2026/03/30 by Philippe Manjakasoa Randriantsoa, Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona +3
#quant-ph
The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature (1,4), we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as Γ= L2/ℓ2 for states that saturate the uncertainty relations, where L and ℓ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit ℓ → 0, the equation reduces to the de Sitter spacetime equation; in the limit L → ∞, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both ℓ → 0 and L → ∞), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.