アブストラクト：

Over a long period of time, there exists the notorius or deserved-toattack

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 Lifshitztype

gravity because it is closely related to the so-called Lifshitz point

in the condensed matter physics. The most amazing thing is its powercounting

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

キーワード：Lifshitz Gravity, Causal Dynamical Triangulation, Spectral Dimension