ua nt - p h / 03 02 08 4 v 1 1 1 Fe b 20 03 P , T , PT , CPT invariance of Hermitian Hamiltonians

Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamil-tonians are P,T,PT,CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the energy-eigenstates are also simultaneous eigenstates of PT. Multipronged investigations supporting this conjecture have been extensively carried out [1-3]. Consequently, the condition of Hermiticity for a Hamiltonian to possess real eigenvalues gets relaxed. It is remarkable that it is discrete symmetries of an Hamiltonian which seem to decide if the eigenvalues will be real.