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Unifying QuantumElectrodynamics and

Many-Body Perturbation TheoryIngvar Lindgren

ingvar.lindgren@physics.gu.se

Department of Physics

University of Gothenburg, Gothenburg, Sweden

Slides with Prosper/LATEX p. 1/48

New Horizons in Physics

Makutsi, South Africa, 22-28 November, 2015

In honor of Prof. Walter Greinerat his 80th birthday

Slides with Prosper/LATEX p. 2/48

CoworkersSten SalomonsonDaniel HedendahlJohan Holmberg

Slides with Prosper/LATEX p. 3/48

Combining MBPT and QED

Quantum physics/chemistry follows mainly

the rules of Quantum Mechanics (QM)

Some effects lie outside: Lamb shift(electron self-energy and vacuum polarization)

qq

s s

Slides with Prosper/LATEX p. 4/48

Combining MBPT and QED

Quantum physics/chemistry follows mainly

the rules of Quantum Mechanics (QM)

Some effects lie outside: Lamb shift(electron self-energy and vacuum polarization)

require Field theory (QED)

Normally these effects are evaluated separatelyFor high accuracy they should be evaluated in acoherent way

QED effects should be included in thewave function

Slides with Prosper/LATEX p. 5/48

Combining MBPT and QED

QM and QED are seemingly

incompatible

QM: single time (t,x1,x2, )

Field theory: individual times (t1,x1; t2,x2, )Consequence of relativistic covariance

Bethe-Salpeter equation is

relativistic covariant

can lead to spurious solutions

(Nakanishi 1965; Namyslowski 1997)Slides with Prosper/LATEX p. 6/48

Combining MBPT and QED

Compromise:

Equal-time approximationAll particles given the same time

makes FT compatible with QM

Some sacrifice of the full covariance

very small effect at atomic energies

Slides with Prosper/LATEX p. 7/48

Controversy

Chantler (2012) claims that there are significantdiscrepancies between theory and experiment forX-ray energies of He-like ionsTheory: Artemyev et al 2005, Two-photon QED

Slides with Prosper/LATEX p. 8/48

Higher-order QED

Higher -order QED can be evaluated by means of theprocedure for

combining QED and MBPT using theGreens operator,

a procedure for time-dependent perturbation theory

Slides with Prosper/LATEX p. 9/48

Time-independent perturbation

H = (H + V ) = E target function

0 = P model function

P projection operator for the model space

= 0 wave operator

Bloch equation

P = (

VW)

P = 1E0H0

W = PVP Effective Interaction

Heff0 = (PH0P +W )0 = (E0 +E)0 Effective Ham.

Slides with Prosper/LATEX p. 10/48

Bloch equation

P = (

VW)

P = 1E0H0

r r +P

VP = r rr r

+

P

r rr rr r

+

P

= [V + V V + V V V + ]P

Singular when intermediate state in model space (P)Singularity cancelled by the term WP

Leads to Bloch equationP = Q

(

VW)

P Q =Q

E0H0

The finite remainder is the

Model-Space Contr. QWPSlides with Prosper/LATEX p. 11/48

Time-dependent perturbation

Standard time-evolution operator

(t) = U(t, t0)(t0)

s s Particles

Time propagates only forwards

Slides with Prosper/LATEX p. 12/48

Time-dependent perturbation

Standard time-evolution operator

(t) = U(t, t0)(t0)

s s Particles

s s

s s

s s

Part.

Holes

Electronpropagators

Electron propagators make evolution operator covariant

Covariant Evolution Operator (UCov)

Slides with Prosper/LATEX p. 13/48

Time-dependent perturbation

Covariant evolution ladder (t = 0, t0 = )

UCov = 1+ r r

E

+ r rr r

+

E

r rr rr r

+

E

UCov = 1 + V + V V + ; =1

E H0

Same as first part of MBPT wave operator

= 1 + (

VW)

Singular when intermediate state in model space

Slides with Prosper/LATEX p. 14/48

Greens operator

The Greens operator is defined

UCov(t) = G(t) PUCov(0)

is the regular part of the Covariant Evolution Oper.

