aa r X i v : . [ qu a n t - ph ] J a n Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2009), 001, 19 pages Three-Hilbert-Space Formulationof Quantum Mechanics ⋆ Miloslav ZNOJILNuclear Physics Institute ASCR, 250 68 ˇReˇz, Czech Republic
E-mail: [email protected]
URL: http://gemma.ujf.cas.cz/ ∼ znojil/ Received October 29, 2008, in final form December 31, 2008; Published online January 06, 2009doi:10.3842/SIGMA.2009.001
Abstract.
In paper [Znojil M.,
Phys. Rev. D (2008), 085003, 5 pages, arXiv:0809.2874]the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics hasbeen revisited. In the present continuation of this study (with the spaces in question denotedas H (auxiliary) and H (standard) ) we spot a weak point of the 2HS formalism which lies in thedouble role played by H (auxiliary) . As long as this confluence of roles may (and did!) lead toconfusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulationof the same theory. As a byproduct of our analysis of the formalism we offer an amendment ofthe Dirac’s bra-ket notation and we also show how its use clarifies the concept of covariancein time-dependent cases. Via an elementary example we finally explain why in certainquantum systems the generator H (gen) of the time-evolution of the wave functions may differ from their Hamiltonian H . Key words: formulation of Quantum Mechanics; cryptohermitian operators of observables;triplet of the representations of the Hilbert space of states; the covariant picture of timeevolution
In contrast to classical mechanics where various formulations of the theory abound, there existnot too many alternative formulations of quantum theory. Moreover, most of their lists (to befound mostly in textbooks and just rarely in review papers like [1]) pay attention just to thehistory of the subject, creating an impression that the formulation of quantum theory does notlead to any interesting theoretical developments.Ten years ago this false impression has been challenged by Bender and Boettcher [2] whosurprised the physics community by a numerically supported conjecture that quite a few one-dimensional quantum potentials V ( x ) may generate bound states ψ n ( x ) with real energies E n even when the potentials themselves are not real. The conjecture seemed to contradict thecurrent experience with quantum mechanics. Using the traditional textbook single-Hilbert-space (1HS) formulation of the theory [3] we usually employ the fact that the separable Hilbertspaces are all unitarily equivalent. This allows us to restrict attention, say, to the exceptionaland most user-friendly representations ℓ or L ( R ) of the Hilbert space of states (cf. Appendix Afor a few additional comments). On this background one can expect a complexification of thespectra whenever the underlying Hamiltonian becomes manifestly non-Hermitian. It is just thisnaive belief which has been shattered by the Bender’s and Boettcher’s illustrative Hamiltonians ⋆ M. Znojilwhich all possessed, as an additional methodical benefit, the most elementary and common form H = p + V ( x ) of the sum of the kinetic and potential “energies”.Obviously, any imaginary component in V ( x ) makes the latter Hamiltonians with real spectrasafely non-Hermitian in L ( R ). This is a paradox which evoked an intensive interest in thefamilies of apparently unphysical Hamiltonians H the Hermiticity of which appeared broken ina theoretically inappropriate but mathematically exceptional and friendly specific representationof an abstract Hilbert space which will be denoted here as H (auxiliary) , H = H † in H (auxiliary) . (1)The progress in this direction of research has been communicated in dedicated conferences(cf. their webpage [4] and proceedings [5] or, better, the recent compact review paper [6]).It became clear that in spite of the undeniable appeal of the models where E n = real while V ( x ) = complex one must treat their “non-Hermiticity” (1) as an ill-conceived concept. Theoperators H with real spectra have been reinterpreted as Hermitian after an appropriate ad hoc redefinition of the Hilbert space H of states. For this purpose one must merely replace theoriginal, false space H (auxiliary) by another, physical space H (standard) of states with the standardquantum-mechanical meaning.A few relevant aspects of such a theoretical and conceptual innovation will be discussed inour present paper. Our introductory Section 2 and Appendix A recollect the main features ofthe currently most popular two-Hilbert-space (2HS) formulation of such a version of quantumtheory which is based on the use of the so called quasi-Hermitian or, better, cryptohermitian representation of observables. In Subsection 2.2, in particular, we return briefly to our recentpaper [9] where the 2HS formalism has been shown suitable for the description of such quasi-Hermitian quantum systems which require the use of manifestly time-dependent operators ofobservables.In Section 3 our present main result is described showing that a thorough simplification ofthe theory can be achieved when one replaces its 2HS formulation by a more appropriate three-Hilbert-space (3HS) reformulation. A few subtleties of the resulting generalized formalism (aswell as of our present amended notation conventions) are illustrated via an elementary time-independent two-dimensional matrix solvable example in Section 4.Section 5 returns again to the results of [9]. We stress there that one of the most remark-able applications of the innovated 3HS formalism can be found in the perceivably facilitatedcovariant construction of certain sophisticated generators of time evolution. For illustration wereturn to the matrix example of Section 4 and we re-analyze its time-dependent generalizedversion in Section 6. We firmly believe that the 3HS description of this and similar examplescan perceivably simplify our understanding of paradoxes which may emerge in quasi-Hermitianmodels.The summary of our results is provided in Section 7. Several apparently anomalous propertiesof the Bender’s and Boettcher’s potentials are discussed there as an inspiration of revisitinga few less rigorous formulations of the first principles of quantum theory. No real necessity ofthe changes of these general principles themselves is encountered. Still, our present modificationof their implementation and of the related conventions in notation appears both desirable andbeneficial. The practical use of phenomenological Hamiltonians H which are apparently non-Hermitian(cf. (1)) does certainly enhance the flexibility of the constructive model-building activities in With the roots dating back to Scholtz et al. [7] or even to Dieudonne et al. [8]. I.e., non-Hermitian in H (auxiliary) but Hermitian in H (standard) . hree-Hilbert-Space Formulation of Quantum Mechanics 3physics even behind the framework of quantum theory (cf. [10]). It also broadens the space for feasible applications of non-local models in a way exemplified by the above-mentioned complex Bender–Boettcher toy potentials [6]. In similar cases, an attachment of the doublet of Hilbertspaces H (auxiliary) and H (standard) to a single quantum system may make good sense. One of