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Why Gravity Cannot Be Quantized Canonically, and What We Can We Do About It

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Why Gravity Cannot Be Quantized Canonically, and What We Can We Do About It WHY GRAVITY CANNOT BE QUANTIZED CANONICALLY, AND WHAT WE CAN WE DO ABOUT IT Philip D. Mannheim Department of Physics University of Connecticut Presentation at Miami 2013, Fort Lauderdale December 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 PT -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 Does gravity actually know about quantum mechanics? 1.1 Experimental considerations Before discussing how one might quantize gravity we need to discuss whether we need to. Macroscopically, there are two established sources of gravity that are intrinsically quantum-mechanical: 3=2 2 (1) Pauli pressure of white dwarf stars, with MCH ∼ (¯hc=G) =mp. 2 4 4 3 3 (2) The energy density ρ = π kBT =15c h¯ and pressure p = ρ/3 of black-body radiation in cosmology. Microscopically, the Colella-Overhauser-Werner (Phys. Rev. Lett. 34, 1472 (1975)) experiment shows that the quantum-mechanical phase of the wave function of a neutron of mass m and velocity v is modified as it traverses 2 the gravitational field g of the earth, viz. ∆φCOW = −mgH =vh¯ where AB = BD = DC = CA = H (see e.g. P. D. Mannheim (Phys. Rev. A 57, 1260 (1998)), yielding an observable fringe shift at D even though H is of the order of centimeters | it is just that mgH2=v is not small on the scale of Planck's constant. C D3 D b C D 1 1 2b D2 z x ¡ A2 A B S 2b b B1 A1 Figure 1 3 1.2 Theoretical considerations and concerns In equations such as the Einstein equations: 0 1 1 µν 1 µν α µν − @R − g R A = T ; (1) 8πG 2 α M if the Einstein tensor is equal to the matter field energy-momentum tensor, then either both sides are classical c-numbers or both sides are quantum-mechanical q-numbers. Otherwise, if the gravity side were to be classical µν while the matter side were to be quantum mechanical, then the quantum TM would have to be equal to a c-number in every single field configuration imaginable. (This is actually not impossible for a specific field configuration since a quantum-mechanical quantity such as the equal time field commutator [φ(t; x¯); π(t; y¯)] = ihZδ¯ 3(x¯ − y¯) is a c-number - it is just impossible if it has to be true for every field configuration.) However, from gravitational experiments described above we know that the source of gravity is quantum-mechanical, and we know that gravity knows it. Hence gravity must be quantized. However, since these gravitational experiments do not involve quantum gravity effects themselves (graviton loops), for phenomenological purposes it is conventional to use a hybrid in which we keep gravity classical but take c-number matrix elements of its source, to give 0 1 1 µν 1 µν α µν − @R − g R A = hT i: (2) 8πG 2 α M µν But hTM i involves products of fields at the same point, so it is not finite. Thus we additionally subtract off the µν µν µν infinite zero-point part and take the source to be the normal-ordered hTM iFIN = hTM i − hTM iDIV instead, to thereby yield: 4 0 1 1 µν 1 µν α µν − @R − g R A = hT i : (3) 8πG 2 α M FIN It is in this subtracted form that the standard applications of gravity are made. Thus in Σ(aya + 1=2)¯h! we keep the Σayah!¯ term but ignore the Σ(1=2)¯h! zero-point energy density term 00 in hTM i. ij Also ignore the zero-point pressure in the spatial hTM i. As of today there is no known justification for doing this, with this subtracted hybrid not having been derived from a fundamental theory or having been shown to be able to survive quantum gravitational corrections. There is no apparent reason why the zero point energy density of the matter sector should not gravitate. While one only needs to consider energy differences in flat space physics, in gravity one has to consider energy itself, with the hallmark of Einstein gravity being that gravity couple to everything. Hence for gravity one cannot ignore zero-point contributions. 5 Moreover, even if one does start with the subtracted hybrid, then the order G contribution to gravity is given µν µν by the flat spacetime hΩjTM jΩiFIN, with Lorentz invariance allowing a finite flat spacetime hΩjTM jΩiFIN to be of the generic form −Λgµν. Thus even if one ignores the matter sector zero-point energy contributions one still has a vacuum energy problem; with the standard strong, electromagnetic, and weak interactions typically then generating a huge such Λ. We thus recognize two types of vacuum problem, zero-point and cosmological constant problems. Moreover, if we take gravity to be quantum-mechanical (as we must), it too will have zero-point contributions. As I have shown in my papers and discussed at Miami 2011, it is the interplay of the gravity and matter field zero-point contributions that leads to a solution to the cosmological constant µν problem, with the problem having arisen entirely due to ignoring how hTM iFIN got to be finite in the first µν µν α µν place, and then using R − (1=2)g R α = −8πGhTM iFIN as the starting point. However, in order to be able to discuss gravitational zero-point contributions, we need to to quantize gravity. As we now show, such quantization cannot be done canonically, with gravity actually being quantized by the source that it is coupled to, rather than by being quantized on its own. 6 2 Canonical quantization of matter fields 2.1 Massless fermions R 4 ¯ µ ¯ µ For massless flat space fermion fields with IM = d xih¯ γ @µ , stationary fields obey δIM/δ = ihγ¯ @µ = 0. We introduce creation and annihilation operators according to 3 Z d k (x; t) = X b(k; s)u(k; s)e−ik·x + dy(k; s)v(k; s)eik·x ; s (2π)3=2 3 Z d k y(x; t) = X by(k; s)uy(k; s)eik·x + d(k; s)vy(k; s)eik·x : (4) s (2π)3=2 y 0 3 0 Quantizing the fermion field according to f α(x; t); β(x ; t)g = δ (x − x )δα,β then requires that its creation and annihilation operators obey y 0 0 3 0 y 0 0 3 0 fb(k; s); b (k ; s )g = δ (k − k )δs;s0; fd(k; s); d (k ; s )g = δ (k − k )δs;s0: (5) In the Fock space vacuum the vacuum matrix element of the energy-momentum tensor is given by 2 3 3 −1=2 δIM M 2¯h Z d k 4 5 ¯ hΩj 2g µν jΩi = hΩjTµνjΩi = hΩj ihγ¯ µ@ν jΩi = − 3 kµkν: (6) δg (2π) k0 and we can identify zero point energy density and pressure terms of the form 3 2 M 2¯h Z 3 M M M 2¯h Z d k k ρM ρM = hΩjT00 jΩi = − 3 d k!k; pM = hΩjT11 jΩi = hΩjT22 jΩi = hΩjT33 jΩi = − 3 = : (2π) (2π) !k 3 3 (7) M M R 3 M Thus even though hΩjT0i jΩi and the total matter vacuum momentum Pi = d xhΩjT0i jΩi are both zero, M M M hΩjT11 jΩi = hΩjT22 jΩi = hΩjT33 jΩi are not, to thus yield a vacuum pressure.
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