We address the issue of how to properly treat, and in a more general setting, the concept of a weak value of a weak measurement in quantum mechanics. We show that for this purpose, one must take in account the effects of the measuring process on the entire phase space of the measuring system. By using coherent states, we go a step further than Jozsa in a recent paper, and we present an example where the result of the measurement is symmetrical in the position and momentum observables and seems to be much better suited for quantum optical implementation.Comment: 07 pages, accepted for PR

Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.

Some algebraic properties of Schwinger's quantum kinematical phase space theory are presented. These properties lead to a definition of the maximum number of degrees of freedom of an arbitrary finite dimensional quantum system which is different from the one originally proposed by Schwinger. 0378-4371/95/$09.50 ~;~ 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371(94)00262-2

We present the Weyl-Wigner (WW) transform theory in a much more compact way than usual, by introducing the basis in an intrinsic form. This permits the derivation of new identities and also leads to generalizations, like the inclusion of ÿnite-dimensional systems in the WW theory, which is also discussed. We show, in this case, some striking di erences in the structure of ÿnite phase space depending on the underlying dimension of quantum space being an even or odd integer.

In memory of our colleague and friend, Maria Carolina Nemes, who was a shining example of excellence and team effort.Received: date / Accepted: date Abstract We address two major conceptual developments introduced by Aharonov and collaborators through a quantum phase space approach: the concept of modular variables devised to explain the phenomena of quantum dynamical non-locality and the two-state formalism for Quantum Mechanics which is a retrocausal timesymmetric interpretation of quantum physics which led to the discovery of 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.