Jean-Pierre Zendri
INFN-Padova
Planck Scale
Quantum Field Theory
General Relativity
Compton wavelength:
Schwarzschild radius:
ℎ
𝜆𝑐 =
𝑚𝑐
2𝐺𝑚
𝑟𝑆 = 2
𝑐
𝑚𝑝 =
ℏ𝑐
𝐺
Is the Planck scale accessible in earth based Experiments ?
𝐿𝑎𝑟𝑔𝑒 𝐻𝑎𝑑𝑟𝑜𝑛 𝐶𝑜𝑙𝑙𝑖𝑑𝑒𝑟
𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑤𝑎𝑣𝑒 𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟𝑠
𝐴𝑡𝑜𝑚𝑖𝑐 𝑐𝑙𝑜𝑐𝑘𝑠
𝐸 ≈ 1013 𝑒𝑉
∆𝑥 ≈ 10−21 𝑚
∆𝑡 ≈ 10−18 𝑠
Limits to distance (D) measurements
D
Body A
Mass:M
Size:<D
A
B
Ligth Pulse
T:Round trip time
𝑐𝑇
𝐷=
2
∆𝑥0
Quantum Mechanics
ℏ
∆𝑝 ~
2 ∆𝑥0
𝛿𝐷 ~ ∆𝑥0 +
𝑇
2 𝑀 ∆𝑥0
𝛿𝐷𝑀𝑖𝑛 ~
ℏ𝑇
=
2𝑀
General Relativity
To avoid a Black-Hole formation 𝐷 ≥ 𝑅𝑆𝑐ℎ𝑤𝑎𝑟𝑧𝑠ℎ𝑖𝑙𝑑
2𝐺𝑀
=
𝑐2
Quantum mechanics + General Relativity
𝜹𝑫𝑴𝒊𝒏 ≥
ℏ𝑮
= 𝑳𝒑
𝟑
𝒄
2ℏ𝐷
𝑐𝑀
→ 0
𝑀=∞
Uncertainty Principle and Gravity
Heisembeg Telescope
1
∆𝑥 ≥
2𝜋𝜔𝑆𝑖𝑛(𝜀)
∆𝑝 ≥ ℎ𝜔𝑆𝑖𝑛(𝜔)
∆𝑥∆𝑝 ≥ ℏ
Newtonian Gravity + Special Relativity +Equivalence principle
E: photon energy
𝐺 (𝐸/𝑐 2 )
𝑎≈
𝑅2
Tint= Interaction time
+Photon unknown direction (e)
(∆𝑝)2
∆𝑥∆𝑝 ≈ ℏ +
(𝐿𝑝 )2
ℏ
(𝐿𝑝 )2 ∆𝑝
∆𝑥 ≈
ℏ
Uncertainty Principle and Gravity
Different approaches to Quantum Gravity
String Theory and loop quantum gravity
Amati, D., Ciafaloni, M. & Veneziano, G.
Superstring collisions at planckian energies. Phys. Lett. B 197, 81-88 (1987)
Gross, D. J. & Mende, P. F. String theory beyond the Planck scale.
Nucl. Phys. B 303, 407-454 (1988)
Thought experiments
Limits to the measurements of BH horizon area
Maggiore, M. A generalized uncertainty principle in quantum gravity.
Phys. Lett. B 304, 65-69 (1993).
Scardigli, F. Generalized uncertainty principle in quantum gravity from micro-black hole
gedanken experiment. Phys. Lett. B 452, 39-44 (1999).
Jizba, P., Kleinert, H. & Scardigli, F. Uncertainty relation on a world crystal and its
applications to micro black holes. Phys. Rev. D 81, 084030 (2010).
Generalized Uncertainty Principle (GUP)
(∆𝑝)2
∆𝑥∆𝑝 ≈ 1 +
(𝑀𝑃 𝑐)2
ℏ
2
The length uncertaincy should be larger than Lp
• Including the clock wave function spread (quantum clock)
R.J.Adler, et all., Phys. Lett. B477, 424 (2000) .
W.A.Christiansen et all., Phys. Rev. Lett. 96, 051301 (2006).
• Including clock rate in the Schwarzschild geometry and holographic principle
E.Goklu, G. Lammerzhal Gen. Rel. and Grav. 43, 2065 (2011).
Y.J.Ng et all. Acad. Sci.755, 579 (1995).
