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Keywords:

  • many-body perturbation theory;
  • quantum;
  • electrodynamics;
  • relativistic quantum chemistry

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography

In the first part, a review is made of the development of the many-body perturbation theory for the last half century. The development of methods for quantum electrodynamics (QED) calculations, which have occurred essentially in parallel, is also briefly reviewed. In the second part, an effort of unifying the two is discussed. The covariant-evolution-operator method, developed primarily for QED calculations by the Gothenburg group, can be combined with atomic/molecular many-body perturbation theory (MBPT) and for the first time making it possible to combine full QED with electron correlation of arbitrary order. The basic idea is to extend the standard procedure of MBPT to a time- or energy-dependent formalism, which makes it possible to include QED perturbations in a rigorous fashion. The procedure has been implemented and as a first test applied to helium like ions. The method can also be applied to dynamical processes. © 2014 Wiley Periodicals, Inc.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography

The procedures of many-body theory (MBPT) and quantum electrodynamics (QED) have been developed essentially independently for more than half a century. No serious efforts seem to have been made to unify the two. One reason might be that the structures of them are seemingly incompatible—MBPT is based upon standard quantum mechanics with a single time, while QED is based upon relativistic covariance with individual time for each particle. In order to combine the two, one way is to sacrifice the full covariance and apply the equal-time approximation. Fortunately, it turns out that this has very small effect on problems in atomic physics and quantum chemistry (see, for instance, the discussion in the Introduction of Ref. [1]).

In the first part of this report, the standard procedures of MBPT and QED and their development will be reviewed. Here, emphasis will be put on the so-called folded contribution of the Bloch equation of MBPT, which represents the remainder after the singularities, due to intermediate model-space states, are eliminated (also referred to as “model-space contribution” or “reference-state contribution”). It turns out that this plays an important role in the generalization of the procedure, needed to include QED and other energy-dependent perturbations.

In the second part, it will be reported how MBPT and QED can be unified, based upon one of the QED approaches, the CEO approach.

Historical Development

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography

Many-body perturbation theory

The perturbation theory was developed to high sophistication long ago, primarily in connection with astronomy. In connection with the development of quantum mechanics in the early part of the previous century particularly two forms of perturbation theory were developed, the Brillouin-Wigner (BWPT) and the Rayleigh-Schrödinger (RSPT) perturbation theories. The former has a simpler structure, but has the disadvantage that it depends on the exact energy of the system and therefore has to be solved in a self-consistent way. RSPT, conversely, depends only of the unperturbed energy, at the expense of a more complex structure.

In 1955, Bloch[2] found that the various orders of RSPT could easily by generated by iterating an equation, later know as the “Bloch equation.” At about the same time, Brueckner[3] discovered that certain terms in the expansion, which he referred to as the unlinked, had a nonlinear dependence on the number of particles. Brueckner conjectured that these terms must vanish for physical reasons, and he was able to show this to fourth-order in a pure algebraic way. Two years later Goldstone[4] could show this to all orders of perturbation theory. Goldstone used a graphical representation to represent the terms in the expansion, now known as Goldstone diagrams and closely related to the famous Feynman diagrams that Feynman introduced in the late 1940's in connection with the development ofQED. It turned out that the unlinked terms found by Brueckner were here represented by diagrams that were disconnected (unlinked), and this led to what in now knows as the “linked-cluster or linked-diagram theorem.” This theorem has simplified the treatment of atomic and molecular systems tremendously. The expansion now has the same simple structure as BWPT but still contained only the unperturbed energy. The RSPT—in its original as well as linked form—has in addition an essential advantage compared to the BWPT that it is size extensive, which implies that its energy increases essentially linearly with the number of particles. This is vital for studying, for instance, the dissociation of molecules. 1

The perturbation expansions developed so far start from a single unperturbed state or from a group of degenerate states forming a model space. In 1967, Brandow,[6] who worked in nuclear theory, demonstrated—by means of a double expansion—the linked-diagram expansion also for a nondegenerate (quasi degenerate) model space. He then found that the elimination of the unlinked diagrams left a finite remainder, which he called “folded” terms. Sandars[7] referred to these terms a “backwards.” In 1974, Lindgren[8] derived a “generalized Bloch equation,” which directly generated the perturbative and linked-diagram expansion for a general multidimensional model space without a double expansion. This equation has ever since been frequently used in atomic and molecular calculations.

