Emergent Einstein Universe under Deconstruction: Self-Consistent Geometry Induced in Theory Space

Abstract

We do not know yet any consistent quantum theory of gravity. While the success of the combined model of three fundamental interactions described using gauge theories was achieved many years ago, only gravitational interaction is left unbound today. An eccentric idea that the gravitational interaction can be induced by quantum fluctuations of matter fields was introduced almost forty years ago. There are many versions of such ‘induced gravity’ (or ‘emergent gravity’).∗∗∗) In most cases, explicit values for fundamental constants cannot be calculated because of the cutoff dependence, even if higher-loop contributions are neglected. In our previous paper,2) it is shown that calculable models of induced gravity can be constructed on the basis of knowledge of spectral graph theory.3) The Newton constant and the cosmological constant can be calculated at one-loop level in flatspace limit. On the basis of this work, we consider self-consistent equations for a ‘graph theory space’ in the present paper. We demonstrate specific results for models based on cycle graphs, which represent the original moose diagram in dimensional deconstruction.4) The present paper is organized as follows. In §2, we will begin by reviewing background matters, i.e., the basic ideas behind the present work and techniques used in the following sections. Our previous work is briefly summarized in §3, for convenience. In §4, it is shown that the heat kernel method is used to evaluate the self-consistent solutions in specific models. The results for the models are shown in §5. We provide a summary with some future perspectives in the last section. We use the metric signature (−+++) throughout this paper.