the first applications of the apparently non-Hermitian Hamiltonians with real spectraappeared in nuclear physics [7]. The correct physical interpretation of the model in H (standard) has been separated there from the facilitated calculations of the spectrum in H (auxiliary) . Furtherphysical application of the same method appeared in Mostafazadeh’s study of the free relativisticKlein–Gordon equation [11] which is traditionally introduced in the Feshbach’s and Villars’[12] unphysical representation space H (auxiliary) = L ( R ) L L ( R ). A reconstruction of theinner product has been offered as a means of recovering the consistent picture of physics. Thesame approach avoiding spurious states or negative probabilities has been extended to the first-quantized models of massive particles with spin one [13].On theoretical level the 2HS reformulation of quantum theory might look almost trivial.Still, the application of the idea to the Bender’s and Boettcher’s elementary examples and theresolution of some of the related puzzles took time [14]. Fortunately, the theory seems to beclarified at present. Its key mathematical feature lies in the Hamiltonian-dependent replacementof the spaces, H (auxiliary) → H (standard) . (2)The detailed description of its mathematical subtleties can be found explained in the availableliterature. The innovative 2HS approach to the description of pure states in a quantum systemcharacterized by an apparently non-Hermitian Hamiltonian can even be presented using thestandard 1HS language (cf., e.g., [15]). In such an approach it is only necessary to introducea rather complicated notation in which the same state is characterized by two different Greekletters (say, Φ and Ψ as recommended in [15]).Some details of this convention are recollected and summarized in Appendix A.2. Here, letus only emphasize that we must remember that although equation (2) does not involve a changeof the underlying vector space V itself, it does modify the inner product in this space. Thus,we must introduce two graphically different Dirac’s bra-vector symbols associated with theindividual Hilbert spaces H (auxiliary / standard) . Of course, this enables us to restore the necessaryphysical Hermiticity of our Hamiltonian, H = H ‡ in H (standard) . In the other words, even if we start from a non-Hermitian model (1), we may update the correctphysical form of the Hilbert space and re-establish, thereby, the validity of all of the standardpostulates of quantum theory.
Although we do not intend to accept the above 1HS notation conventions in our present paper,we would still like to keep our present paper self-contained. For this reason we added furthercomments on the 1HS notation and postponed them to Appendix A. The main reason is that weare persuaded that the consequent 2HS notation which makes an explicit use of the two spacesappearing in (2) is perceivably simpler and less confusing.We have to admit in advance that neither of the two spaces H (auxiliary) and H (standard) offersin fact a conceptually fully satisfactory frame for wave functions of a given quantum system. M. ZnojilIndeed, the former space remains manifestly unphysical while the work in the latter one requiresthe construction and use of a Hamiltonian-dependent metric operator Θ = I . Still, our recentapplication [9] of the 2HS ideas to models with a nontrivial dependence on time re-demonstratedthe mathematical strength as well as physical productivity of the 2HS approach (cf. Table 1). Table 1.
Concise summary of the extended 2HS notation as employed in [9].
Hilbert space ket state its dual its Hamiltonian H (auxiliary)(unphysical) ≡ H ( A ) | Φ i h Φ | H = H † H (standard)(physical) ≡ H ( P ) | Φ i h Ψ | = h ϕ | Ω H = H ‡ H (auxiliary)(physical) ≡ H ( A ) | ϕ i = Ω | Φ i h ϕ | = h Φ | Ω † h = Ω H Ω − = h † Some of the key ideas of [9] were inspired by the transparency of the notation as suggestedin [16]. The core of their efficiency lies in the simultaneous use of two different basis sets in the same friendly Hilbert space H (auxiliary) (denoted as H ( A ) in [9]). This effectively separated theoriginal computing-frame role of this space from its other role of a benchmark physical space.A certain invertible non-unitary transformation Ω : H ( A ) → H ( A ) has been invented as formallyconnecting these two roles of space H (auxiliary) . Section III of paper [9] could be consulted formore details. The correspondence between these two roles is reflected also by the first and lastrow in Table 1.The clarity of the message mediated by Table 1 is weakened by the fact that our notationhas been taken from [15] in spite of its being not too suitable for the given purpose. Indeed, thecomparison of Table 1 with the 1HS Table 3 of Appendix A.2 shows that the separation of thetwo bases is not well reflected by the notation. The necessary use of the third reserved Greekletter ϕ representing the same state only enhances the danger of confusion. A more thoroughlyamended version of the Dirac’s notation is to be offered in the next section. During the proofreading of the text of [9] we imagined that it offers a slightly confusing pictureof cryptohermitian quantum systems, especially due to the use of the imperfect 2HS notationas sampled here in Table 1. As we already noticed, it is rather unfortunate that this notationemploys three different Greek letters (viz., Φ, Ψ and ϕ ) representing the same physical state. Inaddition, this notation also introduces a strange asymmetry between the two Hilbert spaces H ( A ) and H ( P ) .There is in fact no reason why the former one should be treated as a single Hilbert spacebecause its underlying vector space V is in fact being equipped with the two different innerproducts. This is also the driving idea of our present proposal of transition from 2HS to 3HSlanguage. Its mathematical background is virtually trivial as it makes merely use of the wellknown formal unitary equivalence between any two (separable) Hilbert spaces. Once appliedto the two physical spaces of Table 1, we may declare the parallel mathematical and physicalequivalence between the second and the third item of this Table, i.e., between the standard andauxiliary physical Hilbert spaces even if they cease to share the underlying vector space V .The main advantage of the resulting 3HS separation of the constructive definitions of thelatter two representations of the Hilbert space lies in the possibility of the decoupling of thehree-Hilbert-Space Formulation of Quantum Mechanics 5underlying vector spaces, V = V H (standard)(physical) = V H (new auxiliary)(physical) := W (3)(cf. Appendix A for notation). In its turn, such a new, 3HS-specific freedom (3) enables us toget rid of the extremely unpleasant nontriviality of the metric also in the latter, physical Hilbertspace,Θ (new auxiliary)(physical) = I. It is encouraging to see that the only price to be paid for this 3HS freedom lies just in the (non-unitary) generalization of the mapping Ω which will now be acting between the two different vector spaces,Ω :
V −→ W . A more detailed analysis of some other consequences of the new perspective may be found inAppendix B below.