A condition for the schemes of Brandow and Lindgren is that the model space is complete in the sense that it contains all configurations with a certain number of electrons that could be formed by the valence electrons. Later Mukherjee[9] showed that the theorem could under certain circumstances hold also for incomplete model spaces.

There was a parallel development in nuclear physics. Around 1960, Coster and Kümmel[10] developed the “Coulped-Cluster Approach” or “Exponential Ansatz” that was in 1966 introduced into quantum chemistry by Čižek.[11] Here several all-order correlational effects could be coupled in a very efficient way, for instance, most of the quadruple excitations could be accounted for coupling two pair equation. This scheme has been intensively used in various fields of quantum chemistry,[12] including also true (connected) triple and (partly) quadruple clusters (see Ref. [13] for a recent review).

The coupled-cluster scheme was later extended to open-shell (quasidegenerate) systems by several groups,[5, 14-16] first for a complete model space. Here, a serious problem is the appearance of so-called “intruder states,” that is, states outside the group of target states (originating from the model space) that could cross one or several target states as the perturbation is gradually turned on, which destroys the convergence. Various schemes have been developed to avoid or reduce this effect. One is the so-called “intermediate Hamiltonian,” developed by the Toulouse group,[17] another is the use of an incomplete model space, as developed by Mukherjee et al.[5, 9, 16]

Relativistic MBPT

The negative-energy solutions of the relativistic Dirac equation are difficult to handle in a many-body calculation. Performing in a naïve way could lead to what is known an the degeneracy collapse or the Brown–Ravenhall disease,[18] when an intermediate state with zero excitation energy can appear. The standard way to avoid that is to restrict the excitations to positive-energy orbitals by means of projection operators, suggested by Sucher in 1980.[19] This approach is usually referred to as the no-(virtual)-pair approximation (NVPA).

An alternative to the perturbation expansion is the “multiconfiguration Hartree/Dirac-Fock method,” where also electron correlation can be included essentially to all orders[20] in a variational way.

QED effects

The effects beyond NVPA are conventionally defined as the QED corrections, which are of the order inline image or higher. They involve retardation of the electromagnetic interaction, virtual electron-positron pairs, electron self energy and vacuum polarization, photon self energy (a form of vacuum polarization,) and vertex correction (see Fig. 1).

image

Figure 1. Examples of low-order QED effects.

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QED in its modern form was developed in the late 1940's by Feynman, Schwinger, Tomanaga, Dyson, and others. Then they discovered a means of eliminating the disturbing singularities my means of regularization and renormalization. The aim was here to study problems in elementary-particle physics. Some methods of field theory, such as the Feynman diagrams, were introduced in atomic and molecular physics in the 1960's, particularly by Kelly,[21] but QED calculations were not performed in these areas until the 1980's. Several groups were here active (Gothenburg, Paris, NIST, St Petersburg).

Mainly three types of QED procedures were introduced, the standard S-matrix formulation,[22] the Two-times Green's function, developed by the St Petersburg group,[23] and the CEO method, developed by the Gothenburg group.[24] All three methods are limited for computational reasons to second-order (two-photon exchange). This might be quite sufficient for highly charged ions, where the electron correlation is quite weak but less so for lighter elements, where the correlation effect dominates. The S-matrix formulation is limited to a degenerate model space, while the other two also are applicable in the quasidegenerate situation. The S-matrix and two-times Green's function approaches rare restricted to energy calculations, while the CEO method can also be used to evaluate the wave function (wave operator). The latter property make this method a candidate for a unification with MBPT, and this has led to a development of a unified MBPT-QED procedure during the last decade by the Gothenburg group.[1, Ch. 10,25]

Unification with MBPT

The covariant-evolution operator (CEO), which represents the time evolution of the relativistic wave, becomes singular when an intermediate model-space states appears in the expansion. These can be eliminated by counterterms, and the regular part is referred to as the “Green's operator,” due to its analogy with the standard Green's function. This operator can be regarded as a time-dependent generalization of the wave operator of standard MBPT. The Green's operator satisfies a Bloch-type equation, quite similar to that of MBPT, which implies that the procedures are closely related. The only difference lies in the fact that there is an additional term involving the energy derivative of the perturbations that are energy dependent. This leads to a general “time- or energy dependent perturbation theory,” which makes it possible to treat the QED and Coulomb perturbations in the same fashion.

