On the relation of Manin’s quantum plane and quantum Clifford algebras

In a recent work we have shown that quantum Clifford algebras — i.e. Clifford algebras of an arbitrary bilinear form — are closely related to the deformed structures asq-spin groups, Hecke algebras,q-Young operators and deformed tensor products. The question to relate Manin’s approach to quantum Clifford algebras is addressed here. Explicit computations using the CLIFFORD Maple package are exhibited. The meaning of non-commutative geometry is reexamined and interpreted in Clifford algebraic terms. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

New Einstein-Maxwell fields of Levi-Civita’s type

The method based on the Horsky-Mitskievitch conjecture is applied to the Levi-Civita vacuum metric. It is shown, that every Killing vector is connected with a particular class of Einstein-Maxwell fields and each of those classes is found explicitly. Some of the obtained classes are quite new. Radial geodesic motion in constructed space-times is discussed and graphically illustrated in the Appendix.     

On gravity’s role in the genesis of rest masses of classical fields

It is shown that in the Einstein-conformally coupled Higgs–Maxwell system with Friedman–Robertson–Walker symmetries the energy density of the Higgs field has stable local minimum only if the mean curvature of the \(t=\mathrm{const}\) hypersurfaces is less than a finite critical value \(\chi _c\), while for greater mean curvature the energy density is not bounded from below. Therefore, there are extreme gravitational situations in which even quasi-locally defined instantaneous vacuum states of the Higgs sector cannot exist, and hence one cannot at all define the rest mass of all the classical fields. On hypersurfaces with mean curvature less than \(\chi _c\) the energy density has the ‘wine bottle’ (rather than the familiar ‘Mexican hat’) shape, and the gauge field can get rest mass via the Brout–Englert–Higgs mechanism. The spacelike hypersurface with the critical mean curvature represents the moment of ‘genesis’ of rest masses.     

On the implication of Bell’s probability distribution and proposed experiments of quantum measurement

In derivating of Bell’s inequalities, the probability distribution is supposed to be a function only of a hidden variable. We point out that the true implication of the probability distribution of Bell’s correlation function is the distribution of joint measurement outcomes on the two sides. It is therefore a function of both the hidden variable and the settings. In this case, Bell’s inequalities fail. Our further analysis shows that Bell’s locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, and hence violation of Bell’s inequalities cannot rule out the existence of local hidden variable. To explain the results of EPR-type experiments, we suppose that a polarization-entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any nonlocal influence on it. Then several experiments are suggested to reveal the components of the polarization-entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with the same quantum state behave the same under the same measuring conditions.     

Parity effect of the initial capital based on Parrondo’s games and the quantum interpretation

In our previous study [Zhu et al., Quantum game interpretation for a special case of Parrondo’s paradox, Physica A 390 (2011) 579], the capital-dependent Parrondo’s game where one game depends on the capital modulus $M=4$ was shown not to have a definite stationary probability distribution and that payoffs of the game depended on the parity of the initial capital. This paper presents a generalization of these results to even $M$ greater than 4. An intuitive explanation for producing this phenomenon is that the discrete-time Markov chain of the game is divided into two completely unrelated inner and outer rings. The process taking the inner ring or outer ring of the game is determined by the initial capital of parity and then a win or loss of the game is determined. Quantum game theory is used to further analyze the phenomenon. The results show that the explanation of the game corresponding to a stationary probability distribution is that the probability of the initial capital has reached parity.     

Densities of electron’s continuum in gravitational and electromagnetic fields

Relativistic dynamics of distributed mass and charge densities of the extended classical particle is considered for arbitrary gravitational and electromagnetic fields. Both geodesic and field gravitational equations can be derived by variation of the same Lagrange density in the classical action of a nonlocal particle distributed over its radial field. Vector geodesic relations for material space densities are contraction consequences of tensor gravitational equations for continuous sources and their fields. Classical four-flows of elementary material space depend on local electromagnetic fourpotentials for charged densities, as in quantum theory. Besides the Lorentz force, these potentials result in two more accelerating factors vanishing under equilibrium internal stresses within the continuous particle.     

The topological quantum field theory of Riemann’s theta functions

In this paper we prove the existence and uniqueness of a topological quantum field theory that incorporates, for all Riemann surfaces, the corresponding spaces of theta functions and the actions of the Heisenberg groups and modular groups on them.     

