Abstract
With the simplest proof ever, we justify the significance of quantum-gravity in non-relativistic quantum mechanics together with the related theories and experiments. Since the de Broglie wave length is inverse proportional to the mass, it would descend towards and below the Planck scale 10−33 cm for large masses even at slow non-relativistic motion. The tricky relationship between gravity and quantum mechanics —well-known in the relativistic case—shows up in non-relativistic motion of massive objects. Hence the gravity-related modification of their Schrüdinger equation is mandatory. We also recall the option of an autonomous Newtonian quantum-gravity, a theory parametrized by ℏ and G. On cancellation of c from the Newtonian limit of Planck scale metric fluctuations is given a new hint.
Highlights
Much larger masses will push the structure of Ψ(x, t) towards the Planck scales, such role of the mass is what our work aims to emphasize
We argued how quantum-gravity, relativistic by its definition, should influence nonrelativistic quantum mechanics
Model [13, 14] of gravity-related decoherence and wave function collapse in massive degrees of freedom, parametrized again by G and only. It is based on a conjectured ultimate uncertainty of space-time but neither the relationship to the Planck length lPl nor the cancellation of c are explained in concrete forms
Summary
Quantum-gravitational limitations of our very notion of space-time were conjectured by Bronstein in 1936 [1], the same issue was famously characterized by Wheeler’s foamy structure of space-time [2] In the back-ground of quantized matter, the classical notion of space-time continuum can not be maintained at short distances. Direct observation of distances like 10−33 cm would require incredible high precision — beyond imaginations. Their indirect test would be possible around the extreme high Planck energy EPl = 1.2 × 1019 GeV per particle, which existed right after the Big Bang only. Planck scale effects are too small to be testable. [5] arguing for low-energy gravity-related decoherence theories and experiments. There are many discussions of quantum-gravity related effects that might occur at non-relativistically low energies, cf., e.g. What is the reason that the extreme-relativistic Planck scale effects, suppressed by a factor 10−28 at low energies (a number used in [6]), may amplify (or accumulate) to the level of laboratory testability? What is the reason that the extreme-relativistic Planck scale effects, suppressed by a factor 10−28 at low energies (a number used in [6]), may amplify (or accumulate) to the level of laboratory testability? Here we are going to show the trivial answer lies in non-relativistic quantized motion of massive objects, and the argument is no doubt simpler than any earlier ones
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