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arXiv:0707.3219 · Journal of Physical Studies

Unitarity and real properties of the neutral meson complex

K. Urbanowski, J. Jankiewicz

Published Jan 1, 2011• 0 citations
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Abstract

The proof of the Khalfin Theorem for the 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 that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex cannot 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.Using the Khalfin's Theorem it is shown that if such Hamiltonian is time-independent then the evolution operator for the total a system containing the neutral meson complex cannot 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.

DOI
10.30970/jps.15.1004

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The proof of the Khalfin Theorem for the 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 that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex cannot 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.Using the Khalfin's Theorem it is shown that if such Hamiltonian is time-independent then the evolution operator for the total a system containing the neutral meson complex cannot 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.
This paper begins with a discussion of the general form and general CP-and CPT-transformation properties of the Lee-Oehme--Yang (LOY) effective Hamiltonian for the neutral kaon complex.Next, the properties of the exact effective Hamiltonian determined by the properties of the exact transition amplitudes for this complex are discussed.Using the Khalfin Theorem we show that contrary to the standard result of the LOY theory, the diagonal matrix elements of the 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 > t 0 (t 0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant.The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle "1" and its antiparticle "2" need not be equal for t o t 0 in the case of preserved CPT and violated CP symmetries.We also show that there exists an approximation which is more accurate than the LOY, and which leads to an effective Hamiltonian whose diagonal matrix elements posses properties consistent with the conclusions for the exact effective Hamiltonian described above.
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>t0t > t_{0}t>t0​ (t0t_{0}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 K0K_{0}K0​ mass and the barK0{\\bar K}_{0}barK0​ mass as the CPT--symmetry test is wrong.
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>t0t > t_{0}t>t0​ (t0t_{0}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 K0K_{0}K0​ mass and the Kˉ0{\bar K}_{0}Kˉ0​ mass as the CPT--symmetry test is wrong.
This paper begins with a discussion of the general form and general CP-and CPT-transformation properties of the Lee-Oehme--Yang (LOY) effective Hamiltonian for the neutral kaon complex.Next, the properties of the exact effective Hamiltonian determined by the properties of the exact transition amplitudes for this complex are discussed.Using the Khalfin Theorem we show that contrary to the standard result of the LOY theory, the diagonal matrix elements of the 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 > t 0 (t 0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant.The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle "1" and its antiparticle "2" need not be equal for t o t 0 in the case of preserved CPT and violated CP symmetries.We also show that there exists an approximation which is more accurate than the LOY, and which leads to an effective Hamiltonian whose diagonal matrix elements posses properties consistent with the conclusions for the exact effective Hamiltonian described above.
The basic requirement that, in quantum theory, the time evolution of any state is determined by the action of a unitary operator, is shown to be the underlying cause for certain ``exact'' results that have recently been reported about the time dependence of transition rates in quantum theory. Departures from exponential decay, including the ``quantum Zeno effect,'' as well as a theorem by Khalfin about the ratio of reciprocal transition rates, are shown to follow directly from such considerations. At sufficiently short times, unitarity requires that reciprocity must hold, independent of whether T invariance is valid. If T invariance does not hold, unitarity restricts the form of possible time dependence of reciprocity ratios. \textcopyright{} 1996 The American Physical Society.
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.