University of Zielona Gora, Institute of Physics,

Ul. Podgorna 50, 65-246 Zielona Gora, POLAND

E-mail: K.Urbanowski@proton.if.zu.zgora.pl, K.Urbanowski@if.zu.zgora.pl

**CPT symmetry and properties of the exact and approximate
effective Hamiltonians for the neutral kaon complex**

**Abstract:**

We start from 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 we 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
CPT transformation properties, which differ from those of the LOY effective
Hamiltonian. These properties of the mentioned effective Hamiltonians are
compared with the properties of the exact effective Hamiltonian for the
neutral kaon complex. Using the Khalfin Theorem we show that 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 $t > t_{0}$ ($t_{0}$ is the instant of creation
of the pair) when the total system under consideration is CPT invariant
but CP noninvariant. To achieve this, we use the transition amplitudes
of type $<{\bf j}|\exp (-itH)|{\bf k}>$ (j=1,2) for transitions $|{\bf
1}> \rightarrow |{\bf 2}>$, $|{\bf 2}> \rightarrow |{\bf 1}>$ together
with an identity expressing the effective Hamiltonian by these amplitudes
and their derivatives with respect to time $t$. (Here $|{\bf j}>$, $|{\bf
k}>$, $j,k =1,2,$ denote the states of type $|K_{0}>,$ $|{\overline{K}}_{0}>$,
$H$ is the total Hamiltonian of the system considered). This identity must
be fulfilled by any effective Hamiltonian (both approximate and exact)
derived for the two state complex. 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
$t \gg t_{0}$ in the case of preserved CPT and violated CP symmetries.
It appears that the mentioned more accurate approximation than the
LOY, leads to an effective Hamiltonian whose diagonal matrix elements posses
properties consistent with the conclusions for the exact effective Hamiltonian
described above.