ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

Time-Dependent and/or Nonlocal Representations of Hilbert Spaces
in Quantum Theory

M. Znojil

Abstract

A few recent innovations of the applicability of standard textbook Quantum Theory are reviewed. The three-Hilbert-space
formulation of the theory (known from the interacting boson models in nuclear physics) is discussed in its slightly broadened
four-Hilbert-space update. Among applications involving several new scattering and bound-state problems the central role
is played by models using apparently non-Hermitian (often called “crypto-Hermitian”) Hamiltonians with real spectra.
The formalism (originally inspired by the topical need for a mathematically consistent description of tobogganic quantum
models) is shown to admit even certain unusual nonlocal and/or “moving-frame” representations H(S) of the standard
physical Hilbert space of wave functions.

Keywords: Quantum Theory, cryptohermitian operators of observables, stable bound states, unitary scattering, quantum
toboggans, supersymmetry, time-dependent models.

1 Introduction
The Fourier transformation F:ψ(x) → ψ̃(p) of wave
functions converts differential kinetic-energy operator
K ∼ d2/dx2 into a trivialmultiplication by a number,
K̃ = FKF−1 ∼ p2. Thismeans that for certain quan-
tum systems the Fourier transformation offers a sim-
plification of the solution of the Schrödinger equation.
Thegeneralized,nonunitary (oftencalledDyson)map-
pings Ω play the same simplifying role in the context
of nuclear physics [1]. In our present brief review pa-
perwe intend to recall anddiscuss very recentprogress
and, mainly, a few of our own results in this direction.
Our textwill bemore or less self-contained, though

the limitations imposed on its length will force us to
skip all remarks on the history of the subject aswell as
on references and on a broader context. Fortunately,
interested readersmay very easily get acquaintedwith
these aspects of the new theory in several very thor-
ough and extensive reviews [2] and also in our own
recent compact review [3] and/or in our two-years-old
short paper [4].
In section 2 we shall start our discussion from

bound-state models characterized by the loss of ob-
servability of complexified coordinates. In the generic
dynamical scenario where the Riemann surface of
the wave functions can be assumed multisheeted, we
shall define certainmonodromy-sensitivemodels called
quantum toboggans. Our selection of their sample ap-
plications will cover innovativemodels possessing sev-
eral branch points in the complex x-plane and/or ex-
hibiting supersymmetry.
Section 3 will offer information about the specific

cryptohermitianapproachtobound-statemodels char-
acterized by themanifest time-dependence of their op-

erators of observables (cf. paragraph 3.1) or by the
presence of a fundamental length in the theory (cf.
paragraph 3.2).
The twopossiblemechanisms of a return to unitar-

ity in the models of scattering by complex potentials
will be described in Section 4. Via concrete examples
we shall emphasize the beneficial role of a “smearing”
of phenomenological potentials and the necessity of an
appropriate redefinition of the effectivemass in certain
regimes.
Section 5 contains a few concluding remarks. For

the sake of completeness, a few technical remarks con-
cerning the role of the Dyson mapping in the abstract
formulation of Quantum Theory as well as in some of
its concrete applications will be added in the form of
three Appendices.

2 Quantum theories working
with quadruplets of
alternative Hilbert spaces

Within the cryptohermitian approach, a new cat-
egory of models of bound states appeared, a few
years ago, under the name of quantum toboggans [5].
Their introduction extended the class of integra-
tion paths of complexified “coordinates” x = q(s)
in the standard Schrödinger equations to certain
topologically nontrivial complex trajectories. The
Hamiltonians H(T) = p2 + V (T)(x) containing an-
alytic potentials V (T)(x) with singularities (the su-
perscripts (T) stand here for “tobogganic”) were
connected with the generalized complex asymptotic
boundary conditions and specified as operating in
a suitable Hilbert space H(T) of wave functions

