Physical Meaning of Hermiticity and Shortcomings of the Composite (Hermitian + non-Hermitian) Quantum Theory of Günther and Samsonov

In arXiv:0709.0483 Günther and Samsonov outline a “generalization” of quantum mechanics that involves simultaneous consideration of Hermitian and non-Hermitian operators and promises to be “capable to produce effects beyond those of standard Hermitian quantum mechanics.” We give a simple physical interpretation of Hermiticity and discuss in detail the shortcomings of the above-mentioned composite quantum theory. In particular, we show that the corresponding “generalization of measurement theory” suffers from a dynamical inconsistency and that it is by no means adequate to replace the standard measurement theory. PACS number: 03.65.Ca, 11.30.Er, 03.65.Pm, 11.80.Cr In [1] we proved the following theorem. Theorem: The lower bound on the travel time (upper bound on the speed) of unitary evolutions is a universal quantity independent of whether the evolution is generated by a Hermitian or non-Hermitian Hamiltonian. A direct implication of this theorem, which contradicts the main result of [2], is that as far as the Brachistochrone problem is concerned the use of non-Hermitian (in particular PT -symmetric) Hamiltonians that are capable of generating unitary time-evolutions does not offer any advantage over the Hermitian Hamiltonians. This is in complete agreement with the earlier results on the physical equivalence of the pseudo-Hermitian (in particular PT -symmetric) quantum mechanics and the standard (Hermitian) quantum mechanics, [3, 4]. To avoid this equivalence, Günther and Samsonov [5] have recently outlined a composite quantum theory involving both Hermitian and