In the Bender–Boettcher-type bound-state models where H = H † in H (auxiliary) it proved con-venient to factorize their nontrivial, non-Dirac metric in H (standard) , either in the form Θ = CP (where P is parity and C represents a charge [6]) or in the form Θ = PQ (where Q is quasi-parity [17]). Unfortunately, after we turn attention to the other quantum systems with thescattering-admitting Hamiltonians H = T + V [18], we discover that the construction of anappropriate metric Θ = I only remains feasible for certain extremely elementary models ofdynamics [19].In this context, our present 3HS formulation of Quantum Mechanics found one of its sec-ondary sources of inspirations in the possibility of a return from Θ (non-Dirac) to Θ (Dirac) . Thisindicates that the second (in principle, extremely complicated but still norm- and inner-productpreserving) update of the physical Hilbert space H (standard)(non-Dirac) −→ H (new auxiliary)(Dirac) := H ( T ) (4)proves desirable and very natural. It can also be read as an introduction of the third Hilbertspace H ( T ) . Thus, equation (4) complements equation (2) above.All the necessary details and formulae can be found again shifted to Appendix B. Here,let us only summarize that the symbol H ( T ) representing the third space in equation (4) willbe accompanied, in what follows, by the other two abbreviations representing the first and the second Hilbert spaces H (auxiliary)(unphysical) := H ( F ) , H (standard)(physical) := H ( S ) of Table 1, respectively. In summary we can now recommend that the differences in mappingsbetween our three different Hilbert spaces H ( F,S,T ) can be very easily reflected by the differencesin a rationalized Dirac notation where just the graphical form of the bras and kets will be varied.This will enable us to correlate the graphical form of the bras and kets with the three in-dividual Hilbert spaces. At the same time, the same letter (say, ψ ) will always represent thesame physical state. Preliminarily, this pattern of notation is summarized in Table 2. Multipleparallels with Table 1 can be noticed here. M. Znojil Table 2. ψ in three alternative representations. Hilbert space ket-vector bra-vector norm squared H ( F )(friendly) | ψ i ∈ V h ψ | h ψ | ψ iH ( S )(standard) | ψ i ∈ V hh ψ | = ≺ ψ | Ω hh ψ | ψ i = h ψ | Ω † Ω | ψ iH ( T )(textbook) | ψ ≻ = Ω | ψ i ∈ W ≺ ψ | = h ψ | Ω † ≺ ψ | ψ ≻ = h ψ | Ω † Ω | ψ i The explicit use of mappings between Hilbert spaces is quite common in textbooks [3] wherea unitary map (e.g., Fourier transformation Ω) produces the correspondence. In the 3HS contextthe same transition is being postulated, | ψ i ∈ H ( F,S ) = ⇒ | ψ ≻≡ Ω | ψ i ∈ H ( T ) . (5)Nevertheless, the majority of the nontrivial aspects of the present three-Hilbert-space approachto quantum models will only emerge when Ω ceases to be norm-preserving (let us still sayunitary). In such a setting the two physical spaces of states H ( S ) and H ( T ) are accompaniedby their unphysical partner H ( F ) . This partnership can already be perceived as aiming ata reformulation of quantum theory. Another hint lies in the isospectrality of the Hamiltonians,of which h acts in H ( T ) while H ≡ Ω − h Ω acts in H ( F ) or in H ( S ) . This opens a constructivepossibility of the choice of a Hamiltonian which is allowed to be non-Hermitian (cf. equation (1)). In nuce , our present main technical trick is that in place of the unitary transformation ofspaces (4) (i.e., the second option in equation (5)) we intend to achieve the same goal indirectly,by means of the technically less difficult and non-unitary transition between the other twoHilbert spaces [i.e., via the first option in equation (5)], H (auxiliary)(Dirac) (cid:2) = H ( F ) (cid:3) → H (standard)(Dirac) (cid:2) = H ( T ) (cid:3) both of which are equipped with the same and, namely, trivial Dirac metric. The key featuresof the latter idea may be read out of the parallelled Tables 1 and 2.A detailed inspection of Table 2 reveals the coincidence of the kets in the “ F ” and “ S ” doublet.The mapping between the respective “ S ” and “ T ” Hilbert spaces H ( S ) and H ( T ) preserves theinner product and is, in this sense, unitary, ≺ ψ ′ | ψ ≻ = hh ψ ′ | ψ i . Equivalent physical predictions will be obtained in both of the latter spaces. The third pair ofthe “Dirac-metric” spaces with Θ (Dirac) = I and superscripts “ F ” and “ T ” shares the form ofthe Hermitian conjugation.In such a balanced scheme the space H ( T ) is slightly exceptional. Not only by its full com-patibility with the standard textbooks on quantum physics but also by its role of an extremelycomputing-unfriendly (i.e., practically inaccessible) representation space. In both of these rolesits properties are well exemplified by the overcomplicated fermionic Fock space which occurredin the above-mentioned nuclear-physics context [7]. Summarizing, all of the three spaces inhree-Hilbert-Space Formulation of Quantum Mechanics 7Table 2 can be arranged, as vector spaces, in the following triangular ket-vector patternvector space W physics clear in H (T) kets | ψ ≻ = uncomputablemap Ω ր ց map Ω − vector space V mathematics OK in H (F) kets | ψ i = computable map ΩΩ − = I −→ vector space V math . phys . synthesis : H (S) kets | ψ i = the sameIn parallel we have to study the bras. After the transition to the conjugate vector spaces offunctionals the above-indicated pattern gets modified as follows,dual vector space W ′ ≺ ψ | ∈ W ′ constructions prohibitively difficultmap Ω † ր ց map Ωdual vector space V ′ h ψ | ∈ V ′ absent physical meaning map Θ=Ω † Ω = I −→ modified dual space hh ψ | = h ψ | Θ ∈ (cid:2) H ( S ) (cid:3) ′ non-Dirac conjugation In the second row of Table 1 one finds the condition of the Hermiticity of our Hamiltonian h in H ( T ) . The pullback of this condition to H ( F ) gives H † = Θ H Θ − , Θ = Ω † Ω = Θ † > H has been proposedby Mostafazadeh [20]. In a toy space H ( F ) which is just two-dimensional, this Hamiltonian isrepresented by the two-dimensional matrix H = H ( AM ) ( r, β ) = (cid:18) r e i β r − e − i β (cid:19) (7)which is strictly two-parametric, β ∈ (0 , π ) and r ∈ R \ { } . Its condition of quasi-Hermiti-city (6) can be read as four linear homogeneous algebraic equations determining