A new uncertainty principle from quantum gravity and its implications. Saurya Das University of Lethbridge CANADA. Chandrayana 06 January 2011

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1 A new uncertainty principle from quantum gravity and its implications Saurya Das University of Lethbridge CANADA Chandrayana 06 January

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3 Plan: A problem with Quantum Gravity A new uncertainty principle from Quantum Gravity Phenomenology and Predictions: can it be tested in the laboratory? 1 and 3 dimensions Discreteness of Space? Summary and Outlook 1. S. Das, E. C. Vagenas, Phys. Rev. Lett. 101, (2008), arxiv: A. Ali, S. Das, E. C. Vagenas, Phys. Lett. B678, (2009), arxiv: S. Das, E. C. Vagenas, A. Ali, Phys. Lett. B690 (2010) 407, arxiv:

4 Why Quantum Gravity? Why not? (3 other forces are quantum) Classical Gravity + Quantum Fields Information Loss/Non-unitary QM Resolution of singularities Interaction of classical gravity wave with a quantum wavefunction energy/momentum non-conservation 4

5 Information Black Hole Heat little or no information INFORMATION LOSS }{{} ρ 2 t = (Uρ o U )(U ρ 0 U ) = Uρ 2 0 U = Uρ 0 U = ρ t Thus: I Pure density matrix ρ 2 0 = ρ 0 Non-unitary evolution U U 1 Mixed density matrix ρ 2 t ρ t 5

6 Problem with Quantum Gravity Newton s constant G d = 1 M d 2 P l Dimensionless G d s d 2/2, G d t d 2/2 as s, t (s, t (Energy) 2 ) Perturbatively, Gravity is Non-Renormalizable Some candidate theories String Theory, Loop Quantum Gravity, Path Integrals, Causal Sets, Causal Dynamical Triangulations, Non-Commutative Geometry, Supergravity,... 6

7 Too many theories: Another problem with Quantum Gravity String Theory, Loop Quantum Gravity, Non-Commutative Field Theory, Dynamical Triangulations, Causal Sets,... Too few experiments = Zero Why? T ev Quantum Gravity effects expected at the Planck Scale Atomic Physics 10 ev T ev. LHC 10 T ev Difference of orders of magnitude QG Experimental Signatures? First: A New Uncertainty Principle in Quantum Gravity It is my honest opinion that when people try to get hold of the laws of nature by thinking alone, the result is pure rubbish - Max Born to Einstein,

8 Generalized Uncertainty Principle: why? Black Hole Physics String Theory Loop Quantum Gravity, via Polymer Quantization Non-commutative geometry A meeting ground for various theories? 8

9 Heisenberg s Microscope D f φ λ y p cos φ φ Position uncertainty x f λ D λ 2ϕ e x p sin φ x (Minimum Resolving Power) Momentum uncertainty p = p sin ϕ = h λ sin ϕ hϕ λ p x h 2 Heisenberg Uncertainty Principle 9

10 Heisenberg s Microscope with a Black Hole (Extremal RN) D f φ λ y p cos φ φ x p sin φ x xnew = r + (M + M) r + (M) (GM + G M) 2 GQ 2 = G M + r + = GM + (GM) 2 GQ 2 (GM) 2 GQ 2 2G M l2 P l λ ( M = λ h h sin ϕ, p = ) λ l2 P l p h sin ϕ l2 P l p h x + x new h x + β 0 l2 P l p h [ ] p x h l β 2 P l 0 p 2 h 2 Generalized Uncertainty Principle l P l = G h c 3 = m. The new term is effective only when x l P l or p T ev /c M. Maggiore, Phys. Lett. B304, 65 (1993) 10

11 Heisenberg s Microscope with an elementary string D Critical Angle f φ λ y p cos φ φ x p sin φ x x new E p l P l h p [ ] p x h l β 2 P l 0 p 2 h 2 Generalized Uncertainty Principle D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B (1989) 11

12 Minimum Observable Length from the GUP Invert p x h 2 [ 1 + β 0 l 2 P l h 2 p 2 ] p h l P l [1 ± x 2 β 0 l 2 P l ] x β 0 l P l x min (Min length) Note: x min / Space is discrete 12

