Abstract
We consider a quadratic Lagrangian, in both curvature and torsion, with the aim of exploring the possibility that torsion and curvature behave as conjugate variables satisfying the commutation relations. For that proposal we first show that torsion represents a quantum correction to the classical equations of motion. We then observe that we have to introduce the spin in the Einstein theory with two spaces: a real space-time where we describe the curvature with tensors; and a complex space-time, where we describe torsion with spinors. This is not satisfactory, and to overcome this contradiction we describe both mass and spin in a unique manifold, the real space-time, with the use of geometric algebra and geometric product. In that manner we are able to introduce a torsion trivector and curvature trivector that satisfy the commutation relations, opening the road for a quantum description of gravity theory.
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de Sabbata, V., Ronchetti, L. A Hamiltonian Formulation of Gravitational Theory that Allows One to Consider Curvature and Torsion as Conjugate Variables. Foundations of Physics 29, 1099–1117 (1999). https://doi.org/10.1023/A:1018889716186
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DOI: https://doi.org/10.1023/A:1018889716186