the matrixelements of all the eligible positive definite matrices Θ = Θ ( AM ) . The general solution of theseequationsΘ ( AM ) = f · Θ Z , Θ Z = (cid:18) r e i β cos Zre − i β cos Z r (cid:19) (8)depends on two new parameters or, if we ignore the overall factor f , on Z ∈ (0 , π ).Any other observable quantity must be represented by the operator Λ which is also quasi-Hermitian with respect to the same metric,ΘΛ = Λ † Θ . (9)The inverse problem of specification of all of the eligible Λs is, in our schematic example, easilysolvable,Λ = (cid:18) a pe i β qe − i β d (cid:19) . (10) M. ZnojilIn this family of solutions the range of the four real parameters a , p , q and d is only restrictedby the inequality ( a − d ) > pq which guarantees the reality of both the observable eigenvaluesand by the single nontrivial constraint resulting from quasi-Hermiticity equation, p = qr + ( a − d ) r cos Z. (11)Once we fix Z and f > Z = Ω † Z Ω Z of the metriccan be performed, say, in terms of triangular matrices,Ω Z = (cid:18) re i β cos Z r sin Z (cid:19) , Ω − Z = (cid:18) − e i β cot Z / ( r sin Z ) (cid:19) . This definition specifies, finally, the family of eligible selfadjoint Z -dependent Hamiltonians h ( AM ) ∼ h Z = (cid:18) cos Z e i β sin Ze − i β sin Z − cos Z (cid:19) defined by pullback to H ( T ) and isospectral with H ( AM ) . The same pullback mapping must bealso applied to the second observable Λ of course.Marginally, let us mention that in many items of the current literature (well exemplifiedby [11]) the factorization of Θ is only being made in terms of very special Hermitian and positivedefinite mapping operators Ω (herm) = Θ / . In this context our illustrative example shows thatthe choice of the special Ω (herm) is just an arbitrary decision rather than a necessity dictated bythe mathematical framework of quantum theory. We shall see below that a consistent treatmentof the ambiguity of our choice of Ω is in fact very similar to the treatment of the ambiguityencountered [7] during the assignment of the metric Θ to a given Hamiltonian H . In the 3HS formulation of quantum physics we may start building phenomenological modelsinside any item of the triplet H ( F,S,T ) . Still, in a way noticed in [20] and worked out in [9]this freedom may be lost when one decides to admit also the models where the parameterswhich define the system and its properties (i.e., say, the parameters in the quasi-HermitianHamiltonian H of equation (7) and/or in another observable Λ given by equation (10)) becomeallowed to vary with time. Typically, this time-dependence may involve not only the externalforces which control the system but also, in principle, the related measuring equipment.In such an overall setting one has to imagine that the constraints imposed upon the assignmentof a suitable metric Θ, say, to a given H and Λ may prove impossible. In the language of ourillustrative example this danger has been illustrated in [20] where in illustrative example (7)the so called quasistationarity condition Θ = Θ( t ) has been shown to imply that r = r ( t ) and β = β ( t ). The mathematical reason was easy to find since the quasi-stationary time-dependentgeneralization of the set of constraints (6) and (9) proved overcomplete. In [9] we have shown the reasons why the same overrestrictive role is played by the overcomplete-ness also in the generic, model-independent quasi-stationary scenario. In the opposite direction,our study [9] recommended to relax the quasi-stationarity constraint as too artificial. Then, the3HS formalism proved applicable in its full strength. Now, we intend to show that it restrictsthe form of the time-dependence of the operators of observables as weakly as possible.We shall again “teach by example” and demonstrate our statement via the same schematicexample as above. Still, we have to recollect the basic theory first. Thus, we start from thehree-Hilbert-Space Formulation of Quantum Mechanics 9exceptional space H ( T ) which offers the absence of all doubts in the physical interpretation ofany 3HS model. In particular, the time-evolution will be controlled by the textbook Schr¨odingerequation in H ( T ) ,i ∂ t | ψ ( t ) ≻ = h | ψ ( t ) ≻ . This is fully compatible with the textbook concepts of the measurement [3] based on the ideathat the complete physical information about a given system prepared in the so called pure stateis compressed in its time-dependent wave function | ψ ( t ) ≻ .Secondly, in H ( T ) there are also no doubts about the standard postulate that all of the othermeasurable characteristics of the system in question are obtainable as eigenvalues or mean valuesof the other operators of observables exemplified here, for the sake of simplicity, by λ = ΩΛΩ − .The analysis of this aspect of the problem is postponed to the next section here. In a preparatory step of the study of the time-evolution problem let us just be interested in thesingle observable (viz., Hamiltonian) and let us admit that a manifest time-dependence occursin all of the operators, h ( t ) = Ω( t ) H ( t )Ω − ( t ) . An easier part of our task is to write the time-dependent Schr¨odinger equation in H ( T ) ,i ∂ t | ϕ ( t ) ≻ = h ( t ) | ϕ ( t ) ≻ . Its formal solution | ϕ ( t ) ≻ = u ( t ) | ϕ (0) ≻ employs just the usual evolution operator,i ∂ t u ( t ) = h ( t ) u ( t ) (12)which is unitary in H ( T ) so that we can conclude that the norm of the state in question remainsconstant, ≺ ϕ ( t ) | ϕ ( t ) ≻ = ≺ ϕ (0) | ϕ (0) ≻ . In the next step we recollect all our previous considerations and define | ϕ ( t ) i = Ω − ( t ) | ϕ ( t ) ≻ and hh ϕ ( t ) | = ≺ ϕ ( t ) | Ω( t ). We are then able to distinguish between the two formal evolutionrules in the physical space H ( S ) . One of them controls the evolution of kets, | ϕ ( t ) i = U R ( t ) | ϕ (0) i , U R ( t ) = Ω − ( t ) u ( t )Ω(0)while the other one (written here in its H ( F ) -space conjugate form) applies to ketkets |·ii ≡ ( hh·| ) † , | ϕ ( t ) ii = U † L ( t ) | ϕ (0) ii , U † L ( t ) = Ω † ( t ) u ( t ) (cid:2) Ω − (0) (cid:3) † . The pertaining differential operator equations for the two (viz., right and left) evolution opera-tors readi ∂ t U R ( t ) = − Ω − ( t ) [i ∂ t Ω( t )] U R ( t ) + H ( t ) U R ( t )and i ∂ t U † L ( t ) = H † ( t ) U † L ( t ) + (cid:2) i ∂ t Ω † ( t ) (cid:3)(cid:2) Ω − ( t ) (cid:3) † U † L ( t ) , H ( S ) , the conservation of the norm hh ϕ ( t ) | ϕ ( t ) i of states is re-established and paralleledby the same phenomenon inside H ( T ) . One has to solve, therefore, the two equations whichform the doublet of non-Hermitian partners of the standard single evolution equation (12) inthe third and most computation-friendly space H ( F ) .We may conclude that the conservation of the norm of the states which evolve with timein H ( S ) becomes a trivial consequence of the unitary equivalence of the model to its image in H ( T ) (cf. the explicit formulae in Table 1). One can also recommend the abbreviation ˙Ω( t ) ≡ ∂ t Ω( t )which enables us to introduce the time-evolution generator in H ( F ) , H (gen) ( t ) = H ( t ) − iΩ − ( t ) ˙Ω( t ) . Its most remarkable feature is that it remains the same for both the time-dependent Schr¨odingerequations in H ( S ) ,i ∂ t | Φ( t ) i = H (gen) ( t ) | Φ( t ) i , (13)i ∂ t | Φ( t ) ii = H (gen) ( t ) | Φ( t ) ii . (14)A virtually equally remarkable feature of the operator H (gen) ( t ) is that it ceases to be an elemen-tary observable in H ( S ) [21]. This is very natural of course. The reason is that by our assumptionthe time-dependence of the system ceases to be generated solely by the Hamiltonian (i.e., energy-operator). Indeed, the manifest time-dependence of the other operators of observables representsan independent and equally relevant piece of input information about the dynamics. The core of our present message is that after the change of the representation of the Hilbert spaceof states H ( T ) → H ( S ) one should not insist on the survival of the time-dependent Schr¨odingerequation in its usual form where the Hamiltonian acts as the generator of time shift. We haveshown that the doublet of equations (13) + (14) must be used instead. Such a replacementopens the space for the consistent use of a broad class of metrics (i.e., spaces H ( S ) ) which varywith time!The consequences of the new freedom in Θ = Θ( t ) may be illustrated again on our elementarytwo-by-two model (7) where both the parameters r = r ( t ) ∈ R \ { } and β = β ( t ) ∈ R maynow arbitrarily depend on time. In such an exemplification the explicit time-dependence of themetricΘ( t ) = f ( t ) · Θ Z ( t ) , Θ Z ( t ) = (cid:18) r ( t ) e i β ( t ) cos Z ( t ) r ( t ) e − i β ( t ) cos Z ( t ) r ( t ) (cid:19) appears to have two independent sources. Firstly it results from the direct transfer of the mani-fest time-dependence from the Hamiltonian H ( t ). Secondly, a new source of time-dependenceenters via the new free function Z ( t ) of time. Obviously, its time-dependence may be read either as responsible for the time-dependence freedom in the metric Θ( t ) or as a consequence of the time-dependence of the second observable Λ( t ) which is, in principle, independently pre-scribed .Thus, via the existence and role of the function Z ( t ) of time our example illustrates that thetime-variation of both H ( t ) and Λ( t ) must be read as the two components of an input, external information about the dynamics of our system. This information cannot be contradicted by anyadditional constraints upon the metric and, in this sense, this information restricts the freedomin our consistent choice of the metric Θ( t ).hree-Hilbert-Space Formulation of Quantum Mechanics 11For the first time the latter connection between metric Θ( t ) and observables H ( t ) and Λ( t )has been noticed by the authors of [22] and formulated in [20]. One can conclude that insimilar situations we are not allowed to impose additional conditions upon Θ( t ) so that theinner products in our Hilbert space H ( S ) vary with time in general.This looks like a paradox but its essence is sufficiently clearly illustrated by our examplewhere, due to its simplicity, all the one-to-one correspondence between the variations of ob-servables and the metric is mediated by the mere single function Z ( t ). The time-dependenceof Z ( t ) enters the game via an external specification of some particular form of the observ-able Λ( t ). Nevertheless, we already know that this observable must have the above-mentionedfour-parametric structure (10). Its four real parameters a ( t ), p ( t ), q ( t ) and d ( t ) must be mutuallycoupled by the time-dependent version of constraint (11), p ( t ) = q ( t ) r ( t ) + [ a ( t ) − d ( t )] r ( t ) cos Z ( t ) . This formula may easily be reinterpreted as an implicit def inition of the originally arbitraryfunction, Z ( t ) = arccos p ( t ) − q ( t ) r ( t )[ a ( t ) − ad ( t )] r ( t ) . In our illustrative two-by-two example we may conclude that the input information about thetime-dependence of Λ( t ) represents and independent and relevant contribution to the time-dependence of the metric. Thus, even if we just parallel the construction of Section 4 andemploy the triangular factorization of the metric with,Ω Z ( t ) = (cid:18) r ( t ) e i β ( t ) cos Z ( t )0 r ( t ) sin Z ( t ) (cid:19) the difference between the two operators H ( t ) and H (gen) ( t ) (representable by a slightly cum-bersome closed formula) will not vanish in general. The main message of this text is a confirmation of the full compatibility of several recentapplications of quantum theory with its standard form known to us from textbooks. This beingsaid, the present innovative 3HS formulation of the theory can be classified as lying very close,in its spirit if not dictum, to the one offered by Scholtz et al. [7]. Here we just showed thatthere is no need to feel afraid of the use (i) of the 3HS-extended Dirac notation and (ii) of theconcept of covariance in the applications of the theory to the systems where observables happento be manifestly time-dependent.In the literature, depending on the respective authors, the applications in question carry therespective names of quasi-Hermitian quantum mechanics (cf. the oldest two proposals offered,practically independently, in mathematics [8] and physics [7]), PT -symmetric quantum mecha-nics (= the best advertised Carl Bender’s trademark [6]), pseudo-Hermitian quantum mechanics(= perceivably more general concept dating back to M.G. Krein and given second life by AliMostafazadeh [15], with