Does the Sum Rule Hold at the Big Bang?

Abstract

In Dice 2010 Sumati Surya brought up a weaker Quantum sum rule as a biproduct of a quantum invariant measure space. Our question is stated as follows:  Does it make sense to have disjointed sets to give us quantum conditions for a measure at the origin of the big bang? We argue that the answer is no, which has implications as to quantum measures and causal set structure. What is called equation (1) in the text requires a length, and interval, none of which holds at a point in space-time.singularity. What are the reasons? First, measurable spaces allow disjoint sets. Also, that smooth relations alone do not define separability or admit sets Planck’s length, if it exists, is a natural way to get about the ‘bad effects’ of a cosmic singularity at the beginning of space-time evolution, but if a new development is to believed, namely by Stoica in the article, about removing the cosmic singularity as a break down point in relativity, there is nothing which forbids space-time from collapsing to a point. If that happens, the cautions as to no disjoint intervals at a point raise the questions as to the appropriateness of Surya’s quantum measure. Since Stoica’s re scaling of pressure and density involve the cube of the scale factor, a, the differentiability and smoothness issues of the Friedman and acceleration equation vanish, leading to problems with the Hausdorf limiting cases for disjoint open sets, which makes quantum vector meaures not feasibile, due to vanishing of disjoint sets, as we approach a point in space-time. The final conclusion is that the intial singularity has to be embedded into higher dimensions, as in String theory due to 4 dimensional problems with quatum measures which in themselves in four dimensions break down.