Self-adjoint time operator of a quantum field

We study the properties of a quantum field with time as a dynamical variable. Temporal vibrations are introduced to restore the symmetry between time and space in a matter field. The system with vibrations of matter in time and space obeys the Klein–Gordon equation and Schrödinger equation. The energy observed is quantized under the constraint that a particle’s mass is on shell. This real scalar field has the same properties of a zero-spin bosonic field. Furthermore, the internal time of this system can be represented by a self-adjoint operator without contradicting the Pauli’s theorem. Neutrino can be an interesting candidate for investigating the effects of these temporal and spatial vibrations because of its extremely light weight.