• String theory
S.Abel and J.Santiago, J.Phys. G30, 83 (2004)
(∆𝑝)2
∆𝑥∆𝑝 ≈ 1 + 𝛽0
(𝑀𝑃 𝑐)2
At the Planck scale
𝛽0 ≈ 1
ℏ
2
GUP and harmonic oscillator ground state
GUP and AURIGA
Higher ground state energy for a quantum oscillator
Test on low-temperature oscillators set limits
GUP effects expected to scale with the mass m
Massive cold oscillators
(Sub millikelvin cooling of ton-scale oscillator)
The AURIGA detector
•




3m long
Al5056
2200 kg
4.5 K
•
•
•
Strain sensitivity 2 10-21<Shh<10-20
Hz-1/2
over 100 Hz band (FWHM ~ 26 Hz)
Burst Sensitivity hrss ~ 10-20 Hz-1/2
Duty-cycle ~ 96 %
~ 20 outliers/day at SNR>6
Effective mass vs reduced mass
Readout measures the axial displacement of
a bar face corresponding to the first longitudinal mode
xcm1
xcm2
Meff depends on the modal shape and interrogation
point of the readout (e.g. Meff ∞ if the measurement
is performed on a node of the vibration mode)
Really moving mass
1) Modal motion implies an oscillation of each half-bar center-of-mass, to
which is associated a reduced mass M/2
2) The energy associated to the oscillation of the couple of c.m.’s, having a
reduced mass Mred = M/2 is about 80% of that of the modal motion
Active Cooling: principle
𝑥 𝑡 = 𝑋1 (t)∙ 𝐶𝑜𝑠 𝜔𝑡 + 𝑋2 (t) ∙ 𝑆𝑖𝑛 𝜔𝑡
∝𝑇
• FSig Signal Force
•FTh Langevin thermal force
Equipartion theorem
(< 𝑋1
2
>+< 𝑋2
2
>)𝑚𝜔2 = 𝑘𝐵 𝑇
𝑇 = 4.5 𝐾𝑒𝑙𝑣𝑖𝑛
𝐸𝑒𝑥𝑝 ~𝑘𝐵 𝑇 ≈ 6 × 10−23 𝐽𝑜𝑢𝑙𝑒 (3.7 × 10−4 𝑒𝑉)
Not significant for GUT
Active Cooling: principle
𝑥 𝑡 = 𝑋1 (t)∙ 𝐶𝑜𝑠 𝜔𝑡 + 𝑋2 (t) ∙ 𝑆𝑖𝑛 𝜔𝑡
• FSig Signal Force
•FTh Langevin thermal force
• FCD Feedback Force
Equipartion theorem
(< 𝑋1
2
>+< 𝑋2
2
>)𝑚𝜔2 = 𝑘𝐵 𝑇
Active Cooling: principle
𝑥 𝑡 = 𝑋1 (t)∙ 𝐶𝑜𝑠 𝜔𝑡 + 𝑋2 (t) ∙ 𝑆𝑖𝑛 𝜔𝑡
∝ 𝑇/𝛼
∝𝑇
• FSig Signal Force
•FTh Langevin thermal force
• FCD Feedback Force
Equipartion theorem
(< 𝑋1
2
>+< 𝑋2
2
>)𝑚𝜔2 = 𝑘𝐵 𝑇
Cold damped distribution
(< 𝑋1
2
>+< 𝑋2
2
>)𝑚𝜔2 = 𝑘𝐵 𝑇
𝑚𝜔
𝑄𝛼
Cooling down to the ground state
Active or passive feedback cooling of one (few) oscillator mode
Displacement sensitivity improvement
Prepare oscillator in its fundamental state
NO
YES
Back-action force F∝ 𝐸0 × 𝑄 𝑡
MB
KB
MT
KT
MLC
KLC
1) The bar resonator is coupled to a lighter resonator, with the same resonance
frequency to amplify signals
2) A capacitive transducer, converts the differential motion between bar and the
ligher resonator into an electrical current, which is finally detected by a
low noise dc SQUID amplifier
3) The transducer efficiency is further increased by placing the
resonance frequency of the electrical LC circuit close to the mechanical resonance
frequencies, at 930 Hz.
System of three coupled resonators:
the bar and the transducer mechanical resonators and the LC electrical resonator
System of three coupled resonators:
the bar and the transducer mechanical resonators
and the LC electrical resonator
- At increasing feedback gain, the 3
modes of the detector reduce
their vibration amplitude.
- The equivalent temperature of
the vibration was reduced down to
Teff=0.17 mK
AURIGA minimal energy
Modified commutators I
GUP can be associated to a deformed canonical commutator
Planck scale modifications of the energy spectrum of quantum systems
Lamb shift in hydrogen atoms
1S-2S level energy difference
in hydrogen
Lack of observed deviations from
theory at the electroweak scale
Active Cooling: Fundamental limit
𝑇
𝑇𝑒𝑓𝑓 ∝
𝛼
xn:Amplifier additive noise
𝑇𝑒𝑓𝑓𝑀𝑖𝑛 = 𝑇𝑛
Amplifier Noise
temperature
Amplifier Noise
Stiffness
α≫1
0
Fn: Amplifier back-action noise
ℏ𝜔
2 𝑇 𝑘
1+
≤ 𝑇𝑛 ≤
2𝑘𝐵
𝑄 𝑇𝑛 𝑘𝑛
1
𝑇𝑛 =
𝑆
𝑆
𝑘𝐵 𝜔 𝐹𝑛𝐹𝑛 𝑥𝑛 𝑥𝑛
𝑘𝑛 =
𝑆𝐹𝑛 𝐹𝑛
𝑆𝑥𝑛𝑥𝑛
Quantum Mechanics 𝑇 ≤ ℏ𝜔
𝑛
2𝑘𝐵
Auriga possible upgrading
𝑘𝐵 𝑇𝑒𝑓𝑓𝑀𝑖𝑛 𝑘𝐵 𝑇𝑛
2 𝑇 𝑘
𝑛𝑡 =
=
1+
ℏ𝜔
ℏ𝜔
𝑄 𝑇𝑛 𝑘𝑛
Temper. [K]
Tn [quanta]
nt
b0
Auriga Now
4.2
400
25000
<3x1033
Auriga Cooled
0.1
27 *
1000
<1.2x1032
Auriga cooled+new
SQUID
0.1
10 **
600
<7x1031
* P.Falferi et al. APS 88 062505 (2006)
** P.Falferi et al. APS 93 172506 (2008)
Auriga just cooling down
Auriga cooled down +
New SQUID
Further improvements expected
decreasing the LC thermal noise
(but never nt<1/2)
Modified commutators II
Modifications of commutators are not unique
Experiments could distinguish between the various approaches
M. Maggiore Phys. Lett. B 319, 83-86 (1993)
μ0 < 4 x 10 -13
Spacetime granularity (Quantum Foam)
General Relativity
Quantum mechanics
Mass (energy) curves spacetime
Vacuum energy
Energy of the virtual particles gives
space time a "foamy" character at L ≈ Lp
(Wheeler, 1955)
Property of the spacetime geometry and not of physical objects
(Soccer ball problem ?)
