Space, Matter and Interactions in a Quantum Early Universe Part I: Kac–Moody and Borcherds Algebras

Abstract

We introduce a quantum model for the universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac–Moody Lie algebra e9. We investigate Kac–Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.

Space, Matter and Interactions in a Quantum Early Universe. Part II : Superalgebras and Vertex Algebras

“…This is the second of two papers -see also [1] -describing an algebraic model of quantum gravity. In the first paper we have described the basic principles of our model and we have investigated the mathematical structures that may suit our purpose.…”

“…Let X, Y, Z be generators of g u of degree i, j, k ∈ {0, 1} respectively. We remind, from [1], that the generators x α p have degree 0, whereas the generators x α+p have degree α = 0 if α is bosonic and degree α = 1 if α is fermionic. Let [X, Y ] still denote the product of X, Y in g u and X ⊗ e x •Y ⊗ e y the corresponding product in sg u .…”

“…As mentioned in the Introduction of [1], the interactions have a tree structure whose building blocks involve only three particles, and they are expressed by the product in the underlying algebra. The scattering amplitudes are proportional, up to normalization, to the structure constants of the related products.…”

“…However, for the sake of simplicity, in [1] we have chosen to deal with a simpler Lie algebra, g u , that extends e 8 and e 9 . In the present paper, we will start invertigating a physical model for quantum gravity based on this particular rank-12 algebra g u .…”

“…6 and 7 we will define the quantum states, and then we will merge the algebra sg u into a vertex-type algebra, representing the quantum early Universe with its expanding spacetime. 2 The Lie algebra g u We start and consider B + , the Lie subalgebra of the rank-12 Borcherds algebra B 12 introduced in [1] and generated by the Chevalley generators corresponding to positive roots. A further simplification will then give rise to g u , the Lie algebra that acts locally on the quantum state of the Universe [1].…”