Observables and Dynamics in Quantum Gravity

Merging the concepts of quantum mechanics and those of general relativity presents us with one of the most interesting and fundamental contradictions in physics; how should we define time? In quantum mechanics time is a non-dynamical external parameter. In general relativity, time is one of the four dynamical variables of spacetime. How can we put these two different concepts together? What is the proper definition for a quantum relativistic time? This question has been occupying many theoretical physicist for the past eighty years, and despite many advances, and a number of possible candidates for a correct theory of quantum gravity, no unified answer has yet been reached. In this book, we explore this question in detail. We present two of the main candidates which attempt to address it, namely Loop Quantum Gravity and String Theory. We do this in the very simple yet effective context of a two dimensional spacetime or worldsheet. Through this work, the reader can become familiar with the elementary concepts of these theories of quantum gravity, and should see that despite their progress we are yet quite far from any satisfactory answer as to what time is.