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.
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.
In this thesis, we develop a probabilistic description of the time-symmetric formulation of quantum mechanics.We derive a generalized form of the probability distribution of a quantum process which is determined by performing pre-and post-selections on a system.It gives the well-known value for an observable associated to a quantum process, called weak value, as an expectation value.We show that the generalized probability satisfies the law of total probability and admits the transitive form by means of conditional probabilities.This motivates us to step forward to a classical ensemble picture in quantum mechanics.In addition, we present the description of quantum measurement in terms of the generalized probability.It is suggested that the difference between the projective measurement and the weak measurement corresponds to the choice of selections of the forward and backward propagating processes. Consistency of the Probability in the Forward and Backward Processes 6 Conclusion and Discussion Bibliography
In the standard formulation of quantum mechanics, observables are represented by operators in a Hilbert space and states by vectors in it. Quantum field theory too is formulated in terms of field operators. States in field theory include the vacuum state, one particle states, two particle states and so on. One does not measure operators or state vectors, however; measurements involve eigenvalues of operators or transition probabilities between states. It turns out that one can formulate quantum mechanics and field theory without using operators. Indeed, the probability amplitude for a transition from a position q1 at time t1 to a position q2 at time t2 can be expressed as a path integral ∫ dt L(q(t),q˙(t)), (5.1) where q(t) runs over all possible functions satisfying the conditions q(t1) = q1, q(t2) = q2 (5.2) and the integral over q(t) is an integral over all such paths of the exponential evaluated for each path. The measure of integration is not well defined a priori, but it is possible to give prescriptions that make sense. In classical mechanics, positions at two different times determine a path from the second order differential equation of motion, which yields a definite solution if two conditions are imposed. In contrast, quantum mechanics does not allow us to talk about trajectories. The path integral formulation manifestly refers to the myriad paths that may be taken by a particle in going from one point to another. It goes so far as to assign a probability amplitude to each path, and it is related to the classical action corresponding to that path. However, it is complex and it also involves a superposition, characteristic of quantum mechanics.
The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.
We suggest an interpretation of quantum mechanics, inspired by the ideas of Aharonov et al. of a time-symmetric description of quantum theory. We show that a special final boundary condition for the Universe, may be consistently defined as to determine single classical-like measurement outcomes, thus solving the "measurement problem". No other deviation is made from standard quantum mechanics, and the resulting theory is deterministic (in a two-time sense) and local. Quantum mechanical probabilities are recovered in general, but are eliminated from the description of any single measurement. We call this the Two-time interpretation of quantum mechanics. We analyze ideal measurements, showing how the quantum superposition is, in effect, dynamically reduced to a single classical state via a "two-time decoherence" process. We discuss some philosophical aspects of the suggested interpretation. We also discuss weak measurements using the two-time formalism, and remark that in these measurement situations, special final boundary conditions for the Universe, might explain some unaccounted for phenomena.
We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.
According to the Kolmogorovian Censorship Hypothesis, everything that quantum theory says about the world in the language of the quantum mechanical Hilbert space formalism is actually about relationships between ordinary relative frequencies expressible in operational terms using classical Kolmogorovian probability theory. In other words, a quantum theoretical description of a system should in principle be translatable into a purely operational-probabilistic description. However, our goal in this paper is different; we do not want to deal with the problem how to translate the known theory of quantum mechanics into operational terms, or to reconstruct the theory from postulates which can be interpreted in operational terms. Our aim is somewhat broader and points in the opposite direction. We start with a general scheme for the operational description of an arbitrary physical system. The description is based solely on the notion of observable events (measurement operations and measurement results) and on general, empirically established simple laws concerning their relative frequency. These laws are so simple and fundamental that they apply equally to any physical system. In the first part of the paper, we outline the basic elements of such an operational-probabilistic theory. In the second part, we discuss how this operational-probabilistic description compares to the quantum mechanical description and to what extent the standard Hilbert space quantum mechanics can be regarded as a reformulation of the general operational-probabilistic theory.