Conceptual and technical challenges of quantum gravity

The appearance of infinity together with collapsing quantum state due to the observation or interaction, which are two challenging features of quantum field theory, become very serious problems in quantum gravity as well as in quantum geometry of space-time. These problems can be divided into two categories: technical and conceptual parts. In the technical aspect, the biggest problem comes from the definition of the point, which constructs classical geometry. By changing the point definition, the technical problem may be approximately bypassed, and a mathematical formulation of quantum geometry may be found. On the other hand, the conceptual problem comes from the quantum state collapse due to the observation since the entanglement between the observer and gravity cannot be eliminated in the presence of gravity. The conceptual part does not depend on the techniques which are employed. As much as possible, we try to discuss these problems by rediscovering the perception and understanding of a phenomenon for an observer in quantum theory.