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. 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 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 indicating the asymptotic freedom, 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.

- Name: Ken-ji HAMADA
- Affiliation: Institute of Particle and Nuclear Studies, KEK, Tsukuba, Japan
- Research Interests: Renormalizable Quantum Gravity, Conformal Field Theory and Cosmology

- Background Free Quantum Gravity and Cosmology,

K. Hamada, S. Horata and T. Yukawa, ``Focus on Quantum Gravity Research" (Nova Science Publisher, NY, 2006), Chap. 1.Contents Section 1 Introduction Section 2 Renormalizable Quantum Gravity 2.1 Brief Summary of The Model 2.2 Conformal Invariance and Physical States 2.3 Renormalization Section 3 Numerical Quantum Gravity Section 4 Cosmology Section 5 Conclusion

[Abstract] Tracing the history of the universe, spacetime in its early epoch would be totally fluctuating quantum mechanically, and there would be no classical spacetime to describe dynamics. The models of quantum gravity we are going to discuss are aimed to describe such states with background-metric independence and transitions to the conventional spacetime dynamics.

The first model we choose is a renormalizable theory of gravity developed on the basis of the conformal gravity in four dimensions. 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 mode indicates the asymptotic freedom. It implies that at very high energies beyond the Planck mass scale, quantum fluctuations of the conformal field are dominated as a consequence of the background-metric independence. The second model we plan to discuss is a numerical method for quantizing gravity carried on the simplicial spacetime manifold. The theory treats dynamics non-perturbatively within the limited space allowed in a cluster of computers. It is significant that these two theories seem to be in the same universality class so far as the critical exponents are compared. It means that a conformal invariant spacetime can be mapped to a simplicial manifold with varying connectivity of simplices.

The asymptotic freedom also implies that there exists a dynamical scale where the effective coupling of traceless 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 field would be frozen to the classical background. The primordial power spectrum obtained from the two point correlation function of scalar curvature shows a significant character of the theory which is expected to be observed cosmologically. The sharp fall off of the angular power spectra at low multipole components of anisotropies in the cosmic microwave background is a topic we wish to discuss.

- BRST Conformal Symmetry and Quantum Gravity
- Gravitational Counterterms Near Four Dimensions and Conformal Anomalies
- Renormalizable 4D Quantum Gravity and its Cosmological Implications
- Analyzing WMAP Observation by Quantum Gravity

Quantum Gravity Cosmology

- From CFT Spectra to CMB Multipoles in Quantum Gravity Cosmology,

K. Hamada, S. Horata and T. Yukawa, Phys. Rev. D81 (2010) 083533, arXiv:0908.0192[astro-ph]. - Analyzing WMAP Observation by Quantum Gravity,

K. Hamada, S. Horata, N. Sugiyama and T. Yukawa, Prog. Theor. Phys. 119 (2008) 253, arXiv:0705.3490[astro-ph]. - Space-time Evolution and CMB Anisotropies from Quantum Gravity,

K. Hamada, S. Horata and T. Yukawa, Phys. Rev. D74 (2006) 123502, astro-ph/0607586. - CMB Anisotropies Reveal Quantized Gravity,

K. Hamada and T. Yukawa, Mod. Phys. Lett. A20 (2005) 509, astro-ph/0401070. - Baryogenesis by Quantum Gravity,

K. Hamada, A. Minamizaki and A. Sugamoto, Mod. Phys. Lett. A23 (2008) 237, arXiv:0708.2127 [astro-ph].

Quantum Diffeomorphism and Conformal Field Theory

- BRST Analysis of Physical Fields and States for 4D Quantum Gravity on R x S^3,

K. Hamada, Phys. Rev. D86 (2012) 124006, arXiv:1202.4538[hep-th]. - BRST Invariant Higher Derivative Operators in 4D Quantum Gravity based on CFT,

K. Hamada, Phys. Rev. D85 (2012) 124036, arXiv:1203.4894[hep-th]. - Background Free Quantum Gravity based on Conformal Gravity and Conformal Field Theory on M^4,

K. Hamada, Phys. Rev. D85 (2012) 024028, arXiv:1109.6109[hep-th]. - Vertex Operators in 4D Quantum Gravity Formulated as CFT,

K. Hamada, Found. Phys. 41 (2011) 863, arXiv:1005.2453[hep-th]. - Conformal Field Theory on R x S^3 from Quantized Gravity,

K. Hamada, Int. J. Mod. Phys. A24 (2009) 3073--3110, arXiv:0811.1647[hep-th]. - Building Blocks of Physical States in a Non-Critical 3-Brane on R x S^3,

K. Hamada, Int. J. Mod. Phys. A20 (2005) 5353--5398, hep-th/0402136. - Conformal Algebra and Physical States in a Non-critical 3-brane on R x S^3,

K. Hamada and S. Horata, Prog. Theor. Phys. 110 (2003) 1169, hep-th/0307008.

Renormalization

- Physical Cosmological Constant in Asymptotically Background Free Quantum Gravity,

K. Hamada and M. Matsuda, arXiv:1704.03962[hep-th]. - Two-Loop Quantum Gravity Corrections to Cosmological Constant in Landau Gauge,

K. Hamada and M. Matsuda, Phys. Rev. D93 (2016) 064051, arXiv:1511.09161[hep-th]. - RG Analysis for Quantum Gravity with A Single Dimensionless Coupling,

K. Hamada, Phys. Rev. D90 (2014) 084038, arXiv:1407.4532[hep-th]. - Determination of Gravitational Counterterms Near Four Dimensions from RG Equations,

K. Hamada, Phys. Rev. D89 (2014) 104063, arXiv:1403.4354[hep-th]. - Resummation and Higher Order Renormalization in 4D Quantum Gravity,

K. Hamada, Prog. Theor. Phys. 108 (2002) 399, hep-th/0203250. - Background-metric Independent Formulation of 4D Quantum Gravity,

K. Hamada and F. Sugino, Nucl. Phys. B553 (1999) 283, hep-th/9810095.

Two Dimensional Quantum Gravity

- Quantum Theory of Dilaton Gravity in 1 + 1 Dimensions,

K. Hamada, Phys. Lett. B300 (1993) 322, hep-th/9206071. - Physical States, Factorization and Non-linear Structures in Two Dimensional Quantum Gravity,

K. Hamada, Nucl. Phys. B413 (1994) 278, hep-th/9309094. - Ward Identities of W_infinity Symmetry in Liouville Theory coupled to c_M < 1 Matter,

K. Hamada, Phys. Lett. B324 (1994) 278, hep-th/9311047. - Non-linear Structures in Non-critical NSR String,

K. Hamada and H. Ishikawa, Commum. Math. Phys. 176 (1996) 401, hep-th/9405130.

Others

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