Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

No abstract

No abstract

No abstract

We reemphasize that the ratio R sµ ≡B(B s → µμ)/∆M s is a measure of the tension of the Standard Model (SM) with the latest measurements ofB(B s → µμ) that does not suffer from the persistent puzzle on the |V cb | determinations from inclusive versus exclusive b → c ν decays and which affects the value of the CKM element |V ts | that is crucial for the SM predictions of bothB(B s → µμ) and ∆M s , but cancels out in the ratio R sµ . In our analysis, we include higher-order electroweak and QED corrections and adapt the latest hadronic input to find a tension of about 2σ for R sµ measurements with the SM independently of |V ts |. We also discuss the ratio R dµ which could turn out, in particular in correlation with R sµ , to be useful for the search for new physics, when data on both ratios improve. R dµ is also independent of |V cb | or more precisely |V td |.