possible applications reaching far beyond quantum mechanics) or cryp-tohermitian quantum mechanics (by my opinion, the most explanatory name proposed, onlyvery recently, by Andrei Smilga).Although the formalism of quantum theory is most often formulated in the language offunctional analysis and linear algebra (LA, using Hilbert spaces, etc), the concrete realizations of the operators of observables (i.e., Hamiltonians, etc) are very often chosen as fairly elementarydifferential operators. The first two physicists who noticed a deep mathematical relationship2 M. Znojilbetween these two ingredients in phenomenology were probably Bender and Wu [23]. Almostforty years ago they discovered that for the quartic anharmonic oscillator of coupling g > all the spectrum of bound state energies E ( g ) < E ( g ) < · · · is given by the single analyticfunction of complex g .Although this amazingly close correspondence between the analytic and LA aspects of boundstates still eludes a full appreciation, its other manifestation has been revealed by Bender andBoettcher ten years ago [2]. They conjectured that for a family of elementary and analytic com-plex oscillator potentials all the spectrum of bound state energies E < E < · · · is strictly real .Three years later, the purely analytic aspects have been shown decisive in a rigorous proof ofthe reality of the spectrum [24]. One year later, several consequences have been drawn also in a2HS reformulation of the underlying LA formalism of quantum theory [14]. An ad hoc operatorof metric Θ = CP has been introduced there. In LA language, as a indirect consequence of theanalyticity of the Bender’s and Boettcher’s potentials, two non-equivalent Hilbert spaces, i.e.,in our present notation, H ( F,S ) proved needed.A counterintuitive limitation of the 2HS formalism to the models with time-independent met-rics Θ = Θ( t ) has been revealed by A. Mostafazadeh [20]. In [9] we removed this limitation viaan introduction of the third Hilbert space (in our present notation, of H ( T ) ). Unfortunately, wepreserved the Mostafazadeh’s 2HS notation which made the resulting gem of the 3HS formalismclumsy.In our present paper we introduced, therefore, an adequate generalization of the traditionalDirac notation and described the updated 3HS formulation of quantum theory in full detail.We also illustrated this formalism via an elementary solvable two-dimensional matrix example.This example demonstrates some details of the theory and of its applications. In particular,the solvability of this example underlines an easy nature of the transition to the models withtime-dependent metric.In the summary we would like to emphasize that the use of the triplet of Hilbert spaces H ( F,S,T ) and of the related adapted Dirac notation can find its applicability in many apparentlydifferent contexts ranging from computational and variational nuclear physics [7] and analyticperturbation theory [25] through phenomenological field theory [6] up to the first quantizationof Klein–Gordon field [11] or Proca’s field [13]. For the nearest future one could thereforeencourage the use of the time-dependent metric Θ( t ) in all of these physically interesting con-texts. A The standard Dirac notation and its shortcomings
In some elementary introductions to Quantum Mechanics the abstract concept of Hilbert space H is being replaced by its concrete representations, say, ℓ where the elements of the space areidentified with the ordered sets of complex numbers arranged as column vectors. To each ofthese elements one then associates a dual row vector obtained via the so called Hermitian-conjugation operation (i.e., transposition plus complex conjugation). Similarly, the analysisof some concrete physical quantum systems can be based on the restriction of our attentionto the other concrete representations of H like, e.g., the functional space L ( R ) as mentionedin Section 1 or the Klein–Gordon-equation space H (auxiliary) = L ( R ) L L ( R ) as cited at thebeginning of Section 2.One of the most serious weaknesses of such a pedagogically simplified approach is that when-ever we start analyzing the quasi-Hermitian models with property (1) we are forced to turnattention to a modified inner product [7]. This means that in order to avoid confusion one hasto replace the concrete models like ℓ by the more general notion of the abstract Hilbert spaceas given, say, by von Neumann.hree-Hilbert-Space Formulation of Quantum Mechanics 13 A.1 Dif f iculties with variable inner products
The term “Hilbert space” and its symbol H can be assigned the precise mathematical meaningin several ways. One of the most common definitions specifies the abstract Hilbert space H asa vector space V = V H equipped with a suitable inner product. By this “product” a complexnumber c ∈ C is assigned to any pair of elements a and b of V . In the Dirac-inspired notationmany authors write simply c = h a | b i (cf., e.g., [15]).The main advantage of such a von Neumann-inspired approach is that it does not require theknowledge of the more general concept of Banach spaces and that it still enables us to overcomecertain mathematical difficulties in rigorous manner. The most serious ones emerge when thevector space V ceases to be finite-dimensional. Then, the abstract inner-product spaces of thistype are required complete as metric spaces. A dual partner V ′ of the vector space V itself is alsovery easily defined, in this language, as a set of bounded linear maps µ : V → C . Furthermore,the requirement of the self-duality of Hilbert space H can be reformulated as the existence of a (byfar not unique) isomorphism T between vector spaces V and V ′ . The “canonical” isomorphism T (Dirac) : a → µ (Dirac) a with a ∈ V is usually introduced by the formula µ (Dirac) a ( · ) = h a |·i , i.e., itclearly and explicitly depends on our choice of the initial inner product h·|·i .Whenever we decide to work with the single and fixed inner product h·|·i , we are allowed toidentify the elements a of V with the Dirac’s ket-vectors | a i . The parallel identification of thelinear functionals µ (Dirac) b ( · ) ∈ V ′ with the Dirac’s bra-vectors h b | remains equally straightforwardbut it will not survive a change of the inner product. This is an important observation. Wheneverwe follow Bender and Boettcher [2] and turn our attention to a quasi-Hermitian quantum modelwe are forced to endow the given, single vector space V with two different inner products. Someauthors [15] characterize the second, modified inner product by its double bracketing, hh·|·ii . Theprice to be paid for this rather unfortunate decision is that the consistent use of the unmodifiedDirac’s notation ceases to be possible. A.2 Mostafazadeh’s [15] single-Hilbert-space notation conventions