Ultimately, the CEO procedure can be shown to be compatible with the covariant “Bethe-Salpeter equation,” derived in 1951 by Bethe and Salpeter[26, 27] as well as by Gell-Mann and Low,[28] for two-electron systems, which demonstrates its relativistic covariance. It can be regarded as a perturbative procedure for this equation, where certain effects can be treated in higher orders that others (at some expense of the full covariance).

The full CEO generates all QED effects, combined with all-order electron correlation. From this end various approximation can be derived. By considering only Coulomb (and possible instantaneous Breit) interaction, omitting all retardation effects, leads to the “Fock-space procedure,” developed by Kutzelnigg, Liu, and others.[29, 30] Supplemented by a so-called “charge-conjugated contraction” potential this generates all first-order QED effects in this approximation.[31] Another approximation is to consider single-photon effects, which includes full first-order QED (retardation, virtual electron-positron pairs, vacuum polarization, Lamb shift), omitting irreducible multiphoton exchange. This is the approximation we are pursuing, and it is expected to include all relevant effects in most cases.

Formal Development

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography

Non-QED methods

Standard Many-Body Perturbation Theory

As an introduction to the formal treatment, we shall review the standard nonrelativistic many-body perturbation theory (MBPT).[32] We consider a number of “target states,” satisfying the Schrödinger equation,

  • display math(1)

where H is the standard nonrelativistic Hamiltonian used in atomic and molecular calculations. For each target state, there exists a model state, which in intermediate normalization is the projection on the model space, inline image. The model states form a model space, P. A wave operator transforms the model states to the full target states

  • display math(2)

An effective Hamiltonian can be defined, so that it generates the exact energies, operating on the model functions

  • display math(3)

We partition the Hamiltonian in the standard way into a zeroth-order part and a perturbation

  • display math(4)

where H0 is a sum of Schrödinger single-electron operators in an external (mainly nuclear) field and the perturbation is the electron-electron interaction.

The wave operator satisfies a Bloch equation

  • display math(5)

where

  • display math(6)

is the “effective interaction.”

It can be shown (for a complete and in certain cases also for an incomplete model space, see above) that so-called unlinked diagrams, that is, diagrams with a disconnected closed part (with no other free lines than valence lines) do vanish, leading to the linked-diagram theorem.

For a degenerate part of the model space of energy inline image, the Bloch equation yields

  • display math(7)

where inline image is the resolvent.

Omitting the last term of the Bloch equation (7), leads to the expansion

  • display math(8)

This becomes (quasi)singular, when a state (intermediate or final) lies in the model space. The singularities are removed by the last term of the Bloch equation, and there is a finite remainder, which is represented by the linked part of this term. This is referred to as the model-space contribution. It is traditionally drawn in a “folded” or “backward” way, and we shall refer to it as the “folded term.” We shall see that it plays an important role in the generalization to energy-dependent formalism that we are presently considering.

All-order and coupled-cluster methods

In the perturbation expansion certain effects, such as the pair correlation, can be included iteratively to arbitrary order by separating the wave operator by means of second quantization into one-body, two-body … effects

  • display math(9)

This leads to the coupled equations

  • display math(10)

Expanding the two-body part (without singles) leads to the pair function, illustrated in Figure 2.

image

Figure 2. Expanding the two-body part of the wave operator (without singles) leads to the pair function.

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By means of the exponential Ansatz this leads to the coupled-cluster approach,

  • display math(11)

with the cluster operator T being a sum of one-, two- body operators, inline image. This leads to a series of coupled equations

  • display math(12)

The diagrams of the cluster operator are connected. (Note the difference between “linked” and “connected”[32].

For the open-shell systems, it is convenient to introduce the normal-ordered exponential Ansatz,[15] which removes the spurious contractions between valence orbitals,

  • display math(13)

The curly brackets represents here normal ordering. The cluster operator is also separated into valence sectors, depending on the number of valence orbitals (open-shell lines) involved.

Relativistic MBPT

The standard relativistic MBPT is based upon the “Dirac-Coulomb-Breit approximation”[19]

  • display math(14)
  • display math(15)

where the projection operators inline image eliminate negative energy states. This is known as the NVPA.

