The classical-quantum duality of nature including gravity

Abstract

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: [Formula: see text], [Formula: see text] being the Planck scale. This duality is universal. From the gravity (G) and quantum (Q) variables [Formula: see text], we define new quantum gravity (QG) variables [Formula: see text] which include all (classical, semiclassical and QG) domains passing through the Planck scale and the elementary particle domain. The QG variables are more complete than the usual ([Formula: see text], [Formula: see text]) ones which cover only one domain (Q or G). Two [Formula: see text] or [Formula: see text] values [Formula: see text] are needed for each value of [Formula: see text] (reflecting the two dual ways of reaching the Planck scale). We perform the complete analytic extension of the QG variables through analytic (holomorphic) mappings which preserve the light-cone structure. This allows us to reveal the classical-quantum duality of the Schwarzschild–Kruskal spacetime: 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 [Formula: see text]. Spacetime reflections, antipodal symmetry and PT or CPT symmetry are contained in the QG symmetry, which also shed insight on the global properties of the Kruskal manifold and its present renewed interest.