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Acta Polytechnica Vol. 50 No. 3/2010

in which the Hamiltonian itself is manifestly non-
Hermitian.
Practical phenomenological use of any cryptoher-

mitian quantum model requires, firstly, a sufficiently
persuasive demonstrationof the reality of its spectrum
and, secondly, the availability of at least one metric
operator Θ = Θ(H) (cf. Appendices A–C for its defi-
nition). Usually, both of these conditions are nontriv-
ial, so that any form of the solvability of the model
is particularly helpful. Vice versa, once the Hamilto-
nian H proves solvable in Hilbert space H(T), we may
rely upon the availability of the closed solutions of the
underlying Schrödinger equations and on the related
specific spectral representations of the necessary oper-
ators (cf. [6, 7] for more details).
The topological nontrivality of the tobogganic

paths of coordinates running over several Riemann
sheets of wave functions happened to lead to severe
complications in the numerical attempts to compute
the spectra. This difficulty becomes almost insur-
mountable when the wave functions describing quan-
tum toboggans happen to possess two or more branch
points (cf. [8] for an illustrative example). For these
reasons it is recommended to rectify the tobogganic
integration paths via a suitable change of variables in
a preparatory step [9]. Our tobogganic Schroedinger
equations then acquire the generalized eigenvalue-
problem form Hψ = EW ψ of the so called Sturm-
Schroedinger equations with the rectified Hamilto-
nian H �= H† and with a nontrivial weight operator
W �= W † �= I. Both of these operators are defined in
another, transformed, “more friendly” Hilbert space
H(F) of course [10].

2.1 Supersymmetric quantum
toboggans

The introduction of the cryptohermitian and tobog-
ganicmodels proveduseful in the context of supersym-
metry (SUSY).Asampleofpapersdevoted to this sub-
ject is referenced in [2]. The easiest case (called super-
symmetric quantum mechanics) uses just the Hamil-
tonian and the two charge operators generating the
SUSY algebra,

H =
[

H(−) 0

0 H(+)

]
=

[
BA 0

0 AB

]
,

Q =
[
0 0

A 0

]
, Q̃ =

[
0 B

0 0

]
.

For the solvable model of ref. [11] the energy spec-
trum (composed of four families En = a(n), . . . , d(n))
is displayed in figure 1. At γ = −1/2 the singular-
ity vanishes and the (up to the ground state) doubly
degenerate SUSY spectrum becomes strictly equidis-
tant.

The imposition of supersymmetry has been ex-
tended to quantum toboggans in [12]. Both the com-
ponents of the super-Hamiltonian were defined along
topologically nontrivial complex curves which con-
nect several Riemann sheets of the wave function.
The new feature of this generalized model lies in the
non-uniqueness of the map T between “tobogganic”
partner curves. As a consequence, we must redefine
the creation- and annihilation-like operators as fol-
lows,

A = −T
d
dx
+T W(−)(x) ,

B =
d
dx
T −1 + W(−)(x)T −1 .

In contrast to the non-tobogganic cases, the Hermi-
tian-conjugation operator T even ceases to be invo-
lutory (i.e., T �= T −1, cf. paper [12] for more de-
tails).

c(0)

c(1)

c(2)

c(3)

c(4)

d(0)

d(1)

d(2)

d(3)

c(0)
d(0)

c(1)
d(1)

c(2)
d(2)

b(0)

b(1)

b(2)

b(3)

a(0)

a(1)

a(2)

a(3)

a(4)

a(5)

-2 -1 0 1

γ

E

Fig. 1: Spectrumof the singular supersymmetric harmonic
oscillator

2.2 Four-Hilbert-space Quantum
Mechanics

In a way explained in our papers [7], the tobog-
ganic quantum systems with real energies gener-
ated by their apparently non-Hermitian Hamiltoni-
ans may be assigned the entirely standard and con-
sistent probabilistic interpretation. For this pur-
pose the initial Hilbert space H(T) is replaced
by another, friendly Hilbert space H(F) in which
the above-mentioned Sturm-Schroedinger equations
Hψ = EW ψ have to be solved. This forces us to
generalize the three-Hilbert-space scheme of paper [3]
[cf. also Appendices A and B and figure 2] and to
use the following four-Hilbert-space pattern of map-
pings

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Acta Polytechnica Vol. 50 No. 3/2010

tobogganic space H(T)

analytic multivalued ψ[q(s)]

multisheeted paths q(N)(s)

physics in H(P)

h = h† , w = w†

dynamics via topology

↓
(the change of variables)

rectification ↑
(the unitary mapping)

equivalence

feasibility in H(F)

H �= H† , W �= W †

Sturm−Schrödinger eqs.

