On Quantisations, Quasi-probabilities and the Weak Value

Abstract: We propose a general framework of the quantum/quasi-classical transformations by introducing the concept of quasi-joint-spectral distribution (QJSD). Specifically, we show that the QJSDs uniquely yield various pairs of quantum/quasi-classical transformations, including the Wigner-Weyl transform. We also discuss the statistical behaviour of combinations of generally non-commutin quantum observables by introducing the concept of quantum correlations and conditional expectations defined analogously to the classical counterpart. Based on these, Aharonov's weak value is given a statistical interpretation as one realisation of the quantum conditional expectations furnished in our formalism.