Quantum theory with arrow of time: symmetry breaking and non-local spinor realization with non-commuting operators of energy and decay

We present a modified quantum dynamics with a non-Hermitian Hamiltonian H-circumflex≠H-circumflex+, and a complex parameter of evolution. The theory is developed with [H-circumflex+, H-circumflex] = κ2, where κ is the new fundamental constant. It is shown that this theory loses its symmetry in the case of κ ≠ 0. We demonstrate that this theory allows a thermodynamical interpretation and possesses an 'arrow of time' for isothermal and adiabatic regimes of evolution. A realization of this theory is a modified four-dimensional Dirac spinor system, which appears non-local in the physical three-dimensional space.