Abstract
In this work we show that the existence of a complete biorthonormal set of eigenvectors of the effective Hamiltonian governing the time evolution of neutral meson system is a necessary condition for diagonalizability of such a Hamiltonian. We also study the possibility of probing the CPT invariance by observing the time dependence of cascade decays of type P○(P¯○)→{Ma,Mb}X→fX by employing such basis and exactly determine the CPT violation parameter by comparing the time dependence of the cascade decays of tagged P○ and tagged P¯○.
Highlights
In the Wigner-Weisskopf (W-W) approximation [1] the effective Hamiltonian which describes the P ◦ − P ◦ system is not Hermitian
We emphasis that when a non-hermitian and non-normal operator is encountered, use of a complete biorthonormal basis is in order which reduces to orthogonal basis as soon as the operator is considered hermitian and normal
We study the possibility of probing CP T invariance by observation of the time dependence of the cascade decay of the type Bd◦(Bd◦) → J/ψ{KL, KS} → J/ψf by employing the complete biorthonormal basis and introduce new ratios of decay amplitudes
Summary
In the Wigner-Weisskopf (W-W) approximation [1] the effective Hamiltonian which describes the P ◦ − P ◦ system is not Hermitian. A normal operator, in finite dimensions with no extra conditions and in infinite-dimensions with appropriate extra conditions, admit a diagonal metrix representation in some orthogonal basis This is usually called diagonalizability by a unitary transformation. In order to see the equivalence of the existence of a complete biorthonormal set of eigenvectors of Hand its diagonalizability, we note that by definition a diagonalizable Hamiltonian Hsatisfies A−1H A = H◦ for an invertible linear operator Aand a diagonal linear operator H◦, i.e., there is an orthogonal basis {|n } in the Hilbert space and complex numbers En such that H◦ =. In such a case the orthonormality relations between the basis of Hare valid and that the eigenket of Hare discriminant and the biorthonormal basis turn into orthonormal basis automatically and |ψn ’s are the same as |φn ’s
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