Quantum effects in curved spacetime Hongwei Yu

Outline Motivation Lamb shift induced by spacetime curvature Thermalization phenomena of an atom outside a Schwarzschild black hole Conclusion

Quantum effects unique to curved spacetime Motivation Quantum effects unique to curved spacetime 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?

Lamb shift What is Lamb shift? Theoretical result: 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?

Lamb shift induced by spacetime curvature 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

Vacumm fluctuations Radiation reaction

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?

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.

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%. 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!

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.

Who is holding the atom at a fixed radial distance? circular geodesic motion bound circular orbits for massive particles stable orbits How does the circular Unruh effect contributes to the Lamb shift? Numerical estimation

Summary Spacetime curvature affects the atomic Lamb shift. It weakens the Lamb shift! The curvature induced Lamb shift can be remarkably significant outside a compact massive astrophysical body, e.g., the correction is ~25% at r~4M, ~16% at r~10M, ~1.6% at r~100M. The results suggest a possible way of detecting fundamental quantum effects in astronomical observations.

Thermalization of an atom outside a Schwarzschild black hole How a static two-level atom evolve outside a Schwarzschild black hole? Model: A radially polarized two-level atom coupled to a bath of fluctuating quantized electromagnetic fields outside a Schwarzschild black hole in the Unruh vacuum. The Hamiltonian

How – theory of open quantum systems The von Neumann equation (interaction picture) Environment (Bath) System The interaction Hamiltonian The evolution of the reduced system The Lamb shift Hamiltonian The dissipator

The master equation (Schrödinger picture) For a two-level atom The master equation (Schrödinger picture) The spontaneous emission rate The spontaneous excitation rate The time-dependent reduced density matrix The coefficients

The line element of a Schwarzschild black hole The trajectory of the atom The Wightman function The Fourier transform

The summation concerning the radial functions in asymptotic regions The spontaneous excitation rate of the detector The proper acceleration

The effective temperature The equilibrium state The effective temperature The grey-body factor

The geometrical optics approximation Low frequency limit High frequency limit The geometrical optics approximation The grey-body factor tends to zero in both the two asymptotic regions.

For an arbitrary position Near the horizon Spatial infinity For an arbitrary position

A stationary environment out of thermal equilibrium The effective temperature Analogue spacetime? B. Bellomo et al, PRA 87.012101 (2013).

Summary In the Unruh vacuum, the spontaneous excitation rate of the detector is nonzero, and the detector will be asymptotically driven to a thermal state at an effective temperature, regardless of its initial state. The dynamics of the atom in the Unruh vacuum is closely related to that in an environment out of thermal equilibrium in a flat spacetime.

Conclusion The spacetime curvature may cause corrections to quantum effects already existing in flat spacetime, e.g., the Lamb shift. The Lamb shift is weakened by the spacetime curvature, and the corrections may be found by looking at the spectra from a distant astrophysical body. The close relationship between the dynamics of an atom in the Unruh vacuum and that in an environment out of thermal equilibrium in a flat spacetime may provides an analogue system to study the Hawking radiation.

I'd put my money on some other correction Speculation “The size of the Proton”, Nature, 466, 213(2010) =0.84184(67) fm, which differs by 5.0 standard deviations from the CODATA value of 0.8768(69) fm. It implies that either the Rydberg constant has to be shifted by -110 kHz/c (4.9 standard deviations), or the calculations of the QED effects in atomic hydrogen atoms are insufficient. The first thing is to go through the existing calculations carefully ". It could be that an error was made, or that approximations made in existing quantum calculation simply aren't good enough. I'd put my money on some other correction