Approaches to Quantum Gravity

396 J. HensonzyxFig. 21.1. A causal set. The figure shows an example of a Hasse diagram. In sucha diagram, the elements of a causal set are represented by dots, and the relationsnot implied by transitivity are drawn in as lines (for instance, because x ≺ y andy ≺ z, there is no need to draw a line from x to z, since that relation is impliedby the other two). The element at the bottom of the line is to the past of the one atthe top of the line.If x ≺ y then we say “x is to the past of y”, and if two points of the set Care unrelated by ≺ we say they are spacelike (in short, all the normal “causal”nomenclature is used for the partial order).It is this partial order that we choose as fundamental. To achieve discreteness,the following axiom is introduced.(iii) Local finiteness: (∀x, z ∈ C)(card {y ∈ C | x ≺ y ≺ z} < ∞).Here, card X is the cardinality of the set X. In other words, we have required thatthere only be a finite number of elements causally between any two elements in thestructure (the term “element” replaces “point” in the discrete case). A locally finitepartial order is called a causal set or causet, an example of which is illustrated infigure 21.1. Many researchers have independently been led to the same hypothesis[24; 25; 1]: that the causal set should be the structure that replaces the continuummanifold.21.1.3 The continuum approximationIn all standard quantum theories, be they direct quantisations of a classical theoryor discrete approximations, there is an approximate correspondence between atleast some of the underlying histories and those of the limiting classical theory,needed in order to relate the quantum theory to known physics. 2 (Similarly, in thestate vector formulation, there must be a correspondence between configurations ata particular time, each assigned a basis vector in the Hilbert space of the quantumtheory, and the allowed configurations in the classical theory – something thoughtto be true even in loop Quantum Gravity [26]). The view taken in causal set theory2 In standard theories, this statement is dependent upon the choice of a Hilbert space basis, e.g. the positionbasis for a Shrödinger particle. The path integral can be expressed using other Hilbert space bases, but it isnevertheless always true that there exists some basis (usually a class of them) in which the above statement istrue.