Mean Field Approximation of IIB Matrix Model and Emergence of Four Dimensional Space-Time

For the purpose of analyzing non-perturbative dynamics of string theory, Nishimura and Sugino have applied an improved mean field approximation (IMFA) to IIB matrix model. We have extracted the essence of the IMFA and obtained a general scheme, an improved Taylor expansion, that can be applied to a wide class of series which is not necessarily convergent. This approximation scheme with the help of the 2PI free energy enables us to perform higher order calculations. We have shown that the value of the free energy is stable at higher orders, which supports the validity of the approximation. Moreover, the ratio between the extent of ``our space-time and that of the internal space is found to increase rapidly as we take the higher orders into account. Our results suggest that the four dimensional space-time emerges spontaneously in IIB matrix model.