TPAC: 2525-007, Vol. 2, No 2, p. 9–21
Quantization of complex harmonic oscillator and the zero-point energy vanishing effect
Abstract
The system of two harmonic oscillators with different sign frequencies is presented as a positive-frequency oscillator with a complex generalized coordinate where there are a global U(1) symmetry and a charge conjugation symmetry (C-symmetry). It is shown that two pairs of ladder operators, appearing at the frequency decomposition of canonical variables, are not mutually charge-conjugate and their standard interpretation as operators of charge-conjugate quanta breaks the C-symmetry. Bilinear operator identities for the ladder operators are found allowed the theory to obey the C-symmetry restrictions and to express observables through the mutually charge-conjugate operators. These identities hold for C-symmetric interactions also. It is shown that C-symmetry eliminates a ground state zero-point charge and a zero-point energy. The uncertainty relations are generalized for non-hermitian canonical variables and it is shown that the charge-conjugation symmetric ground states do not quantized. It is shown that the C-symmetric interactions do not contribute to the ground state energy in all orders of perturbation theory.
PACS: 03.65.Ge, 11.30.Er, 1130.Ly, 11.90.+t
Key words: Hamiltonian dynamics, discrete symmetries, quantization, oscillator, uncertainty relations
Vol. 2, No 2, p. 9–21, 30 November 2007.