−→
(metric is introduced)

hermitization

standard space H(S)

H = H‡ , W = W ‡

ad hoc metric Θ �= I

The analyticity of the original wave function ψ[q(s)]
along the given tobogganic integration path with pa-
rameter s ∈ (−∞, ∞) is assumed. The rectification
transition between Hilbert spaces H(T) and H(F) is
tractable as an equivalence transformation under this
assumption [10]. In the subsequent sequence of maps
F → S and F → P one simply follows the old
three-Hilbert-spacepatternofAppendixC [3] inwhich
just the nontrivial weight operators W and/or w are
added and appear in the respective generalizedSturm-
Schrödinger equations.
Marginally, let us add that various, suitably mod-

ified spectral representations of the eligible metric op-
erators may be used, say, in the form derived in [7].
The purely kinematical and exactly solvable topolog-
ically nontrivial “quantum knot” example of ref. [13]
can also be recalled here as an exactly solvable illus-
tration in which the confining role of the traditional
potential is fully simulated by the mere topologically
nontrivial shape of the complex integration path.

3 Bound-state theories
working with the triplets of
alternative Hilbert spaces

3.1 Quantum models admitting the
time-dependence of their
cryptohermitian Hamiltonians

In our review [3] of the three-Hilbert-space (3HS) for-
malism we issued a warning that some of the conse-
quences of the enhancedflexibility of the languageand
definitions may sound like new paradoxes. For illus-
tration, let us mention just that in the 3HS approach
the generator H(gen) = H(gen)(t) of the time-evolution
of wave functions is allowed to be different from the
Hamiltonian operator H = H(t) of the system in ques-
tion [14].
The key to the disentanglement of the similar puz-

zles is easily found in the explicit specification of the
Hilbert space in which we define the Hermitian con-
jugation. We showed in [14] that the use of the

full triplet of spaces of figure 2 becomes unavoid-
ablewhenever our cryptohermitian observables are as-
sumed time-dependent because their variations may
and must be matched by the time-dependence of the
representation of the physical ad hoc Hilbert space
H(S). Its nontrivial inner product is capable of play-
ing the role of a “moving frame” image of the origi-
nal physical Hilbert space H(P). Although our “true”
Hamiltonian (i.e., operator h(t) in H(P)) is the gener-
ator of the time evolution in H(P), the time-evolution
of the wave functions in H(S) is controlled not only by
the “dynamical” influence of H = H(t) itself but also
by the “kinematical” influence of the time-dependence
of the “rotating” Dyson mapping Ω = Ω(t). Thus,
the existence of any other given and manifestly time-
dependent observable o(t) in H(P) will leave its trace
in Dyson map Ω(t), i.e., in metric Θ(t), i.e., in the
time-dependence of the “moving frame” Hilbert space
H(S).
This circumstance implies the existenceof twopull-

backs of the evolution law fromH(P) to H(S), with the
recipe |ϕ(t)〉 = Ω−1(t) |ϕ(t) � being clearly different
from the complementary recipe 〈〈ϕ(t) | =≺ ϕ(t) |Ω(t).
The same Dyson mapping leads to the two different
evolution operators, viz., to the evolution law for kets,

|ϕ(t)〉 = UR(t) |ϕ(0)〉 , UR(t)=Ω−1(t)u(t)Ω(0)

and to the different evolution law for brabras,

|ϕ(t)〉〉 = U †L(t) |ϕ(0)〉〉, U
†
L(t)=Ω

†(t)u(t)
[
Ω−1(0)

]†
.

