The Generalized Uncertainty Principle and Quantum Gravity Phenomenology

0.1 MB·English

by  Ahmed Farag Ali

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The Generalized Uncertainty Principle and Quantum Gravity
Phenomenology
AhmedFaragAli,SauryaDas
Dept. of Physics, Universityof Lethbridge, 4401 UniversityDrive,
0 Lethbridge, Alberta, Canada T1K 3M4
1 E-mails: [email protected], [email protected]
0 http://directory.uleth.ca/users/ahmed.ali, http://people.uleth.ca/∼saurya.das
2

n EliasC.Vagenas
a Research Centerfor Astronomy & Applied Mathematics,
J Academy of Athens,

8 Soranou Efessiou 4, GR-11527, Athens, Greece
1 E-mail: [email protected]
http://users.uoa.gr/∼evagenas

]
h
t In this article we examine a Generalized Uncertainty Principle which differs from the
-
p HeisenbergUncertaintyPrinciplebytermslinearandquadraticinparticlemomenta,as
e proposedbytheauthorsinanearlierpaper.WeshowthatthisaffectsallHamiltonians,
h and in particular those which describe low energy experiments. We discuss possible
[ observational consequences. Further, we also show that this indicates that space may
bediscreteatthefundamental level.
2
v Keywords: QuantumGravityPhenomenology
2 String Theory,1 certain other approachesto Quantum Gravity,as well as Black
4 HolePhysics2 suggestamodificationoftheHeisenberg’sUncertaintyPrinciplenear
6
thePlanckscaletoaso-calledGeneralizedUncertaintyPrinciple(GUP)oftheform
2
.
1 ~ ℓ2
0 ∆p ∆x≥ 2 (cid:20)1+β0 ~P2l∆p2(cid:21) (1)
0
1
v: where ℓPl = qGc3~ = 10−35m is the Planck length and β0 is a constant, normally
i assumed to be of order unity. Evidently, the new second term on the RHS of (1)
X
is important only when x,∆x ℓ or p,∆p p 1016TeV/c (the Planck mo-
Pl Pl
r ≈ ≈ ≈
mentum), i.e. at very high energies/small length scales. Inverting Eq.(1), we get
a
∆p ~ ∆x ∆x2 β ℓ2 , implying the existence of a minimum measur-
able≤lenβ0gℓt2Phl h∆x ±∆px − √0βPℓli . It can be shown that the above GUP can be
min 0 Pl
≥ ≡
derived from a modified Heisenberg algebra3
β ℓ2
[x ,p ]=i~[δ + 0 Pl(p2δ +2p p )] . (2)
i j ij ~2 ij i j
On the other hand, Doubly Special Relativity (DSR) theories4 suggest yet another
modified algebra between position and momenta5
[x ,p ]=i~[(1 ℓ p~)δ +ℓ2 p p ] (3)
i j − Pl| | ij Pl i j
as well as the existence of a maximum observable momentum ∆p ∆p
max
≤ ≈
M c.UsingtheJacobiidentity[[x ,x ],p ]+[[x ,p ],x ]+[[p ,x ],x ]=0andthe
Pl i j k j k i k i j
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assumptionthatspacecommuteswithspaceandmomentawithmomenta,algebras
(2) and (3) can be reconciled as limits of a single algebra of the form6 ∗
p p
[x ,p ]=i~ δ α pδ + i j +α2(p2δ +3p p ) . (4)
i j ij ij ij i j
(cid:20) − (cid:18) p (cid:19) (cid:21)
Here α = α0 = α0ℓPl. Again, α is normally assumed to be of order unity. The
MPlc ~ 0
abovealgebrapredictbotha∆x anda∆p .Italsoimpliesthe followingrep-
min max
resentationof the momentum operator in position space p =p 1 αp +2α2p2
j 0j − 0 0
where p = i~ ∂ is canonical(but unphysical)and satisfies the(cid:0)usualcommuta(cid:1)-
tor [x ,p0j ]=−i~δ∂x.jCorrespondingly,a non-relativisticHamiltonian takes the form
i 0j ij
H = p2 +V(~r) = p20 +V(~r) i~3α d3 where the last term can be considered
2m 2m − m dx3
as a Quantum Gravity induced perturbation in the time-dependent Schr¨odinger
Equation
~2 d2 α~3 d3 ∂ψ
[H +H ]ψ = +V(x) i ψ =i~ .
0 1 (cid:20)−2mdx2 − m dx3(cid:21) ∂t
The above equation admits of a new conservedcurrent J = ~ ψ⋆dψ ψdψ⋆ +
2mi(cid:16) dx − dx (cid:17)
α~2 d2|ψ|2 3dψdψ⋆ and charge ρ = ψ 2 , such that ∂J + ∂ρ = 0. The effect of
m (cid:16) dx2 − dx dx (cid:17) | | ∂x ∂t
the perturbation can be found for example on a simple harmonic oscillator, with
V = mω2x2/2, for which the shift in the ground state energy eigenvalues is, using
second order perturbation theory ∆EGUP(0) ~ωmα2 .
E0 ∼
Concerning Landau Levels, for a particle of mass m, charge e in a constant
magnetic field B~ = Bzˆ 10T, A~ = Bxyˆ and cyclotron frequency ω = eB/m,
