Unifying Quantum Electrodynamics and Many-Body Perturbation Theory

Transcription

1 Slides with rosper/lat X p. 1/48 Unifying Quantum Electrodynamics and Many-Body erturbation Theory Ingvar Lindgren ingvar.lindgren@physics.gu.se Department of hysics University of Gothenburg, Gothenburg, Sweden

2 Slides with rosper/lat X p. 2/48 New Horizons in hysics Makutsi, South Africa, November, 2015 In honor of rof. Walter Greiner at his 80 th birthday

3 Slides with rosper/lat X p. 3/48 Coworkers Sten Salomonson Daniel Hedendahl Johan Holmberg

4 Slides with rosper/lat X p. 4/48 Combining MBT 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)

5 Slides with rosper/lat X p. 5/48 Combining MBT 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 separately For high accuracy they should be evaluated in a coherent way QED effects should be included in the wave function

6 Combining MBT and QED QM and QED are seemingly incompatible QM: single time Ψ(t,x 1,x 2, ) Field theory: individual times Ψ(t 1,x 1 ;t 2,x 2, ) Consequence of relativistic covariance Bethe-Salpeter equation is relativistic covariant can lead to spurious solutions (Nakanishi 1965; Namyslowski 1997) Slides with rosper/lat X p. 6/48

7 Slides with rosper/lat X p. 7/48 Combining MBT and QED Compromise: Equal-time approximation All particles given the same time makes FT compatible with QM Some sacrifice of the full covariance very small effect at atomic energies

8 Slides with rosper/lat X p. 8/48 Controversy Chantler (2012) claims that there are significant discrepancies between theory and experiment for X-ray energies of He-like ions Theory: Artemyev et al 2005, Two-photon QED

9 Slides with rosper/lat X p. 9/48 Higher-order QED Higher -order QED can be evaluated by means of the procedure for combining QED and MBT using the Green s operator, a procedure for time-dependent perturbation theory

10 Slides with rosper/lat X p. 10/48 Time-independent perturbation HΨ = (H +V)Ψ = EΨ target function Ψ 0 = Ψ model function projection operator for the model space Bloch equation Ψ = ΩΨ 0 Ω wave operator Ω = Γ ( VΩ ΩW ) Γ = 1 E 0 H 0 W = V Ω Effective Interaction H eff Ψ 0 = (H 0 +W)Ψ 0 = (E 0 + E)Ψ 0 Effective Ham.

11 Slides with rosper/lat X p. 11/48 Bloch equation Ω = Γ ( VΩ ΩW ) Γ = 1 E 0 H 0 + ΓVΩ = + = [ΓV +ΓVΓV +ΓVΓVΓV + ] + Singular when intermediate state in model space () Singularity cancelled by the term ΓΩW Leads to Bloch equation Ω = Γ Q ( VΩ ΩW ) The finite remainder is the Model-Space Contr. Γ Q ΩW ΓQ = Q E 0 H 0

12 Slides with rosper/lat X p. 12/48 Time-dependent perturbation Standard time-evolution operator Ψ(t) = U(t,t 0 )Ψ(t 0 ) articles Time propagates only forwards

13 Slides with rosper/lat X p. 13/48 Time-dependent perturbation Standard time-evolution operator Ψ(t) = U(t,t 0 )Ψ(t 0 ) articles art. Holes Electron propagators Electron propagators make evolution operator covariant Covariant Evolution Operator (U Cov )

14 Slides with rosper/lat X p. 14/48 Time-dependent perturbation Covariant evolution ladder (t = 0, t 0 = ) U Cov = E E E U Cov = 1+ΓV +ΓV ΓV + ; Γ = 1 E H 0 Same as first part of MBT wave operator Ω = 1+Γ ( VΩ ΩW ) Singular when intermediate state in model space

15 Slides with rosper/lat X p. 15/48 Green s operator The Green s operator is defined U Cov (t) = G(t) U Cov (0) is the regular part of the Covariant Evolution Oper.

