Dear Colleagues,

For around hundred years, quantum mechanics has demonstrated an ability to be useful in understanding atomic and nuclear phenomena and the behavior of elementary particles and explaining processes in solid-state physics. At the beginning of the last century, Schrödinger introduced the notion of the state of a quantum system, which is a complex wave function.

Classical physics includes an intuitively, absolutely clear notion of state of particles which is characterized by position and velocity (or momentum) evolving according to Newton laws. Even in the case of the classical particle moving inside of the environment like a molecule moving in the room at a given temperature, the notion of the state of the system intuitively is clearly described, taking into account fluctuations of molecule position and momentum. The only change in the notion of the state is using the probability distribution function of the random position and momentum to identify it within the system state. This picture we have in classical statistical mechanics and the notion of probability intuitively are absolutely clear due to the everyday experience of people. For example, we know from our childhoods the game of tossing coins with probabilities to get either head or tale. For wave function, only its modulus has the interpretation of probability distribution, but the phase of the wave function does not have a probabilistic interpretation. Since the very early days of quantum mechanics, there have been attempts to find a formulation of this theory which is closer to classical one.

Wigner in 1932 introduced the notion of Wigner function depending on position and momentum, and this function is similar, in some respects, to probability distribution determining the particle state in classical statistical mechanics. However, this function takes negative values, and due to this, it is not the probability distribution. There were other attempts, such as the independent introduction by Glauber and Sudarshan of the quasidistribution function used intensively in quantum optics. At the end of century, the experimental study of photon states using quantum optical tomography with homodyne detectors demonstrated that a tool to obtain the Wigner function of the photon state means to measure optical tomographic probability distribution and reconstruct the Wigner function by means of integral Radon transform. The idea appeared that the tomographic probability distribution can be considered as a primary notion of the photon state, and all the quantum properties of the state are contained in the probability distribution. This created attempts to develop the tomographic approach and to formulate the probability representation of quantum states of all the systems, including spin systems (qubits, quidits, N-level atoms). All the interesting phenomena, including entanglement, quantum correlations, and other fundamental aspects of quantum mechanics, can be formulated using the notion of system states identified with a probability distribution.

The suggested Special Volume is dedicated to fundamental aspects of quantum theory with a focus on discussing the different representations, especially probability representations of the quantum states, and a closer relation with the classical statistical mechanics. In this representation, quantum evolution equations take the form of kinetic, classical-like equations for probability distributions. The discussion of the new probability representation of quantum mechanics will attract the interest to old discussions of Bohr and Einstein as well as other pioneers of quantum theory because for long periods of quantum theory history, the possibility of existence of the probability description of quantum states was considered as very improbable. We invite researchers interested in the foundations of quantum mechanics in connection, also with the development of new technologies, to submit contributions to this volume.

Prof. Vladimir I. Manko
Guest Editor

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