Quantum cosmology and stationary states

A model for quantum gravity, in which the conformal part of the metric is quantized using the path integral formalism, is presented. Einstein's equations can be suitably modified to take into account the effects of quantum conformal fluctuations. A closed Friedman model can be described in terms of well-defined stationary states. The “ground state” sets a lower bound (at Planck length) to the scale factor preventing the collapse. A possible explanation for matter creation and quantum nature of matter is suggested.     

The vanishing likelihood of space-time singularity in quantum conformal cosmology

A general formalism is developed for studying the behavior of quantized conformal fluctuations near the space-time singularity of classical relativistic cosmology. It is shown that if the material contents of space-time are made of massive particles which obey the principle of asymptotic freedom and interact only gravitationally, then it is possible to estimate the quantum mechanical probability that, of the various possible conformal transforms of the classical Einstein solution, the actual model had a singularity in the past. This probability turns out to be vanishingly small, thus indicating that within the regime of quantum conformal cosmology it is extremely unlikely that the universe originated out of a space-time singularity.     

Quantum fluctuations in the conformally flat and the schwarzschild space-times

A general technique is described for dealing with the quantum fluctuations between conformally flat space-times. The second part of the paper deals with the Schwarzschild spacetime. It is shown there that this space-time is stable against fluctuations of mass, but transitions between two space-times of different masses can be obtained via conformai fluctuations. Purely conformal fluctuations of the Schwarzschild metric are, however, damped at the event horizon. Similar conclusions are drawn about the Reissner-Nordstrom space-time.     

Zero modes of quantum fluctuations for the instanton

All possible zero eigenvalues for quantum fluctuations in the presence of the BPST instanton are treated in the light of the spontaneous breakdown of conformal symmetry. The zero eigensolution in the fermion equation is automatically given as a consequence of the breakdown of the conformal super-symmetry.     

Quantum fluctuations and nonavoidance of the singularity in Bianchi type I cosmology

An effective metric is defined and used for analyzing the quantum fluctuations in a classical geometry. Earlier work showing that quantum (conformal) fluctuations avoid the classical singularity in the case of spherically symmetric collapse is briefly reviewed. It is shown that this result doesnot extend to anisotropic Bianchi type I cosmology. Here the dispersion in the fluctuations increases too slowly to quench the classical singularity. The singularity persists in the space-time described by the effective metric.     

Quantum gravity and the “flatness problem” of the standard big bang universe

It is shown that, if the universe originated through quantum conformal fluctuations from the empty Minkowski space, then it is most likely to be spatially flat.     

Quantum conformal fluctuations near the classical space-time singularity

This paper investigates the behavior of conformal fluctuations of space-time geometry that are admissible under the quantized version of Einstein's general relativity. The approach to quantum gravity is via path integrals. It is shown that considerable simplification results when only the conformal degrees of freedom are considered under this scheme, so much so that it is possible to write down a formal kernel in the most general case where the space-time contains arbitrary distributions of particles with no other interaction except gravity. The behavior of this kernel near the classical space-time singularity then shows that quantum fluctuations inevitably diverge near the singularity. It is shown further that the root cause of this divergence lies in the fact that the Green's function for the conformally invariant scalar wave equation diverges at the singularity. The limitations on the validity of classical general relativity near the space-time singularity are discussed and it is argued that the notion of singularity itself needs to be radically modified once the quantum effects are taken into account.On leave of absence from the Tata Institute of Fundamental Research, Bombay, India     

Conformal Fluctuations of the Interior Schwarzschild Solution

The basic formalism for conformal fluctuations of the gravitational field is presented. After developing a master propagator for the interior Schwarzschild solution, the time development of the gravitational wave function is considered. The effect of the two classical singularities (resp. pseudo-singularities) of the Schwarzschild solution on the quantum wave function for the gravitational field is studied using a wave function initially localized on the classical solution. While the true singularity at r = 0 imparts consequences on the wave function that cannot be ignored, the pseudo-singularity at the event horizon does not seem to cause any divergences on the interior fluctuations of the Schwarzschild solution.     

Making the Case for Conformal Gravity

We review some recent developments in the conformal gravity theory that has been advanced as a candidate alternative to standard Einstein gravity. As a quantum theory the conformal theory is both renormalizable and unitary, with unitarity being obtained because the theory is a PT symmetric rather than a Hermitian theory. We show that in the theory there can be no a priori classical curvature, with all curvature having to result from quantization. In the conformal theory gravity requires no independent quantization of its own, with it being quantized solely by virtue of its being coupled to a quantized matter source. Moreover, because it is this very coupling that fixes the strength of the gravitational field commutators, the gravity sector zero-point energy density and pressure fluctuations are then able to identically cancel the zero-point fluctuations associated with the matter sector. In addition, we show that when the conformal symmetry is spontaneously broken, the zero-point structure automatically readjusts so as to identically cancel the cosmological constant term that dynamical mass generation induces. We show that the macroscopic classical theory that results from the quantum conformal theory incorporates global physics effects that provide for a detailed accounting of a comprehensive set of 138 galactic rotation curves with no adjustable parameters other than the galactic mass to light ratios, and with the need for no dark matter whatsoever. With these global effects eliminating the need for dark matter, we see that invoking dark matter in galaxies could potentially be nothing more than an attempt to describe global physics effects in purely local galactic terms. Finally, we review some recent work by ’t Hooft in which a connection between conformal gravity and Einstein gravity has been found.     

