Special Issue “Pseudo-Hermitian Hamiltonians in Quantum Physics in 2014”
Int J Theor Phys
Special Issue “Pseudo-Hermitian Hamiltonians in Quantum Physics in 2014”
Miloslav Znojil 0
0 M. Znojil ( ) Nuclear Physics Institute ASCR , 250 68 Rˇ ezˇ , Czech Republic
To some physicists the title “Pseudo-Hermitian Hamiltonians in quantum physics” (PHHQP) of the present special issue referring to a rather specific mathematical concept might sound enigmatic, so the name probably deserves a brief introductory comment first of all. For the guest-editor's making such an introductory clarification there exist several options. Besides a purely formal reference to the series of international conferences on this particular subject (carrying the same PHHQP name - see their common webpage [1]) or to the related proceedings and/or special issues of various international scientific Journals [2] one could accept an alternative strategy of recalling at least a few most characteristic publications in the field (in this setting, the Daniel Hook's dedicated webpage [3] seems to be the best “bookkeeping” of the current related publication activities). Thirdly (and let me adopt this, less formal approach here) one could merely remind the readers about a few, randomly selected historical roots of the ideas which seem shared by the participants of the above-mentioned traditional conferences and/or by the authors of the related recent publications. For the purpose let me try to proceed in a parallel to the introductory phrase of ref. [4] and classify the roots as three - not entirely independent - research territories and evolutionary branches. Out of them, the oldest one (let me call it a “historical tendency A”) originates in quantum many-body physics. Its basic ideas could be attributed to Freeman Dyson [5]. The mathematical essence of the Dyson's approach to the problem of the construction of bound states may be briefly characterized as an isospectral-mapping transition from a known, “realistic”, Hermitian but, alas, prohibitively user-unfriendly Hamiltonian h of the system in question to its auxiliary, non-Hermitian but much friendlier representation
-
H =
−1h
where the invertible “intertwiner” should be assumed, for the sake of non-triviality of the
analysis, non-unitary. In opposite direction, precisely this assumption broadened the scope
of the theory. In other words, the broadened possibilities of the judicious choice of the
non-unitary Dyson’s maps offer also an explanation of the practical computational
success of the “operator preconditioning” (1) (cf., e.g., its “interacting boson” applications in
nuclear physics as reviewed by Scholtz et al. [6]).
In contrast to recipe “A”, the other, much younger historical tendency “B” proceeds in
an opposite direction reconstructing, whenever asked for, the complicated (and initially
unknown) physical Hamiltonian h from an input ansatz (or rather a trial-and-error choice) of
a sufficiently elementary form of some elementary non-Hermitian right-hand-side operator
H = H † of (1). In this alternative approach, extremely successfully introduced and
advocated by Carl Bender with multiple co-authors [7], one must first translate the obligatory,
physical Hermiticity condition h = h† into the language of H s. This yields the Dieudonne´’s
[8] relation
H †P = P H
=
Let us abbreviate † = calling this product a “new Hilbert-space metric” [6, 9].
Now, approach “B” requires that one re-factorizes the metric back into its Dyson-mapping
factors, completing the picture and reconstructing the “missing” textbook Hamiltonian h
whenever necessary.
In 1993, a very nice and mathematically rigorous illustration of the latter,
tworepresentation relationship (1) between alternative Hamiltonian-operator representations
has been provided by Buslaev and Grecchi. Indeed, today, their originally half-forgotten
paper [10] may be read as one of the first successful derivations of a closed-form
Hermitian h from a given non-Hermitian input H . Unfortunately, this analytic and exact,
non-numerical model proved so exceptional that it took more than ten years before a
similar realistic physical sample of the correspondence (1) has been found and described, by Ali
Mostafazadeh, in his remarkable innovative application of tendency “B” to first-quantized
Klein-Gordon fields [11].
The climax and main success of the Dyson’s philosophy of simplification h → H
emerged between the years 1998 - 2004 during which Bender et al proposed and defended
their ideas of further simplification of the formalism which we might call here “the youngest
historical tendency C”. Incidentally, the authors themselves already made these ideas well
known under the nickname of P T −symmetric quantum mechanics. The essence of their
contribution may be briefly characterized as based on the most intuitive perception of the
breakdown of spatial parity P of quantum systems and of an entirely new and enormously
inspiring requirement of restoration of the conservation of symmetry after the multiplication
of P by an antiline (...truncated)