Quantum Mechanics, Gravity and Modified Quantization Relations

In this paper we investigate a possible energy scale dependence of the quantization rules and in particular, from a phenomenological point of view, an energy scale dependence of an effective $\hbar$. We set a bound on the deviation from the value of $\hbar$ at the muon scale from its usual value using measurements of the anomalous magnetic moment of the muon. Assuming that inflation took place, we can conclude that nature is described by a quantum theory at least up to an energy scale of about $10^{16}$ GeV.

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