A Multi Gravity Approach to Space-Time Foam Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano Low Energy Quantum Gravity York,

2 Outline  The Cosmological Constant Problem  The Wheeler-De Witt Equation  Aspects of Space Time Foam (STF)  Multi Gravity Approach to STF  Conclusion  Problems and Outlook

3 The Cosmological Constant Problem  At the Planck era For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), astro-ph/ ; N. Straumann, The history of the cosmological constant problem gr-qc/ ; T.Padmanabhan, Phys.Rept. 380, 235 (2003), hep-th/ Recent measuresRecent measures A factor of

4 Action in 3+1 dimensions  The action in 4D with a cosmological term  Introducing lapse and shift function (ADM) The scalar curvature separates into Conjugate momentum

5 Action in 3+1 dimensions  The action in 4D with a cosmological term Legendre transformation Gravitational Hamiltonian Classical constraint +E ADM

6 Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967).  G ijkl is the super-metric, 8G and  is the cosmological constant  R is the scalar curvature in 3-dim.   can be seen as an eigenvalue   [g ij ] can be considered as an eigenfunction

7 Re-writing the WDW equation Where

8 Eigenvalue problem Quadratic Approximation Let us consider the 3-dim. metric g ij and perturb around a fixed background, g ij = g S ij + h ij

9 Form of the background N(r)  Lapse function b(r)  shape function for example, the Ricci tensor in 3 dim. is

10 Canonical Decomposition  h is the trace  (L ij is the longitudinal operator  h  ij represents the transverse-traceless component of the perturbation  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).

11 Integration rules on Gaussian wave functionals 12345

12 Graviton Contribution W.K.B. method and graviton contribution to the cosmological constant

13 Regularization Zeta function regularization  Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect

14 Isolating the divergence

15 Renormalization  Bare cosmological constant changed into The finite part becomes

16 Renormalization Group Equation  Eliminate the dependance on  and impose   must be treated as running

17 Energy Minimization (  Maximization)  At the scale     has a maximum for with Not satisfying

18 Why Spacetime Foam?  J. A. Wheeler established that: The fluctuation in a typical gravitational potential is The fluctuation in a typical gravitational potential is [ J.A. Wheeler, Phys. Rev., 97, 511 (1955)] It is at this point that Wheeler's qualitative description of the quantum geometry of space time seems to come into play [J.A. Wheeler, Ann. Phys., 2, 604 (1957)] On the atomic scale the metric appears flat, as does the ocean to an aviator far above. The closer the approach, the greater the degree of irregularity. Finally, at distances of the order of l P, the fluctuations in the typical metric component, g , become of the same order as the g  themselves. This means that we can think that the geometry (and topology) of space might be constantly fluctuating

19 Motivating Multigravity 1) In a foamy spacetime, general relativity can be renormalized when a density of virtual black holes is taken under consideration coupled to N fermion fields in a 1/N expansion [L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.]. [L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.]. 2) When gravity is coupled to N conformally invariant scalar fields the evidence that the ground-state expectation value of the metric is flat space is false [J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.]. [J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.]. Merging of point 1) and 2) with N gravitational fields (instead of scalars and fermions) leads to multigravity Hope for a better Cosmological constant computation

20 First Steps in Multigravity Pioneering works in 1970s known under the name strong gravity strong gravity or f-g theory (bigravity) [C.J. Isham, A. Salam, and J. Strathdee, Phys Rev. D 3, 867 (1971), A. D. Linde, Phys. Lett. B 200, 272 (1988).]

21 Structure of Multigravity T.Damour and I. L. Kogan, Phys. Rev.D 66, (2002). A.D. Linde, hep-th/ N massless gravitons gravitons

22 Multigravity gas For each action, introduce the lapse and shift functions Choose the gauge Define the following domain No interaction Depending on the structure You are looking, You could have a ‘ideal’gas of geometries. Our specific case: Schwarzschild wormholes

23  Wave functionals do not overlap Additional assumption The single eigenvalue problem turns into problem turns into

24 And the total wave functional becomes The initial problem changes into 

25 Further trivial assumption R. Garattini - Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/ N w copies of the same gravity Take the maximum

26 Bekenstein-Hawking arguments on the area quantization From Multi gravity we need a proof that

27 Conclusions  Wheeler-De Witt Equation  Sturm-Liouville Problem.  The cosmological constant is the eigenvalue.  Variational Approach to the eigenvalue equation (infinites).  Eigenvalue Regularization with the Riemann zeta function  Casimir energy graviton contribution to the cosmological constant.  Renormalization and renormalization group equation.  Generalization to multigravity.  Specific example: gas of Schwarzschild wormholes.

28 Problems and Outlook  Analysis to be completed.  Beyond the W.K.B. approximation of the Lichnerowicz spectrum.  Discrete Lichnerowicz spectrum.  Specific examples of interaction like the Linde bi-gravity model or Damour et al.  Possible generalization con N ‘different gravities’?!?!  Use a distribution of gravities!! Massive graviton?!?  Generalization to f(R) theories (Mono-gravity case S.Capozziello & R.G., C.Q.G. (2007))