Quantum particles from classical statistics

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being “classical” or “quantum” ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross‐fertilization between classical statistics and quantum physics.     

Quantum groups and generalized statistics in integrable models

The paper deals with the integrable massive models of quantum field theory. It is shown that generalized statistics of physical particles is closely connected with the invariance under quantum groups. This invariance provides the possibility to construct quasi-local operators (parafermions) possessing generalized statistics which interpolates the physical particles. For the particular examples of SG, RSG models and scaling 3-state Potts model the parafermions are described completely (all their matrix elements in the space of states are presented).     

Quantum theory for a special kind of hard-core particles with intermediate statistics

Starting from a specific definition of fractional statistics for hard-core particles, we find a set of intermediate statistics systems, in which a single-particle quantum state can be effectively occupied by an integer number of identical particles — M-ons, as in the case for fermions and bosons. A quantum statistical theory of an ideal M-on gas is formulated exactly, and the associated distribution is explicitly represented in a simple analytical form. A possible application to the fractional quantum Hall effect is also briefly discussed.     

Fock spaces with generalized statistics

Fock representations for quantum fields which obey generalized statistics are explicitly constructed. The main features of these representations are investigated.     

Topological quantum liquids with quaternion non-Abelian statistics

Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).     

Quantum linguistic statistics

Formulas specifying the Zipf law for frequency dictionaries are presented. Principles from which these formulas follow are discussed.     

Quantum counterfactuality with identical particles

Quantum self-interference enables the counterfactual transmission of information, whereby the transmitted bits involve no particles traveling through the channel. In this work, we show how counterfactuality can be realized even when the self-interference is replaced by interference between identical particles. Interestingly, the facet of indistinguishability called forth here is associated with first-order coherence, and is different from the usual notion of indistinguishability associated with ...     

Full counting statistics of chiral Luttinger liquids with impurities

We study the statistics of charge transfer through an impurity in a chiral Luttinger liquid (realized experimentally as a quantum point contact in a fractional quantum Hall edge state device). Taking advantage of the integrability we present a procedure for obtaining the cumulant generating function of the probability distribution to transfer a fixed amount of charge through the constriction. Using this approach we analyze in detail the behavior of the third cumulant C3 as a function of applied voltage, temperature, and barrier height. We predict that C3 can be used to measure the fractional charge at temperatures, which are several orders of magnitude higher than those needed to extract the fractional charge from the measurement of the second cumulant. Moreover, we identify the component of C3, which carries the information about the fractional charge.     

Quantum mechanics as relativistic statistics. II: The case of two interacting particles

It is shown that non-relativistic quantum mechanics of two particles interacting with external electromagnetic field or between each other can be considered as statistics of two-dimensional surfaces. These surfaces represent the relativistical state of two indeterministic particles in the eight-dimensional space which is a tensor product of space-times for each of the two particles.     

Fock representations of quantum fields with generalized statistics

We develop a rigorous framework for constructing Fock representations of quantum fields obeying generalized statistics. The main features of these representations are investigated. Various aspects of the underlying mathematical structure are illustrated by means of explicit examples.     

Quantum crystal with jumps of particles

The relation between quasi-averages and averages in the case of spontaneous breakdown of translational symmetry is analysed. A hybrid system with virtual breakdown and restoration of translational symmetry is examined. The definitions of phase probabilities and phase transitions are given. The probabilities of the formation of the localized and nonlocalized phases are found and the existence of solid-liquid phase transition is proved. The calculated melting temperatures of the crystals of the inert group are in good agreement with experimental data. The collective excitation spectrum of the hybrid crystal is determined. An explanation of the high temperature anomaly of the specific heat in crystalline helium is proposed.     

Quantum statistics of anti-stokes luminescence

Anti-Stokes luminescence is theoretically described from the quantum statistical point of view. It is shown that this phenomenon depends on the statistical properties of the input fields. The cases of cooperative and sequential sensitization mechanisms are separately considered and the anti-Stokes generation rate terms are studied and discussed for both cases.     

Quantum statistics of parametric amplification

Quantum statistical properties of the parametric amplification and generation are investigated. The losses are included and the pump field is treated classically. It is shown that the normal counting generating function decomposes into the product of two functions which are the generating functions for the superposition of coherent and chaotic field and that the outputting field exhibits the anticorrelation effect when the phase of incident fields fulfil a certain condition.The authors thank Mrs A.Mikulecká from Laboratory of Computer Science of Palacký University for performing computations.     

Quantum statistics of reaction-diffusion systems

A strictly quantum-statistical theory of inhomogeneous reactions is presented. The treatment is based on the theory of multi-channel reactive scattering. For the configuration probabilities of the reactants a system of reaction-diffusion equations is obtained. The expressions for the diffusion tensor and the reaction-rate coefficients are established in terms of microscopic parameters characteristic of the reacting species.     

Quantum particles and classical particles

The relation between wave mechanics and classical mechanics is reviewed, and it is stressed that the latter cannot be regarded as the limit of the former as 0. The motion of a classical particle (or ensemble of particles) is described by means of a Schrödinger-like equation that was found previously. A system of a quantum particle and a classical particle is investigated (1) for an interaction that leads to stationary states with discrete energies and (2) for an interaction that enables the classical particle to act as a measuring instrument for determining a physical variable of the quantum particle.     

Quantum statistics in parametric generation

The optical parametric generation is investigated in connection with the photon statistics and squeezing which are calculated in a short time approximation up to the fourth power of the interaction time.     

Quantum statistics and isospin invariance

The level density of an ideal Bose or Fermi gas is written in terms of usual phase-space integrals taking isospin conservation into accoutt. This cluster decomposition includes quantum statistics corrections between equal as well as unequal charge states of the considered particles. The isospin weights are given. These results are used to formulate a simple isospin-invariant statistical bootstrap model with Bose statistics. In the framework of this model the production of neutral pions in e⁺e^? and $\text{N}\text{N}$ annihilation is investigated.