Probability distribution and Bell’s inequality for the quantum qubit state

The problem of Bell’s inequality violation for a particle with spin 1/2 is studied within the tomographic approach. Two possible methods for constructing the distribution functions associated with the qubit quantum state are presented. The Bell parameter maximum is studied for each proposed distribution.     

From Schrödinger’s equation to the quantum search algorithm

The quantum search algorithm is a technique for searching N possibilities in only O(√N) steps. Although the algorithm itself is widely known, not so well known is the series of steps that first led to it, these are quite different from any of the generally known forms of the algorithm. This paper describes these steps, which start by discretizing Schrödinger’s equation. This paper also provides a self contained introduction to quantum computing algorithms from a new perspective.     

A classification of quantum GL(2,C)’s

We show that the two-parameter standard quantum GL(2, C) (except for roots of unity) and the Jordanian quantum GL(2, C) have the “same” representation theory as the (ordinary) group GL(2, C), and that they are the only quantum groups with this property. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Efficient universal quantum channel simulation in IBM’s cloud quantum computer

The study of quantum channels is an important field and promises a wide range of applications, because any physical process can be represented as a quantum channel that transforms an initial state into a final state. Inspired by the method of performing non-unitary operators by the linear combination of unitary operations, we proposed a quantum algorithm for the simulation of the universal single-qubit channel, described by a convex combination of “quasi-extreme” channels corresponding to four Kraus operators, and is scalable to arbitrary higher dimension. We demonstrated the whole algorithm experimentally using the universal IBM cloud-based quantum computer and studied the properties of different qubit quantum channels. We illustrated the quantum capacity of the general qubit quantum channels, which quantifies the amount of quantum information that can be protected. The behavior of quantum capacity in different channels revealed which types of noise processes can support information transmission, and which types are too destructive to protect information. There was a general agreement between the theoretical predictions and the experiments, which strongly supports our method. By realizing the arbitrary qubit channel, this work provides a universally- accepted way to explore various properties of quantum channels and novel prospect for quantum communication.     

On Ohm’s law in the thin current sheets of the Earth’s magnetosphere

The analytical and numerical dependences of the total transverse current on an electric field, the normal component of a magnetic field and the ion and electron temperatures are obtained using analytical approximation of numerical results provided by a self-consistent model of the magnetospheric thin sheet. The dependence of current on the parameters ?, T _i , b _n is shown to be nonlinear. The relative contributions of different plasma components into the total current are estimated.     

Extension of Haag’s theorem in the case of the lorentz invariant noncommunitative quantum field theory in a space with arbitrary dimension

The generalized Haag theorem was proven in SO(1, k) invariant quantum field theory. Apart from the k + 1 variables, an arbitrary number of additional coordinates, including noncommutative ones, can occur in the theory. In SO(1, k) invariant theory new corollaries of the generalized Haag theorem are obtained. It has been proven that the equality of four-point Wightman functions in the two theories leads to the equality of elastic scattering amplitudes and thus to the equality of the total cross sections in these theories. It was also shown that at k > 3 the equality of (k + 1) point Wightman functions in the two theories leads to the equality of the scattering amplitudes of some inelastic processes. In the SO(1, 1) invariant theory it was proven that if in one of the theories under consideration the S-matrix is equal to unity, then in another theory the S-matrix equals unity as well.     

Dynamics of the Earth’s radiation belts

In this article, we provide a short review of the current state of the field of the radiation belts of the Earth. The main attention is given to the variations of energetic particle fluxes during geomagnetic storms. Electron and proton acceleration mechanisms in the terrestrial magnetosphere are discussed. The possibility of predicting various space weather parameters using the data on relativistic electrons of the outer radiation belt is discussed.     

Thermodynamic theory of the Tolman’s length

There still exists controversy on the sign and magnitude of the Tolman’s length and the Tolman’s gap.Further experimental,computational and theoretical investigations on them are needed to solve this problem.In 2006,Blokhuis and Kuipers obtained a rigorous relationship between the Tolman’s length and other thermodynamic quantities for the single-component liquid-vapour system.In the present paper,we derive two general relationships between the Tolman’s length and other thermodynamic quantities for the single-component liquid-vapour system.The relationship derived by Blokhuis and Kuipers and an earlier result turn out to be two special cases of our results.