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

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

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?

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

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?

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

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.

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

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

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

Vacuum fluctuations Radiation reaction Vacuum fluctuations Radiation reaction

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

—— 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

Symmetric operator ordering uncertain? Symmetric operator ordering

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

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

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:

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.

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

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

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

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.

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

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!

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.

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:

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,

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

The dashed lines represents and the solid represents .

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

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.

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.

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.

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.

Thank you!