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arXiv:0707.0635 · Physica A Statistical Mechanics and its Applications

Possible connection between probability, spacetime geometry and quantum mechanics

Enrique Canessa

Published Jun 14, 2007• 2 citations
arXivHTMLPDF
DOI
10.1016/j.physa.2007.06.006

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Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a nonlinearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.
Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a non-linearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.
What i Really RepresentsFrom Spacetime Action to Quantum Phase: The Imaginary Unit as aMarker of Action-Phase GeometryPeter Davies, v1.0230th April, 2026AbstractQuantum mechanics differs from classical probability theory by assigning complex amplitudes rather than real probabilities to physical alternatives. The appearance of the imaginary unit i in the Schr¨odinger equation, unitary time evolution, path integrals, gauge transformations, and geometric formulations of quantum mechanics suggests that i is not merely a calculational artifact.This paper argues that i is best understood as the algebraic marker of action-phase geometry: the phase-rotation structure through which action, measured in units of ℏ, becomes a probability-preserving quantum amplitude. In this interpretation, spacetime geometry determines the action accumulated along possible histories, while quantum mechanics maps this action into phase through eiS/ℏ. The constants ℏ and h acquire complementary interpretations: ℏ is action per radian of phase, while h = 2πℏ is action per full phase cycle. Observable probabilities arise only after interference among phase-weighted histories, with the final quadratic product converting the surviving amplitude structure into measurable probability. The paper develops this interpretation through phase gradients, action periodicity, proper-time phase, gauge connections, gravitational phase modulation, light–matter phase matching, and multi-photon interference. The proposed contribution is not a replacement of quantum mechanics, quantum field theory, or general relativity, but a conceptual synthesis: the repeated appearance of i, ℏ, h, action, proper time, and interference may be understood as different aspects of a single action-phase structure linking spacetime and quantum mechanics. The proposal is interpretive rather than a Bell-local hidden-variable completion of quantum mechanics. It does not assign pre-existing local outcome values to measurement events, and it is intended to remaincompatible with the nonclassical correlation structure required by Bell-type results. - Preprint version v1.02. This manuscript presents a conceptual framework and has not yet undergone peer review. Later versions may incorporate corrections, expanded references, and feedback on framing.