The discovery of the cosmic microwave background radiation (CMB) by Penzias and Wilson has proven that the universe was once a melting pot of high energy reactions at its early stage, and since then it has been cooling as it expands. Valuable information on physical processes taking place in the expanding universe has been recorded in the CMB as tiny anisotropies. Angular power spectra of the anisotropies, recently observed by the Cosmic Background Explorer (COBE) and the Wilkinson Microwave Anisotropy Probe (WMAP), are, roughly speaking, projection of the history for the period between its birth to the present. It gives us a amazing hope that if we believe the idea of inflationary universe describing a extremely rapid expansion without global thermalization, the long-distance correlation in observed anisotropies can provide the information about dynamics of the period before the universe grew to the Planck scale. We are now on the stage of revealing and verifying the quantum aspect of universe.
After passing many theoretical as well as experimental tests, the general theory of relativity has been established as the fundamental theory of gravity capable of describing the evolution of universe. Many questions in the mysteries of universe have arose expecting that the theory can give answers, such as, how the universe is created, why it is expanding, how it will evolve in the future, and so on. In order to answer these questions, we need to know the initial condition of the universe that would be set at the Planck time. However, it is the period when quantum mechanics rules the law of universe, and it is beyond the power of the classical general relativity. If we wish to apply the Einstein theory for the Planck scale phenomena, it has fatal difficulties, such as the black-hole singularity and divergences in the canonical quantization procedure. Existence of these problems makes us wonder whether the Planck scale is the upper limit of the energy beyond which nature cannot be described in terms of physical laws. It is our ultimate task to construct the quantum theory which breaks the wall of Planck scale and creates a new frontier.
Renormalizable quantum theory of gravity developed on the basis of the conformal gravity in four dimensions provides an evolutional scenario of the universe which starts from quantum states with conformal invariance, passing through the inflationary era, and then makes transition to the conventional Einstein space-time without adding any artificial field by hands. In this theory the metric tensor field is written as a product of the conformal factor and the traceless tensor, and the conformal mode is treated non-perturbatively while the traceless tensor field is handled as a perturbation. The renormalized coupling of the traceless tensor mode indicates the asymptotic freedom. It implies that at very high energies beyond the Planck mass scale, quantum fluctuations of the conformal mode are dominated as a consequence of the background-metric independence, and space-time is non-singular. This nature is called ``asymptotic background freedom''
to distinguish from the normal asymptotic freedom. The power-law spectrum obtained from the two point correlation function of scalar curvature shows a significant
character of the conformally invariant field theory which is expected to be observed cosmologically.
The asymptotic background freedom also implies that there exists a dynamical scale where the effective coupling of traceless tensor mode diverges, and the conformal symmetry loses its validity. About this point the universe is expected to make a transition from the quantum spacetime to the classical Einstein spacetime, and fluctuations of the conformal mode would be frozen to the classical background. Such a dynamical scale is given by the order of 10^17GeV below the Planck scale. The sharp fall off of the angular power spectra at low multipole components of anisotropies in the CMB is expected to be an appearance of dynamical scale.
The Einstein theory defined by the Ricci scalar is known to be not renormalizable, but it does not implies that diffeomorphism invariance conflicts with renormalization theory. Renormalizable quantum gravity presented here is defined by the following action determined by imposing the conditions of diffeomorphism invariance and Wess-Zumino integrability:
The first term is the Weyl action and "t" is the dimensionless coupling constant with negative beta function, which take care of the traceless tensor field. On the other hand, the conformal factor in the metric field is treated non-perturbatively without introducing its own coupling constant. The path integral over the metric field is described as
Here, S is the Wess-Zumino action induced from the path integral measure. It is necessary to preserve diffeomorphism invariance when we rewrite the path integral using the practical measure defined on the background metric. Especially, the action S that appears at the UV fixed point of t=0 is called the Riegert-Wess-Zumino action, which is given by
At the UV fixed point, the theory is described as a certain conformal field theory (CFT). Exact conformal symmetry comes out as a part of quantum diffeomorphism invariance, realized by taking the sum over all possible conformally flat configurations. The background-metric independence is then represented as a gauge equivalency under conformal transformations, called BRST conformal symmetry. On the other hand, the coupling t measures a degree of devitation from this CFT. With the increase of it, spacetime will make a transition from quantum gravity phase to classical Einstein phase, as mentioned above.
The inverse of the dynamical energy scale mentioned above gives the correlation length of quantum gravity. If shorter than this length, we cannot measure the distance, which is due to the background-metric independence where the concept of time and space is lost. Thus, it represents the minimal length of space-time we can measure physically. In this sense, space-time is practically quantized by this length unit without descretizing it. The energy of excitations in quantum gravity would be then given by the dynamical scale. If there is a stable excitation, it becomes a candidate for dark matter.
|Section 2||Renormalizable Quantum Gravity|
|2.1 Brief Summary of The Model|
|2.2 Conformal Invariance and Physical States|
|Section 3||Numerical Quantum Gravity|
Quantum Gravity Cosmology
Quantum Diffeomorphism and Conformal Field Theory
Two Dimensional Quantum Gravity
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