13 . p x < [x, p] > HUP GUP Principle p i x i h 2 p i x i h 2 [ 1 + β 0 l 2 P l h 2 p 2 ] Algebra [x i, p j ] = i hδ ij [x i, p j ] = i h[δ ij + β 0 h 2 l2 P l(p 2 δ ij + 2p i p j )] }{{} New A. Kempf, G. Mangano, R. B. Mann, Phys. Rev. D (1995) 13

14 Problems T ev, m (Planck scale/qg) in whose frame?? Problem of Lorentz Covariance of QG Effects l 2 P l m 2 Bad for phenomenology Is there a GUP linear in l P l? Postulate a linear term Look for a linear term 14

15 New [x, p] algebra from Doubly Special Relativity Theories (One way to solve the QG in whose frame problem) [J i, K j ] = ϵ ijk K k, [K i, K j ] = ϵ ijk J k But K i = L i 0 + l P l p i p a p a v Z Z p 0 = p z = p x = p y = γ(p 0 vpz ) 1+l P l (γ 1)p 0 l P l γvpz γ(pz vp 0 ) 1+l P l (γ 1)p 0 l P l γvpz γ(pz vp 0 ) 1+l P l (γ 1)p 0 l P l γvpz γ(pz vp 0 ) 1+l P l (γ 1)p 0 l P l γvpz 15

16 E 2 p 2 m 2 E 2 f(e, p, l P l ) p 2 g(e, p, l P l ) = m 2 ϵ 2 π 2 [x i, p j ] = i h p i π j f = p E, g = 1 1 l P l p (massless) [x i, p j ] = i h[(1 l P l p )δ ij + l 2 P l p ip j ] J. Magueijo, L. Smolin, Phys. Rev. Lett (2002), J. L. Cortes, J. Gamboa, Phys. Rev. D (2005) 16

17 So now we have two new algebras/gups [x i, p j ] = i h[1 + β 0 l2 P l h 2 (p 2 δ ij + 2p i p j )] Quadratic & x l P l [x i, p j ] = i h[(1 l P l p )δ ij + l 2 P l p i p j ] Linear & quadratic & p M P l c Can we make them compatible? Try [x i, p j ] = i h[δ ij + δ ij α 1 p + α 2 p i p j p + β 1δ ij p 2 + β 2 p i p j ] Use Jacobi identity with commuting coordinates and momenta ] ] ] [[x i, x j ], p k = [[x j, p k ], x i + [[p k, x i ], x j = 0 [x i, p j ] = i h [ ( ) δ ij α pδ ij + p ip j p ] + α 2 (p 2 δ ij + 3p i p j ) x ( x) min α 0 l P l, p ( p)max M P l c α 0 α = α 0 M P l c = α 0 l P l h. α 0 = O(1) (normally) Although current experiments α α 1 10 T ev /c 17

18 [x i, p j ] = i h [ ( δ ij α pδ ij + p i p j p But define: ( ) p j = p 0j 1 αp0 + 2α 2 p 2 0 [x i, p j ] =... is satisfied Consider any Hamiltonian H = p2 2m + V ( r) = 1 2m = p2 0 2m + V ( r) α }{{} m p3 0 + O(α 2 ) }{{} H 0 H 1 Consequences ) ( )] + α 2 p 2 δ ij + 3p i p j p j i h x pos n space i with [x i, p 0j ] = i hδ ij, p 0j = i h x j ( ( )) p0j 1 αp0 + 2α 2 p V (r) = p2 0 2m + V ( r) Universal Quantum Gravity Effect! i h3 α d 3 } m {{ dx 3 } position space 18

19 Schrödinger Equation [H 0 +H 1 ]ψ = [ h2 2m d 2 + V (x) iα h3 dx2 m d 3 ] dx 3 ψ = i h ψ t Two Consequences 1. New Perturbed Solutions and New Conserved Current ( ) ( ) J = h 2mi ψ dψ dx ψ dψ dx + α h2 d 2 ψ 2 m dx 3 dψ dψ 2 dx dx ρ = ψ 2 J, x + ρ t = 0 New Reflection/Transmission Currents 2. New Non-Perturbative solution e ix/l P l Discreteness of space 19