Apparatus independent (not based on a specific QG model)
AURIGA: re-interpretation
AURIGA is not the “coolest” oscillator, but is the most motionless
Xrms= (kT/mω2)1/2 = (Eexp/mω2)1/2 ≈ 6 X 10-19 m
Macroscopic oscillators in their quantum ground state
(ω0 = 1 GHz
T ≈ 50 mK)
Experimental proposals I/1
A sequence of 4 pulse is applied to the mechanical oscillator such that
in the mechanical moves in the phase space around a loop:
+Xm –Pm - Xm +Pm
Quantum mechanics: [Xm ,Pm]≠0
Experimental proposals Ib
Experimental set-up
Critical Points
• Classical phase rotation
• Short cavity is chalangin (10-6 m !)
Experimental proposals IIa
𝑀, 𝑛
1) The photon has to discard momentum
into the crystal
2) This momentum will be returned to the
photon upon exit.
3) The crystal moves for a distance scaling
with the energy of the incoming photon
ℎ𝜈/𝑐
v=0
ℎ𝜈/𝑛𝑐
v≠0
ℎ𝜈/𝑐
∆𝑥 ≈ 𝐿
v=0
ℎ𝜈(𝑛 − 1)
𝑀𝑐 2
Experimental proposals IIb
If the energy of the photon is so low that the crystal should move less than the Planck
length, the photon cannot cross the crystal, leading to a decrease in the transmission
Critical points
Thermal noise
Optical dissipations
Experimental proposals III
1. Space time is described with a wave function frequency limited (the Plank
frequency).
2. If one end point of the particle position is limited by an aperture with size D
the uncertaincy of the other points at distance L is limited by diffraction to be
lpL/D
3. The possible orientation are minimized for DlpL/D thus D(LpL)1/2
There is an unavoidable transverse uncertaincy
∆𝑋1 ∆𝑋2 ≈ 𝐿𝑝 × 𝐿
In a Michelson interferometer the effect appears as noise that resembles a random
Planckian walk of the beam splitter for durations up to the light-crossing time.
Measurable (according to Hogan) using two cross-correlated two nearly
coolacated ng Michelson interferometers with arms length of about L=40
meters
Conclusions
•The Auriga detector constrained the plank scale for a macorscopic body. Further
improvements are possible but still far from the “traditional” plank scale.
•Recent experiments on macroscopic mechanical oscillators showed that they
behaves as quantum oscillator.
•Put in prespective a new generation of experiment with macroscopic body should be
abble to approach the plank scale.
•It is not clear if these experiments would be abble to constrain the quantum gravity
because the describtion of the macroscopic objects in the framework of quantum
gravity models is still lacking.
•A INFN group is in charge to examine the feasibility of the new proposed
experiments and in case to propose new and more realistic set-up
Heisenberg
Uncertainty
Measured with
Opto-mechanical
Resonator
HUMOR
Soccer ball problem
Minimal length
Problems with standard special relativity (SR)
Doubly Special Relativity
Amelino-Camelia, G. Int. J. Mod. Phys. D 11, 1643-1669 (2002).
Modified SR with two invariants: speed of light c, and minimal length Lp
Extremely huge for particles,
but small for planets, stars and ... soccer balls
Problem of macroscopic
(multiparticle) bodies
1) Physical momenta sum nonlinearly
2) Correspondence principle
Possible (ad hoc) solutions
1) Effects scale as the number of constituent
(Atoms, quarks... ?)
Quesne, C. & Tkachuk, V. M. Phys. Rev. A 81, 012106 (2010).
Hossenfelder, S. Phys. Rev. D 75, 105005 (2007) and Refs. therein
2) Decoherence ?
Magueijo, J. Phys. Rev. D 73, 124020 (2006)