THE ORIGIN OF THE COSMOLOGICAL CONSTANT PROBLEM AND A POSSIBLE SOLUTION TO IT Philip D. Mannheim Department of Physics University of Connecticut Presentation at the Tales of Lambda Conference University of Nottingham, July 2013 1 GHOST PROBLEMS, UNITARITY OF FOURTH-ORDER THEORIES AND PT QUANTUM MECHANICS 1. P. D. Mannheim and A. Davidson, Fourth order theories without ghosts, January 2000 (arXiv:0001115 [hep-th]). 2. P. D. Mannheim and A. Davidson, Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Phys. Rev. A 71, 042110 (2005). (0408104 [hep-th]). 3. P. D. Mannheim, Solution to the ghost problem in fourth order derivative theories, Found. Phys. 37, 532 (2007). (arXiv:0608154 [hep-th]). 4. C. M. Bender and P. D. Mannheim, No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett. 100, 110402 (2008). (arXiv:0706.0207 [hep-th]). 5. C. M. Bender and P. D. Mannheim, Giving up the ghost, Jour. Phys. A 41, 304018 (2008). (arXiv:0807.2607 [hep-th]) 6. C. M. Bender and P. D. Mannheim, Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart, Phys. Rev. D 78, 025022 (2008). (arXiv:0804.4190 [hep-th]) 7. C. M. Bender and P. D. Mannheim, PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues, Phys. Lett. A 374, 1616 (2010). (arXiv:0902.1365 [hep-th]) 8. P. D. Mannheim, PT symmetry as a necessary and sufficient condition for unitary time evolution, Phil. Trans. Roy. Soc. A. 371, 20120060 (2013). (arXiv:0912.2635 [hep-th]) 9. C. M. Bender and P. D. Mannheim, PT symmetry in relativistic quantum mechanics, Phys. Rev. D 84, 105038 (2011). (arXiv:1107.0501 [hep-th]) CONFORMAL GRAVITY AND THE COSMOLOGICAL CONSTANT AND DARK MATTER PROBLEMS 10. P. D. Mannheim, Alternatives to dark matter and dark energy, Prog. Part. Nuc. Phys. 56, 340 (2006). (arXiv:0505266 [astro-ph]). 11. P. D. Mannheim, Why do we believe in dark matter and dark energy – and do we have to?, in ”Questions of Modern Cosmology – Galileo’s Legacy”, M. D’Onofrio and C. Burigana (Eds.), Springer Publishing Company, Heidelberg (2009). 12. P. D. Mannheim, Conformal Gravity Challenges String Theory, Proceedings of the Second Crisis in Cosmology Conference, CCC-2, Astronomical Society of the Pacific Conference Series Vol. 413. (F. Potter, Ed.), San Francisco (2009). (arXiv:0707.2283 [hep-th]) 13. P. D. Mannheim, Dynamical symmetry breaking and the cosmological constant problem, Proceedings of ICHEP08, eConf C080730. (0809.1200 [hep-th]). 14. P. D. Mannheim, Comprehensive solution to the cosmological constant, zero-point energy, and quantum gravity problems, Gen. Rel. Gravit. 43, 703 (2011). (arXiv:0909.0212 [hep-th]) 15. P. D. Mannheim, Intrinsically quantum-mechanical gravity and the cosmological constant problem, Mod. Phys. Lett. A 26, 2375 (2011). (arXiv:1005.5108 [hep-th]). 16. P. D. Mannheim and J. G. O’Brien, Impact of a global quadratic potential on galactic rotation curves, Phys. Rev. Lett. 106, 121101 (2011). (arXiv:1007.0970 [astro-ph.CO]). 17. P. D. Mannheim and J. G. O’Brien, Fitting galactic rotation curves with conformal gravity and a global quadratic potential, Phys. Rev. D 85, 124020(2012). (arXiv:1011.3495 [astro-ph.CO]) 18. P. D. Mannheim, Making the case for conformal gravity, Found. Phys. 42, 388 (2012). (arXiv:1101.2186 [hep-th]). 19. J. G. O’Brien and P. D. Mannheim, Fitting dwarf galaxy rotation curves with conformal gravity. MNRAS 421, 1273 (2012). (arXiv:1107.5229 [astro-ph.CO]) 20. P. D. Mannheim, Cosmological perturbations in conformal gravity, Phys. Rev. D. 85, 124008 (2012). (arXiv:1109.4119 [gr-qc]) 21. P. D.Mannheim Astrophysical Evidence for the Non-Hermitian but P T -symmetric Hamiltonian of Conformal Gravity, Fortschritte der Physik - Progress of Physics 61, 140 (2013). (arXiv:1205.5717 [hep-th]) 22. P. D. Mannheim and J. G. O’Brien, Galactic rotation curves in conformal gravity, Journal of Physics: Conference Series 437, 012002 (2013). (arXiv:1211.0188 [astro-ph.CO]) 2 1 The Einstein Equations: What Kind of Operator Equations are They? 1 µν 1 µν α µν − R − g R α = TM . (1) 8πG 2 Classical equals classical? Quantum equals quantum? Einstein gravity is not renormalizable, so classical equals classical? Not so, since known gravitational sources include the intrinsically quantum-mechanical Pauli pressure 3/2 2 2 4 4 3 3 of white dwarf stars (with MCH ∼ (¯hc/G) /mp) and the energy density ρ = π kBT /15c h¯ and pressure p = ρ/3 of black-body radiation in cosmology. However, since cannot