General properties of eigenvectors and eigenvalues for an effective Hamiltonian governing time evolution in a two state subspace of the state space of the total system under consideration are discussed. The Lee, Oehme and Yang (LOY) theory of time evolution in such a subspace and CP-- and CPT--symmetry properties of the LOY effective Hamiltonian are considered. The CPT transformation properties of the exact effective Hamiltonian for two state subspace are discussed also. Using the Khalfin Theorem we show that contrary to the properties of the LOY theory, the diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for
t>t0 (
t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. Using the more accurate approximation than the LOY one we show that the interpretation of the tests measuring the difference between the
K0 mass and the
barK0 mass as the CPT--symmetry test is wrong.
We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-Hermitian Hamiltonian
Heff, in accord with the Wigner-Weisskopf approximation, and an additional term of the Lindblad form, the so-called dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-Hermitian part of
Heff---the ``particle decay''---can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the formulation of the time evolution on the enlarged space has a Hermitian Hamiltonian and is probability conserving. The equivalence of the formulation with the original one demonstrates that the time evolution which is governed by a non-Hermitian Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with Hermitian Hamiltonians.
The Lee, Oehme and Yang (LOY) theory of time evolution in two state subspace of states of the complete system is discussed. Some inconsistencies in assumptions and approximations used in the standard derivation of the LOY effective Hamiltonian, H(LOY), governig this time evolution are found. Eliminating these inconsistecies and using the LOY method, approximate formulae for the effective Hamiltonian, H(eff), governing the time evolution in this subspace (improving those obtained by LOY) are derived. It is found, in contradistinction to the standard LOY result, that in the case of neutral kaons ( - ), cannot take the zero value if the total system the preserves CPT--symmetry. Within the use of the method mentioned above formulae for H(eff) acting in the three state (three dimensional) subspace of states are also found.
Approximate formulas are given for the effective Hamiltonian ${\mathit{H}}_{\mathrm{\ensuremath{\parallel}}}$(t) governing the time evolution in a subspace ${\mathit{scrH}}_{\mathrm{\ensuremath{\parallel}}}$ of the state space scrH. It is proved that this approximation is correct for any Hamiltonian H of the system under consideration at the early-time period. The approximate form of the survival amplitude for a given state improving Fleming's estimation in the short-time region is found and the properties of a decay rate for small, intermediate, and long times are discussed.
General properties of eigenvectors and eigenvalues for an effective Hamiltonian governing time evolution in a two state subspace of the state space of the total system under consideration are discussed. The Lee, Oehme and Yang (LOY) theory of time evolution in such a subspace and CP-- and CPT--symmetry properties of the LOY effective Hamiltonian are considered. The CPT transformation properties of the exact effective Hamiltonian for two state subspace are discussed also. Using the Khalfin Theorem we show that contrary to the properties of the LOY theory, the diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for
t>t0 (
t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. Using the more accurate approximation than the LOY one we show that the interpretation of the tests measuring the difference between the
K0 mass and the
Kˉ0 mass as the CPT--symmetry test is wrong.
We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude
a(t) of the state considered is found. Our attention is focused on the late time properties of
a(t) and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as
t goes to the infinity more slowly than any exponential function of
t. This effect can be described using a time-dependent decay rate
γ(t) and then the Khalfin result means that this
γ(t) is not a constant but at late times it tends to zero as
t goes to the infinity. It appears that the energy
E(t) of the unstable state behaves similarly: It tends to the minimal energy
Emin of the system as
t→∞. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of
E(t) and
γ(t). We discuss also possible implications of such a late time asymptotic properties of
E(t) and
γ(t) and cases where these properties may manifest themselves.