c
≈
the Hamiltonian is H = 1 p~ eA~ 2 α p~ eA~ 3 = H √8mαH23 and the
2m(cid:16) 0− (cid:17) −m(cid:16) 0− (cid:17) 0− 0
energyshifts are ∆EnE(GnUP) =−√8mα(~ωc)21(n+ 12)21 ≈−10−27α0 ,fromwhichwe
concludethatifα 1,then ∆En(GUP) is toosmallto measure.Onthe otherhand,
0 ∼ En
with current measurement accuracy of 1 in 103, one obtains the following upper
bound on the GUP parameter: α <1024.
0
Similarly for a Hydrogen atom with standard Hamiltonian H0 = 2pm20 − kr and
perturbing Hamiltonian H = αp3, it can be shown that the GUP effect on the
1 −m 0
Lamb Shift is ∆En(GUP) =2∆|ψnlm(0)| α 4.2×104E0 10−24 α . Again, if α 1,
∆En ψnlm(0) ≈ 0 27MPlC2 ≈ 0 0 ∼
then ∆En(GUP) is too small, whereas with current measurement accuracy of 1 in
En
1012, we infer α < 1012. For some other examples, we refer the reader to our
0
earlier papers7,8 .
Finally, we consider the free-particle Schr¨odinger equation for a particle in a
box of length L,6 with the solution ψ(x) = Aeik′x +Be−ik′′x +Ce2iαx~. Note the
appearance of a new oscillatory term. Here k′ = k(1+kα~) , k′′ = k(1 kα~) (to
−
leading order in α). The boundary condition ψ(0)=0 implies A+B+C =0, and
∗Wealsocitereference[6]formorereferencestoearlierworks.
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in addition to the boundary condition ψ(L)=0, this yields
2iAsin(kL)= C e−i(kL+θC) ei(L/2α~−θC) + (α2) , (5)
| |h − i O
where C = C e−iθC. Taking real parts of both sides (assuming A is real, without
| |
loss of generality),we get cos L θ =cos(kL+θ )=cos(nπ+θ +ǫ) which
2α~ − C C C
has the solutions (cid:0) (cid:1)
L L
= =nπ+2qπ+2θ or = nπ+2qπ [n,q N] . (6)
2α~ 2α ℓ C − ∈
0 Pl
Fromthe aboveweconcludethataparticlecanbeconfinedonlyinboxesofcertain
discrete lengths, and further speculate that this might indicate that all measurable
lengths are quantized, since measurement of lengths require at least one particle,
possibly many.We think that this resultcanbe generalizedto relativistic particles,
as well as to the quantization of areas and volumes9 .
In summary, in this article we have shown that a single GUP exists, which is
consistent with the predictions of Black Hole Physics, String Theory, DSR etc.,
and that this induces perturbations to all Hamiltonians. Applying this to a few
concreteexamplessuchastheHarmonicOscillator,LandauLevelsandLambShift,
we have computed corrections due to this perturbation. From these, we concluded
that if the GUP parameter α is of order unity, these corrections are probably
0
too small to be measured at present. On the other hand, current experimental
accuracies impose upper bounds on the GUP parameter. Finally, by solving the
GUP corrected Schr¨odinger equation for a particle in a box, we have shown that
boundaryconditionsrequiretheboxlengthtobequantized,suggestingquantization
of measurable lengths, and possibly of surfaces and volumes as well. We hope to
report further on these elsewhere.
ThisworkissupportedbytheNaturalSciencesandEngineeringResearchCoun-
cil of Canada and the Perimeter Institute for Theoretical Physics.
References
1. D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216, 41 (1989).
2. M. Maggiore, Phys. Lett.B 304, 65 (1993) [arXiv:hep-th/9301067].
3. A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52, 1108 (1995) [arXiv:hep-
th/9412167].
4. G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002) [arXiv:gr-qc/0012051];
J.MagueijoandL.Smolin,Phys.Rev.Lett.88,190403(2002)[arXiv:hep-th/0112090].
J. Magueijo and L. Smolin, Phys. Rev.D 71, 026010 (2005) [arXiv:hep-th/0401087].
5. J. L. Cortes and J. Gamboa, Phys.Rev. D 71, 065015 (2005) [arXiv:hep-th/0405285].
6. A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett. B 678, 497 (2009) [arXiv:0906.5396
[hep-th]].
7. S.DasandE.C.Vagenas,Phys.Rev.Lett.101,221301 (2008) [arXiv:0810.5333 [hep-
th]].
8. S. Das and E. C. Vagenas, Can. J. Phys. 87, 233 (2009) [arXiv:0901.1768 [hep-th]].
9. A. F. Ali, S. Das, E. C. Vagenas, in preparation.