__Hou Yau__

## Restoring Time and Space Symmetry in a Quantum Field

Despite nature’s preference for symmetries, the treatment of time and space in quantum theory is not symmetrical. To restore the symmetry, we introduce an additional degree of freedom allowing matter to vibrate not only in the spatial directions but also in the temporal direction. We find that a system with matters vibrating in space and time obeys the Klein-Gordon equation and Schrödinger equation. The energy in this system must be quantized under the constraint that a particle’s mass is on shell. There is only a probability of finding a particle at a given location. In the non-relativistic limit, the vibrations in time and space can be related to the wave function in quantum mechanics but with an arbitrary phase difference. The emerged quantum wave can have physical vibrations despite the overall phase for the wave function is unobservable. The system can also be transformed to a quantum field via canonical quantization. This real scalar field has the same basic properties of a zero-spin bosonic field. Furthermore, the internal time of this system can be represented by a self-adjoint operator. The spectrum of this operator spans the entire real line despite the Hamiltonian of the system is bounded from below without contradicting the Pauli’s theorem. By restoring the symmetry between time and space, we reconcile the quantum properties of a bosonic field.