Unitarity and real properties of the neutral meson complex

TL;DR

This paper analyzes the implications of the Khalfin Theorem for the unitarity and properties of the neutral meson complex, revealing constraints on effective Hamiltonians and time evolution.

Contribution

It demonstrates that unitarity of the total system's evolution ensures the Khalfin Theorem holds and explores its consequences for effective Hamiltonians in neutral meson systems.

Findings

Diagonal elements of the effective Hamiltonian cannot be equal if CPT holds and CP is violated.

Time-independent effective Hamiltonians imply non-unitary evolution of the total system.

Graphical model illustrates how the Khalfin's Theorem constrains the system's properties.

Abstract

The proof of the Khalfin Theorem for neutral meson complex is analyzed. It is shown that the unitarity of the time evolution operator for the total system under considerations assures that the Khalfin's Theorem holds. The consequences of this Theorem for the neutral mesons system are discussed: it is shown, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex can not be equal if CPT symmetry holds and CP symmetry is violated. Properties of time evolution governed by a time--independent effective Hamiltonian acting in the neutral mesons subspace of states are considered. Using the Khalfin's Theorem it is shown that if such Hamiltonian is time--independent then the evolution operator for the total system containing the neutral meson complex can not be a unitary operator. It is shown graphically for a given specific model how the Khalfin's Theorem works. It is also shown for this model how the difference of the mentioned diagonal matrix elements of the effective Hamiltonian varies in time.