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

Completeness of non-normalizable modes

We establish the completeness of some characteristic sets of non-normalizable modes by constructing fully localized square steps out of them, with each such construction expressly displaying the Gibbs phenomenon associated with trying to use a complete basis of modes to fit functions with discontinuous edges. As well as being of interest in and of itself, our study is also of interest to the recently introduced large extra dimension brane-localized gravity program of Randall and Sundrum, since the particular non-normalizable mode bases that we consider (specifically the irregular Bessel functions and the associated Legendre functions of the second kind) are associated with the tensor gravitational fluctuations which occur in those specific brane worlds in which the embedding of a maximally four-symmetric brane in a five-dimensional anti-de Sitter bulk leads to a warp factor which is divergent. Since the brane-world massless four-dimensional graviton has a divergent wave function in these particular cases, its resulting lack of normalizability is thus not seen to be any impediment to its belonging to a complete basis of modes, and consequently its lack of normalizability should not be seen as a criterion for not including it in the spectrum of observable modes. Moreover, because the divergent modes we consider form complete bases, we can even construct propagators out of them in which these modes appear as poles with residues which are expressly finite. Thus even though normalizable modes appear in propagators with residues which are given as their finite normalization constants, non-normalizable modes can just as equally appear in propagators with finite residues too -- it is just that such residues will not be associated with bilinear integrals of the modes.Comment: 34 pages, 6 figures. Revte

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

Implications of Cosmic Repulsion for Gravitational Theory

In this paper we present a general, model independent analysis of a recently detected apparent cosmic repulsion, and discuss its potential implications for gravitational theory. In particular, we show that a negatively spatially curved universe acts like a diverging refractive medium, to thus naturally cause galaxies to accelerate away from each other. Additionally, we show that it is possible for a cosmic acceleration to only be temporary, with some accelerating universes actually being able to subsequently recontract.Comment: RevTeX, 13 page

Comprehensive Solution to the Cosmological Constant, Zero-Point Energy, and Quantum Gravity Problems

We present a solution to the cosmological constant, the zero-point energy, and the quantum gravity problems within a single comprehensive framework. We show that in quantum theories of gravity in which the zero-point energy density of the gravitational field is well-defined, the cosmological constant and zero-point energy problems solve each other by mutual cancellation between the cosmological constant and the matter and gravitational field zero-point energy densities. Because of this cancellation, regulation of the matter field zero-point energy density is not needed, and thus does not cause any trace anomaly to arise. We exhibit our results in two theories of gravity that are well-defined quantum-mechanically. Both of these theories are locally conformal invariant, quantum Einstein gravity in two dimensions and Weyl-tensor-based quantum conformal gravity in four dimensions (a fourth-order derivative quantum theory of the type that Bender and Mannheim have recently shown to be ghost-free and unitary). Central to our approach is the requirement that any and all departures of the geometry from Minkowski are to be brought about by quantum mechanics alone. Consequently, there have to be no fundamental classical fields, and all mass scales have to be generated by dynamical condensates. In such a situation the trace of the matter field energy-momentum tensor is zero, a constraint that obliges its cosmological constant and zero-point contributions to cancel each other identically, no matter how large they might be. Quantization of the gravitational field is caused by its coupling to quantized matter fields, with the gravitational field not needing any independent quantization of its own. With there being no a priori classical curvature, one does not have to make it compatible with quantization.Comment: Final version, to appear in General Relativity and Gravitation (the final publication is available at http://www.springerlink.com). 58 pages, revtex4, some additions to text and some added reference

Light deflection in Weyl gravity: critical distances for photon paths

The Weyl gravity appears to be a very peculiar theory. The contribution of the Weyl linear parameter to the effective geodesic potential is opposite for massive and nonmassive geodesics. However, photon geodesics do not depend on the unknown conformal factor, unlike massive geodesics. Hence light deflection offers an interesting test of the Weyl theory. In order to investigate light deflection in the setting of Weyl gravity, we first distinguish between a weak field and a strong field approximation. Indeed, the Weyl gravity does not turn off asymptotically and becomes even stronger at larger distances. We then take full advantage of the conformal invariance of the photon effective potential to provide the key radial distances in Weyl gravity. According to those, we analyze the weak and strong field regime for light deflection. We further show some amazing features of the Weyl theory in the strong regime.Comment: 20 pages, 9 figures (see published version for a better resolution, or online version at stacks.iop.org/CQG/21/1897

