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.
Highlights
This is the first of two papers—see [1]—describing an algebraic model of quantum gravity.The intrinsic difficulty of quantizing gravity, encountered in the most acclaimed approaches of string theory and loop quantum gravity, has led us to this attempt of thinking outside the box and exploiting only the most fundamental principles of quantum mechanics and general relativity, as we believe that they should apply in the extreme conditions of a hot dense universe in its early stages
Borcherds algebras are a generalization of Kac–Moody algebras obtained by releasing the condition on the diagonal elements of the Cartan matrix, which are allowed to be non-positive, as well as by restricting the Serre relations to the generators associated to positive norm simple roots [34,35]
By applying to r0 Weyl reflections, we stay within e9, since every Kac–Moody algebra is invariant under the Weyl group; r is a real root of e9, namely r = α + mδ, α ∈ e8 and mδ is light-like; 4
Summary
This is the first of two papers—see [1]—describing an algebraic model of quantum gravity. As in the theory of fields, the interactions may only occur locally, point-by-point in the expanding spacetime, which can be viewed as a parameter on which the algebra product depends; In agreement with the theory of a big bang, strongly supported by the current observations, we assume the existence of an initial quantum state, mathematically represented by an element of the universal enveloping algebra of gu. Such an element is made of generators that can all interact among themselves, yielding the first geometrical interpretation: that of a point where particles may interact; A particle has a certain probability amplitude to interact and not to interact, in which case, it expands, as described in Section 1.2; Particles are quantum objects, their existence through interactions occurs with certain amplitudes. The steady state of such a particle is a superposition of states with opposite 3-momenta, representing an object that moves simultaneously in opposite directions, where, by 3-momentum, we denote the spatial component of 4-momentum
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