set classical equals quantum try: 1 µν 1 µν α µν − R − g R α = hTM i. (2) 8πG 2 µν But hTM i is not finite. So subtract off infinite part and try: 1 µν 1 µν α µν − R − g R α = hTM iFIN. (3) 8πG 2 Where does this equation come from? As yet not derived from a fundamental theory or known to be able to survive quantum gravitational corrections. Also it violates the hallmark of Einstein gravity, namely that gravity couple to energy rather than to energy difference. Thus in Σ(a†a +1/2)¯hω keep the Σa†ahω¯ term but ignore the Σ(1/2)¯hω zero-point energy density term. Also ignore zero-point pressure. And when all this is µν done, in flat spacetime Lorentz invariance still allows a finite hΩ|TM |ΩiFIN to be of the generic form −Λgµν, with the standard strong, electromagnetic, and weak interactions typically then generating a huge such Λ. So the zero-point problem exists even in the absence of gravity, and thus how can gravity be expected to solve a µν µν problem it did not cause. Also while a finite hΩ|TM |ΩiFIN only has one term, before it is made finite hΩ|TM |Ωi has two independent infinities to deal with, with there thus being both a zero-point energy density problem and µν a zero-point pressure problem. The cosmological constant problem arises entirely due to ignoring how hTM iFIN µν µν α µν got to be finite in the first place, and then using R − (1/2)g R α = −8πGhTM iFIN as the starting point. 3 2 The Zero-Point Energy Density and Zero-Point Pressure Problems in More Detail Consider a free fermion of mass m in flat, four-dimensional spacetime. With kµ = ((k2 + m2/h¯2)1/2, k¯) the µν ¯ µ ν vacuum expectation value of the matter field energy-momentum tensor TM = ih¯ψγ ∂ ψ evaluates to µ ν 2¯h ∞ k k hΩ|T µν|Ωi = − d3k . (4) M (2π)3 Z−∞ k0 00 11 We recognize a zero-point energy density ρM = hΩ|TM |Ωi and a zero point pressure pM = hΩ|TM |Ωi = 22 33 µν µν hΩ|TM |Ωi = hΩ|TM |Ωi with ηµνhΩ|TM |Ωi =3pM − ρM, and can writehΩ|TM |Ωi in the form of a perfect fluid µν µ ν µν hΩ|TM |Ωi =(ρM + pM)U U + pη , (5) h¯ m2K2 m4 4¯h2K2 m4 4 ρM = − 2 K + 2 − 4 ln 2 + 4 , 4π h¯ 4¯h m 8¯h h¯ m2K2 3m4 4¯h2K2 7m4 4 pM = − 2 K − 2 + 4 ln 2 − 4 . (6) 12π h¯ 4¯h m 8¯h µν We note that hΩ|TM |Ωi is not in the form of cosmological constant where one would have pM = −ρM, and µν indeed could not be since in the m = 0 limit TM is traceless and pM = ρM/3. Also we note three separate divergences: quartic, quadratic, and logarithmic. If we introduce a set of covariant Pauli-Villars regulator masses Mi with each such regulator contributing 2 2 an analogous term as multiplied by some overall factor ηi, the choice 1 + ηi = 0, m + ηiMi = 0, 4 4 P P m + ηiMi = 0 will not only then lead to finite regulated ρREG and pREG, it will give them the values P h¯ 4 2 4 2 ρREG = −pREG = − m lnm + ηiMi lnMi . (7) 16π2 X µν The regulated and thus now finite hΩ|TM |Ωi is indeed in the form of a cosmological constant, but it is completely uncontrollable, being as large as the regulator masses. 4 3 The Additional Problems due to Symmetry Breaking: Scalar Field Case In flat space in a normal vacuum |Ni where φN = hN|φ|Ni =0or hN|ψψ¯ |Ni = 0 we already have divergent zero-point energy density and pressure contributions, and regulating these divergences leads to a cosmological constant term, one as large as the scale associated with the regulator cut-off scale, being possibly as large as the Planck scale. Moreover, if for the strong, electromagnetic and weak interactions we change the vacuum by spontaneous breakdown to a vacuum |Si where φS = hS|φ|Si 6= 0 or hS|ψψ¯ |Si 6= 0, we then induce an additional contribution to the cosmological constant. For the scalar field case we note that if we have a double-well potential 4 2 2 V (φ) = λφ − µ φ + V0, 2 1/2 with a local maximum in which φN = 0 and a global minimum where φS = (µ /2λ) , then not only do we induce a contribution 4 V (φS) − V (φN) = −µ /4λ due to the change in vacuum, in addition we do not know where the actual global minimum is since V0 is unconstrained. Thus in flat space we only measure the energy difference V (φS) − V (φN), but gravity couples to energy itself, i.e. to 4 V (φS) = −µ /4λ + V0. 4 Even though both hS|φ|Si and V (φS) = −µ /4λ+V0 are finite (though large), since φ is a fundamental scalar field it has an uncontrollable quadratically divergent self-energy, and should thus have a mass at the Planck cut-off scale rather than at the now observed 126 GeV.