Euclidean and Lorentzian quantum gravity -- lessons from two dimensions
In this article some recent progress in understanding of two-dimensional quantum gravity is reviewed. In particular, the relation between the Euclidean and Lorentzian sectors of the quantum theory are discussed. It is shown that it is indeed possible to formulate a background-independent theory of quantum gravity and still maintain some analogues of concepts from classical gravity. This holds true at least in the case of one-plus-one dimensional gravity. The authors show that the generic two-dimensional geometry is fractal with intrinsic Hausdorff dimension four. They conclude that this situation is reminiscent of the random walk representation for a free particle. The authors also discuss the relation between Euclidean quantum gravity, defined via a non-pertubative path integral, and a similar construction in the Lorentzian case. In the latter case they obtain a continuum limit different from that of Euclidean quantum gravity. The authors conclude that this difference can be traced to the presence or absence of topology changes of space. In fact, it is shown that an agreement with the theory of Euclidean quantum gravity can only be obtained when spatial topology changes are allowed.
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