1. Department of Physics, Hunan Normal University, Changsha, Hunan, China
Spacetime curvature induced corrections to Lamb shift --A quantum effect from the sky?
Hongwei Yu
Department of Physics, Hunan Normal University, Changsha, Hunan, China

2. OUTLINE Why-- Test of Quantum effects How -- DDC formalism
Modes outside a Massive body
Curvature induced correction to Lamb shift
Conclusion

3. Quantum effects unique to curved space
Why
Quantum effects unique to curved space
Hawking radiation
Gibbons-Hawking effect
Particle creation by GR field
Unruh effect
Challenge: Experimental test.
Q: How about curvature induced corrections to those already existing in flat spacetimes?

4. What is Lamb shift? Theoretical result: Experimental discovery:
The Dirac theory in Quantum Mechanics shows: the states, 2s1/2 and 2p1/2 of hydrogen atom are degenerate.
Experimental discovery:
In 1947, Lamb and Rutherford show that the level 2s1/2 lies about 1000MHz, or 0.030cm-1 above the level 2p1/2. Then a more accurate value 1058MHz.
The Lamb shift

5. Physical interpretation
The Lamb shift results from the coupling of the atomic electron to the vacuum electromagnetic field which was ignored in Dirac theory.
Important meanings
In the words of Dirac (1984), “ No progress was made for 20 years. Then a development came, initiated by Lamb’s discovery and explanation of the Lamb shift, which fundamentally changed the character of theoretical physics. It involved setting up rules for discarding … infinities…”
The Lamb shift and its explanation marked the beginning of modern quantum electromagnetic field theory.
Q: What happens when the vacuum fluctuations which result in the Lamb shift
are modified?

6. Our interest If modes are modified, what would happen?
1. Casimir effect
2. Casimir-Polder force
How spacetime curvature affects the Lamb shift? Observable?

7. fluctuating electromagnetic fields
How
Bethe’s approach, Mass Renormalization (1947)
A neutral atom
fluctuating electromagnetic fields
Propose “renormalization” for the first time in history! (non-relativistic approach)
Relativistic Renormalization approach (1948)
The work is done by N. M. Kroll and W. E. Lamb;
Their result is in close agreement with the non-relativistic calculation by Bethe.

8. fluctuating electromagnetic fields
Welton’s interpretation (1948)
The electron is bounded by the Coulomb force and driven by the fluctuating vacuum electromagnetic fields — a type of constrained Brownian motion.
Feynman’s interpretation (1961)
It is the result of emission and re-absorption from the vacuum of virtual photons.
Interpret the Lamb shift as a Stark shift
A neutral atom
fluctuating electromagnetic fields

9. C. Cohen-Tannoudji 1997 Nobel Prize Winner
DDC formalism (1980s)
J. Dalibard
J. Dupont-Roc
C. Cohen-Tannoudji Nobel Prize Winner

10. Reservoir of vacuum fluctuations
a neutral atom
Reservoir of vacuum fluctuations
Field’s variable
Free field
Source field
Atomic variable
0≤λ ≤ 1

11. Vacuum fluctuations
Radiation reaction
Vacuum fluctuations
Radiation reaction

12. Model: a two-level atom coupled with vacuum scalar field fluctuations.
How to separate the contributions of vacuum fluctuations and radiation reaction?
Model: a two-level atom coupled with vacuum scalar field
fluctuations.
Atomic operator

13. —— corresponding to the effect of vacuum fluctuations
Atom + field Hamiltonian
Heisenberg equations for the field
Heisenberg equations for the atom
Integration
The dynamical equation of HA
—— corresponding to the effect of vacuum fluctuations
—— corresponding to the effect of radiation reaction

14. Symmetric operator ordering
uncertain?
Symmetric operator ordering

15. For the contributions of vacuum fluctuations and radiation reaction to the atomic level ,
with

16. Application:
1. Explain the stability of the ground state of the atom;
2. Explain the phenomenon of spontaneous excitation;
3. Provide underlying mechanism for the Unruh effect; …
4. Study the atomic Lamb shift in various backgrounds

17. Waves outside a Massive body
A complete set of modes functions satisfying the Klein-Gordon equation:
Radial functions
Spherical harmonics
outgoing
ingoing
with the effective potential
and the Regge-Wheeler Tortoise coordinate:

18. Positive frequency modes → the Schwarzschild time t.
reflection coefficient
transmission coefficient
The field operator is expanded in terms of these basic modes, then we can define the vacuum state and calculate the statistical functions.
Positive frequency modes → the Schwarzschild time t.
Boulware vacuum:
D. G. Boulware, Phys. Rev. D 11, 1404 (1975)
It describes the state of a spherical massive body.

19. Is the atomic energy mostly shifted near r=3M?
For the effective potential:
Is the atomic energy mostly shifted near r=3M?

20. Lamb shift induced by spacetime curvature
For a static two-level atom fixed in the exterior region of the spacetime with a radial distance (Boulware vacuum),
with

21. Analytical results In the asymptotic regions:
P. Candelas, Phys. Rev. D 21, 2185 (1980).

22. The revision caused by spacetime curvature.
The grey-body factor
The Lamb shift of a static one in Minkowski spacetime with no boundaries. —
It is logarithmically divergent , but the divergence can be removed by exploiting a relativistic treatment or introducing a cut-off factor.

23. Consider the geometrical approximation:
Vl(r)
The effect of backscattering of field modes off the curved geometry.

24. Discussion:
1. In the asymptotic regions, i.e., and , f(r)~0, the revision is negligible!
Near r~3M, f(r)~1/4, the revision is positive and is about 25%! It is potentially observable.
The spacetime curvature amplifies the Lamb shift!
Problematic!

25. position
sum
Candelas’s result keeps only the leading order for both the outgoing and ingoing modes in the asymptotic regions. 1.
The summations of the outgoing and ingoing modes are not of the same order in the asymptotic regions. So, problem arises when we add the two. We need approximations which are of the same order! 2.
Numerical computation reveals that near the horizon, the revisions are negative with their absolute values larger than 3.

26. Numerical computation
Target:
Key problem:
How to solve the differential equation of the radial function?
In the asymptotic regions, the analytical formalism of the radial functions:

27. Set:
with
The recursion relation of ak(l,ω) is determined by the differential of
the radial functions and a0(l,ω)=1, ak(l, ω)=0 for k<0,

28. They are evaluated at large r!
For the outgoing modes,
with
They are evaluated at large r!
Similarly,

29. The dashed lines represents and the solid represents .

30. For the summation of the outgoing and ingoing modes:
4M2gs(ω|r) as function of ω and r.

31. For the relative Lamb shift of a static atom at position r,
The relative Lamb shift F(r) for the static atom at different position.

32. Conclusion:
The relative Lamb shift decreases from near the horizon until the
position r~4M where the correction is about 25%, then it grows
very fast but flattens up at about 40M where the correction is still
about 4.8%. 1.
F(r) is usually smaller than 1, i.e., the Lamb shift of the atom at an arbitrary r is usually smaller than that in a flat spacetime. The spacetime curvature weakens the atomic Lamb shift as opposed to that in Minkowski spacetime! 2.

33. Discussion：
What about the relationship between the signal emitted from the static atom and that observed by a remote observer?
It is red-shifted by gravity.

34. Conclusion It weakens the Lamb shift!
1. Spacetime curvature affects the atomic Lamb shift.
It weakens the Lamb shift!
2. It may become appreciable in certain circumstances
such as outside a very compact neutral star or a black hole.

35. Thank you!