1 Theoretical Physics, Astrophysics and Cosmology Vol. 1, No 1 , p. 1 – 10, v1, 22 September 2006 Online: TPAC: 2091-001 v2, 28 September 2012 ISSN 1819-9135; ISSN 1991-3915 (е) © 2006, 2012 CTPA. All rights reserved. DOI: 10.9751/TPAC.2091-001 Symmetries of harmonic oscillator generating the zero-point and negative energies Zahid Zakir * 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 Content Introduction ................................................................................................................................ 1 1. A symmetry leading to the zero-point energy ........................................................................ 2 1.1. The zero-point energy for an oscillating particle ................................................................... 2 1.2. The zero-point energy for quantized fields ............................................................................ 3 2. The extended space of states for harmonic oscillator ............................................................ 4 2.1. The time-reversal and the negative-frequency modes .......................................................... 4 2.2. A symmetry between positive- and negative-frequency modes ........................................... 5 2.3. The extended space of states and generalized ladder operators .......................................... 7 2.4. Reinterpretation of negative-frequency states ..................................................................... 9 Conclusion .................................................................................................................................. 9 References ................................................................................................................................ 10 Introduction A quantized harmonic oscillator plays a key role in modern physics since various systems, particularly free relativistic fields, are quantized by analogy with it. Two properties of the harmonic oscillator have been exploited in quantum field theory (QFT) – a formalism of ladder operators and a zero-point energy for ground states. * Centre for Theoretical Physics and Astrophyics, Tashkent, Uzbekistan; zahidzakir@theor-phys.org