## Archive

Archive for the ‘2011 Volume 6’ Category

**Rotatory quantization of charge-conjugation symmetric systems****.**

**3. Relativistic fields.**

*Zahid Zakir* [1]

**Abstract**

Quantum theory of complex fields with rotational modes based on a harmonic rotator model is constructed. For purely rotational modes the energy spectrum is equidistant, observables are automatically normal-ordered and there is no zero-point vacuum energy and zero-point charge. Frequencies of quanta are angular speeds of rotating field vectors (in real or field spaces). States of two signs of the helicity (particle-antiparticle) are related through the crossing symmetry. The well-known examples are photon field with circular polarization and complex fields. The spin and isospins of particles appear as related to their frequencies, representing angular momenta of rotations of field vectors with these frequencies. It is shown that the standard covariant perturbation theory is constructed in fact for description on the basis of harmonic rotators, where the recipe of transition from oscillatory to the rotatory representations of modes earlier it has been found empirically as normal-ordering of operators. The vacuum energy vanishes for free Hamiltonians and C-symmetric interactions.

*PACS: 03.70. +**k, 11.30.Er*

Vol. **6**, No 3, p. 48 – 63, v1, 14 December 2011

Online: TPAC: 4000-023 v2, 28 September 2012; DOI: 10.9751/TPAC.4000-023

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

zahidzakir@theor-phys.org

**Rotatory quantization of charge-conjugation symmetric systems****.**

**1. Harmonic** **oscillators**

* **Zahid Zakir* [1]

**Abstract**

In a system of a particle and antiparticle in the harmonic potential, represented as an oscillator with a complex generalized coordinate, there is a global U(1) symmetry and the charge conjugation (C) symmetry. It is shown that two pairs of ladder operators, introduced at the frequency decomposition of canonical variables, are not mutually charge-conjugate and that, therefore, their standard interpretation as operators of the charge-conjugate quanta breaks C-symmetry. Operator identities between bilinear products of the ladder operators are discovered, allowing expressing observables through charge-conjugate operators and it is correct to take into account C-symmetry. It is shown that these identities are maintained and at insert of the C-symmetric interactions. In a Lagrangian unsymmetrized and symmetrized orderings of complex conjugate operators of a momentum lead to different charge operators and are not equivalent at interaction with the gauge field. It is shown that due to C-symmetry conditions a zero-point charge does not arise in both orderings and in the first case a zero-point energy disappears also. The contribution of interaction with the gauge field and anharmonic potentials in higher orders of perturbation theory is considered. The same system also can be presented as a particle with positive and negative frequencies and, if to consider that a sign of mass of the particle coincides with a sign of its frequency, then the norm of negative frequency states remains positive.

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

*Key words: Hamiltonian dynamics, discrete symmetries, quantization*

Online: TPAC: 3900-021 v2, 28 September 2012; DOI: 10.9751/TPAC.3900-021

**Download pdf** 426 kb

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

zahidzakir@theor-phys.org

**Rotatory quantization of charge-conjugation symmetric systems****.**

**2. Harmonic** **and magneto-harmonic rotators**

*Zahid Zakir* [1]

**Abstract**

For soft rotators the lack of a radial component of velocity is a defining property and it allows to simplify quantization of harmonic and magneto-harmonic rotators. Operators of observables of soft rotators are normal ordered due to symmetries of the system, energy spectrum is linear under frequency and equidistant, and in the ground state there is no zero-point energy from rotational modes. It coincides with a generalization of the uncertainty relations for systems with non-hermitian canonical variables where the restrictions on fluctuations depend on state’s charge. Applications of the new formalism to quantization of waves at collective rotations of one-dimensional chain of harmonic rotators allows to model fields with charge-conjugation and gauge symmetries. For the rotating modes there is a crossing symmetry between states with opposite rotation directions, and arising of negative-frequency modes are positive-frequency states of antiquanta with replaced initial and final states. The commutators and causal correlators (propagators) of generalized coordinates of the harmonic rotator are derived.

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

*Key words: discrete symmetries, rotations, charge-conjugation symmetry, Landau levels, chain of rotators, propagators*

Vol. **6**, No 2, p. 31 – 47, v1, 5 September 2011

Online: TPAC: 3900-022 v2, 28 September 2012; DOI: 10.9751/TPAC.3900-022

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

zahidzakir@theor-phys.org

**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

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