Sep
22

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

4 Responses to “TPAC: 2091-001 v2, Vol. 1, p. 1–10”

  1. Zahid Zakir
    October 8th, 2012 at 15:23 | #1

    1. New result:
    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.

  2. Zahid Zakir
    October 8th, 2012 at 15:25 | #2

    2. New result:
    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.

  3. Zahid Zakir
    October 8th, 2012 at 15:26 | #3

    3. New result:
    For quantized fields, canonically conjugate variables of which enter into the Hamiltonian unsymmetrically, the zero-point energy could not arise.

  4. Zahid Zakir
    October 8th, 2012 at 15:28 | #4

    4. New result:
    For the charge-conjugation symmetric system of oscillators an extended space of states with generalized operators is constructed.

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