Symmetry breaking in de Sitter spacetime by antipodal symmetric and antisymmetric fields

The symmetry behaviour of a ϵø⁴ model in the identified de Sitter space is considered. The different boundary conditions imposed on the quantum field correspond to antipodally symmetric and antisymmetric cases. For both cases, the variance of the quantum fluctuations around a constant background field is evaluated using a result on the conformal transformation of the effective action.     

Nonabelions in the fractional quantum hall effect

Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks in certain rational conformal field theories. Using this point of view a hamiltonian is constructed for electrons for which the ground state is known exactly and whose quasihole excitations have nonabelian statistics; we term these objects “nonabelions”. It is argued that universality classes of fractional quantum Hall systems can be characterized by the quantum numbers and statistics of their excitations. The relation between the order parameter in the fractional quantum Hall effect and the chiral algebra in rational conformal field theory is stressed, and new order parameters for several states are given.     

New classical solutions with fermion in conformal invariant field theories

New instanton type solutions for coupled non-linear equations of scalar and fermion are given. Invariance properties of the solutions under the six-dimensional conformal group are studied. Quantum significances are discussed, and the equations of motion for quantum fluctuations turn out to be the eigenvalue equations for the Casimir operators of the O(5) group.     

Quantum effects in a homogeneous dust cloud collapse

The status of a classical space-time singularity, when quantum effects are taken into account, has remained a matter of intense interest ever since the epochmaking paper of DeWitt [1] on quantum gravity. We examine here the evolution of quantum fluctuations in the vicinity of the singularity arising out of the classical collapse of a homogeneous dust cloud. As opposed to the pathintegral method used to quantize the conformal degree of freedom (see, e.g., [3] or [4]), we use here the traditional operator approach to the quantum theory which is much more direct and appealing while achieving an additional generalization that the wave function of the system is assumed to have a completely general form. It is shown that the quantum uncertainty diverges in the limit of approach to the classically singular epoch and that nonsingular, nonclassical states can occur with finite probability.     

Quantum noncommutative gravity in two dimensions

We study quantisation of noncommutative gravity theories in two dimensions (with noncommutativity defined by the Moyal star product). We show that in the case of noncommutative Jackiw–Teitelboim gravity the path integral over gravitational degrees of freedom can be performed exactly even in the presence of a matter field. In the matter sector, we study possible choices of the operators describing quantum fluctuations and define their basic properties (e.g., the Lichnerowicz formula). Then we evaluate two leading terms in the heat kernel expansion, calculate the conformal anomaly and the Polyakov action (as an expansion in the conformal field).     

Scaling and enhanced symmetry at the quantum critical point of the sub-ohmic Bose-Fermi Kondo model

We consider the finite-temperature scaling properties of a Kondo-destroying quantum critical point in the Ising-anisotropic Bose-Fermi Kondo model (BFKM). A cluster-updating Monte Carlo approach is used, in order to reliably access a wide temperature range. The scaling function for the two-point spin correlator is found to have the form dictated by a boundary conformal field theory, even though the underlying Hamiltonian lacks conformal invariance. Similar conclusions are reached for all multipoint correlators of the spin-isotropic BFKM in a dynamical large-N limit. Our results suggest that the quantum critical local properties of the sub-Ohmic BFKM are those of an underlying boundary conformal field theory.     

On the nonlinear energy transfer in the unidirected adiabatic surface waves

The results of numerical simulation of the adiabatic evolution of waves are presented. The model is based on the fully nonlinear 1D equations of potential waves written in conformal coordinates. It is shown that a wave spectrum is subject to strong fluctuations. Most of such fluctuations are reversible, however a residual effect of the fluctuations causes downshifting of the spectrum. The rate of downshifting depends on nonlinearity.     

Conformal Anomaly and Large-ScaleGravitational Coupling

We present a model in which the breakdown of conformal symmetry of a quantumstress-tensor due to the trace anomaly is related to a cosmological effect in agravitational model. This is done by characterizing the traceless part of thequantum stress-tensor in terms of the stress-tensor of a conformal invariantclassical scalar field. We introduce a conformal frame in which the anomaloustrace is identified with a cosmological constant. In this conformal frame weestablish the Einstein field equations by connecting the quantum stress-tensorwith the large-scale distribution of matter in the universe.     

Geometric phase under the Unruh effect with intermediate statistics

Utilizing the geometric phase (GP) acquired in a quantum evolution, we manifest the thermality and quantum nature of the Unruh effect of an accelerating detector. We consider an UDW detector coupling to a conformal field in Minkowski spacetime, whose response spectrum exhibits an intermediate statistics of (1+1) anyon field. We find that comparing to an inertial moving detector, the GP in accelerating frame is modified after the nonunitary evolution of the detector due to the Unruh effect. We show that such modification can distinguish the different thermalizing ways of the detector, which depends on the scaling dimension of the conformal primary field. Finally, we estimate the difference between the GP under the Unruh radiation and that in a thermal bath for a static observer, which reveals the quantum origin of the Unruh effect rather than a conventional thermal noise.     

Construction of exponential Liouville field operators for closed string models

Operators for arbitrary exponentials exp(λφ) of a periodic Liouville field φ(τ,σ) are represented iteratively by an infinite power series in terms of a periodic scalar free field. Necessary quantum corrections of the Liouville operators with respect to their classical expressions are fixed by conformal covariance and locality. Canonical commutation relations for the Liouville field quantities are valid when the canonical quantization of the scalar free field is imposed. A quantum correction of the energy momentum tensor can be avoided thus preserving the conformal invariance of the Liouville theory.