Next, one can introduce the left-invariant metric, g b ( L b a x, L b a y ) = g a ( x, y ), enriching the manifold 𝔐 with a metric structure. The family of local loops constructed in this way satisfies some natural algebraic identities and uniquely defines the space with affine connection. The operation L x a y is uniquely defined through the parallel transport of the geodesic 〈 ay〉, connecting the point a with the point y, along the geodesic 〈 ax〉. Then, in a neighborhood of an arbitrary point a ∈ 𝔐 on one can introduce the geodesic local loop Q a with a smooth partial ternary operation, L x a y, a being the neutral element of Q a, and a, x, y ∈ 𝔐. Let 𝔐 be a manifold with an affine connection. The main algebraic structures emerging in nonassociative geometry may be described as follows. In the case of smooth manifolds it is equivalent to conventional differential geometry. Nonassociative geometry, being based on the theory of quasigroups and loops, provides a unified algebraic description of manifolds. These approaches are based on the hypothesis that spacetime is discrete and causality is a fundamental principle. Among advanced models that propose discreteness, two that will be relevant to our work are Causal Sets (CS) and Causal Dynamical Triangulations (CDT). In Nesterov and Sabinin, we have proposed a new unified approach, based on nonassociative geometry, for describing both continuum and discrete spacetime. This discreteness is motivated by several heuristic arguments, but so far has not been deduced rigorously. The proposal that spacetime may be fundamentally discrete has been adopted by numerous strategies. This means that at the Planck scale the standard concept of spacetime must be replaced by some discrete structure. In the second approach, a formulation of the quantum theory may require omitting use of continuum concepts a priori. The second strategy assumes that the classical structures are emergent from the other theory, which is more fundamental theory from the very beginning. The first strategy consists in quantizing a classical structure, that later is recovered as a limit of the quantum theory. One can distinguish two general strategies to achieve this goals. As part of the general problem of time, one of the most challenging and open problems is the origin of irreversibility in our universe, also known as the problem of the arrow of time.
Any attempt to create a quantum gravitation theory faces the challenge of comprehension the concept of quantizing of spacetime and of describing the quantum nature of space and time.
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