The CEO approach to QED

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

  • display math(16)

Here, inline image are the electron-field operators, inline image the Dirac operator, and inline image the electro-magnetic field.

Green's function

The single-particle field-theoretical Green's function can in the single-reference case be defined[33]

  • display math(17)

where T is the Wick time-ordering operator and inline image are the electron-field operators in the Heisenberg representation. The state inline image 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.

Green's operator

The standard, nonrelativistic evolution operator represents the time evolution of the nonrelativistic wave function or state vector

  • display math(18)

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.

image

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

  • display math(19)

using the same vacuum expectation as in the definition of the Green's function. In the interaction picture this leads to the expansion

  • display math(20)

coupled to a one-body operator. Note that a contraction of TWO interactions of inline image 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

  • display math(21)

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)]

  • display math(22)
image

Figure 4. Ladder diagrams in the pair approximation.

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Intermediate model-space states will make the CEO (quasi-)singular. We shall refer to the regular part of CEO as the Green's operator, inline image, defined by

  • display math(23)

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

  • display math(24)

But inline image is the model function, inline image, showing that the Green's operator acts as a time-dependent wave operator

  • display math(25)

We shall in the following always assume that the initial time is inline image, 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

  • display math(26)

or

  • display math(27)

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 ( inline image).

For time t = 0 the Green's operator becomes equivalent to the standard MBPT wave operator (2)

  • display math(28)

The covariant effective interaction is found to be

  • display math(29)

The Green's operator satisfies the Bloch-type equation

  • display math(30)

inline image is the zeroth-order Green's function, which becomes unity for t = 0. The asterisk indicates that the differentiation is restricted to the last factor of inline image. This relation can be rewritten as

  • display math(31)

with differentiation of the last interaction only. This equation can be compared with the standard Bloch equation (7), and we see that the essential difference lies in the extra model-space term, involving the energy derivative of the interaction. The equation holds also when the interactions are different, provided that V in the second term is the last interaction and that inline image in the remaining terms is formed by the last interactions and W of the remaining ones.

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.

image

Figure 5. Iteration of the Bloch equation (30) can generate reducible diagrams (defined in the footnote).

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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).

image

Figure 6. Examples of irreducible multiphoton diagrams.

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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.

Implementation

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography

The CEO approach to QED, described earlier, has been implemented in our group and applied to helium like systems. QED effects, like retardation, virtual pairs and electron self-energy, have for the first time been evaluated together with high-order electron correlation. These results, which go beyond what has so far been obtained, will be published separately. It has been found that the procedure is also applicable to dynamical processes, like “radiative electron recombination,” and work along this line is now in progress.

Summary and conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography

What has been presented here can be summarized in the following way:

  • A review is given of the development of standard MBPT and QED calculations.
  • A procedure is described for unifying the two procedures, based upon the CEO method, which is now being implemented.
  • This is the only process known today, where QED effects can be combined with electron correlation in a complete and rigorous manner.
  • The process has primarily been applied to determine the ground-state energy of He-like ions.
  • The procedure can also be applied to dynamical processes.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography

The author wants to express his gratitude to his coworkers, Sten Salomonson, Johan Holmberg, and Wenjian Liu, for valuable collaboration and to Professor Werner Kutzelnigg and Professor Bogumil Jeziorski for stimulating discussions.

  • 1

    This concept should not be confused with that of size consistency, which requires that also the wave function separates at the dissociation (see, for instance, Ref. [5] for a detail account of the concepts.)

Biography

  1. Top of page
  2. Abstract
  3. Introduction
  4. Historical Development
  5. Formal Development
  6. Implementation
  7. Summary and conclusions
  8. Acknowledgments
  9. Biography
  • Image of creator

    Ingvar Lindgren was born in Uppsala (Sweden) in May 1931, and obtained PhD in Physics in his hometown university in 1959. He has been appointed as professor in physics at Chalmers University of Technology, Gothenburg, in 1966, he has been the member of the Royal Swedish Academy of Sciences since 1975 and of the Nobel Prize Committee for physics between 1978 and 1991, serving as its chairman from 1989 to 1991. He has served as dean of the Faculty of Mathematics and Natural Sciences at University of Gothenburg between 1988 and 1993, and Managing Director of the Swedish Foundation for Strategic Research from 1994 to 1998.