The Classical-Quantum Duality of Nature. New Variables for Quantum Gravity
La Dualité Classique-Quantique de la Nature. Nouvelles Variables por la Gravitation Quantique
Résumé
The classical-quantum duality at the basis of quantum theory is here extended to the Planck scale domain. The classical/semiclassical gravity (G) domain is dual (in the precise sense of the classical-quantum duality) to the quantum (Q) elementary particle domain: O Q = o 2 P O −1 G , o P being the Planck scale. This duality is universal. From the gravity (G) and quantum (Q) variables (O G , O Q), we define new (QG) quantum gravity variables O QG = (1/2)(O G + O Q) which include all (classical, semiclassical and quantum gravity) domains passing by the Planck scale and the elementary particle domain. The QG variables are more complete than the usual (O Q , O G) ones which cover only one domain (Q or G).Two O G or O Q values (±) are needed for each value of O QG (reflecting the two different and dual ways of reaching the Planck scale). We perform the complete analytic extension of the QG variables through analytic (holomorphic) mappings which preserve the ligth-cone structure. This allows us to reveal the classical-quantum duality of the Schwarzschild-Kruskal space-time: exterior regions are classical or semiclassical while the interior is totally quantum: its boundaries being the Planck scale. Exterior and interior lose their difference near the horizon: four Planck scale hyperbolae border the horizons as a quantum dressing or width:l'horizon habillé. QG variables are naturally invariant under O G → O Q. Space-time reflections, antipodal symmetry and PT or CPT symmetry are contained in the QG symmetry, which also shed insight into the global properties of the Kruskal manifold and its present renewed interest. Further results are presented in another paper. Norma.Sanchez@obspm.fr https://chalonge-devega.fr/sanchez
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