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The quantum weak value draws many attentions recently from theoreticalcuriosity to experimental applications. Now we design an unusual weak measuringprocedure as the pre-selection, mid-selection and post-selection to study thecorrelation function of two weak values, which we called the weak correlationfunction. In this paper, we proposed an weak measurement experiment to measurethe canonical commutator $[\hat{x},\hat{p}]=i\hbar$ in quantum mechanics.Furthurmore, we found the intriguing equivalence between the canonicalcommutation relation and Riemann hypothesis, and then obtained the weak valueof nontrivial Riemann zeros. Finally, as an nontrivial example of weakcorrelations, we also passed successfully a testing on the (anti-)commutatorsof Pauli operators, which followed the experimental setup of the landmark paperof Aharonov, et al. in 1988. Our proposed experiments could hopefully test thefundamental canonical relationship in quantum worlds and trigger more testingexperiments on weak correlations.

How to measure the canonical commutation relation $\mathbf{[\hat{x},\hat{p}]=i\hbar}$ in quantum mechanics with weak measurement?