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arXiv:physics/9803030

Improved Lee, Oehme and Yang approximation

K. Urbanowski, Jarosław Piskorski

Published Mar 23, 1998• 5 citations
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Abstract

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.

DOI
10.48550/arxiv.physics/9803030

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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.
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.
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 Kˉ0{\bar K}_{0}Kˉ0​ mass as the CPT--symmetry test is wrong.
Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.
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.