20 Applications to Quantum Mechanical Systems Simple Harmonic Oscillator Landau Levels Lamb Shift Scanning Tunneling Microscope Particle in a box - new non-perturbative solution Precision required to test GUP: 1 part in S. Das, E. C. Vagenas, Phys. Rev. Lett. 101, (2008), arxiv:

21 Simple Harmonic Oscillator) H = p2 0 2m mωx2 }{{} [ H 0 ψn = 1 2 n n! ( mω π h ) 1 4 e + α m p3 + 3α2 2 p4 }{{} H 1 mωx 2 2 h Hn( mω h x) ] E GUP = ψ n H 1 ψ n (p 4 term) + k n <k 0 H 1 n 0 > 2 E 0 n E0 k (p 3 term) E GUP (0) E 0 = 9 2 h2 ω 2 mα hωmα 2 E GUP (0) α 2 Not so good 21

22 Landau Levels H 0 = 1 2m Particle of mass m, charge e in constant B = Bẑ, i.e. A = Bxŷ, ωc = eb/m ( p 0 e A) 2 = p 2 0x H = 1 2m p 2 2m + 0y 2m }{{} h 2 k 2 /2m ( p 0 ea eb m xp 0y + e2 B 2 2m x2 = p2 0x 2m ) 2 α m ( p 0 ea) 3 = H0 3 8mαH mω2 c ( x E n(gup ) E n = 8mα( hω c ) 1 2 (n ) α (B = 10 T ) Conclude ) 2 mωc hk α 1 and E n(gup ) E n is too small, or Measurement accuracy of 1 in 10 3 in STM α 0 <

23 Lamb Shift [ ][ ] H 0 = p2 0 2m k r, H 1 = m α p3 0 = (α 8m) H 0 + k H r 0 + k 1 2 r ) En = 4α2 3m 2 (ln 1 α ψ nlm (0) 2 (Lamb Shift) ψ 100 ( r) = E n(gup ) En ψ nlm(0) = 2 ψ nlm (0) n m l H 1 nlm {n l m r n } ={nlm} En E l m = 10 3 αe n 0 ψ 200 ( r) E n(gup ) E n = 2 ψ nlm(0) ψ nlm (0) α E 0 27M P lc α 0 Conclude α 1 and E n(gup ) E n is too small, or Measurement accuracy of 1 in α 0 < (better!) 23

Potential Barrier (Scanning Tunneling Microscope) E V >E 0 Quantum Tunneling Transmission Current x=0 Barrier x=a TIp Sample [H 0 + H 1 ]ψ = Eψ [H 0 + H 1 ]ψ = (V 0 E)ψ [H 0 + H 1 ]ψ = Eψ ψ 1 = Ae ik

24 Potential Barrier (Scanning Tunneling Microscope) E V >E 0 Quantum Tunneling Transmission Current x=0 Barrier x=a TIp Sample [H 0 + H 1 ]ψ = Eψ [H 0 + H 1 ]ψ = (V 0 E)ψ [H 0 + H 1 ]ψ = Eψ ψ 1 = Ae ik x + Be ik ix x + P e 2α h k = 2mE h 2, k 1 = 2m(V 0 E) h 2 ψ 2 = F e k 1 x + Ge k 1 x ix + Qe 2α h ψ 3 = Ce ik ix x + Re 2α h k = k(1 + α hk), k = k(1 α hk), k 1 = k 1 (1 iα hk 1 ), k 1 = k 1 (1 + iα hk 1 ) 24

25 Take into account Continuity of ψ, ψ, ψ at each boundary (cannot set P, Q, R = 0) New current Transmission Current T = J R JL = C A 2 2α hk B A 2. [ = T 0 1+2α hk(1 T 1 0 ) ], T 0 = 16E(V 0 E) V 2 0 e 2k 1 a = usual m = me = 0.5 MeV /c 2, E V 0 = 10 ev a = m, I = 10 9 A, G = 10 9, δi GUP I 0 I T = δt GUP T 0 = 10 26, δi GUP = GδI GUP = A, α 0 = 1, T 0 = 10 3 τ = e δi GUP = 10 7 s a month 25

