Approaches to Quantum Gravity

394 J. Hensonall the other subjects covered, and new results, which solve this problem in somesituations for the first time, are mentioned.Besides the present work, there are many other reviews available. One of themost recent is [5], while motivation and earlier work is covered in [2; 3; 4]. Aphilosophically oriented account of the conception of the causal set idea is givenin [6], and there is a recent review which introduces some of the core concepts ofcausal set kinematics and dynamics [7]. Many of these articles, and other causalset resources, are most easily found at Rafael Sorkin’s web site [8].21.1 The causal set approachThis program is a development of “path-integral” or sum-over-histories (SOH) typeapproaches (for reasons to adopt this framework in Quantum Gravity, see [9; 10]).In such approaches, a space of histories is given, and an amplitude (or more generallya quantum measure), is assigned to sets of these histories, defining a quantumtheory in analogy with Feynman’s path integrals. A basic question, then, is whatthe space of histories should be for Quantum Gravity. Should they be the continuousLorentzian manifolds of general relativity – or some discrete structure to whichthe manifold is only an approximation?21.1.1 Arguments for spacetime discretenessA number of clues from our present theories of physics point towards discreteness.The problematic infinities of general relativity and quantum field theory arecaused by the lack of a short distance cut-off in degrees of freedom; although therenormalisation procedure ameliorates the problems in QFT, they return in naiveattempts to quantise gravity (see [11] and references therein). Secondly, technicalproblems arise in the definition of a path-integral on a continuous historyspace that have never been fully resolved. On top of this, the history space ofLorentzian manifolds presents special problems of its own [12]. A discrete historyspace provides a well defined path integral, or rather a sum, that avoids theseproblems.Perhaps the most persuasive argument comes from the finiteness of black holeentropy. With no short-distance cut-off, the so called “entanglement entropy” ofquantum fields (the entropy obtained when field values inside a horizon are tracedout) seems to be infinite (see [13; 14], and [15; 16] for some debate). If this entropyis indeed included in the black hole entropy, as many expect, a short distance cut-offof order the Planck scale must be introduced to allow agreement with the wellknownsemiclassical results. This, and similar analysis of the shape degrees offreedom of the black hole horizon [17; 18] lead to the conclusion that Planck scale