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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