26 Apparent barrier height Φ A V 0 E ΦA = h d ln I 8m da (T 0 = 16E(V 0 E) V 0 2 e 2k 1 a ) α h2 (k 2 +k 2 1 )2 8m(kk 1 ) e 2k 1a New ΦA vs d ln I da 26

27 GUP effects on Atomic/Molecular/Condensed Matter Systems Stark Effect, Zeeman Effect, Berry s Phase, Bohm-Aharonov effect, Dirac Quantization, Anomalous Magnetic Moment of Electron, Quantum Hall Effect, Anderson Localization, Superconductivity, Coherent States, Lasers,... Statistical Mechanical Systems Bose-Einstein Condensation, Fermi Levels, Chandrasekhar Limit,... Normally forbidden processes Atomic Transitions 27

28 Look at New Non-perturbative solution of the Cubic Schrödinger Equation 28

29 Particle in a Box n=1 x=0 x=l n=2 h2 2m d 2 ψ dx 2 = Eψ ψ(x) = Ae ikx + Be ikx (k 2mE/ h 2 ) ψ(0) = 0 A + B = 0 ψ(x) = 2iA sin(kx) ψ(l) = 0 kl = nπ E n = k2 h 2 2m = n2 π 2 h 2 2mL 2 29

30 Particle in a Box with GUP New n=1 x=0 x=l [ h2 2m n=2 d 2 ψ dx i α h3 2 m n=1 1 d 3 dx 3 ] = Eψ ψ(x) = Ae ik x + Be ik x + Ce ix Choose Real 2α h New Solution (Cannot just set it to zero!) k = k(1+kα h), k = k(1 kα h) 30

31 ψ = 2iA sin(kx)+c 2iA sin(kl) = C C = C e iθ C etc ψ(0) = 0 A + B + C = 0 [ e ikx + e ix/2α h] α hk 2 x ψ(l) = 0 [ i Ce ikx + 2A sin(kx) [ )] [ ] e i(kl+θ C ) e i(l/2α h θ C + α hk 2 L i C e i(kl+θ C ) + 2A sin(kl) }{{} = C [e )] i(kl+θ C ) e i(l/2α h θ C Take real parts of both sides (A = Real) cos( L 2α h θ C ) = cos(kl + θ C ) = cos(nπ + θ C + ϵ) Solution ] O(α higher power ) L 2α h = L = nπ + 2qπ + 2θ C n 1 π + 2θ C 2α 0 l P l L 2α h = L = nπ + 2qπ n 1 π 2α 0 l P l n 1 2q ± n N. 31

32 n=1 x=0 n=2 n=1 1 x=l New Only certain Ls can fit both sin(kl) and cos( L 2α h ) Box Length is Quantized! Need at least one particle for measuring lengths Perhaps all measured lengths are quantized? A. Ali, S. Das, E. C. Vagenas, Phys. Lett. B678, (2009), arxiv:

33 Relativistic Wave Equations Klein-Gordon For Stationary States: 2mE E 2 m 2, k k E 2mc 2 mc2 2E L Quantization Unchanged Problems with KG Most elementary particles are fermions How to generalize to 2 and 3 dimensional box? p = p 0 αp 0 p = p 0 α p 2 0x + p2 0y + p2 0z p p 0 α h dx 2 d2 dy 2 d2 dz 2 }{{} Non-local d2 p 33

34 Dirac Equation Z Y p 0 α p X Hψ = ( α p + βmc 2) ψ = ( α p 0 α( α p 0 )( α p 0 ) + βmc 2) ψ = Eψ ψ e i t r ( χ ρ σχ ) Hψ = e i t r ( ((m αt 2 ) + t r + i σ ( t ρ))χ ( ) t (m + α 2 t 2 ) ρ σχ ) = Eψ, ψ = e i k r ( χ ), ψ = e i ˆq r α h ( χ ) rˆk σχ ˆq σχ 34

35 Confining Wavefunction Superposition of 2 d + 1 eigenfunctions (d=1,2,3) ψ = d [ d j=1 i=1 [ d (e )) ] ik i x i + e i(k i x i δ i + F e i ˆq r α h χ i=1 ( e ik i x i + ( 1) δ ij e i(k i x i δ i ) ) rˆk j +F e i ˆq r α h ] qˆ j σ j χ MIT Bag Boundary Conditions (Zero flux through boundaries) ψγ µ ψ = 0 ± iβα l ψ = ψ 35

