Quantum Geometrodynamics of Higher Derivative Theories with and without Conformal Symmetry
Branislav Nikolic
备注：
Doctoral dissertation (University of Cologne), 273 pages, Identical to the final published version except for minor formatting improvements and corrected typos
The main goal of this thesis is to quantize the Einstein-Hilbert action extended by the quadratic curvature terms within the canonical quantization approach, thus formulating quantum geometrodynamics of the higher derivative theories. The motivation is to provide an alternative to the standard canonical quantization based on the Einstein-Hilbert action alone, because the latter does not generate the quadratic curvature terms in the semiclassical limit. A particular formulation of a semiclassical approximation scheme is employed which ensures that the effects of the quadratic curvature terms become perturbative in the semiclassical limit. This leaves the classical General Relativity intact, while naturally giving rise to its first semiclassical corrections. Another topic of interest is a classical theory where the quadratic Ricci scalar and the Einstein-Hilbert term are absent from the action, which then enjoys the symmetry with respect to the conformal transformation of fields (local Weyl rescaling). We pay a special attention to this case, since it provides a natural setting for the absence of the notion of a physical length scale. Certain useful model-independent tools are also constructed in this thesis. Firstly, dimensionless coordinates and the unimodular decomposition of the metric are used to expose the only conformally variant degree of freedom, making the geometrical origin of the physical length scale apparent. With such an approach several earlier results become much more transparent. Secondly, using unimodular-conformal variables a model-independent generator of conformal field transformations is constructed in terms of which a reformulation of the definition of conformal invariance is given. Thirdly, it is argued that a canonical quantization scheme makes more sense to be based on the quantization of generators of relevant transformations, than on first class constraints.
The reduced vacuum Hamiltonian equations of conformal geometrodynamics of compact manifolds in extrinsic time are written. This is achieved by generalizing the theorem of implicit function derivative to the functional analysis.
The global time is defined in covariant form under the condition of a constant mean curvature slicing of spacetime. The background static metric is taken in the tangent space. The global intrinsic time is identified with the logarithmic function of the mean value of the ratio of the square root of the metric determinants. The procedures of the Hamiltonian reduction and deparametrization of dynamical systems are implemented. The Hamiltonian system appears to be non-conservative. The Hamiltonian equations of motion of gravitational field in the global time are written.
Within the framework of the Einstein's standard equations of the general theory of relativity, flat galactic rotational curves of galaxies cannot be explained without hypothesis attracting the dark matter, the particles of which had not yet been identified. The vacuum central-symmetric solution of the equations of conformal gravitation is well known as metrics of Mannheim-Kazanas, on the basis of which these curves receive purely geometrical explanation. We show in our work that the metrics of Mannheim-Kazanas is the solution of not only Bach equations received from conformal-invariant Weyl La…
An exact solution was produced for one of the versions of conformal -invariant gravitation theories (conformal geometrodynamics - CG) for the body with a mass and an electric charge. The solution is analogous to the Reissner-Nordström solution in the general relativity theory. Its similar solution was earlier produced by Mannheit and Kazanas for the different version of the conformal-invariant gravitation theory -- Weyl conformal gravitation theory (Weyl CGt). Similar solutions do not comply. Issues of conformity with regard to the similar solutions are discussed. A conclusion is made that onl…
We present a new formulation for the canonical approach to conformal (Weyl-squared) gravity and its extension by the Einstein-Hilbert term and a nonminimally coupled scalar field. For this purpose we use a unimodular decomposition of the three-metric and introduce unimodular-conformal canonical variables. The important feature of this choice is that only the scale part of the three-metric and the rescaled trace part of the extrinsic curvature change under a conformal transformation. This significantly simplifies the constraint analysis and manifestly reveals the conformal properties of a theor…
An intrinsic local time in Geometrodynamics is obtained with using a scaled Dirac's mapping. By addition of a background metric, one can construct a scalar field. It is suitable to play a role of intrinsic time. Cauchy problem was successfully solved in conformal variables because they are physical ones. First, the intrinsic time as a logarithm of determinant of spatial metric, was applied to a cosmological problem by Misner. A global time is exist under condition of constant mean curvature slicing of spacetime. A volume of hypersurface and the so-called mean York's time are canonical conjugat…
There exist several ways of constructing general relativity from `first principles': Einstein's original derivation, Lovelock's results concerning the exceptional nature of the Einstein tensor from a mathematical perspective, and Hojman-Kuchař-Teitelboim's derivation of the Hamiltonian form of the theory from the symmetries of spacetime, to name a few. Here I propose a different set of first principles to obtain general relativity in the canonical framework without presupposing spacetime in any way. I first require consistent propagation of scalar spatially covariant constraints. I find that u…
In four dimensions complexified General Relativity (GR) can be non-trivially deformed: There exists an (infinite-parameter) set of modifications all having the same count of degrees of freedom. It is trivial to impose reality conditions that give versions of the deformed theories corresponding to Riemannian and split metric signatures. We revisit the Lorentzian signature case. To make the problem tractable, we restrict our attention to a four-parameter set of deformations that are natural extensions of Ashtekar's Hamiltonian formalism for GR. The Hamiltonian of the later is a linear combinatio…
An intrinsic time in Geometrodynamics is obtained with using a scaled Dirac's mapping. By addition of a background metric, one can construct a scalar field. It is suitable to play a role of intrinsic time. Cauchy problem was successfully solved in conformal variables because they are physical ones. First, the intrinsic time as a logarithm of determinant of spatial metric, was applied to a cosmological problem by Misner. A global time is exist under condition of constant mean curvature slicing of spacetime. The volume of hypersurface and so-called mean York's time are canonical conjugated pair.…
After an overview of variational principles for discrete gravity, and on the basis of the approach to conformal transformations in a simplicial PL setting proposed by Luo and Glickenstein, we present at a heuristic level an improved scheme for addressing the gravitational (Euclidean) path integral and geometrodynamics.
