Quantum conformal fluctuations and stationary states

Conformal fluctuations serve as a powerful tool to study the nature of quantum gravity. They lead, in a natural fashion, to the concept of stationary states for the quantum geometry. We attempt to incorporate the effect of conformal fluctuations into the background metric and matter. A modified set of equations, including the effect of conformal fluctuations, is presented and the solutions are discussed. It is shown that matter-free vacuum is unstable to conformal fluctuations. A scenario for creation of matter is indicated.     

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.     

The conformal structure of space-time in a constructive axiomatics based on elements of quantum mechanics

A constructive axiomatic approach to the geometry of space-time in the domain of classical and quantum physics can be founded on basic experiences made with quantum objects. The corresponding matter fields are described by linear systems of partial differential equations. Their characteristics are studied. Postulates concerning the related jump cones and jump amplitudes lead to a unique conformal structure attributed to the space-time geometry.     

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.     

Superpotentials for Gauge and Conformal Supergravity Backgrounds

Effective superpotentials obtained by integrating out matter in super Yang-Mills and conformal supergravity backgrounds in SUSY theories are considered. The pure gauge and supergravity contributions (generalizing Veneziano-Yankielowicz terms) are derived by considering the case with matter fields in the fundamental representation of the gauge group. These contributions represent quantum corrections to the tree-level Yang-Mills and conformal supergravity actions. The classical equations of motion following from the conformal supergravity action require the background to be (super)conformally flat. This condition is unchanged by quantum corrections to the effective superpotential, irrespective of the matter content of the theory.     

Gravitational coupling,dynamical changes of units and the cosmological constant problem

A spacetime interval connecting two neighbouring points can be measured in different unit systems.For instance,it can be measured in atomic unit defined in terms of fundamental constants existing in quantum theories.It is also possible to use a gravitational unit which is defined by the use of properties of macroscopic objects.These two unit systems are usually regarded as indistinguishable up to a constant conversion factor.Here we consider the possibility that these two units are related by an epoch-dependent conversion factor.This is a dynamical changes of units.Regarding a conformal transformation as a local unit transformation,we use a gravitational model in which the gravitational and the matter sectors are given in different conformal frames(or unit systems).It is relevant to the cosmological constant problem,namely the huge discrepancy between the estimated and the observational values of the cosmological constant in particle physics and cosmology,respectively.We argue that the problem arises when one ignores evolution of the conversion factor relating the two units during expansion of the Universe.Connection of the model with violation of equivalence principle and possible variation of fundamental constants are also discussed.     

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. In our approach 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.     

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.     

The Conformal Metric Associated with the U(1) Gauge of the Stueckelberg–Schrödinger Equation

We review the relativistic classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelbert–Schrödinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields with compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann–Robertson–Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson–Walker scale factor. Using the Einstein equations, one see that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann–Robertson–Walker dynamical model and the configurations in the conformal coordinates.     

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).     

Renormalizable 4D Quantum Gravity as a Perturbed Theory from CFT

We study the renormalizable quantum gravity formulated as a perturbed theory from conformal field theory (CFT) on the basis of conformal gravity in four dimensions. The conformal mode in the metric field is managed non-perturbatively without introducing its own coupling constant so that conformal symmetry becomes exact quantum mechanically as a part of diffeomorphism invariance. The traceless tensor mode is handled in the perturbation with a dimensionless coupling constant indicating asymptotic freedom, which measures a degree of deviation from CFT. Higher order renormalization is carried out using dimensional regularization, in which the Wess-Zumino integrability condition is applied to reduce indefiniteness existing in higher-derivative actions. The effective action of quantum gravity improved by renormalization group is obtained. We then make clear that conformal anomalies are indispensable quantities to preserve diffeomorphism invariance. Anomalous scaling dimensions of the cosmological constant and the Planck mass are calculated. The effective cosmological constant is obtained in the large number limit of matter fields.     

Lifting a conformal field theory from d-dimensional flat space to (d+1)-dimensional AdS space

A quantum field theory on anti-de Sitter space can be constructed from a conformal field theory on its boundary Minkowski space by an inversion of the holographic mapping. The resulting theory is defined by its Green functions and is conformally covariant. The structure of operator product expansions is carried over to AdS space. We show that this method yields a higher spin field theory HS(4) from the minimal conformal O(N) sigma model in three dimensions.     

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.     

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.     

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.     

Interacting entropy-corrected new agegraphic dark energy in Brans–Dicke cosmology

Motivated by a recent work of one of us (Sheykhi in Phys Rev D 81: 023525, 2010), we extend it by using quantum (or entropy) corrected new agegraphic dark energy in the Brans–Dicke cosmology. The correction terms are motivated from the loop quantum gravity which is one of the competitive theories of quantum gravity. Taking the non-flat background spacetime along with the conformal age of the universe as the length scale, we derive the dynamical equation of state of dark energy and the deceleration parameter. An important consequence of this study is the phantom divide scenario with entropy-corrected new agegraphic dark energy. Moreover, we assume a system of dark matter, radiation and dark energy, while the later interacts only with dark matter. We obtain some essential expressions related with dark energy dynamics. The cosmic coincidence problem is also resolved in our model.     

Derivation of the Dirac Equation by Conformal Differential Geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.     

Quantum cosmological model with a scalar field as source

It is shown that if both the conformal factor and scalar field are quantized on a background spacetime, a singularity-free cosmological model can be obtained and the quantum states must be zero-energy ones.     

The gravitational anomaly: An elementary,coordinate-space approach

Alvarez-Gaumé and Witten have shown that energy-momentum conservation must be violated in certain parity-violating quantum field theories involving gravity. In two dimensions this effect can be studied without the aid of Feynman diagrams or calculations in momentum space. The arguments parallel those for the conformal (trace) anomaly; as in that case, there are two kinds of arguments, one based on the conservation equations themselves with some very general assumptions, and the other based on explicit calculations and renormalization in a model theory with a linear field equation. The basic point is that if matter is created at all by the gravitational field, it must appear in both left-moving and right-moving modes if the conservation law is to hold always.This essay received an honourable mention (1985) by the Gravity Research Foundation—Ed.