casimir energy, the cosmological constant and massive gravitons remo garattini università di...
TRANSCRIPT
Casimir Energy, the Casimir Energy, the Cosmological Constant andCosmological Constant and
massivemassive gravitonsgravitons
Remo GarattiniRemo Garattini
Università di BergamoUniversità di Bergamo
I.N.F.N. - Sezione di MilanoI.N.F.N. - Sezione di Milano
Cala Gonone, 12-9-2005
22
The Cosmological Constant The Cosmological Constant ProblemProblem
At the Planck eraAt the Planck era
For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 6161, 1 (1989)., 1 (1989).For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys.For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys.D 9D 9, 373 (2000), astro-ph/9904398; N. Straumann, , 373 (2000), astro-ph/9904398; N. Straumann, The history of the cosmologicalThe history of the cosmologicalconstant problemconstant problem gr-qc/0208027; T.Padmanabhan, Phys.Rept. gr-qc/0208027; T.Padmanabhan, Phys.Rept. 380380, 235 (2003),, 235 (2003),hep-th/0212290.hep-th/0212290.
47110 GeVC •Recent measuresRecent measures
44710 GeVC
A factor of 10118
33
Wheeler-De Witt Equation Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.B. S. DeWitt, Phys. Rev.160160, 1113 (1967)., 1113 (1967).
GGijklijkl is the super-metric, is the super-metric, 88G and G and cc is the is the cosmological constantcosmological constant
R is the scalar curvature in 3-dim.R is the scalar curvature in 3-dim. cc can be seen as an eigenvalue can be seen as an eigenvalue
ioneigenfunctan as considered becan ijg
66
Quadratic ApproximationQuadratic Approximation
Let us consider the 3-dim. metric Let us consider the 3-dim. metric ggijij and and perturb around a fixed background, (e.g. perturb around a fixed background, (e.g. Schwarzschild)Schwarzschild)
ggij= ij= ggSSij+ ij+ hhijij
77
Canonical DecompositionCanonical Decomposition
h is the traceh is the trace (L(Lijij is the longitudinal operator is the longitudinal operator hh
ij represents the transverse-traceless ij represents the transverse-traceless component of the perturbation component of the perturbation graviton graviton
M. Berger and D. Ebin, J. Diff. Geom.3, 379 (1969). J. W. York Jr., J. Math. Phys., 14, 4 (1973); Ann. Inst. Henri Poincaré A 21, 319 (1974).
ijijijij hLhgh 3
1
88
Integration rules on Gaussian wave functionals
11
22
33
ijijij hxhxh
ij
ij
ij hxh
ix
klijij hhh 2*121 D
1010
Graviton ContributionGraviton Contribution
operator czLichnerowi modified theis 2
r)(Propagato
2:,
klij
yhxhyxK ijkl
iakl
a
jijkl xxKxxKGgxdV
ijkl ,2
1,2
4
1ˆ2
,13
W.K.B. method and graviton contribution to the cosmological constant
1111
Regularization Regularization
i
rm
ii
ii d
rmi
2
2
122
2
216,
Mm curvature
Mmr
MGrm
Mmr
MGrm
20
203
0
22
203
0
21
mass 3
3
• Riemann zeta function Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect
2
12ln2ln
1
256,
2
2
2
4
rm
rm
i
ii
1313
RenormalizationRenormalization
Bare cosmological constant changed intoBare cosmological constant changed into
div 0
The finite part becomes
rG
TTeff ,
8 210
1414
Renormalization Group EquationRenormalization Group Equation
Eliminate the dependance on Eliminate the dependance on and impose and impose
d
rd
G
TTeff ,
8
1 0
must be treated as running
0
42
41000 ln
16,,
rmrm
Grr
1515
Energy Minimization Energy Minimization (( Maximization) Maximization)
At the scale At the scale
2
1
4ln
16,
20
204
0000 Mm
MmG
r
has a maximum for
32,
40
000
Gr
e
Mm 1
4 20
20
1616
If gravitons are massive….If gravitons are massive….
Pauli e Fierz (M. Fierz and W. Pauli, Proc. Roy. Soc. Pauli e Fierz (M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173, 211 (1939)) introduce a mass term Lond. A 173, 211 (1939)) introduce a mass term in the gravitational action.in the gravitational action.
The term does not introduce neither ghosts nor The term does not introduce neither ghosts nor instabilities, but….instabilities, but….
We need to introduce the Pauli-Fierz term
1717
It breaks the gauge invariance It breaks the gauge invariance of the formof the form
2hh
Gauge invariance can be recovered with the Stückelberg methodE.C.G. Stückelberg (Helv. Phys. Acta 30, 209 (1957).
1818
No mass in General RelativityNo mass in General Relativity
D.G. Boulware and S. Deser, Phys. D.G. Boulware and S. Deser, Phys. Rev. Rev. D 12D 12 3368 (1972). 3368 (1972).
1919
Rubakov ProposalRubakov Proposal
If we chooseIf we choose
We recover the Pauli-Fierz term
V.A. Rubakov, Lorentz-Violating Graviton Masses: getting around ghosts, low strong coupling scale and VDVZ discontinuity. hep-th/0407104.
0 0 224
23
22
21
20 mmmmmm
iijjii
ijij
ii
m hhmhhmhhmhhmhhmgxdS 0024
23
220
02100
0020
4 228
1
2020
A very particular choice
ijij
gijij
m hhmgxdhhmgxdS 2422
4
8
1
8
1
In terms of the linearized Hamiltonian
ijijg
m hhgxNdm
H 32
8
2121
The effective mass becomesThe effective mass becomes
Mmmr
MGmrm
Mmmr
MGmrm
gg
gg
20
23
0
222
20
23
0
221
3
3
3 Case3 Case
Mmmg20
2 a) Case
Mmmg20
2 b) Case
Mmmg20
2 c) Case
2222
Repeating the procedure as in the Repeating the procedure as in the mass-less casemass-less case
Case a)Case a)
• Case b)
• Case c)
2323
ConclusionsConclusions Wheeler-De Witt Equation Wheeler-De Witt Equation Sturm-Liouville Problem. Sturm-Liouville Problem. The cosmological constant is the eigenvalue.The cosmological constant is the eigenvalue. Variational Approach to the eigenvalue equation (infinites).Variational Approach to the eigenvalue equation (infinites). Eigenvalue Regularization with the Riemann zeta function Eigenvalue Regularization with the Riemann zeta function
Casimir energy graviton contribution to the Casimir energy graviton contribution to the cosmological constant.cosmological constant.
Renormalization and renormalization group equation.Renormalization and renormalization group equation. In the mass-less graviton case, gravity contributes with a In the mass-less graviton case, gravity contributes with a
‘’‘’curvature masscurvature mass’’’’ A massive graviton (in this specific case), behaves as a A massive graviton (in this specific case), behaves as a
cosmological constant.cosmological constant. 3 different cases.3 different cases. Case mCase mgg=m=m00(M) gives the highest contribution to (M) gives the highest contribution to C.C.
2424
ProblemsProblems
Analysis to be completed.Analysis to be completed. Beyond the W.K.B. approximation of the Beyond the W.K.B. approximation of the
Lichnerowicz spectrum.Lichnerowicz spectrum. Discrete Lichnerowicz spectrum.Discrete Lichnerowicz spectrum. Trace Contribution to be included in the Trace Contribution to be included in the
massive graviton case.massive graviton case. A CURIOUS Thing A CURIOUS Thing
Factor