jean-pierre zendri infn- padova
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Jean-Pierre Zendri INFN- Padova. Planck Scale. Compton wavelength:. Quantum Field Theory. Schwarzschild radius:. General Relativity. Is the Planck scale accessible in earth based Experiments ? . Limits to distance (D) measurements. D. T:Round trip time. A. B. B ody A - PowerPoint PPT PresentationTRANSCRIPT
Jean-Pierre Zendri INFN-Padova
Planck Scale
Quantum Field Theory Compton wavelength:
General Relativity Schwarzschild radius:
𝜆𝑐=h𝑚𝑐
𝑟 𝑆=2𝐺𝑚𝑐2
𝑚𝑝=√ℏ𝑐𝐺
𝐿𝑎𝑟𝑔𝑒𝐻𝑎𝑑𝑟𝑜𝑛𝐶𝑜𝑙𝑙𝑖𝑑𝑒𝑟 𝐸≈ 1013𝑒𝑉𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙𝑤𝑎𝑣𝑒𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟𝑠 ∆ 𝑥≈ 10−21𝑚
𝐴𝑡𝑜𝑚𝑖𝑐𝑐𝑙𝑜𝑐𝑘𝑠 ∆ 𝑡≈ 10−18 𝑠
Is the Planck scale accessible in earth based Experiments ?
Limits to distance (D) measurements
Quantum Mechanics
General Relativity
Quantum mechanics + General Relativity
∆ 𝑝 ℏ2 ∆𝑥0
𝛿𝐷 ∆𝑥0+𝑇
2𝑀∆ 𝑥0𝛿𝐷𝑀𝑖𝑛 √ ℏ𝑇2𝑀=√ 2ℏ𝐷
𝑐𝑀→𝑀=∞
0
To avoid a Black-Hole formation 𝐷≥𝑅 h h𝑆𝑐 𝑤𝑎𝑟𝑧𝑠 𝑖𝑙𝑑=2𝐺𝑀𝑐2
𝜹𝑫𝑴𝒊𝒏≥√ ℏ𝑮𝒄𝟑 =𝑳𝒑
Body A Mass:MSize:<DBA D T:Round trip time
𝐷=𝑐𝑇2
∆ 𝑥0
Ligth Pulse
Uncertainty Principle and Gravity
Newtonian Gravity + Special Relativity +Equivalence principle
𝑎≈𝐺(𝐸 /𝑐2)𝑅2 ∆ 𝑥 ≈
(𝐿𝑝❑)2 ∆𝑝ℏ
∆ 𝑥 ≥ 12𝜋𝜔𝑆𝑖𝑛(𝜀) ∆ 𝑝≥ h𝜔𝑆𝑖𝑛(𝜔)
∆ 𝑥 ∆𝑝≥ℏ
Heisembeg Telescope
∆ 𝑥 ∆𝑝≈ℏ+¿¿
E: photon energy Tint= Interaction time
+Photon unknown direction (e)
Uncertainty Principle and Gravity
Different approaches to Quantum GravityString Theory and loop quantum gravityAmati, 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 experimentsLimits 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 holegedanken 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).
∆ 𝑥 ∆𝑝≈ ¿
Generalized Uncertainty Principle (GUP)
• 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)
∆ 𝑥 ∆𝑝≈ ¿At the Planck scale 𝛽0 ≈ 1
The length uncertaincy should be larger than Lp
GUP and harmonic oscillator ground state
GUP and AURIGAHigher 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)
• 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
The AURIGA detector
3m long Al5056 2200 kg 4.5 K
Effective mass vs reduced mass
Readout measures the axial displacement of a bar face corresponding to the first longitudinal mode
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
xcm1 xcm2
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
• FSig Signal Force• FTh Langevin thermal
force
(t)
(
Equipartion theorem
Active Cooling: principle
∝𝑇
()
𝑇=4.5𝐾𝑒𝑙𝑣𝑖𝑛
Not significant for GUT
• FSig Signal Force• FTh Langevin thermal
force• FCD Feedback Force
(t)
(
Equipartion theorem
Active Cooling: principle
• FSig Signal Force• FTh Langevin thermal
force• FCD Feedback Force
(t)
(
Equipartion theorem
Active Cooling: principle
Cold damped distribution
¿
∝𝑇
/
Cooling down to the ground state
Active or passive feedback cooling of one (few) oscillator mode
Displacement sensitivity improvement
Prepare oscillator in its fundamental stateNOYES
KBKT
KLC
MB MT MLC
System of three coupled resonators: the bar and the transducer mechanical resonators and the LC electrical resonator
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 amplifier3) 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.
F Back-action force
- 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
System of three coupled resonators: the bar and the transducer mechanical resonators
and the LC electrical resonator
AURIGA minimal energy
Modified commutators IGUP can be associated to a deformed canonical commutator
Planck scale modifications of the energy spectrum of quantum systems
Lack of observed deviations fromtheory at the electroweak scale
Lamb shift in hydrogen atoms
1S-2S level energy differencein hydrogen
Active Cooling: Fundamental limit𝑇𝑒𝑓𝑓 ∝ 𝑇𝛼
𝑇𝑒𝑓𝑓 𝑀𝑖𝑛=𝑇𝑛√1+ 2𝑄𝑇𝑇 𝑛𝑘𝑘𝑛
≤𝑇𝑛
𝑇 𝑛=1𝑘𝐵𝜔
√𝑆𝐹𝑛𝐹 𝑛𝑆𝑥𝑛𝑥𝑛
𝑘𝑛=√ 𝑆𝐹𝑛𝐹 𝑛𝑆𝑥𝑛𝑥𝑛
𝑇 𝑛≤ ℏ𝜔2𝑘𝐵Amplifier Noise temperature
Amplifier Noise Stiffness
Quantum Mechanics→
≤ ℏ𝜔2𝑘𝐵
xn:Amplifier additive noise Fn: Amplifier back-action noise
𝑛𝑡=𝑘𝐵𝑇𝑒𝑓𝑓 𝑀𝑖𝑛ℏ𝜔 =
𝑘𝐵𝑇 𝑛ℏ𝜔 √1+ 2
𝑄𝑇𝑇 𝑛𝑘𝑘𝑛
Auriga possible upgrading
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)
μ0 < 4 x 10 -13
Modified commutators IIModifications of commutators are not unique
Experiments could distinguish between the various approaches
M. Maggiore Phys. Lett. B 319, 83-86 (1993)
Spacetime granularity (Quantum Foam)
Property of the spacetime geometry and not of physical objects(Soccer ball problem ?)
Apparatus independent (not based on a specific QG model)
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)
AURIGA: re-interpretationAURIGA 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 set-up
Experimental proposals Ib
• Classical phase rotation• Short cavity is chalangin (10-6 m !)
Critical Points
h𝜈 /𝑐
h𝜈 /𝑐
Experimental proposals IIa
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
v=0
v=0
v ≠ 0
∆ 𝑥 ≈ 𝐿 h𝜈 (𝑛−1)𝑀𝑐2
𝑀 ,𝑛
h𝜈 /𝑛𝑐
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
transmissionCritical points
Thermal noiseOptical 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 DlpL/D thus D(LpL)1/2
There is an unavoidable transverse uncertaincy ∆ 𝑋 1 ∆ 𝑋 2≈ 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
HUMORHeisenberg
Uncertainty
Measured with
Opto-mechanical
Resonator
•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 quantumgravity 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
Soccer ball problem
Possible (ad hoc) solutions1) 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)
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
Minimal length Problems with standard special relativity (SR)
1) Physical momenta sum nonlinearly2) Correspondence principle