16 Slides with rosper/lat X p. 16/48 Green s operator t = 0 : First order: G (1) = U (1) Cov = Γ QV = Ω (1) Second order: G (2) = Γ Q VG (1) + δg(1) δe W(1) ; Γ Q = Q E H 0 = Γ Q VG (1) Γ Q G (1) W (1) +Γ Q δv δe W (1) Ω (2) = Γ Q VΩ (1) Γ Q Ω (1) W (1) Time- or energy-dependent perturbations can be included in the wave function

17 Slides with rosper/lat X p. 17/48 QED effects Non-radiative Retardation Virtual pair Radiative El. self-energy Vertex correction Vacuum polarization

18 Slides with rosper/lat X p. 18/48 QED effects are time dependent

19 QED effects are time dependent Can be combined with electron correl. Continued iterations Mixing time-independent and time-dependent perturbat. Combining QED and MBT Slides with rosper/lat X p. 19/48

20 Slides with rosper/lat X p. 20/48 Radiative QED Dimensional regularization in Coulomb gauge Developed in the 1980 s for Feynman gauge Formulas for Coulomb gauge derived by Atkins in the 80 s Workable procedure developed by Johan Holmberg in 2011 First applied by Holmberg and Hedendahl

21 First calculation of self-energy in Coulomb gauge Slides with rosper/lat X p. 21/48 Self-energy of hydrogen like ions Hedendahl and Holmberg, hys. Rev. A 85, (2012) Z Coulomb gauge Feynman gauge (3) (1) (2) (8) (3) (1) (1) (2) E = α π (Zα) 4 mc 2 n 3 F(Zα)

22 Slides with rosper/lat X p. 22/48 He-like systems Johan Holmberg s hd thesis Holmberg, Salomonson and Lindgren, hys. Rev. A 92, (2015) Q Q (A) (B) (C) (D) (E) (B) and (E) are DIVERGENT Divergence cancels due to Ward identity

23 Slides with rosper/lat X p. 23/48 He-like Argon (Z=18) Second order Irreducible Q 1621 mev Q Feynman gauge Coulomb gauge

24 Slides with rosper/lat X p. 24/48 He-like Argon (Z=18) Second order Irreducible Model-space contr. + Vertex correction Q Q 1621 mev Feynman , Coulomb

25 Slides with rosper/lat X p. 25/48 He-like Argon (Z=18) Second order Irreducible Model-space contr. + Vertex correction Q Q Q 1621 mev Feynman , Coulomb -1.1

26 Gauge independen Slides with rosper/lat X p. 26/48 He-like Argon (Z=18) Second order Irreducible Model-space contr. + Vertex correction Q Q Q 1621 mev Feynman , Coulomb Large cancellations in Feynman gauge

27 Slides with rosper/lat X p. 27/48 He-like Argon (Z=18) Third order Irreducible Model-space contr. + Vertex correction Q Q 71 Feynm Coul.

28 Slides with rosper/lat X p. 27/48 He-like Argon (Z=18) Third order Irreducible Model-space contr. + Vertex correction Q Q Feynm Coul

29 Slides with rosper/lat X p. 27/48 He-like Argon (Z=18) Third order Irreducible Model-space contr. + Vertex correction Q Q Feynm ??? Coul ???

30 Slides with rosper/lat X p. 28/48 He-like Argon (Z=18) Third order Irreducible Model-space contr. + Vertex correction Q Q Feynm. Coul ?????? First calculation of radiative QED beyond second order

31 Slides with rosper/lat X p. 29/48 He-like Argon (Z=18) Third order Irreducible Model-space contr. + Vertex correction Q Q Feynm. Coul ?????? First calculation of radiative QED beyond second order Has to be performed in Coulomb gauge Holmberg, Salomonson, Lindgren, RA 92, (2015)

32 Slides with rosper/lat X p. 30/48 Summary QED He-like gr. state Non-radiative and radiative (in mev) Z Two-photon Higher orders Non-radiative Radiative Non-radiative Radiative

33 Slides with rosper/lat X p. 31/48 Summary QED He-like gr. state Higher-order QED (in mev) Z Holmberg 2015 (calc) Artemyev 2005 (est d) (2) (3) (5) (8) (2) 7.7(50) Chantler 500 mev

