Khalfin's Theorem and neutral mesons subsystem

TL;DR

This paper examines Khalfin's Theorem in the context of neutral meson systems, revealing that certain properties of the effective Hamiltonian and time evolution are constrained by CPT and CP symmetries, with numerical illustrations.

Contribution

It provides a detailed analysis of Khalfin's Theorem's implications for neutral mesons, including conditions on the effective Hamiltonian and the non-unitarity of evolution operators under specific assumptions.

Findings

Diagonal matrix elements cannot be equal if CPT holds and CP is violated.

Time-independent effective Hamiltonian implies non-unitary evolution operator.

Numerical model illustrates Khalfin's Theorem and time variation of Hamiltonian differences.

Abstract

We analyze the proof of the Khalfin Theorem for neutral meson complex. The consequences of this Theorem are discussed: using this Theorem we find, 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. The Properties of time evolution governed by a time--independent effective Hamiltonian acting in the neutral mesons subspace of states are considered. By means of the Khalfin's Theorem we show that if such Hamiltonian is time--independent then the evolution operator for the total system containing neutral meson complex can not be a unitary operator. Within a given specific model we examine numerically the Khalfin's Theorem. We show for this model in a graphic form how the Khalfin's Theorem works. We also show for this model how the difference of the mentioned diagonal matrix elements of the effective Hamiltonian varies in time.