Open Questions in Classical Gravity

We discuss some outstanding open questions regarding the validity and uniqueness of the standard second order Newton-Einstein classical gravitational theory. On the observational side we discuss the degree to which the realm of validity of Newton's Law of Gravity can actually be extended to distances much larger than the solar system distance scales on which the law was originally established. On the theoretical side we identify some commonly accepted but actually still open to question assumptions which go into the formulating of the standard second order Einstein theory in the first place. In particular, we show that while the familiar second order Poisson gravitational equation (and accordingly its second order covariant Einstein generalization) may be sufficient to yield Newton's Law of Gravity they are not in fact necessary. The standard theory thus still awaits the identification of some principle which would then make it necessary too. We show that current observational information does not exclusively mandate the standard theory, and that the conformal invariant fourth order theory of gravity considered recently by Mannheim and Kazanas is also able to meet the constraints of data, and in fact to do so without the need for any so far unobserved non-luminous or dark matter.Comment: UCONN-93-1, plain TeX format, 22 pages (plus 7 figures - send requests to [email protected]). To appear in a special issue of Foundations of Physics honoring Professor Fritz Rohrlich on the occasion of his retirement, L. P. Horwitz and A. van der Merwe Editors, Plenum Publishing Company, N.Y., Fall 199

On photohadronic processes in astrophysical environments

We discuss the first applications of our newly developed Monte Carlo event generator SOPHIA to multiparticle photoproduction of relativistic protons with thermal and power law radiation fields. The measured total cross section is reproduced in terms of excitation and decay of baryon resonances, direct pion production, diffractive scattering, and non-diffractive multiparticle production. Non--diffractive multiparticle production is described using a string fragmentation model. We demonstrate that the widely used `$\Delta$--approximation' for the photoproduction cross section is reasonable only for a restricted set of astrophysical applications. The relevance of this result for cosmic ray propagation through the microwave background and hadronic models of active galactic nuclei and gamma-ray bursts is briefly discussed.Comment: 9 pages including 4 embedded figures, submitted to PAS

Dirac Quantization of the Pais-Uhlenbeck Fourth Order Oscillator

As a model, the Pais-Uhlenbeck fourth order oscillator with equation of motion $d^4q/dt^4+(\omega_1^2+\omega_2^2)d^2q/dt^2 +\omega_1^2\omega_2^2 q=0$ is a quantum-mechanical prototype of a field theory containing both second and fourth order derivative terms. With its dynamical degrees of freedom obeying constraints due to the presence of higher order time derivatives, the model cannot be quantized canonically. We thus quantize it using the method of Dirac constraints to construct the correct quantum-mechanical Hamiltonian for the system, and find that the Hamiltonian diagonalizes in the positive and negative norm states that are characteristic of higher derivative field theories. However, we also find that the oscillator commutation relations become singular in the $\omega_1 \to \omega_2$ limit, a limit which corresponds to a prototype of a pure fourth order theory. Thus the particle content of the $\omega_1 =\omega_2$ theory cannot be inferred from that of the $\omega_1 \neq \omega_2$ theory; and in fact in the $\omega_1 \to \omega_2$ limit we find that all of the $\omega_1 \neq \omega_2$ negative norm states move off shell, with the spectrum of asymptotic in and out states of the equal frequency theory being found to be completely devoid of states with either negative energy or negative norm. As a byproduct of our work we find a Pais-Uhlenbeck analog of the zero energy theorem of Boulware, Horowitz and Strominger, and show how in the equal frequency Pais-Uhlenbeck theory the theorem can be transformed into a positive energy theorem instead.Comment: RevTeX4, 20 pages. Final version, to appear in Phys. Rev.