## Archive

Archive for September, 2006

**Quantum field theory without divergences**** at a correct temporal integration**

*Zahid Zakir* [1]

**Abstract**

Quantum mechanical trajectories for particles or fields, especially in path integrals, have not temporal differentials and the path integrals are defined only on a finite slice time lattice. Therefore, in QFT all integrals on energy should be taken before turning to the limit of very small times when they are finite. The renormalizable theories are invariant under the dilatations of the time lattice slice. It is a new space-time symmetry extending the Poincaré group and it has been discovered earlier in the momentum representation as the renormgroup. Thus, quantum mechanics requires turning to small times only after summation over all alternatives, i.e. energy integrations in loops, and this fact leads to natural regularization of loop integrals without additional hypotheses. These justify all effective methods of regularization as various realizations of the natural temporal regularization following from the fractal nature of paths. The covariant Planck time appears as a smallest time interval due to the gravitational proper time dilation and redshift of proper frequencies near the Planck energy source. Therefore, the standard QFT with such covariant and finite regularization becomes mathematically rigorous and physically consistent.

*PACS: 11.10.Gh, 11.10.Hi, 11.15.Ha, 11.10.Ly, 12.20.-m*

* Key words: quantum mechanics, path integral, quantization, renormalization, lattice regularization, symmetries, vacuum energy, cosmological constant*

Vol. **1**, No 2, p. 28 – 41, v1, 29 September 2006

Online: TPAC: 2098-003 v2, 28 September 2012; DOI: 10.9751/TPAC.2098-003

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

zahidzakir@theor-phys.org

*T***– ****и**** ***CP***–****symmetric quantization of fields**** without zero-point energy**

*Zahid Zakir* [1]

**Abstract**

The standard replacement of negatively-frequency operators of relativistic fields to operators of antiparticles leads to incomplete time-reversal in field’s observables. In fact, a correct transition to the operators of antiparticles occurs only at a full* T- *or* CP-*transformation of matrix elements. Then the operators of observables of *T* – or *CP*-symmetric fields with standard Lagrangians appear as normal-ordered and the zero-point energy and the zero-point charge do not arise. The rules* *for the *CP-*symmetric quantization of fields are formulated and also the theorem about vanishing of the zero-point energy is proved. The theorem is in agreement with all known experiments since in fact the contributions of real sources are observed only. The divergent zero-point energy and zero-point charge appear only at using of the Lagrangians symmetrized under the complex-conjugate field operators, while the standard Lagrangians without such symmetrization lead to the required vanishing vacuum energy. At spontaneous symmetry breaking the vacuum zero-point energy and zero-point charge do not appear only if the remained scalar field is complex one and in this case the Standard Model predicts a pair boson-antiboson with hypercharge.

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

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

Vol. **1**, No 1 , p. 11 – 27, v1, 22 September 2006

Online: TPAC: 2091-002 v2, 28 September 2012; DOI: 10.9751/TPAC.2091-002

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

zahidzakir@theor-phys.org

**Symmetries of harmonic oscillator generating**** the zero-point and negative energies**

*Zahid Zakir* [1]

**Abstract**

The Hamiltonian of harmonic oscillator is symmetric under the replacement of canonically conjugate variables and a canonical transformation to ladder operators maintains this symmetry. The Hamiltonian is expressed through a symmetrized product of the ladder operators and, as a result, at quantization there arise a zero-point energy. Therefore, for quantized fields, canonically conjugate variables of which enter into the Hamiltonian unsymmetrically, the zero-point energy could not arise. A new symmetry of harmonic oscillator is found: a wave equation and its solutions do not vary at a joint changing of signs of frequency, energy and mass of a particle. It is shown that the problem of the negative norm for negative-frequency states appears at taking positive mass at negative energy and, contrary, the problem disappears at taking the same sign of mass and energy as it is required by relativistic kinematics. In the nonrelativistic theory, considered as a limiting case of relativistic theory, the states of a particle with negative frequency, energy and mass are described consistently as evolving only backward in time and representing the states of its antiparticle with positive frequency, energy and mass evolving forward in time. For such charge-conjugation symmetric system of oscillators an extended space of states with generalized operators is constructed.

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

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

Vol. **1**, No 1 , p. 1 – 10, v1, 22 September 2006

Online: TPAC: 2091-001 v2, 28 September 2012; DOI: 10.9751/TPAC.2091-001

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

zahidzakir@theor-phys.org

## Recent Comments