Can Spacetime curvature induced corrections
to Lamb shift be observable?
Hongwei Yu
Ningbo University and Hunan Normal
University
Collaborator:
Wenting
Zhou (Hunan Normal)
OUTLINE
Why

Test of Quantum effects
How

DDC formalism
Curvature induced correction to Lamb shift
Conclusion
Q
uantum effects unique to curved space
Hawking radiation
Gibbons

Hawking effect
Why
Unruh effect
Challenge: Experimental test.
Q: How about curvature induced corrections to
those already existing in flat spacetimes?
Particle creation by GR field
What is Lamb shift?
Theoretical result:
Experimental discovery:
In 1947, Lamb and Rutherford show that the level 2s
1/2
lies about
1000MHz, or 0.030cm

1
above the level 2p
1/2.
Then a more accurate
value 1058MHz.
The Dirac theory in Quantum Mechanics shows: the states, 2s1/2
and 2p1/2 of hydrogen atom are degenerate.
The Lamb shift
Important meanings
Physical interpretation
The Lamb shift results from the coupling of the atomic electron to
the vacuum electromagnetic field which was ignored in Dirac theory.
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?
What happens if vacuum fluctuations are modified?
How spacetime curvature
affects the Lamb shift? Observable?
If modes are modified, what would happen?
2. Casimir

Polder force
1.
Casimir effect
How
Bethe’s approach, Mass Renormalization (1947)
A neutral atom
fluctuating electromagnetic fields
P
A
H
I
Relativistic Renormalization approach (1948)
Propose “renormalization” for the first time in history!
(non

relativistic approach)
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.
Interpret the Lamb shift as a Stark shift
A neutral atom
fluctuating electromagnetic fields
E
d
H
I
Feynman’s interpretation (1961)
It
is
the
result
of
emission
and
re

absorption
from
the
vacuum
of
virtual
photons
.
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
.
J. Dalibard
J. Dupont

Roc
C. Cohen

Tannoudji
1997 Nobel Prize Winner
DDC formalism (1980s)
a neutral atom
Reservoir of vacuum fluctuations
)
(
I
H
)
(
N
)
(
)
1
(
)
(
)
(
N
t
t
A
t
A
t
)
(
)
(
N
t
A
t
Atomic
variable
Field’s
variable
)
(
N
)
(
t
t
A
0
≤λ ≤
1
)
(
)
(
)
(
t
A
t
A
t
A
s
f
Free field
Source field
Vacuum
fluctuations
Radiation
reaction
Vacuum
fluctuations
Radiation
reaction
Model: a two

level atom coupled with vacuum scalar
field
fluctuations.
Atomic operator
)
(
)
(
3
0
R
H
A
))
(
(
)
(
)
(
2
x
R
H
I
d
dt
a
a
k
d
H
k
k
k
F
3
)
(
How to separate the contributions of vacuum fluctuations and
radiation reaction?
Heisenberg equations
for the field
Heisenberg equations
for the atom
The dynamical
equation of
H
A
Integration
s
f
E
E
E
Atom + field Hamiltonian
I
F
A
system
H
H
H
H
——
corresponding to the effect of vacuum fluctuations
f
E
——
corresponding to the effect of radiation reaction
s
E
uncertain
?
Symmetric operator ordering
For the contributions of vacuum fluctuations and radiation reaction
to the atomic level ,
b
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
2
2
2
2
2
1
2
2
)
/
2
1
(
)
/
2
1
(
d
Sin
d
r
dr
r
M
dt
r
M
ds
A complete set of modes functions satisfying the Klein

Gordon equation:
outgoing
ingoing
Spherical
harmonics
Radial
functions
,
0
)

(
)
(
2
2
2
r
R
r
V
dr
d
l
),
1
2
/
ln(
2
*
M
r
M
r
r
and the Regge

Wheeler Tortoise coordinate:
with the effective potential
.
2
)
1
(
2
1
)
(
3
2
r
M
r
l
l
r
M
r
V
)
(
)
(
l
l
A
A
2
2
2
)
(
)
(
1
)
(
1
l
l
l
B
A
A
The field operator is expanded in terms of these basic modes, then we can
define the vacuum state and calculate the statistical functions.
It describes the state of a spherical massive body.
Positive frequency modes → the Schwarzschild time
t.
Boulware vacuum:
D. G. Boulware, Phys. Rev. D
11
, 1404 (1975)
reflection coefficient
transmission coefficient
0
)
(
dr
r
dV
M
r
3
0
)
(
3
2
2
M
r
dr
r
V
d
2
2
max
27
2
/
1
)
(
M
l
r
V
Is the atomic energy
mostly shifted near r=3M
?
For the effective potential:
3
2
2
)
1
(
2
1
)
(
r
M
r
l
l
r
M
r
V
For a static two

level atom fixed in the exterior region of the spacetime with a
radial distance (Boulware vacuum),
B
2
2
64
with
Lamb shift induced by spacetime curvature
rr
vf
In the asymptotic regions:
P. Candelas, Phys. Rev. D 21, 2185 (1980).
Analytical results
The Lamb shift of a static one in Minkowski spacetime with no boundaries.
M
—
It is logarithmically divergent , but the divergence can be removed by exploiting
a relativistic treatment or introducing a cut

off factor.
M
The revision caused by
spacetime curvature.
The grey

body factor
Consider the geometrical approximation:
3M
r
2M
V
l
(r)
,
max
2
V
;
1
~
l
B
,
max
2
V
.
0
~
l
B
The effect of backscattering of field modes off the curved geometry.
2.
Near r~3M, f(r)~1/4, the revision is positive and is about 25%
!
It is
potentially observable.
1. In the asymptotic regions, i.e., and , f(r)~0, the revision
is negligible!
M
r
2
r
Discussion:
The spacetime curvature
amplifies
the Lamb shift!
Problematic!
M
r
2
r
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.
(
2
l
1
)
R
l
(
r
)
l
0
2
4
2
1
2
M
/
r
(
2
l
1
)
R
l
(
r
)
l
0
2
1
4
M
2
(
2
l
1
)
B
l
(
)
l
0
2
1
r
2
(
2
l
1
)
B
l
(
)
l
0
2
4
2
1
2
M
/
r
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:
M
r
s
2
Set:
with
The recursion relation of a
k
(l,
ω
)
is determined by the differential of
the radial functions and a
0
(l,
ω
)=1, a
k
(l,
ω
)=0 for k<0
,
with
Similarly,
They are evaluated
at large r
!
For the outgoing modes,
r
The dashed lines represents and the solid represents .
2
)
(
l
A
2
)
(
l
B
4M
2
g
s
(
ω
r
) as function of
ω
and r.
For the summation of the outgoing and ingoing modes:
The relative Lamb shift F(r) for the static atom at different position.
For the relative Lamb shift of a static atom at position r,
Conclusion:
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.
The
relative
Lamb
shift
decreases
from
near
the
horizon
until
the
position
r~
4
M
where
the
correction
is
about
25
%
,
then
it
grows
very
fast
but
flattens
up
at
about
40
M
where
the
correction
is
still
about 4.8%.
1.
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.
F(r)
: observed by a static observer
at the position of the atom
F′(r)
: observed by a distant observer
at the spatial infinity
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.
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