Quantum Gravity : discrete vs. continuum

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Over a long period of time, there exists the notorius or deserved-to-attack problem in quantizing the (3+1)-dimensional gravity, uncontrollable ultraviolet divergences. Against this open problem, a brand-new breakthrough has been proposed by P. Horava, which we call the Lifshitz-type gravity because it is closely related to the so-called Lifshitz point in the condensed matter physics. The most amazing thing is its power-counting renormalizability in 3+1 dimensions. What is paid in return is the requirement of the strong anisotropy between space and time in high energies. However this theory is expected to flow naturally to the relativistic general relativity in low energies, which is not confirmed (up to January 2010). In this thesis, we review the Lifshitz-type gravity, and try to uncover some aspects embedded in this profound theory. Further we would like to discuss the possibility that this Lifshitz-type gravity in 3+1 dimensions may be the continuum limit of the (3+1)-dimensional causal dynamical triangulation of the space-time-Lorentzian manifold, which has been pointed out by P. Horava. To carry it out, we model on the discrete-vs.-continuum structure in the 2-dimensional quantum gravity. In this line of thought, we review the 2-dimensional dynamical triangulation and its matrix model dual, and we would like to see how its continuum limit becomes the bosonic Liouville field theory with c=0. Then we discuss the (1+1)-dimensional causal dynamical triangulation which poses the causality on the dynamical triangulation, and discuss its continuum. Taking advantage of the intriguing structure of the 2-dimensional quantum gravity, we investigate the relation between the Lifshitz-type gravity in 3+1 dimensions and the causal dynamical triangulation in 3+1 dimensions via the spectral dimension.
2010-08-20

Quantum Gravity : discrete vs. continuum

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