It is not too easy to find an appropriate notation when a given phenomenological Hamiltonian isquasi-Hermitian, i.e., manifestly non-Hermitian in the sense of equation (1). The main difficultyis that in order to get the states properly normalized, one has to move from the initial, naiveHilbert space H (auxiliary) to the physically correct H (standard) .In similar situations, one often feels afraid of using the Dirac’s notation or its analogs . Inpractice, such a loss of contact between the Hermitian and quasi-Hermitian observables wouldbe rather unpleasant. For this reason people often try to preserve at least part of this nota-tion. Mostafazadeh [15] offers one of the latter, modified notation conventions which becamerather popular in the related literature. Let us briefly recollect some of its principles and rules,therefore.First of all we have to imagine that the violation of the Hermiticity of a given Hamiltonian H in H (auxiliary) implies that the eigenstates of H and of its conjugate H † (as defined in H (auxiliary) )will be different in general. This observation has led to the recipe of [15] where one constructsthe two independent series of the respective eigenstates pertaining to the same energy eigenva-lues E n . These states (numbered by n = 0 , , . . . ) must be denoted by two dif ferent symbols (say,by the “reserved” respective Greek letters Φ n and Ψ n ) when interpreted as ket- and bra-vectorsin H (standard) , h Φ | = ( | Φ i ) † ∈ (cid:2) H (auxiliary) (cid:3) ′ , h Ψ | := h Φ | Θ = ( | Φ i ) ‡ ∈ (cid:2) H (standard) (cid:3) ′ . This is a wide-spread opinion, especially among mathematicians. I.e., two series of ket-vector elements of the same vector space V H (auxiliary) . H of states, two different definitions of the operation of the Hermitian conjugation emerged in ourconsiderations. Table 3. not to be used here. inner product elements of V their duals their norms Hamiltonians H h·|·i (in H (auxiliary) ) | Φ i h Φ | = ( | Φ i ) † h Φ | Φ i “pseudo-Hermitian” hh·|·ii (in H (standard) ) | Φ i h Ψ | = ( | Φ i ) ‡ h Ψ | Φ i “quasi-Hermitian”We see that even without switching to the full-fledged 2HS or 3HS language , one must keeptrace of the change of the inner product via an appropriate modification of the notation. Inthis manner one formally re-establishes the rigorous meaning of the necessary Hermiticity of theHamiltonian in the updated, second, physical Hilbert space, H = H ‡ in H (standard) . As we already mentioned, the so called metric operator (denoted by symbol Θ here) can be alsoused in order to make the underlying definitions less implicit, O ‡ ≡ Θ − O † Θ . In this manner one establishes the new, generalized, Θ-dependent Hermitian conjugation oper-ation which applies to any operator O acting in H (standard) . B A few remarks on the simplif ied Dirac notationin both the 2HS and 3HS approaches
As long as the dual-vector definition is inner-product dependent, the very operation of the(generalized) Hermitian conjugation is metric-dependent. We recommend that the single-crosssuperscript † stays reserved for its use in H (auxiliary) where Θ = I and that it becomes paralleledby its double-cross analogue ‡ in H (standard) where Θ = I . B.1 Metric operator Θ and the 2HS language
A change of the inner product does not necessarily require a simultaneous change h a | b i → hh a | b ii of both the Dirac’s bra and ket graphical symbols. This point of view will be advocated in whatfollows. Although, formally, its formulation could start from the idea of selfduality and be madeequally rigorous as in the approach presented in the preceding two subsections, we shall skip thedetails here.Some of the practical merits of such an approach have been discussed and advocated, e.g.,in the study [16] of quantum knots. An even less formal notation convention has been shownworking in one of the oldest papers on the subject of quasi-Hermiticity [7]. There one findsno double brackets and only the so called metric operator Θ appears there as a key to thetransition (2) between Hilbert spaces. I.e., not speaking openly about the two or three different
Hilbert spaces H . The original symbol T is “translated in Greek” here since it would interfere with the time-reversal symbol. hree-Hilbert-Space Formulation of Quantum Mechanics 15Even in the 1HS language, the temporary return to Table 3 and the update hh a | b ii −→ h a | Θ | b i of the graphical representation of the lower inner product seems fairly well suited for empha-sizing the inner-product dependence of the canonical linear functionals. Although such a 1HSconvention seems insufficient in more complicated models, its 2HS amendment of [9] provedalready sophisticated enough to clarify the emergence of Θ = I and to render it interpreted asan introduction of a new dual space V ′ (non-Dirac) with modified elements (i.e., functionals) µ (non-Dirac) b ( · ) = h Θ b |·i . This version of the 2HS approach can be complemented by the proposal of a more compactnotation as presented, say, in [16]. In essence, one just abbreviates µ (non-Dirac) b ( · ) −→ hh b |·i . B.2 A few details of the transition to the 3HS picture