Slides with Prosper/LATEX p. 15/48

Greens operator

t = 0 : First order: G(1) = U (1)Cov = QV = (1)

Second order:

G(2) = QV G(1) + G

(1)

EW (1) ; Q =

QEH0

= QV G(1) Q G

(1)W (1) + QVE

W (1)

(2) = QV (1) Q

(1)W (1)

Time- or energy-dependent perturbations canbe included in the wave function

Slides with Prosper/LATEX p. 16/48

QED effects

Non-radiative

rr

Retardation

r

r

r r

r r

r

r

Virtual pair

Radiative

r

r

El. self-energy

r

rr r

Vertex correction

r r

Vacuum polarization

r r r r

Slides with Prosper/LATEX p. 17/48

QED effects are time dependent

rr

Slides with Prosper/LATEX p. 18/48

QED effects are time dependent

Can be combined with electron correl.

rr

r rr rr r

Continued iterations

rr

r rr rr r

r rr r

rr

r rr rr r

r rr rr

r

Mixing time-independent and time-dependent perturbat.

Combining QED and MBPTSlides with Prosper/LATEX p. 19/48

Radiative QED

Dimensional regularization in Coulomb gauge

Developed in the 1980s for Feynman gauge

Formulas for Coulomb gauge derived by Atkins in the 80s

Workable procedure developed by Johan Holmberg in2011

First applied by Holmberg and Hedendahl

Slides with Prosper/LATEX p. 20/48

Self-energy of hydrogen like ions

Hedendahl and Holmberg, Phys. Rev. A 85, 012514(2012)

Z Coulomb gauge Feynman gauge

18 1.216901(3) 1.21690(1)

54 50.99727(2) 50.99731(8)

66 102.47119(3) 102.4713(1)

92 355.0430(1) 355.0432(2)

E =

(Z)4mc2

n3F (Z)

First calculation of self-energy in Coulomb gaugeSlides with Prosper/LATEX p. 21/48

He-like systems

Johan Holmbergs PhD thesisHolmberg, Salomonson and Lindgren, Phys. Rev. A 92, 012509

(2015)

r

r

r rP

Q

P

(A)

r

r

r rP

P

P

(B)

r

rr r

P

Q

P

(C)

r

rr r

P

P

P

(D)

r

rr r

P

P

(E)

(B) and (E) are DIVERGENTDivergence cancels due to Ward identity

Slides with Prosper/LATEX p. 22/48

He-like Argon (Z=18) Second order

Irreducible

r

r

r rP

Q

P

1621 meV

-115.8

r

rq

r rP

Q

P

-1707

11.6

Feynman gauge

Coulomb gauge

Slides with Prosper/LATEX p. 23/48

He-like Argon (Z=18) Second order

Irreducible Model-space contr. + Vertex correction

r

r

r rP

Q

P

1621 meV

-115.8

r

rq

r rP

Q

P

-1707

11.6

r

r

r rP

P

P

3819

-24,8

r

rr r

P

-3653

16.2

P

Feynman

Coulomb

Slides with Prosper/LATEX p. 24/48

He-like Argon (Z=18) Second order

Irreducible Model-space contr. + Vertex correction

r

r

r rP

Q

P

1621 meV

-115.8

r

rq

r rP

Q

P

-1707

11.6

r

r

r rP

P

P

3819

-24,8

r

rr r

P

-3653

16.2

P

Feynman

Coulomb

r

rq

r rP

Q

P

-192

-1.1

r

rqr r

P

P

Slides with Prosper/LATEX p. 25/48

He-like Argon (Z=18) Second order

Irreducible Model-space contr. + Vertex correction

r

r

r rP

Q

P

1621 meV

-115.8

r

rq

r rP

Q

P

-1707

11.6

r

r

r rP

P

P

3819

-24,8

r

rr r

P

-3653

16.2

P

Feynman

Coulomb

r

rq

r rP

Q

P

-192

-1.1

r

rqr r

P

P

113

113.8

Gauge independent

Large cancellations in Feynman gauge

Slides with Prosper/LATEX p. 26/48

He-like Argon (Z=18) Third order

Irreducible Model-space contr. + Vertex correction

r

r

r rr rP

Q

P

-142

4.79

r

rq

r rr rQP

P

71

-0.55

Feynm.

Coul.

Slides with Prosper/LATEX p. 27/48

He-like Argon (Z=18) Third order

Irreducible Model-space contr. + Vertex correction

r

r

r rr rP

Q

P

-142

4.79

r

rq

r rr rQP

P

71

-0.55

Feynm.

Coul.

r

r

r rr rP

P

P

-24

1.16

r

rr r

r rP

P

54

-0.73

Slides with Prosper/LATEX p. 27/48

He-like Argon (Z=18) Third order

Irreducible Model-space contr. + Vertex correction

r

r

r rr rP

Q

P

-142

4.79

r

rq

r rr rQP

P

71

-0.55

Feynm.

Coul.

r

r

r rr rP

P

P

-24

1.16

r

rr r

r rP

P

54

-0.73

r

rq

r rr rP

P

P

???

???

r

rqr r

r rP

P

Slides with Pro