Spatial curvature effects on molecular transport by diffusion

For a substance diffusing on a curved surface, we obtain an explicit relation valid for very small values of the time, between the local concentration, the diffusion coefficient, the intrinsic spatial curvature and the time. We recover the known solution of Fick's law of diffusion in the flat space limit. In the biological context, this result would be useful in understanding the variations in the diffusion rates of integral proteins and other molecules on membranes.Comment: 10 page

The Dispersion of Newton's Constant: A Transfer Matrix Formulation of Quantum Gravity

A transfer matrix formalism applicable to certain reparametrization invariant theories, including quantum gravity, is proposed. In this formulation it is found that every stationary state in quantum gravity satisfies a Wheeler-DeWitt equation, but each with a different value of the Planck mass; the value $m_{Planck}^4$ turns out to be proportional to the eigenvalue of the evolution operator. As a consequence, the fact that the Universe is non-stationary implies that it is not in an eigenstate of Newton's constant.Comment: 24 pages, plain LaTeX, NBI-HE-93-5

Green's function for gravitational waves in FRW spacetimes

A method for calculating the retarded Green's function for the gravitational wave equation in Friedmann-Roberson-Walker spacetimes, within the formalism of linearized Einstein gravity is developed. Hadamard's general solution to Cauchy's problem for second-order, linear partial differential equations is applied to the FRW gravitational wave equation. The retarded Green's function may be calculated for any FRW spacetime, with curved or flat spatial sections, for which the functional form of the Ricci scalar curvature $R$ is known. The retarded Green's function for gravitational waves propagating through a cosmological fluid composed of both radiation and dust is calculated analytically for the first time. It is also shown that for all FRW spacetimes in which the Ricci scalar curvatures does not vanish, $R \neq 0$, the Green's function violates Huygens' principle; the Green's function has support inside the light-cone due to the scatter of gravitational waves off the background curvature.Comment: 9 pages, FERMILAB-Pub-93/189-

Gauge Independent Trace Anomaly for Gravitons

We show that the trace anomaly for gravitons calculated using the usual effective action formalism depends on the choice of gauge when the background spacetime is not a solution of the classical equation of motion, that is, when off-shell. We then use the gauge independent Vilkovisky-DeWitt effective action to restore gauge independence to the off-shell case. Additionally we explicitly evaluate trace anomalies for some N-sphere background spacetimes.Comment: 19 pages, additional references and title chang

Noise Kernel in Stochastic Gravity and Stress Energy Bi-Tensor of Quantum Fields in Curved Spacetimes

The noise kernel is the vacuum expectation value of the (operator-valued) stress-energy bi-tensor which describes the fluctuations of a quantum field in curved spacetimes. It plays the role in stochastic semiclassical gravity based on the Einstein-Langevin equation similar to the expectation value of the stress-energy tensor in semiclassical gravity based on the semiclassical Einstein equation. According to the stochastic gravity program, this two point function (and by extension the higher order correlations in a hierarchy) of the stress energy tensor possesses precious statistical mechanical information of quantum fields in curved spacetime and, by the self-consistency required of Einstein's equation, provides a probe into the coherence properties of the gravity sector (as measured by the higher order correlation functions of gravitons) and the quantum nature of spacetime. It reflects the low and medium energy (referring to Planck energy as high energy) behavior of any viable theory of quantum gravity, including string theory. It is also useful for calculating quantum fluctuations of fields in modern theories of structure formation and for backreaction problems in cosmological and black holes spacetimes. We discuss the properties of this bi-tensor with the method of point-separation, and derive a regularized expression of the noise-kernel for a scalar field in general curved spacetimes. One collorary of our finding is that for a massless conformal field the trace of the noise kernel identically vanishes. We outline how the general framework and results derived here can be used for the calculation of noise kernels for Robertson-Walker and Schwarzschild spacetimes.Comment: 22 Pages, RevTeX; version accepted for publication in PR

