μ0 ,ε0→ c
Gfermi ~ α
MW2
V = mgh
| Gravitational force between any two bodies, masses M1 and M2 separated by distance R is given by Newton's Law of universal gravitation | V = GM1M2
R |
G=6.67x10-11 N m2 kg-2 is a universal constant
This gave us:
Nature and Nature's Laws lay hid in Night
God said "Let Newton be" and All was Light
Alexander Pope
But Einstein told us that in fact gravity is a by-product of mass curving space, which then
This gave us:
Cost: General Relativity is very non-intuitive
Nature and Nature's Laws lay hid in Night
God said "Let Newton be" and All was Light
It did not last: the Devil howling "Ho
Let Einstein be!" restored the status quo.
J. C. Squire
No known conflicts between GR and experiment, so maybe we should quit while we are ahead ......... The problems:
An interesting comparison: electric and gravitational forces have exactly the same 1/R2 dependence: what is the ratio of the electric and gravitational forces between two electrons?
F = k q2 R2
G me2 R2
~ 4.2 *1042
(Hitchhiker's Guide fans please note!). Why is this ratio so large?
We have no good theory of quantum gravity: either so weak that the corrections are infinitesimal or so strong that it is non-linear and theories diverge
Unified theories:
Roughly:
| Electricity and Magnetism → Electromagnetism | μ0 ,ε0→ c |
| Electromagnetism and Weak Interactions → Electroweak theory |
Gfermi ~ α
MW2
|
| Electroweak and Strong (QCD) → Grand Unified Theory |
|
But gravity has a long way to go.
| Planck Mass: mass at which grav. interactions become as strong as the rest
MPlanck ~ √1042 melectron
Better: construct mass from G,  and c
MPlanck ~ √(Hope of supergravity theory is to unify all forces |
 |
The Higgs mesonMH ≈ 200 GeVgives mass to all the other particles. However the Higgs interacts with everything, and its mass gets corrections. MH = M0 + C me which is totally irrelevant. |
|
However, Higgs can also interact with any new particles: any new theory of gravity must have new particles at around Planck Mass.
Hence
MH = M0 + C mplanckand if C = 10-15 (say) the masses we calculate are meaningless. |
 |
Options
and this leads us to extra-dimensional theories: many models but take ADD as most interesting: If we live in a 3+n dimensional space, gravitational force goes as
c/G) ~ 1019 GeV ~ 10 kg (!)
F = G3+n m1m2
Rn+2
but we don't!
However, if we have the extra dimensions only showing up at short distances r < r0, it can work: i.e.
F = G3 m1m2 r>r0
R2
F = G3+n m1m2 r<r0
Rn+2
So obviously
G3+n = G3 r0n
This means we can arrange to have Planck mass at much lower energy: e.g. 1-100 TeV
Major consequences:
| n (Number of Extra Dimensions | r0 Size of Extra Dimension |
| 1 | 1013 m (so ruled out) |
| 2 | 1 mm |
| 3 | 10-9 m |
| 4 | 10-11 m |
Picture from Greg Landsberg
| Hence it is very interesting to look at gravity at distances of microns. Washington experiment: confirms Newton down to ~ 150μ |  |
Ultra-Cold Neutrons: Neutrons with E ≤ 10-5 eV are totally reflected from several metal surfaces, including Be. Note this is a quasi-classical phenomenon (coherent interaction of neutron with many atoms)
Hence the "neutron bounce" state: a bound state of the neutron produced by the hard surface and gravity.
Bounce eigenfunctions Zn(z)
|
This can be converted to a dimensionless form via the substitution
y = βz-yn
β = (2mσ)1/3
2
Totally reflecting "ground" ⇒Z(0) = 0.
d2Zn(y) +y Zn( y ) = 0 dy2 Zn(-yn) = 0
| Equation for the Airy function, yn is n-th zero. |  |
energy En
"bounce height" zn |
En = |
For neutron,
| n | En | zn |
| 1 | 1.41 peV | 13.7 μ |
| 2 | 2.46 peV | 24.0 μ |
| 3 | 3.32 peV | 32.5 μ |
| 4 | 4.08 peV | 39.9 μ |
ν2→1 = 254 Hz (∼ middle C!): implies
(Nature 415,297 (2002).)
