On the reality of spin and helicity

The possibilities of a realistic interpretation of quantum mechanics are investigated by means of a statistical analysis of experiments performed on the simplest type of quantum systems carrying spin or helicity. To this end, fundamental experiments, some new, for measuring polarization are reviewed and (re)analyzed. Theunsharp reality of spin is essential in the interpretation of some of these experiments and represents a natural motivation for recent generalizations of quantum mechanics to a theory incorporating effect-valued measures as unsharp observables and generalized systems of imprimitivity.     

The generalized Fényes-Nelson model for free scalar field theory

The generalized Fényes-Nelson model of quantum mechanics is applied to the free scalar field. The resulting Markov field is equivalent to the Euclidean Markov field with the times scaled by a common factor which depends on the diffusion parameter. This result is consistent with Guerra's earlier work on stochastic quantization of scalar fields. It suggests a deep connection between Euclidean field theory and the stochastic interpretation of quantum mechanics. The question of Lorentz covariance is also discussed.     

Zur Frage der Gültigkeitsgrenzen der Quantenmechanik

In recent discussions two different views concerning the rôle of processes of observation and measurement in quantum mechanics have been put forward. One of these (Ludwig) assumes the theory of measurement to be connected with macrophysics, and macrophysics to stand still outside the frame of pure quantum mechanics. Other considerations too supporting the idea that quantum mechanics is not yet the last step in necessary generalisations of classical physics the author reports shortly about his endeavour to recognise mathematical possibilities for further generalisations, starting especially fromNeumann-Birkhoff's “quantum logic”.     

Entangled solitons and stochastic q-bits

Stochastic realization of the wave function in quantum mechanics with the inclusion of soliton representation of extended particles is discussed. Two-soliton configurations are used for constructing entangled states in generalized quantum mechanics dealing with extended particles, endowed with nontrivial spin S. Entangled solitons construction being introduced in the nonlinear spinor field model, the Einstein-Podolsky-Rosen (EPR) correlation is calculated and shown to coincide with the quantum mechanical one for the 1/2-spin particles. The concept of stochastic qubits is used for quantum computing modeling. The text was submitted by the authors in English.     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

On Schrödinger's Search for a Stochastic Interpretation of Quantum Mechanics

Following Schrödinger a stochastic interpretation of quantum mechanics is given based on the introduction of an intermediate probability in diffusion processes. The Schrödinger equation is derived following Nelson's approach and following a variational approach. Some problems of the quantum theory of measurement are discussed.     

Poor decay of correlations in inhomogeneous fluids and solids and their relevance for the physics of phase boundaries

We study the poor decay of correlations for equilibrium states of inhomogeneous fluids and solids, in the regimes of both classical and quantum statistical mechanics. Our main observation is the usefulness of the statistical mechanical expression of thestress tensor and its long-range correlations with the particle density. From this we are able to infer a very slow decay of correlations for the various molecular distribution functions under discussion. The derived results are of relevance both for completely inhomogeneous systems such as quasicrystals or granular structures and for the slightly more regular cases of, e.g., phase separating layers in fluids and solids, ideal crystals, etc. As one of the byproducts we prove the nonexistence of planequantum interfaces in two dimensions (thus extending earlier results of Requardt to the quantum regime). The results hold for arbitrary potentials of not too long range.     

p-adic quantum mechanics

An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.     

A quaternion quantum system

The (generalized) quantum mechanics of the two-dimensional quaternionic Hilbert space is constructed explicitly. In this simple example the unconventional properties of such a system as compared to complex quantum mechanics are discussed.     

Quantum mechanics from self-interaction

We explore the possibility thatzitterbewegung is the key to a complete understanding of the Dirac theory of electrons. We note that a literal interpretation of thezitterbewegung implies that the electron is the seat of an oscillating bound electromagnetic field similar to de Broglie's pilot wave. This opens up new possibilities for explaining two major features of quantum mechanics as consequences of an underlying physical mechanism. On this basis, qualitative explanations are given for electron diffraction, the existence of quantized radiationless states, the Pauli principle, and other features of quantum mechanics.     

The Operator Formulation of Classical Mechanics and Semiclassical Limit

The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h ₀. For the later of these two extreme values, introduced operator algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operator framework.     

Consistent histories and operational quantum theory

In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach, nonrelativistic quantum mechanics is most generally formulated in terms of effects, states, and operations. We formulate a generalized consistent histories theory using the concepts and the terminology which have proven useful in the operational formulation of quantum mechanics. The logical rule of the logical interpretation of quantum mechanics is generalized to the present context. The algebraic structure of the generalized theory is studied in detail.     

The Nambu mechanics as a class of singular generalized dynamical formalisms

In a recent work Nambu has proposed ac-number dynamical formalism which can allow an odd numbern of canonical variables. Naturally associated to this new mechanics there exists ann-linear bracket whose study opens interesting possibilities. The purpose of this work is to show that besides this bracket another one which is bilinear and in fact a Lie bracket can also be associated with the Nambu mechanics. For anyn, however, this bracket is singular. In a sense previously used by the present author, this result exhibits the Nambu mechanics as an interesting class of singular generalized dynamical formalisms irrespective of the number of phase space variables. Reasons are given suggesting that such singular formalisms would be, within our context, the only ones capable of describing classical analogues of generalized quantum systems.     

Quantization in classical mechanics and its relation to the Bohmian Ψ-field

Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of Hamilton systems in the form of the Schrödinger equation.It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with the Bohm “quantum” potential. Within the frame-work of Bohmian quantum mechanics supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Schrödinger equation is equivalent semantically and syntactically to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters. The conditions for the correctness of trajectory interpretation of quantum mechanics are discussed.     

Finite-dimensional quantum mechanics of a particle. III. The Weylian quantum mechanics of confined quarks

A finite-dimensional analog of Weyl's formulation of quantum kinematics of a physical system through irreducible Abelian groups of unitary ray rotations in system space offers many possibilities for the quantum mechanics of confined particles. This paper is devoted to the expansion of the recently developed framework of such Weylian finite-dimensional quantum mechanics which may provide a new way of thinking about the characteristics of quark physics.     

Generalized classical mechanics

Generalized classical mechanics has been introduced and developed as a classical counterpart of the fractional quantum mechanics. The Lagrangian of generalized classical mechanics has been introduced, and equation of motion has been obtained. Lagrange, Hamilton and Hamilton-Jacobi frameworks have been implemented. Oscillator model has been launched and solved in 1D case. A new equation for the period of oscillations of generalized classical oscillator has been found. The interplay between the energy dependency of the period of classical oscillations and the non-equidistant distribution of the energy levels for fractional quantum oscillator has been discussed. We discuss as well, the relationships between new equations of generalized classical mechanics and the well-known fundamental equations of classical mechanics.     

Black hole remnants due to GUP or quantum gravity?

Based on the micro-black hole gedanken experiment as well as on general considerations of quantum mechanics and gravity the generalized uncertainty principle (GUP) is analyzed by using the running Newton constant. The result is used to decide between the GUP and quantum gravitational effects as a possible mechanism leading to the black hole remnants of about Planck mass.     

从贝尔不等式到量子信息"热"

介绍了贝尔不等式的建立及20世纪70年代以来对它的实验验证．对贝尔不等式与量子力学基本解释的关系的深入研究,加速了“纠缠态”及其品性的发现,推动了量子信息“热”的兴起。