The construction of a general inner product in non-Hermitian quantum theory and some explanation for the nonuniqueness of the C operator in PT quantum mechanics
Most recently it has been observed e.g. by Bender and Klevansky (arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetric non-Hermitian Hamilton operator is not unique. Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT-symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operator it is shown that - by a suitable choice of the norm of its eigenvectors - the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.
Most recently it has been observed e.g. by Bender and Klevansky (arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetric non-Hermitian Hamilton operator is not unique. Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT-symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operator it is shown that - by a suitable choice of the norm of its eigenvectors - the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)
A series of recent papers ``Faster than Hermitian Quantum Mechanics'' and related articles made a point of the possibility of a non-Hermitian, but PT-symmetric, operator to play the role of a Hamiltonian. In particular, they show that with an appropriate choice of an inner product, the evolution generated by such an operator will conserve the norm and scalar product. Here we observe that if one chooses such an inner product then the Hamiltonian in question is actually Hermitian, and the whole exercise is to a certain degree redundant.
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C d , it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for Swanson’s PT-symmetric Hamiltonian, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized Swanson’s non-Hermitian (not necessarily PT-symmetric) Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Swanson’s non-Hermitian Hamiltonian and su(1,1)
If a HamiltonianH has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT . The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT -symmetric Hamiltonian H = 12p + 1 2x 2 + i x using perturbative techniques. It is also shown how to construct the operator C for H = 12p 2 + 12x 2 − x using nonperturbative methods [1].