Astrophysical Evidence for the Non-Hermitian but PTPT-symmetric Hamiltonian of Conformal Gravity

In this review we discuss the connection between two seemingly disparate topics, macroscopic gravity on astrophysical scales and Hamiltonians that are not Hermitian but $PT$ symmetric on microscopic ones. In particular we show that the quantum-mechanical unitarity problem of the fourth-order derivative conformal gravity theory is resolved by recognizing that the scalar product appropriate to the theory is not the Dirac norm associated with a Hermitian Hamiltonian but is instead the norm associated with a non-Hermitian but $PT$-symmetric Hamiltonian. Moreover, the fourth-order theory Hamiltonian is not only not Hermitian, it is not even diagonalizable, being of Jordan-block form. With $PT$ symmetry we establish that conformal gravity is consistent at the quantum-mechanical level. In consequence, we can apply the theory to data, to find that the theory is capable of naturally accounting for the systematics of the rotation curves of a large and varied sample of 138 spiral galaxies without any need for dark matter. The success of the fits provides evidence for the relevance of non-diagonalizable but $PT$-symmetric Hamiltonians to physics.Comment: LaTex, 15 pages, 21 figures. Expanded version of talks given at the International Seminar and Workshop "Quantum Physics with Non-Hermitian Operators", Dresden, June 2011 and the Symposium "PT Quantum Mechanics 2011", Heidelberg, September 2011. Prepared for a Special Issue of Fortschritte der Physik - Progress of Physics on "Quantum Physics with Non-Hermitian Operators: Theory and Experiment

Conformal cosmology with a positive effective gravitational constant

The conformal cosmological model presented by Mannheim predicts a negative value for the effective gravitational constant, G. It also involves a scalar field, S, which is treated classically. In this paper we point out that a classical treatment of S is inappropriate, because the Hamiltonian is non-Hermitean, and the theory must be developed in the way pioneered by Bender and others. When this is done, we arrive at a Hamiltonian with an energy spectrum that is bounded below, and also a G that is positive. The resulting theory closely resembles the conventional cosmology based on Einstein relativity

Solution to the ghost problem in fourth order derivative theories

We present a solution to the ghost problem in fourth order derivative theories. In particular we study the Pais-Uhlenbeck fourth order oscillator model, a model which serves as a prototype for theories which are based on second plus fourth order derivative actions. Via a Dirac constraint method quantization we construct the appropriate quantum-mechanical Hamiltonian and Hilbert space for the system. We find that while the second-quantized Fock space of the general Pais-Uhlenbeck model does indeed contain the negative norm energy eigenstates which are characteristic of higher derivative theories, in the limit in which we switch off the second order action, such ghost states are found to move off shell, with the spectrum of asymptotic in and out S-matrix states of the pure fourth order theory which results being found to be completely devoid of states with either negative energy or negative norm. We confirm these results by quantizing the Pais-Uhlenbeck theory via path integration and by constructing the associated first-quantized wave mechanics, and show that the disappearance of the would-be ghosts from the energy eigenspectrum in the pure fourth order limit is required by a hidden symmetry that the pure fourth order theory is unexpectedly found to possess. The occurrence of on-shell ghosts is thus seen not to be a shortcoming of pure fourth order theories per se, but rather to be one which only arises when fourth and second order theories are coupled to each other.Comment: 36 pages, revtex. Prepared for the proceedings of the 2006 Biennial Meeting of the International Association for Relativistic Dynamics Version 2 contains an expanded discussion of the path integral quantization of the Pais-Uhlenbeck fourth order oscillator theor

The democratic origins of the term "group analysis": Karl Mannheim's "third way" for psychoanalysis and social science.

It is well known that Foulkes acknowledged Karl Mannheim as the first to use the term ‘group analysis’. However, Mannheim’s work is otherwise not well known. This article examines the foundations of Mannheim’s sociological interest in groups using the Frankfurt School (1929–1933) as a start point through to the brief correspondence of 1945 between Mannheim and Foulkes (previously unpublished). It is argued that there is close conjunction between Mannheim’s and Foulkes’s revision of clinical psychoanalysis along sociological lines. Current renderings of the Frankfurt School tradition pay almost exclusive attention to the American connection (Herbert Marcuse, Eric Fromm, Theodor Adorno and Max Horkheimer) overlooking the contribution of the English connection through the work of Mannheim and Foulkes

Neutrinos from Gamma Ray Bursts

The observed fluxes of cosmic rays and gamma rays are used to infer the maximum allowed high-energy neutrino flux allowed for Gamma Ray Bursts (GRBs), following Mannheim, Protheroe, and Rachen (2000). It is shown that if GRBs produce the ultrahigh-energy cosmic rays, they should contribute (a) at least 10% of the extragalactic gamma ray background between 3 MeV and 30 GeV, contrary to their observed energy flux which is only a minute fraction of this flux, and (b) a cumulative neutrino flux a factor of 20 below the AMANDA (Neutrino 2000) limit on isotropic neutrinos. This could have two implications, either GRBs do not produce the ultrahigh energy cosmic rays or that the GRBs are strongly beamed and emit most of their power at energies well above 100 GeV implausibly increasing the energy requirements, but consistent with the marginal detections of a few low-redshift GRBs by MILAGRITO, HEGRA-AIROBICC, and the Tibet-Array. All crucial measurements to test the models will be available in the next few years. These are measurements of (i) high-energy neutrinos with AMANDA-ICECUBE or an enlarged ANTARES/NESTOR ocean detector, (ii) GRB redshifts from HETE-2 follow-up studies, and (iii) GRB spectra above 10 GeV with low-threshold imaging air Cherenkov telescopes such as MAGIC and the space telescopes AGILE and GLAST.Comment: 8 pages, 1 figure, to appear in the Proc. of the Heidelberg International Symposium on High Energy Gamma-Ray Astronomy, Heidelberg, June 26-30, 2000, ed. by H.J. Voelk and F. Aharonian, AIP Conf. Pro

Local and global gravity

Our long experience with Newtonian potentials has inured us to the view that gravity only produces local effects. In this paper we challenge this quite deeply ingrained notion and explicitly identify some intrinsically global gravitational effects. In particular we show that the global cosmological Hubble flow can actually modify the motions of stars and gas within individual galaxies, and even do so in a way which can apparently eliminate the need for galactic dark matter. Also we show that a classical light wave acquires an observable, global, path dependent phase in traversing a gravitational field. Both of these effects serve to underscore the intrinsic difference between non-relativistic and relativistic gravity.Comment: LaTeX, 20 pages plus three figures in two postscript files. To appear in a special issue of Foundations of Physics honoring Professor Lawrence Horwitz on the occasion of his 65th birthday; A. van der Merwe and S. Raby, Editors, Plenum Publishing Company, N.Y., 199