Proof of existence
| Neutron absorber is lowered, extinguishing signal. Classical prediction is for signal to vanish smoothly, Q.M. is for sudden cutoff at height h |  |
| Compare classical prediction (UCN's of all bounce heights, but M-B distrib) to quantum prediction (no bounce height ≤ 13 μ) | 
|
| Blow-up of lowest part of plot | 
|
So it exists: what do we do next?
The basic concept
|
 |
Does this work in Q-Mechs?: 3-D Schrodinger equation which describes the neutron:
-
subject to the boundary conditions Ψ(surface) = 0;
Solve:
| initial probability P0(r,z) = | Ψ0(r,z) |2 |  |
| probability P100(r,z) after 100 passes. |  |
Purity of the ground state. The total flux is also shown
| Can measure average height after a number of passes: lowest bounce state has <Height> ∼ 9.5μ |  |
These are for l = 1: works better for large l
Note device is entirely passive: should separate lowest bounce state very effectively.
( courtesy of Mike Pendlebury)
δV = δVR+iδVi where (for Be) δVR = 252neV,δVi = 1.26peV |
|
Real shift:
δEn = PnEb∼0.01peV ∼ 1%
Vi gives finite lifetime: arises from neutron absorption and inelastic interactions with H in the walls.
τ∼1.4×105s
Neutron halflife ∼885 s, so shouldn't be a problem
e.g could apply physical oscillation, see Felber et al
| Static mag. field produces polarized neutron states Spin ↑ and ↓ would have different bounce heights |
Fmag = Fgrav dB = mg ∼ 1.7 Tm-1 dz dmag |
Varying mag. field would allow excitation of state
B(z,r) = B0(z,r)sin(ωt)
Matrix elements vanish if the field is spatially uniform, but easy to arrange for varying field
Tn = <n|dmag.B|1>
The probability for resonant transition to the first excited state
P = Ω2 sin2( γt )
γ2
where
γ2 = Ω2+δω2,
Ω2 = Tn2
2
δω2 = (ω1-ω2)2- ω2
The frequency γ must satisfy
γ << | ω1-ω2 | = 254Hz
which implies a maximum magnetic field of a few milligauss.
"Natural" size of neutron EDM in absence of CP is 10-13 e cm.
| Expected values are model dependent: hence improving limit is important constraint (diagram taken from Ramsey) |  |
Can use analog of mag. mom. argument
dmag.B ⇒ del.E, E(z,r) = E0(z,r)*sin(ωt)
However process is dominated by neutron lifetime |
|
The gravitational potential is taken to be
|
λ in m-1, so we would expect the state to be sensitive to λ ∼ 105. Current expts (variations of Cavendish expt.) give λ<104 m-1for K∼G.
Putting slab of dense material below apparatus would shift energy levels:
Extra interaction is
|
 |
However we could oscillate slab (amplitude z0, frequency ω0)to produce an extra pot. term
δV(z,t)=sin(ωt)2πKρmNe-λ(z+z0) ((λ(2z+z0)+4)I1(λz0)-2λz0I0(λz0))
λ2
Transition time as a function of λ for K = 1. Could probably set limits for K ∼ 10-5 for λ ∼ 10 μ-1 better than anyone else, but not useful for K ∼ G |
 |
A final thought:
What is magic about the neutron?
could we use a bouncing atom (e.g. He3) or molecule (e.g. buckyball)?
Advantages
Disadvantages
so maybe we can construct a quantum pendulum sensitive to λ ~ 106 m-1
References: