## Archive

Archive for the ‘Current information’ Category

**Models of soft rotators**** and the theory of a harmonic rotator**

*Zahid Zakir* [1]

**Abstract**

The states of a planar oscillator are separated to a vibrational mode, containing a zero-point energy, and a rotational mode without the zero-point energy, but having a conserved angular momentum. On the basis of the analysis of properties of models of rigid and semirigid rotators, the theory of soft rotators is formulated where the harmonic attractive force is balanced only by the centrifugal force. As examples a Coulomb rotator (the Bohr model) and a magneto-harmonic rotator (the Fock-Landau levels) are considered. Disappearance of the radial speed in the model of a magneto-harmonic rotator is taken as a defining property of a pure rotational motion in the harmonic potential. After the exception of energies of the magnetic and spin decompositions, specific to magnetic fields, one turns to a simple and general model of a plane harmonic rotator (circular oscillator without radial speed) where kinetic energy is reduced to the purely rotational energy. Energy levels of the harmonic rotator have the same frequency and are twice degenerate, the energy spectrum is equidistant. In the ground state there is no zero-point energy from rotational modes, and the zero-point energy of vibrational modes can be compensated by spin effects or symmetries of the system. In this case the operators of observables vanish the ground state, i.e. are “strongly” normally ordered. In a chain of harmonic rotators collective rotations around a common axis lead to transverse waves, at quantization of which there appear quasi-particles and holes carrying an angular momentum. In the chain SU (2) appears as a group of symmetry of a rotator.

*PACS*: 03.65.Ge, 11.30.Er, 1130.Ly, 11.90. + t

*Key words: quantization, zero-point energy, vibrations, rotations, discrete symmetries*

Vol. **6**, No 1, p. 1 – 13, v1, 28 May 2011

Online: TPAC: 3800-020 v2, 18 September 2012; DOI: 10.9751/TPAC.3800-020

**Download pdf** 320 kb

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Four-index equations for gravitation**** and the gravitational energy-momentum tensor** [1]

* **Zahid Zakir* [2]

**Abstract**

A new treatment of the gravitational energy on the basis of 4-index gravitational equations is reviewed. The gravitational energy for the Schwarzschild field is considered.

*PACS*: 04.20.Cv, 04.20.Fy, 11.10.-z

*Key words: gravitational energy, curvature tensor, vacuum energy *

Vol. **5**, No 2, p. 22 – 25, v1, 9 November 2010

**Four-index energy-momentum tensors for gravitation and matter** [1]

* **Zahid Zakir* [2]

**Abstract**

The 4-index energy-momentum tensors for gravitation and matter are analyzed on the basis of new equations for the gravitational field with the Riemann tensor. Some properties of such defined gravitational energy are discussed.

*PACS*: 04.20.Cv, 04.20.Fy, 11.10.-z

*Key words: gravitational energy, curvature tensor, vacuum energy *

Vol. **5**, No 2, p. 18 – 21, v1, 9 November 2010

Online: TPAC: 3600-018 v2, 28 September 2012; DOI: 10.9751/TPAC.3600-018

**New equations for gravitation with the Riemann tensor and**** 4-index energy-momentum tensors for gravitation and matter** [1]

* **Zahid Zakir* [2]

**Abstract**

A generalized version of the Einstein equations in the 4-index form, containing the Riemann curvature tensor linearly, is derived. It is shown, that the gravitational energy-momentum density outside a source is represented across the Weyl tensor vanishing at the 2-index contraction. The 4-index energy-momentum density tensor for matter also is constructed.

*PACS*: 04.20.Cv, 04.20.Fy, 11.10.-z

*Key words: gravitational energy, curvature tensor, vacuum energy*

Vol. **5**, No 2, p. 14 – 17, v1, 9 November 2010

Online: TPAC: 3600-017 v2, 28 September 2012; DOI: 10.9751/TPAC.3600-017

**Are strings thermostrings? **[1]

* **Zahid Zakir* [2]

**Abstract **

In the method of thermostring quantization the time evolution of point particles at finite temperature kT is described in a geometric manner. The temperature paths of particles are represented as closed (thermo)strings, which are swept surfaces in a space-time-temperature manifold. The method makes it possible a new physical interpretation of superstrings IIA and heterotic strings as point particles in a thermal bath with Planck temperature.

*PACS*: 11.10.Wx, 11.10.Kk, 11.25.-w, 11.25.Uw, 11.25.Wx

*Key words: quantization, finite temperature, extra dimensions, strings, branes *

Vol. **5**, No 1, p. 8 – 13, v1, 20 May 2010

Online: TPAC: 3400-016 v2, 28 September 2012; DOI: 10.9751/TPAC.3400-016

**Download pdf** 182 kb

**Thermostring quantization. **

**An interpretation of strings as particles at finite temperatures** [1]

* **Zahid Zakir* [2]

**Abstract **

In a space-temperature configurational manifold an instantaneous temperature path of a point particle can be represented as a string of length L=1/kT (thermostring). The thermostring swepts a surface in the space-time-temperature manifold at its temporal evolution. The thermostring is closed, its points can be rearranged and the charge is distributed along the length. Some predictions of this method for statistical mechanics and string theories are discussed.

*PACS*: 11.10.Wx, 11.10.Kk, 11.25.-w, 11.25.Uw, 11.25.Wx

*Key words: quantization, finite temperature, extra dimensions, strings, branes*

Vol. **5**, No 1, p. 1 – 7, v1, 20 May 2010

Online: TPAC: 3400-015 v2, 28 September 2012; DOI: 10.9751/TPAC.3400-015

**The theory of stochastic space-time. **

**2. Quantum theory of relativity****[1]**

*Zahid Zakir* [2]

**Abstract**

Nelson’s stochastic mechanics is derived as a consequence of the basic physical principles such as the principle of relativity of observations and the invariance of the action quantum. The unitary group of quantum mechanics is represented as the transformations of the systems of perturbing devices. It is argued that the physical spacetime has a stochastic nature and that quantum mechanics in Nelson’s formulation correctly describes this stochasticity.

*PACS*: 04.20.Cv, 03.65.Ta, 05.40.Jc, 04.62.+v

*Key words: stochastic mechanics, quantum fluctuations, measurements*

Vol. **4**, No 2, p. 10 – 16, v1, 5 October 2009

**Download pdf** 266 kb

[2] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Gravitation as a Quantum Diffusion ****[1]**

*Zahid Zakir* [2]

**Abstract**

Inhomogeneous Nelson’s diffusion in flat spacetime with a tensor of diffusion can be described as a homogeneous one in a Riemannian manifold with this tensor of diffusion as a metric tensor. The influence of matter to the energy density of the stochastic background (vacuum) is considered. It is shown that gravitation can be represented as inhomogeneity of the quantum diffusion; the Einstein equations for the metrics can be derived as the equations for the corresponding tensor of diffusion.

*PACS*: 04.20.Cv, 03.65.Ta, 05.40.Jc, 04.62.+v

*Key words: stochastic mechanics, tensor of diffusion, quantum fluctuations, gravitation, curvature*

Vol. **4**, No 1, p. 6 – 9, v1, 19 March 2009

Online: TPAC: 3000-013 v2, 28 September 2012; DOI: 10.9751/TPAC.3000-013

**Download pdf** 145 kb

[2] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**The Theory of Stochastic Space-Time.**

**1. Gravitation as a Quantum Diffusion****[1]**

*Zahid Zakir* [2]

**Abstract**

The Nelson stochastic mechanics of *inhomogeneous* quantum diffusion in flat spacetime with a tensor of diffusion can be described as a *homogeneous* one in a Riemannian manifold where this tensor of diffusion plays the role of a metric tensor multiplied to the diffusion coefficient. It is shown that such diffusion accelerates both a sample particle and local inertial frames such that their mean accelerations do not depend on their mass. This fact, explaining the principle of equivalence, allows one to represent the curvature and gravitation as consequences of the inhomogeneity of the quantum fluctuations. In this diffusional treatment of gravitation it can be naturally explained the fact that the energy density of the instantaneous Newtonian interaction is negative defined (with respect to a point at spatial infinity).

*PACS*: 04.20.Cv, 03.65.Ta, 05.40.Jc, 04.62.+v

*Key words: stochastic mechanics, tensor of diffusion, quantum fluctuations,** gravitation, curvature*

Vol. **4**, No 1, p. 1 – 5, v1, 19 March 2009

Online: TPAC: 3000-012 v2, 28 September 2012; DOI: 10.9751/TPAC.3000-012

**Download pdf** 259 kb

[2] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Non-dissipative diffusion in classical and quantum systems**

*Zahid Zakir* [1]

**Abstract**

The theory of non-dissipative diffusion is constructed on an example of diffusion of a light particle in a dilute medium of heavy particles and it is shown that in low dissipative systems there are specific effects of non-dissipativity similar to quantum effects. In a non-dissipativity region mean energy of the light particle is conserved and processes are described by two non-linear diffusion equations with forward and backward time derivatives. Then these two diffusion equations are linearized and give one linear Schrödinger equation for complex amplitudes of probability. As a result, in the non-dissipative classical diffusion should be added probability amplitudes and there holds the superposition principle for these amplitudes. A mean square length of free passage and a mean square momentum define an elementary phase volume and a diffusion constant and they obey the uncertainty relations. It is shown that the formalism of quantum mechanics describes the classical non-dissipative diffusion with a homogeneous diffusion constant and that quantum mechanics is only a particular case when the elementary phase volume of free passage is universal and equal to the Planck constant.

*PACS*: 02.50.Ey, 03.65.Ta , 05.40.Jc,

*Key words: stochastic processes, **quantum mechanics, Brownian motion*

Vol. **3**, No 3, p. 16 – 35, v1, 29 August 2008

Online: TPAC: 2798-011 v2, 28 September 2012; DOI: 10.9751/TPAC.2798-011

**Download pdf** 932 kb

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

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