Theoretical Physics, Astrophysics & Cosmology

 Vol. 1, No. 1, p. 1-11.    22 September  2006   version 1  

The Symmetries of the Harmonic Oscillator

Generating the Zero-Point and Negative Energies

Zahid Zakir 

Centre for Theoretical Physics and Astrophysics

(Tashkent, Uzbekistan)

Abstract

The Hamiltonian of a harmonic oscillator is symmetric under the interchanging of the canonically conjugated variables and the canonical transformations to the ladder operators preserve this symmetry. The resulting Hamiltonian contains a symmetrized product of the ladder operators and, as the result, at the quantization there appears a zero-point energy. Therefore, the zero-point energies disappear in the field-theoretic models the Hamiltonians of which are not symmetric under the canonically conjugated variables. It is shown also that there exists a new symmetry for the harmonic oscillator: the wave equations and their solutions are invariant under the combined changing the signs on a frequency, energy and mass of the oscillating particle. The appeared negative-frequency modes, after the time-reversal, describe oscillations of an antiparticle in the non-relativistic limit. For the such charge-symmetric system of oscillators the extended space of states with corresponding generalized operators is constructed.

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

    Key words: Hamiltonian dynamics, discrete symmetries, time 

                     reversal, antiparticles, quantization

 Full text:     pdf (117 Kb)          zip (82 Kb)

Vol. 1, No. 1, p. 12-34.    22 September  2006   version 1 

T- and CP-Symmetric Quantization of Fields

without Zero-Point Energies
 

Zahid Zakir

Centre for Theoretical Physics and Astrophysics

(Tashkent, Uzbekistan)

Abstract

Usual replacement of the negatively-frequency operators of relativistic fields to the operators of antiparticles leads to incomplete time-reversal in field’s observables. In fact, a correct transition to the operators of anti-particles occurs only at the full T- or CP-transformation of matrix elements. As it is shown, in the standard Hamiltonians the operators of antiparticles there appear as the normal-ordered ones without zero-point vacuum energies. The rules for the T-symmetric quantization of fields are formulated, and also the theorem about the zeroing of the zero-point energy is proved. The theorem is in agreement with all known experiments since in fact only the contributions of the real sources have been observed. The divergent zero-point energies appear at the using the symmetrized Lagrangians which, therefore, are in contradiction with the experiments, while QFT with the standard Lagrangians predicts the required vanishing vacuum energy. 

PACS:   03.70.+k;  11.10.-z;  11.30.Er

Key words:  vacuum energy, vacuum fluctuations, Casimir effect,   time reversal, charge conjugation, cosmological constant.

      Full text: pdf  (190 Kb)       zip (148 Kb)      3/04/2006

                              pdf  (186 Kb)                zip (146 Kb)            22/09/2006