Let us assume that the operators of observables Λ , Λ , . . . are admitted non-Hermitian andthat, via a certain non-unitary map Ω, the naive, ill-chosen Hilbert space H (auxiliary) is replacedby another, physical Hilbert space H (standard) where the usual probabilistic interpretation of thesystem in question is being restored. At this point, we shall require that each one of the two rolesof H (auxiliary) will be carried by a separate Hilbert space. The first incarnation of H (auxiliary) will be denoted by the symbol H ( F ) . It will keep playing the auxiliary role of a mathematicallyfriendly space without any immediate physical contents. The second, unitarily nonequivalentspace will be treated as the truly physical Hilbert space H ( T ) which is, by assumption, neithermathematically easily accessible nor user-friendly in the context of physics. Its only comparativeadvantage will be assumed to lie in a return to the simplicity of the metric, Θ ( T ) = I .After such an absolutely minimal extension of the current Dirac conventions we may nowdemand that the third Hilbert space H ( T ) is built over a new , independent vector space W 6 = V with the elements a to be marked by the specific, say, spiked version of the Dirac’s kets, a → | a ≻ .The dual space forms again a vector space W ′ with elements b = ≺ b | . As long as, by definition,Hilbert spaces are self-dual the two vector spaces W and W ′ must be isomorphic . B.2.1 Marking the coexistence of two conjugations
The two Hilbert spaces H ( F,S ) were defined here over the same vector space of kets | ψ i . In themost common applications this means that all of these kets may be treated as linear superpo-sitions of the right eigenkets | n i of the same upper-case Hamiltonian H . As mentioned above,the difference between H ( F ) and H ( S ) only emerges during the introduction of the dual spaceof functionals. In the former case the elements of the dual space are defined by the “standard”(i.e., Dirac’s) Hermitian conjugation. In the language of the so called PT -symmetric quantummechanics [6] this definition is provided by an antilinear operator T = T ( F ) from the vectorspace H ( F ) in its dual space (cid:0) H ( F ) (cid:1) ′ . It transforms each ket-vector into the “usual” bra-vector, T ( F ) : | ψ i −→ h ψ | , | ψ i ∈ H ( F ) . (15)Inside the second Hilbert space H ( S ) , in contrast, another antilinear operation T = T ( S ) definesthe different Hermitian-conjugation operation. At this point one must pay an enhanced attentionto the notation, underlying the necessity of a clear distinction between the “old” (i.e., current,Dirac’s) Hermitian conjugation (15) employed inside the first Hilbert space H ( F ) and the entirelynew, very nonstandard conjugation T ( S ) which is activated inside the second Hilbert space, T ( S ) : | ψ i −→ hh ψ | , | ψ i ∈ H ( S ) . (16) Without any loss of generality we may work again with the canonical form of the isomorphism denoted, say,by the symbol T ( T ) and meaning, in the Dirac’s notation, that T ( T ) : | ψ ≻→≺ ψ | . hh ψ | literally as linear functionals in H ( S ) .An easy guide to the acceptance of the alternative, non-Dirac conjugation (16) appears whenwe notice that in contrast to the Dirac’s inner product h a | b i between | a i ∈ H ( F ) and | b i ∈ H ( F ) ),the new Hilbert space H ( S ) simply allows us to use the new, perceivably less usual, non-Diracinner product h a | Θ | b i between its elements | a i ∈ H ( S ) and | b i ∈ H ( S ) . Of course, in our presentnotation we have hh ψ | ψ i ≡ h ψ | Θ | ψ i . This means that while working in H ( S ) the explicit remarks concerning the nontrivial metric Θare redundant. They must only be made when, for some reason, we must or wish to turn ourexplicit attention to the former Hilbert space H ( F ) . B.2.2 Simplifying spectral formulae
Besides the most usual spectral formula representing, say, the Hamiltonian in H ( T ) , h = ∞ X n =0 | n ≻ E n ≺ n | a perceivably more complicated relation emerges for its isospectral partner which is an operatorin H ( F ) or H ( S ) , H = ∞ X n =0 Ω − | n ≻ E n ≺ n | Ω . (17)Together with the natural innovation of the basis kets in H ( F,S ) , | n i := Ω − | n ≻ we may also introduce another set of “ketket” vectors in both these spaces, | n ii := Ω † | n ≻≡ Ω † Ω | n i ≡ Θ | n i ∈ H ( F,S ) . Formula (17) then acquires a particularly compact form in H ( S ) , H = ∞ X n =0 | n i E n hh n | and an explicitly metric-dependent form in H ( F ) , H = ∞ X n =0 | n i E n h n | Θ . In the same spirit we may deduce that ≺ m | n ≻ = δ m,n = hh m | n i = h m | Θ | n i , m, n = 0 , , . . . . This means that whenever these sets are perceived as bases, they should be called orthogonal in H ( T,S ) and biorthogonal in H ( F ) .hree-Hilbert-Space Formulation of Quantum Mechanics 17 B.2.3 Making use of the maps between spaces
Let us now change and broaden the perspective used during our discussion of the illustrativeexample in Section 4 and assume that we start from the knowledge of the lower-case Hamil-tonian h acting in physical H ( T ) and from the choice of an arbitrary orthogonal basis in H ( S ) composed of its brabras hh m | and kets | m i which are connected by conjugation (16). Withoutany loss of generality we may further use some suitable unitary transformations and representour mapping operator in a simplified single-series formΩ = ∞ X n =0 | n ≻ µ n hh n | . All µ n ∈ C \ { } are independent free parameters. By insertion we may immediately verify thatthese variable parameters enter also the metric operator,Θ = Ω † Ω ≡ ∞ X n =0 | n ii µ ∗ n µ n hh n | . This operator [7, 17] must be Hermitian, positive definite and invertible, i.e., the operatorsΩ − = ∞ X n =0 | n i µ − n ≺ n | , Θ − = ∞ X n =0 | n i µ ∗ n µ n h n | must be assumed to exist. Under this assumption one could have an impression that, in principleat least, we might be able to get rid of the use of the rather exotic concepts of the non-HermitianHamiltonians and/or of the biorthogonal bases in H ( F ) . In practice, this impression proveswrong. In the majority of applications as reviewed in [6] one virtually always works solely inside H ( F ) employing both the other two spaces H ( T,S ) as purely auxiliary theoretical constructs.Nevertheless, one should keep in mind that whenever we try to complement the results of ourcalculations by their correct probabilistic interpretation, the explicit use of the two-step mapping H ( F ) → H ( S ) → H ( T ) proves unavoidable. Acknowledgements
In various stages of development the work has been supported by Institutional Research PlanAV0Z10480505, by the MˇSMT “Doppler Institute” project Nr. LC06002, by GA ˇCR, grant Nr.202/07/1307 and by the hospitality of Universidad de Santiago de Chile. Last but not least,three anonymous referees should be acknowledged for their constructive commentaries.
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