36 ) e iδ k (1 + irˆk k = e i(2k k L k δ k ) ( 1 + irˆk k ) = ( ) irˆk k 1 [F k 1 + ˆqk 2 F, θk arctan ˆqk ] ( ) irˆk k 1 + f 1 k F k e iθ k (x k = 0) ) + f 1 k F i k e ( ˆqk L k α h +θ k e i(k k L k δ k ) (x k = L k ) Comparing k k L k = δ k = arctan ( ) hk k + O(α) mc [ ] ˆq k = 1/ d L k α 0 l P l = (2p k π 2θ k ) d, p k N A N N k=1 L k α 0 l P l = d N/2 N k=1 (2p kπ 2θ k ), p k N. Measurable Lengths, Areas and Volumes are Quantized! 36

37 Dirac equation: Spherical Cavity Z R U=0 U= οο Y ( σ p 0 ) χ 2 + ( σ p 0 ) χ 1 ( ( mc 2 + U mc 2 + U ) ) X χ 1 αp 2 0 χ 1 = Eχ 1 χ 1 αp 2 0 χ 2 = Eχ 2. ψ = ( ) ( χ 1 = χ 2 gκ(r)y j 3 jl (ˆr) ifκ(r)y j 3 jl (ˆr) ) ; gκ(r) = Nj l (p 0 r), fκ(r) = N j l (p 0r) + O(a) f N κ = N e ir/α h, g N κ = in e ir/α h }{{}, New Boundary Conditions j l (pr) = j l (pr), tan(±r/α) = 1. R α = ± π 4 + 2pπ, p N Radii, Areas, Volumes discrete 37

38 Curved Spacetime: Schrödinger Equation General Solution for any (linear) potential ψ = Aψ 1 + Bψ 2 + C(α) ψ 3 }{{} e ix/2α h ψ(0) = 0 = ψ(l) e il/2α h = 1 + O(α) L 2α h = quantized Results hold in Curved Spacetime (small lengths = linear gravitational potential) Curved Spacetime: Dirac Equation? 38

39 Spacetime near the Planck Scale S. Das, E. C. Vagenas, A. Ali, to appear in Phys. Lett. B, arxiv:

40 A note on Planck scale and reference frames Sph. Symm. Hamiltonian H = p2 2m + V (r) E S Classical Solution (Breaks Spherical Symm) Quantum Mech Solution (l=m=s=0) (Uncertainty restores Spherical Symm) 40

41 A preferred frame/ Aether would solve the QG in whose frame? problem - Planck scale in that frame! But aether breaks Lorentz Symm (one p µ in the light cone) But If Ψ }{{} aether = N dω p Ψ p Superpos n of all p µ within the light cone t p Y X (Again, uncertainty restores Lorentz Symmetry) Preferred frame chosen momentarily when a measurement is made! Ψ Ψ = N 2 4π MP l 0 p 2 dp = N 2 M 2 (also normalizable) 2 p 2 +m 2 P l So, a Lorentz covariant aether may solve the problem Dirac, Nature

42 Summary and Conclusions One GUP seems to fit Black Holes, String Theory, DSR,... Planck Scale in whose frame? DSR, Quantum aether à la Dirac? GUP affects all QM Hamiltonians. At least 1 part in precision required for measuring effects Space Quantized near the Planck scale. But, Discreteness at m observable effects at m? Gravity waves, photons, LHC And can do calculus Statistical Mechanical Systems, Normally forbidden Processes, α 1/n Effects Optimistic Scenario: A Low Energy Window to Quantum Gravity Phenomenology? All our results hold so long as there is a O(α) term in the GUP 42

Ahmed Farag Ali. Department of Physics, Benha University, EGYPT. Erice, June 29 th, 2012

Ahmed Farag Ali. Department of Physics, Benha University, EGYPT. Erice, June 29 th, 2012 Ahmed Farag Ali Department of Physics, Benha University, EGYPT Erice, June 29 th, 2012 Generalized Uncertainty Principle (GUP) from String theory and Black hole physics. Testing quantum gravity effects.