34 Slides with rosper/lat X p. 32/48 Summary QED He-like gr. state Higher-order QED (in mev) Z Holmberg 2015 (calc) Artemyev 2005 (est d) (2) (3) (5) (8) (2) 7.7(50) The higher-order QED has previously been underestimated but still much too small to correspond to the Chantler discrepances

35 Slides with rosper/lat X p. 33/48 Dynamical processes The Green s operator can also be used in dynamical processes

36 Slides with rosper/lat X p. 34/48 Free particles Scattering amplitude free particles q p q S p = 2πiδ(E p E q )τ(p q) Optical theorem for free particles 2Im p is p = q ] 2 2πδ(E p E q )τ(p q) p q p Forward scattering p q q p Cross section

37 Slides with rosper/lat X p. 35/48 Free particles Scattering amplitude free particles q p q S p = 2πiδ(E p E q )τ(p q) Optical theorem for free particles 2Im p is p = q ] 2 2πδ(E p E q )τ(p q) The imaginary part of the forward scattering amplitude is proportional to the total cross section

38 Slides with rosper/lat X p. 36/48 Bound particles S = U(, ) = U Cov (, ) S-matrix becomes singular for bound states with intermediate model-space states Optical theorem for bound particles 2Im p ig(, ) p = q ] 2 2πδ(E p E q )τ( q) G(, ) is identical to the S-matrix, if there are no intermediate model-space states G always regular: "S-matrix cleaned from singularities"

39 Slides with rosper/lat X p. 37/48 Bound particles ig(, ) = 2πδ(E in E out )W 2Im p ig(, ) p = q ] 2 2πδ(E p E q )τ(p q) 2Im p W p = q 2πδ(E p E q )τ(p q) 2

40 ig(, ) = 2πδ(E in E out )W 2Im p ig(, ) p = q ] 2 2πδ(E p E q )τ(p q) 2Im p H eff p = q 2πδ(E p E q )τ(p q) 2 H eff = H 0 +W Optical theorem for free and bound particles Lindgren, Salomonson, Holmberg, RA 89, (2014) Slides with rosper/lat X p. 38/48

41 Radiative recombination Lindgren, Salomonson, Holmberg, RA 89, (2014) Shabaev et al, RA 61, (2000) a p First-order amplitude p p a p a a p Forward scattering Cross section Slides with rosper/lat X p. 39/48

42 Radiative recombination Self-energy insertion a p First-order amplitude a p p Forward scattering a a p p Cross section Slides with rosper/lat X p. 40/48

43 Slides with rosper/lat X p. 41/48 Radiative recombination Self-energy insertion leads to singularity that is taken care of in the Green s operator.

44 Radiative decay b a a Forward scattering a b a b ε p Cross section a a b r a r a b b Self-energy insertion (MSC) Slides with rosper/lat X p. 42/48

45 Slides with rosper/lat X p. 43/48 Radiative decay 1s 2p 1/2 transition in H-like Uranium Magnetic quadrupole to electrical dipole ampitude ratio M2/E1 Dirac QED Expt t: Stöhlker et al. RL 105, (2010) JB 48, (2015) Theory: Holmberg, Artemyev, Surzhykov, Yerohkin, Stöhlker, SSRA 92, (2015)

46 Slides with rosper/lat X p. 44/48 Conclusions and Outlook The Green s operator is a time-dependent wave operator Can combine time-dependent and time-independent perturbations, unifying QED and MBT Can be used for stationary as well as dynamical problems (real and imaginary parts, respectively) Improves the accuracy of theoretical estimates one order of magnitude

47 Slides with rosper/lat X p. 45/48 Conclusions and Outlook The Green s operator has been used to evaluate QED beyond second order, employing Coulomb gauge for radiative QED applied to dynamical problems to derive the Optical Theorem for bound systems to evaluate the QED effect in radiative recombination and in radiative decay (together with GSI, Jena)

48 Slides with rosper/lat X p. 46/48 This work has been supported by The Swedish Science Reseach Council The Humboldt Foundation Helmholtz Association Gesellschaft für Schwerionenforschung

49 Thank you! Slides with rosper/lat X p. 47/48

50 Slides with rosper/lat X p. 48/48