A world-volume model of a non-critical 3-brane is quantized in a strong coupling phase in which fluctuations of the conformal mode become dominant. This phase, called the conformal-mode dominant phase, is realized at very high energies far beyond the Planck mass scale. We separately treat the conformal mode and the traceless mode and quantize the conformal mode non-perturbatively, while the traceless mode is treated in a perturbative method that is renormalizable and asymptotically free. In the conformal-mode dominant phase, the coupling of the traceless mode vanishes, and the world-volume dynamics are described by a four-dimensional conformal field theory (CFT 4 ). We canonically quantize this model on R × S 3 , where the dynamical fields are expanded in spherical tensor harmonics on S 3 , which include both positive-metric and negative-metric modes. Conformal charges and a conformal algebra are constructed. They yield strong constraints on physical states. We find that all negative-metric modes are related to positive-metric modes through the charges, and thus negative-metric modes are themselves not independent physical modes. Physical states satisfying the conformal invariance conditions are given by particular combinations of positive-metric and negativemetric modes. An infinite number of such physical states are constructed. In the appendices, we construct spherical vector and tensor harmonics on S 3 in practical forms using the Wigner D functions and the Clebsch-Gordan coefficients and calculate the integrals of three and four products of these harmonics over S 3 . † J− 1 2 M 2 .(5.3)Thus, only ϕ † 00 commutes with Q M . Consider the operator with the level H = 2J + 2,The triangular condition for the standard Clebsch-Gordan coefficients implies that this coefficient vanishes unless the triangular conditions |J 1 − with integer J + J 1 + J 2 , and the requirement M = M 1 + M 2 are satisfied. Also, the D coefficients satisfy the relations

We propose 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 spacetime. The space-time dynamics is derived from the renormalizable higherderivative quantum gravity on the basis of a conformal gravity in four dimensions. Based on the linear perturbation theory in the inflationary background, we simulate evolutions of gravitational scalar, vector and tensor modes, and evaluate the spectra at the transition point located at the beginning of the big bang. The obtained spectra cover the range of the primordial spectra for explaining the anisotropies in the homogeneous CMB. PACS: 98.80.Cq, 98.80.Qc, 98.70.Vc

The phase structure of four-dimensional simplicial quantum gravity coupled to a U (1) gauge field is investigated numerically with the dynamical triangulation method. A smooth phase is found in the region between the crumpled phase and the branched polymer phase. This new phase has a negative string susceptibility exponent. The phase transition between the smooth phase and the crumpled phase is studied using the finite size scaling method. From the numerical results we conclude that this model has a phase transition that is higher than first-order. Numerical results suggest a possibility of the existence of the continuum limit at the critical point. at University of Manchester on March 23, 2015 http://ptp.oxfordjournals.org/ Downloaded from where l ij denotes the link between vertices i and j, and t ijk denotes the triangle with vertices i, j and k. Here, o(t ijk ) is the number of 4-simplices sharing the triangle

We study the inflation process of universe based on the renormalizable quantum gravity formulated as a conformal field theory (CFT). We show that the power-law CFT spectrum approaches to that of the Harrison-Zel'dovich-Peebles type as the amplitude of gravitational potential gradually reduces during the inflation. The non-Gaussanity parameter is preserved within order of unity due to the diffeomorphism invariance. Sharp fall-off of the angular power spectrum of cosmic microwave background (CMB) at large scale is understood as a consequence of the existence of dynamical scale of the quantum gravity Λ QG (≃ 10 17 GeV). The angular power spectra are computed and compared with the WMAP5 and ACBAR data with a quality ofRecent observations of anisotropies in the cosmic microwave background (CMB) by various groups such as the cosmic background explore (COBE) [1], the Wilkinson microwave anisotropy probe (WMAP) [2,3], and the arcminute cosmology bolometer array receiver (ACBAR) [4] have provided a refined picture of the history of universe after the big bang. Cosmological parameters are determined with high accuracy based on the cosmological perturbation theory [5,6,7,8], assuming only the primordial spectrum close to that of the Harrison-Zel'dovich-Peebles [9,10,11]. We believe that one of the important problems remained in the study of inflation [12,13,14,15,16,17,18,19,20] is to clarify dynamics producing such a scaleinvariant spectrum from the fundamental theory rather than introducing an artificial field by hands just for the phenomenological purpose.As the fundamental theory, we will employ the renormalizable quantum gravity formulated based on the conformal field theory (CFT) in four dimensions [21,22,23,24]. It predicts that quantum fluctuations of the conformal mode in gravitational fields become so large at very high energies beyond the Planck scale, and a conformally invariant space-time is realized as a consequence of background metric independence. It then produces a powerlow spectrum and a non-Gaussian fluctuation distribution for the theoretical generation of CMB spectrum. Evolution of the early universe can be regarded as a violating process of conformal invariance [25,23,26,27]. The conformal symmetry starts to be broken at the Planck scale, and the space-time dynamics shifts to the inflationary epoch with the expansion time constant about the Planck mass m pl (= 1/ √ G). The conformal invariance is completely broken at the dynamical scale of quantum gravity Λ QG , which is expected to be 10 17 GeV. At this energy scale, the inflation terminates, and the universe turns to the classical Friedmann universe.The purpose of this paper is to clarify how the Harrison-Zel'dovich-Peebles spectrum is prepared for the initial condition of the cosmological