based on a “ covariant
evolution operator method ” [ 2 – 5 ], that will make it possible to handle energy-dependent QED perturbations very
much in the same way as the energy-independent ones in
MBPT.
2 Covariant evolution operator
The time-evolution operator for the Schrödinger wave function transforms the wave function from one time to another
In the interaction picture the wave function has the time
dependence
(1)
Ψ (t) = U(t, t 0 )Ψ (t 0 ).
Abstract A procedure for energy-dependent perturbation expansion has been developed, based upon the covariant evolution operator method. This makes it possible to
treat energy-dependent perturbations very much like the
energy-independent ones in standard many-body perturbation theory. This has been applied to the non-radiative QED
perturbations (retardation and virtual electron–positron
pairs) as well as the radiative ones (electron self-energy,
vacuum polarization and vertex correction). The combination of QED and electron correlation, beyond two-photon
exchange, has been evaluated, using the Coulomb gauge. It
turned out that in that gauge the extremely time-consuming
model-space contributions of the self-energy and vertex
corrections do not have to be evaluated in full. In the Feynman gauge no sensible results could be obtained in this
way, as is demonstrated by the numerical results.
Keywords Perturbation theory · Quantum
electrodynamics · Electron correlation · Electron
self-energy · Green’s operator · Covariant evolution
operator
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Ingvar Lindgren
ingvar.lindgren@physics.gu.se
1
Department of Physics , University of Gothenburg ,
Gothenburg , Sweden
2
Physikalisches Institut , Universität Heidelberg , Heidelberg ,
Germany
3
Reprinted from the journal
Theor Chem Acc (2015) 134:123
DOI 10.1007/s00214-015-1707-6
REGULAR ARTICLE
Combination of many-body perturbation theory and quantum
electrodynamics
Ingvar Lindgren
1
· Johan Holmberg
2 · Sten Salomonson
1
Received: 22 May 2015 / Accepted: 25 July 2015 / Published online: 6 October 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
The procedures for many-body perturbation calculations
(MBPT) for atomic and molecular systems are nowadays
very well developed, and the dominating electrostatic as
well as magnetic perturbations can be taken to essentially
all orders of perturbation theory (see, for instance, [ 1 ]).
Less pronounced, but in many cases still quite signifi -
cant, are the quantum electrodynamical (QED) perturbations—retardation, virtual pairs, electron self-energy,
vacuum polarization and vertex correction. Sophisticated
procedures for their evaluation have also been developed,
but for practical reasons such calculations are prohibitive beyond second order (two-photon exchange). Pure
QED effects beyond that level can be expected to be very
small, but the combination of QED and electrostatic perturbations (electron correlation) can be signifi cant. However, none of the previously existing methods for MBPT
or QED calculations is suited for this type of calculation.
We have during the past decade developed an energydependent perturbation procedure, based on a “ covariant
evolution operator method ” [ 2 – 5 ], that will make it possible to handle energy-dependent QED perturbations very
much in the same way as the energy-independent ones in
MBPT.
2 Covariant evolution operator
The time-evolution operator for the Schrödinger wave function transforms the wave function from one time to another
In the interaction picture the wave function has the time
dependence
(1)
Ψ (t) = U(t, t 0 )Ψ (t 0 ).
Abstract A procedure for energy-dependent perturbation expansion has been developed, based upon the covariant evolution operator method. This makes it possible to
treat energy-dependent perturbations very much like the
energy-independent ones in standard many-body perturbation theory. This has been applied to the non-radiative QED
perturbations (retardation and virtual electron–positron
pairs) as well as the radiative ones (electron self-energy,
vacuum polarization and vertex correction). The combination of QED and electron correlation, beyond two-photon
exchange, has been evaluated, using the Coulomb gauge. It
turned out that in that gauge the extremely time-consuming
model-space contributions of the self-energy and vertex
corrections do not have to be evaluated in full. In the Feynman gauge no sensible results could be obtained in this
way, as is demonstrated by the numerical results.
Keywords Perturbation theory · Quantum
electrodynamics · Electron correlation · Electron
self-energy · Green’s operator · Covariant evolution
operator
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Ingvar Lindgren
ingvar.lindgren@physics.gu.se
1
Department of Physics , University of Gothenburg ,
Gothenburg , Sweden
2
Physikalisches Institut , Universität Heidelberg , Heidelberg ,
Germany
3
Reprinted from the journal