PT SYMMETRY, QUANTUM GRAVITY, AND THE METRICATION OF THE FUNDAMENTAL FORCES PHILIP D. MANNHEIM DEPARTMENT OF PHYSICS UNIVERSITY OF CONNECTICUT PRESENTATION AT MIAMI 2014 FORT LAUDERDALE FLORIDA, DECEMBER 2014 1 UBIQUITY OF PT SYMMETRY 1. PT SYMMETRY AS A DYNAMICS 2. HIDDEN SECRET OF PT SYMMETRY{UNITARITY/NO GHOSTS 3. PT SYMMETRY AND THE LORENTZ GROUP 4. PT SYMMETRY AND THE CONFORMAL GROUP 5. PT SYMMETRY AND GRAVITY 6. PT SYMMETRY AND GEOMETRY 7. PT SYMMETRY AND THE DIRAC ACTION i 8. PT SYMMETRY AND THE METRICATION OF Aµ AND Aµ 9. THE UNEXPECTED PT SURPRISE 10. ASTROPHYSICAL EVIDENCE FOR THE PT REALIZATION OF CONFORMAL GRAVITY 2 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. (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 - 61, 140 (2013). (arXiv:1205.5717 [hep-th]) 22. P. D. Mannheim and J. G. O'Brien, Galactic rotation curves in conformal gravity, J. Phys. Conf. Ser. 437, 012002 (2013). (arXiv:1211.0188 [astro-ph.CO]) 3 TORSION 1. L. Fabbri and P. D. Mannheim, Continuity of the torsionless limit as a selection rule for gravity theories with torsion, May 2014, Phys. Rev. D 90, 024042 (2014). (arXiv:1405.1248 [gr-qc]). 2. P. D. Mannheim and J. J. Poveromo, Gravitational Analog of Faraday's Law via Torsion and a Metric with an Anti- symmetric Part, General Relativity and Gravitation 46, 1795 (2014). (arXiv:1406.1470 [gr-qc]). 3. P. D. Mannheim, Torsion, Magnetic Monopoles and Faraday's Law via a Variational Principle, June 2014. (arXiv:1406.2265 [hep-th]). 4. P. D. Mannheim, PT Symmetry, Conformal Symmetry, and the Metrication of Electromagnetism, July 2014. (arXiv:1407.1820 [hep-th]). 4 1 PT SYMMETRY AS A DYNAMICS Pioneered by Bender (1998). HERMITICITY is only SUFFICIENT to secure reality of eigenvalues. PT SYMMETRY of a Hamiltonian is the NECESSARY condition. Instead of considering the eigenvector equation Hj i = Ej i, focus on the eigenvalue equation f(λ) = jH − λIj = 0 instead. Theorem: The condition [H; P T ] = 0 is both a SUFFICIENT (Bender, Berry and Mandilara (2002)) and NECESSARY (Bender and Mannheim (2010)) condition for the reality of f(λ). However three realizations: 1. All λi are real and set of all j ii is complete. H is then either already Hermitian or by a non-unitary similarity transform on the eigenvector basis can be brought to a basis in which ∗ Hij = Hji. Thus Hermitian in disguise. Example (Bender, Brody and Jones (2003)): r cos θ + ir sin θ s H = r cos θσ + ir sin θσ + sσ = =6 Hy: (1) 0 3 1 s r cos θ − ir sin θ Here P = σ1, T = K (complex conjugation) and [H; P T ] = 0. Energies E± = r cos θ ± (s2 − r2 sin2 θ)1=2. If s2 − r2 sin2 θ = s sin α is positive, both energies are real. 1 + sin α1=2 1 − sin α1=2 G±1 = σ ± σ ; GHG−1 = r cos θσ + σ (s2 − r2 sin2 θ)1=2 2 sin α 0 2 2 sin α 0 1 (2) 5 ∗ 2. λi come in complex conjugate pairs (T : E ! E ) and set of all j ii is complete. Thus have decaying states (loss) and growing states (gain), and need both (Mannheim (2013)) for unitarity { as the population of a state is depleted the population of its complex conjugate partner grows. Example: s2 − r2 sin2 θ is negative, energies in a complex conjugate pair. 3. The λi are all equal (and thus all real) but eigenvectors incomplete - Jordan block realization. Example: The above H with s2 − r2 sin2 θ = 0 (G no longer exists), or 1 1 M = =6 M y (3) 0 1 TWO real solutions to jM − λIj = 0, viz. λ1 = 1, λ2 = 1 (Trace = 2, Determinant =1). But only ONE eigenstate { 1 1 p p + q p 1 = = ; q = 0; (4) 0 1 q q q 0 Eigenvalues degenerate since they have to share just the one eigenvector. Missing eigenvectors are not stationary. Nonetheless theory is unitary (Bender and Mannheim (2008)) since stationary plus non-stationary states combined form a complete basis. SUMMARY: PT SYMMETRY OF A HAMILTONIAN IS A DIAGNOS- TIC FOR PROPERTIES OF ITS EIGENVALUES, AND IS A BOTH NEC- ESSARY AND SUFFICIENT CONDITION FOR UNITARITY. 6 2 HIDDEN SECRET OF PT SYMMETRY { UNITARITY/NO GHOSTS Once the Hamiltonian is not Hermitian one can no longer use the Dirac norm. Given a right eigenvector jRii of H, which obeys HjRii = EjRii, the Dirac norm hRi(t)jRi(t)i = −iHyt iHt hRi(t = 0)je e jRi(t = 0)i is not equal to hRi(t = 0)jRi(t = 0)i, with the Dirac norm not being preserved in time in the non-Hermitian case. However, if we instead introduce a left eigenvector, which obeys hLijH = hLijE, then the norm hLi(t)jRi(t)i = hLi(t = −iHt iHt 0)je e jRi(t = 0)i = hLi(t = 0)jRi(t = 0)i is preserved in time. With left eigenvectors being related to the PT conjugates of the right eigenvectors, in the non-Hermitian but PT symmetric case one should instead use the PT theory norm, with unitary time evolution then being secured. In those cases where the Dirac norm is found to be negative, this does not necessarily mean that there are ghost states in the theory, since when the theory can be reinterpreted as a PT theory, the PT theory norm would be positive { case (1) above , where the theory Hamiltonian is Hermitian in disguise, and thus necessarily ghost free, with scalar products not being able to change sign under a similarity transformation. 7 Two cases of this effect have been identified in the literature. The Lee model (Lee 1954), and the fourth-order Pais-Uhlenbeck oscillator (Pais and Uhlenbeck (1950)) with action, equation of motion and propagator of the form: γ Z I = dt[z¨2 − (!2 + !2)z_2 + !2!2z2]; PU 2 1 2 1 2 d4z d2z + (!2 + !2) + !2!2z = 0; dt4 1 2 dt2 1 2 1 1 1 1 G(E) = 2 2 2 2 = 2 2 2 2 − 2 2 : (5) (E − !1)(E − !2) !1 − !2 E − !1 E − !2 In both cases the Dirac norm is found to be negative, but both turn out to be non-Hermitian but PT theories, with their PT theory norms being positive.