The recovery of locality for causal sets and related topics
Date of Award
Doctor of Philosophy (PhD)
Elementary Particles and Fields and String Theory
The theory of causal sets is an attempt at a successful quantum theory of gravity. It is widely expected that any theory of quantum gravity should give rise to a discrete structure to spacetime geometry. This supposition rests upon three features of successfully quantized theories. The first is that quantization usually leads to some form of discrete object, or quantum, which is the source of the term "quantization". The second is the need for renormalization of field theories (which is not alleviated by including gravity, using the usual perturbative approach to quantization, as had once been hoped). An ultraviolet cutoff is introduced, which can be imagined to be either the mathematical manifestation of an underlying theory that contains new structures at some scale (for example the "GUT" fields, strings, etc.) or simply the result of the breakdown of the manifold picture at this same scale. Thirdly, quantized fields exhibit the phenomenon of virtual particles of unlimited energies at short length scales. The coupling of gravity to such quantized fields should induce infinite curvatures of the manifold at short scales, thereby dynamically prohibiting a manifold structure. The theory of causal sets begins with a discrete structure. The classical spacetime manifold is expected to emerge as an approximation (in a large grain limit) to the class of causal sets which dominate a sum over causal sets, each causal set being weighted by the exponentiation of an appropriately defined "action functional".
In order for this theory to be successful, several questions need to be addressed. Some of these are directed at the relationship between the causal set theory and the geometry of classical spacetime. Other questions are related to the quantization of the causal set theory and inclusion of quantum fields into the theory's framework. This work addresses the relation between causal sets and classical gravity (specifically the issues of embeddability of a certain class of causal sets known as the binomial partial orders into manifolds), and the recovery of locality for those causal sets which are embeddable into classical manifolds, a necessity for the successful behavior of the theory in the classical regime as well as the inclusion of quantum fields.
This work is divided into five main parts. The first gives an overview of the theory of causal sets. The second deals with the issue of the embeddability of the binomial partial order into manifolds. The third part discusses the locality issue. The fourth treats the still unsolved problem of enumerating causal sets consisting of a finite number of elements. The fifth contains a discussion of some open questions and speculations on the further progress of the theory. The work finishes with two appendices listing the computer codes used.
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Daughton, Alan Ronald, "The recovery of locality for causal sets and related topics" (1993). Physics - Dissertations. 97.