Time-symmetric formulation of quantum mechanics

We explore further the suggestion to describe a pre- and postselected system by a two-state, which is determined by two conditions. Starting with a formal definition of a two-state Hilbert space and basic operations, we systematically recast the basics of quantum mechanics---dynamics, observables, and measurement theory---in terms of two-states as the elementary quantities. We find a simple and suggestive formulation that "unifies" two complementary observables: probabilistic observables and nonprobabilistic "weak" observables. Probabilities are relevant for measurements in the "strong-coupling regime." They are given by the absolute square of a two-amplitude (a projection of a two-state). Nonprobabilistic observables are observed in sufficiently weak measurements and are given by linear combinations of the two-amplitude. As a subclass they include the "weak values" of Hermitian operators. We show that in the intermediate regime, one may observe a mixing of probabilities and weak values. A consequence of the suggested formalism and measurement theory is that the problems of nonlocality and Lorentz noncovariance, of the usual prescription with a "reduction," may be eliminated. We exemplify this point for the Einstein-Podolsky-Rosen experiment and for a system under successive observations.