The Hamiltonian formulation of conformally invariant Weyl-squared higher derivative theory teaches us that conformal symmetry is expressed through particular first class constraints related to the absence of the three-metric determinant and the trace of the extrinsic curvature from the theory. Any term depending on them which is added to this theory breaks conformal invariance and turns these constraints into second class ones. Such second class constraints are missing in the standard canonical formulation of the conformally non-invariant Einstein-Hilbert theory. It is demonstrated that such c…
In any quantum theory of gravity, it is of the utmost importance to recover the limit of quantum theory in an external spacetime. In quantum geometrodynamics (quantization of general relativity in the Schrödinger picture), this leads in particular to the recovery of a semiclassical (WKB) time which governs the dynamics of non-gravitational fields in spacetime. Here, we first review this procedure with special emphasis on conceptual issues. We then turn to an alternative theory - Weyl (conformal) gravity, which is defined by a Lagrangian that is proportional to the square of the Weyl tensor. We…
We derive the functional Schrodinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a WKB-type expansion of the wave functional in powers of the squared Planck mass. The functional Schrodinger equation that we obtain exhibits a functional time derivative that completes the usual definition of WKB time for curved spacetime, and the usual Schrodinger-type evolution is recovered in Minkowski spacetime.
Motivated by well-known obstacles to quantum gravity, I look for the most general geometrodynamical symmetries compatible with a reduced physical configuration space for metric gravity. I argue that they lead either to a completely static Universe, or one embodying spatial conformal diffeomorphisms. Demanding locality for an action compatible with these principles determines the sort of terms one can add, both for the purely gravitational part as well as matter couplings. The symmetries guarantee that there are two gravitational propagating physical degrees of freedom (and no explicit refoliat…
By extending the notion of Lie derivative to distribution-valued tensor fields of order $m$, Lie derivatives with respect to $C^k$ vector fields, $k\geqslant m+1$, can be shown to be well defined. Geometric symmetries, definable in terms of these Lie derivatives, can then be considered. In particular, for spacetimes of low regularity, geometric symmetries generated by vector fields of regularity as low as $C^1$ are definable. We find that the conformal algebra of the impulsive plane wave spacetime (with $+$ polarization) described by the continuous Baldwin-Jeffery-Rosen form of the metric has …
The BRST quantizations of worldsheet gravity corresponding to final more acceptable derivative gauge and the standard conformal gauge are studied. We establish a mapping between these two gauges utilizing FFBRST formulation in standard way. Therefore, we are able to declare that the problems associated with Virasoro constraints are the gauge artifact.
In this paper we establish the correspondence between ghost-free bimetric theory and a class of higher derivative gravity actions, including conformal gravity and New Massive Gravity. We also characterize the relation between the respective equations of motion and classical solutions. We illustrate that, in this framework, the spin-2 ghost of higher derivative gravity is an artifact of the truncation to a 4-derivative theory. The analysis also gives a relation between the proposed partially massless (PM) bimetric theory and conformal gravity, showing, in particular, the equivalence of their eq…
We show that the primordial gravitational wave with scale-invariant spectrum might emerge from a nearly Minkowski space, in which the gravity is asymptotic-past free. We illustrate it with a model, in which the derivative of background scalar field nonminimally couples to gravity. We also show that since here the tensor perturbation is dominated by its growing mode, mathematically our slowly expanding background is conformally dual to the matter contraction, but there is no the anisotropy problem.
We first find the linear approximation of the second plus fourth order derivative massive conformal gravity action. Then we reduce the linearized action to separated second order derivative terms, which allows us to quantize the theory by using the standard first order canonical quantization method. It is shown that quantum massive conformal gravity is renormalizable but has ghost states. A possible decoupling of these ghost states at high energies is discussed.
This letter is dedicated to providing proof of two statements concerning the gradient expansion of relativistic hydrodynamics. The first statement is that \textit{the ordering of transverse derivatives is irrelevant in the gradient expansion of a non-conformal fluid}. The second statement is that \textit{the longitudinal projection of the Weyl covariant derivative can be eliminated in the gradient expansion of a conformal fluid}. This second statement does not apply to curvature tensors. In the conformal case, we know that the ordering of Weyl covariant derivatives is irrelevant in the gradien…
