Weak Values and Modular Variables From a Quantum Phase Space Perspective
  We address two major conceptual developments introduced by Aharonov and collaborators through a \textit{quantum phase space} approach: the concept of \textit{modular variables} devised to explain the phenomena of quantum dynamical non-locality and the \textit{two-state formalism} for Quantum Mechanics which is a retrocausal time-symmetric interpretation of quantum physics which led to the discovery of \textit{weak values.} We propose that a quantum phase space structure underlies these profound physical insights in a unifying manner. For this, we briefly review the Weyl-Wigner and the coherent state formalisms as well as the inherent symplectic structures of quantum projective spaces in order to gain a deeper understanding of the weak value concept.   We also review Schwinger's finite quantum kinematics so that we may apply this discrete formalism to understand Aharonov's modular variable concept in a different manner that has been proposed before in the literature. We discuss why we believe that this\ is indeed the correct kinematic framework for the modular variable concept and how this may shine some light on the physical distinction between quantum dynamical non-locality and the kinematic non-locality, generally associated with entangled quantum systems. 
Augusto C. Lobo, Yakir Aharonov, Jeff Tollaksen, Elizabeth M. Berrigan   and Clyffe A. Ribeiro
quant-ph