Particles with Negative Energies in Nonrelativistic and Relativistic Cases

States of particles with negative energies are considered for the nonrelativistic and relativistic cases. In the nonrelativistic case it is shown that the decay close to the attracting center can lead to the situation similar to the Penrose effect for a rotating black hole when the energy of one of the fragments is larger than the energy of the initial body. This is known as the Oberth effect in the theory of the rocket movement. The realizations of the Penrose effect in the non-relativistic case in collisions near the attracting body and in the evaporation of stars from star clusters are indicated. In the relativistic case similar to the well known Penrose process in the ergosphere of the rotating black hole it is shown that the same situation as in ergosphere of the black hole occurs in rotating coordinate system in Minkowski space-time out of the static limit due to existence of negative energies. In relativistic cases differently from the nonrelativistic ones, the mass of the fragment can be larger than the mass of the decaying body. Negative energies for particles are possible in the relativistic case in cosmology of the expanding space when the coordinate system is used with a nondiagonal term in metrical tensor of the space-time. Friedmann metrics for three cases: open, close and quasieuclidian, are analyzed. The De Sitter space-time is shortly discussed.

Noncommutative space–time and Hausdorff dimension

We study the Hausdorff dimension of the path of a quantum particle in noncommutative space–time. We show that the Hausdorff dimension depends on the deformation parameter [Formula: see text] and the resolution [Formula: see text] for both nonrelativistic and relativistic quantum particle. For the nonrelativistic case, it is seen that Hausdorff dimension is always less than 2 in the noncommutative space–time. For relativistic quantum particle, we find the Hausdorff dimension increases with the noncommutative parameter, in contrast to the commutative space–time. We show that noncommutative correction to Dirac equation brings in the spinorial nature of the relativistic wave function into play, unlike in the commutative space–time. By imposing self-similarity condition on the path of nonrelativistic and relativistic quantum particle in noncommutative space–time, we derive the corresponding generalized uncertainty relation.

Solution of Effective-Mass Dirac Equation with Scalar-Vector and Pseudoscalar Terms for Generalized Hulthén Potential

We find the exact bound state solutions and normalization constant for the Dirac equation with scalar-vector-pseudoscalar interaction terms for the generalized Hulthén potential in the case where we have a particular mass functionm(x). We also search the solutions for the constant mass where the obtained results correspond to the ones when the Dirac equation has spin and pseudospin symmetry, respectively. After giving the obtained results for the nonrelativistic case, we search then the energy spectra and corresponding upper and lower components of Dirac spinor for the case of PT-symmetric forms of the present potential.

Proof of the spin-statistics theorem in the relativistic regimen by Weyl’s conformal quantum mechanics

The traditional standard theory of quantum mechanics is unable to solve the spin-statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle” but by the adoption of the complex standard relativistic quantum field theory. In a recent paper [E. Santamato and F. D. De Martini, Found. Phys. 45 (2015) 858] we presented a complete proof of the spin-statistics problem in the nonrelativistic approximation on the basis of the “Conformal Quantum Geometrodynamics” (CQG). In this paper, by the same theory, the proof of the spin-statistics theorem (SST) is extended to the relativistic domain in the scenario of curved spacetime. No relativistic quantum field operators are used in the present proof and the particle exchange properties are drawn from rotational invariance rather than from Lorentz invariance. Our relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. As in the nonrelativistic case, we find once more that the “intrinsic helicity” of the elementary particles enters naturally into play. It is therefore this property, not considered in the standard quantum mechanics (SQM), which determines the correct spin-statistics connection observed in Nature.

Relativistic persistent currents in ideal Aharonov–Bohm rings

The exact solutions of the complete Dirac equation for fermions moving in ideal Aharonov–Bohm rings are used for deriving the exact expressions of the relativistic partial currents. It is shown that as in the nonrelativistic case, these currents can be related to the derivative of the fermion energy with respect to the flux parameter. A specific relativistic effect is the saturation of the partial currents for high values of the total angular momentum. Based on this property, the total relativistic persistent current at T = 0 is evaluated giving its analytical expression and showing how this depends on the ring parameters.

Lorentz-violating effects in the Bose–Einstein condensation of an ideal bosonic gas

We have studied the effects of Lorentz-violation in the Bose–Einstein condensation (BEC) of an ideal boson gas, by assessing both the nonrelativistic and ultrarelativistic limits. Our model describes a massive complex scalar field coupled to a CPT-even and Lorentz-violating background. We first analyze the nonrelativistic case, at this level by using experimental data, we obtain upper-bounds for some LIV parameters. In the sequel, we have constructed the partition function for the relativistic ideal boson gas which to be able of a consistent description requires the imposition of severe restrictions on some LIV coefficients. In both cases, we have demonstrated that the LIV contributions are contained in an overall factor, which multiplies almost all thermodynamical properties. An exception is the fraction of the condensed particles.

Realistic Approach of the Relations of Uncertainty of Heisenberg

Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.

Optical Transitions and Charge-Exchange in Highly Charged Quasi-Molecules

The interaction between quasimolecular states produces not only nonadiabatic transitions but also some exotic features in the wings of the spectral profiles emitted by the ions in collision. Although this concept has been fruitfully used for neutral species, some new highlighted experimental data on quasimolecular optical transitions in hot dense plasma have renewed the interest to the concept in the recent years. The present review deals with highly charged quasimolecules and it is dedicated specifically to quasimolecules formed by two bare nuclei and one bound electron. The reason for this choice is that, for such quasimolecules, the energy terms and the dipole moments of the optical transitions can be obtained straightforwardly in nonrelativistic case without any approximation that are typical for neutrals. Although the results obtained in the frame of the approach developed here are directly applicable to the case of single collisions or to low-density plasmas, they form a reasonable initial approximation for the problem of optical profiles in hot dense plasmas and can be regarded as a safe framework for qualitative discussions of profiles in those environments.