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• W1587633043 abstract "AbstractAlgebraic systems called the local geodesic loops and their tangent Akivis algebras areconsidered. Their possible role in theory of gravity is considered. Quantum conditions forthe infinitesimal quantum events are proposed. 1 Introduction Spacetime and event are two fundamental concepts of relativistic physics. According to theoryof general relativity, totality of all events or the classical (pre-quantum) spacetime can be repre-sented by a (1+3)-dimensional Lorentzian manifold. Evolution of the Lorentzian metric tensoris prescribed by the Einstein equations, constraining the Einstein and energy-momentum tensorsof spacetime to be proportional. On the classical level, gravitation reveals itself through curva-ture of spacetime. In this sense the general relativity may be proclaimed to be a geometricaltheory of the gravitational fieldThere have been several attempts to quantize gravity. In fact, quantization of the graviata-tional field is one of the most intriguing problems in theoretical physics. In a sense, quantizationis first of all an algebraic method for one looks for the quantum observables which must imitatethe algebraic properties of the classical ones. So every algebraic aspect of the classical spacetimemust be thoroughly taken into account as well.In this paper we outline an idea of the geodesic quantization, based on construction of thelocal geodesic multiplication (GM) of spacetime points (Sec. 2). For the Minkowski spacetimeof special relativity, the GM in fact coincides with the common vector addition rule. Hence, inthis (Minkowskian) case the GM turns out to be a globally defined commutative and associativeoperation. The Abelian property of the GM is in fact due to the globally vanishing torsion andcurvature of the Minkowski space.Geodesic multiplication [1, 2, 3] on manifolds with an affine connection is mathematicallyrather elaborated but is still not widely known for physicists. Nevertheless, it deserves attentionas well.Via the GM on can treat spacetime algebraically, as a collection of the algebraic systemscalled the local geodesic loops. In general relativity, the GM acquires dynamical meaning.Nonassociativity of GM is an algebraic manifestation of the curvature of spacetime (Sec. 3).Tangent spaces at spacetime points turn out to be binary-ternary algebras called the Akivisalgebras. Based on this fact, the geodesic quantum conditions are proposed for the infinitesimalquantum events (Sec. 4)." @default.
• W1587633043 created "2016-06-24" @default.
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• W1587633043 date "2008-03-08" @default.
• W1587633043 modified "2023-09-27" @default.
• W1587633043 title "Geodesic multiplication as a tool for classical and quantum gravity" @default.
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