Nonequilibrium Quantum Dynamics of Second Order Phase Transitions

We use the so-called Liouville-von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schr\"{o}dinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal and coherent-thermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Due to the spinodal instability of long wavelength modes the two-point correlation functions lead to the $t^{1/4}$-scaling relation during the quench and the Cahn-Allen scaling relation $t^{1/2}$ after the completion of quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.Comment: discussion on back-reaction added, typos corrected, references added, final version for PR

Quantum mechanical path integrals and thermal radiation in static curved spacetimes

The propagator of a spinless particle is calculated from the quantum mechanical path integral formalism in static curved spacetimes endowed with event-horizons. A toy model, the Gui spacetime, and the 2D and 4D Schwarzschild black holes are considered. The role of the topology of the coordinates configuration space is emphasised in this framework. To cover entirely the above spacetimes with a single set of coordinates, tortoise coordinates are extended to complex values. It is shown that the homotopic properties of the complex tortoise configuration space imply the thermal behaviour of the propagator in these spacetimes. The propagator is calculated when end points are located in identical or distinct spacetime regions separated by one or several event-horizons. Quantum evolution through the event-horizons is shown to be unitary in the fifth variable.Comment: 22 pages, 10 figure

GUTs in Curved Spacetime: Running Gravitational Constants, Newtonian Potential and the Quantum Corrected Gravitational Equations

The running coupling constants (in particular, the gravitational one) are studied in asymptotically free GUTs and in finite GUTs in curved spacetime, with explicit examples. The running gravitational coupling is used to calculate the leading quantum GUT corrections to the Newtonian potential, which turn out to be of logarithmic form in asymptotically free GUTs. A comparison with the effective theory for the conformal factor ---where leading quantum corrections to the Newtonian potential are again logarithmic--- is made. A totally asymptotically free $O(N)$ GUT with quantum higher derivative gravity is then constructed, using the technique of introducing renormalization group (RG) potentials in the space of couplings. RG equations for the cosmological and gravitational couplings in this theory are derived, and solved numerically, showing the influence of higher-derivative quantum gravity on the Newtonian potential. The RG-improved effective gravitational Lagrangian for asymptotically free massive GUTs is calculated in the strong (almost constant) curvature regime, and the non-singular De Sitter solution to the quantum corrected gravitational equations is subsequently discussed. Finally, possible extensions of the results here obtained are briefly outlined.Comment: LaTeX, 27 pages, 2 uu-figure

Stochastic Behavior of Effective Field Theories Across Threshold

We explore how the existence of a field with a heavy mass influences the low energy dynamics of a quantum field with a light mass by expounding the stochastic characters of their interactions which take on the form of fluctuations in the number of (heavy field) particles created at the threshold, and dissipation in the dynamics of the light fields, arising from the backreaction of produced heavy particles. We claim that the stochastic nature of effective field theories is intrinsic, in that dissipation and fluctuations are present both above and below the threshold. Stochasticity builds up exponentially quickly as the heavy threshold is approached from below, becoming dominant once the threshold is crossed. But it also exists below the threshold and is in principle detectable, albeit strongly suppressed at low energies. The results derived here can be used to give a quantitative definition of the `effectiveness' of a theory in terms of the relative weight of the deterministic versus the stochastic behavior at different energy scales.Comment: 32 pages, Latex, no figure

Bose-Einstein condensation as symmetry breaking in compact curved spacetimes

We examine Bose-Einstein condensation as a form of symmetry breaking in the specific model of the Einstein static universe. We show that symmetry breaking never occursin the sense that the chemical potential $\mu$ never reaches its critical value.This leads us to some statements about spaces of finite volume in general. In an appendix we clarify the relationship between the standard statistical mechanical approaches and the field theory method using zeta functions.Comment: Revtex, 25 pages, 3 figures, uses EPSF.sty. To be published in Phys. Rev.