Wehave no space here for the detailed reproduction of
the whole flow of this argument as presented in [14].
Its final outcome is the definition of the common time-
evolution generator

H(gen)(t)= H(t)− iΩ−1(t)Ω̇(t) .

entering the final doublet of time-dependent Schrödin-
ger equations

i∂t|Φ(t)〉 = H(gen)(t) |Φ(t)〉 , (1)
i∂t|Φ(t)〉〉 = H(gen)(t) |Φ(t)〉〉 . (2)

This ultimately clarifies the artificial character and
redundancy of the Mostafazadeh’s conjecture [15] of

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Acta Polytechnica Vol. 50 No. 3/2010

quasistationarity, i.e., of the requirement of time-
independence of the inner products and of the metric,
i.e., ipso facto, of Hilbert space H(S).

3.2 Systems admitting a controllable
nonlocality

In a way emphasized by Jones [16] the direct observ-
ability of coordinates x is lost for the majority of the
parity-times-time-reversal-symmetric (or, briefly, PT -
symmetric) quantum Hamiltonians. In the context of
scattering, this forced us to admit a non-locality of
the potentials in [17]. Fortunately, in the context of
bound states the loss of the observability of coordi-
nates is much less restrictive since we do not need to
prepare any asymptotically free states. The admissi-
ble Hilbert-space metrics Θ may be chosen as moder-
ately non-local acquiring, in the simplest theoretical
scenario as proposed in our paper [18], the form of a
short-ranged kernel in a double-integral normalization
or in the inner products of the wave functions. The
standard Dirac’s delta-function kernel is simply reob-
tained in the zero-range limit.
In refs. [17, 19] we proposed several bound-state

toymodels exhibiting, in a confined-motiondynamical
regime, various forms of an explicit control of themea-
sure θ of their dynamicallygeneratednon-locality. The
exact solvability of some of these models even allowed
us to assigneachHamiltonian the complete menuof its
hermitizing metricsΘ=Θθ distinguished by their op-
tional fundamental lengths θ ∈ (0, ∞). In this setting
the localmetrics reappear at θ =0 while certain stan-
dard hermitizations only appeared there as infinitely
long-ranged, with θ = ∞.

4 Scattering theories using
pairs of Hilbert spaces
H(P) �= H(F)

In our last illustrative application of 3HS formal-
ism, let us select just two non-equivalent Hilbert
spaces H(F,S) and turn to scattering theory where
one assumes that the coordinate is certainly measur-
able/measured at large distances. This means that
we may employ the operators in coordinate represen-
tation and accept only such models where the metric
operator remains asymptotically proportional to delta
function, 〈x|Θ|x′〉 ∼ δ(x− x′) at |x| ! 1 and |x′| ! 1.
A few concrete models of this type were described
in refs. [17, 19] using minimally nonlocal, “smeared”
point interactions of various types (which were, in
the latter case, multi-centered). The use of nonper-
turbative discretization techniques rendered possible
the construction of the (incidentally, unique)metricΘ
compatible with the required asymptotic locality.
The resulting physical picture of scattering was

unitary and fully compatible with our intuitive expec-

tations. In our last paper [20] the scope of the theory
has further been extended to the generalized scatter-
ing models, where the matrix elements 〈x|Θ|x′〉 of the
metric were allowed operator-valued.
A slightly different approach to scatteringhas been

initiated in paper [21] where we studied the analytic
and “realistic” Coulombic cryptohermitian potentials
defined along U-shaped complex trajectories circum-
venting the origin in the complex x plane from below.
Unfortunately, this model was unstable with respect
to perturbations. A few years later we clarified, in
paper [22], that a very convenient stabilization of the
modelmaybe based onaminus-sign choice of the bare
mass in theSchrödiner equation. Very soonafterwards
we also revealed that the scattering by the amended
Hamiltonian is unitary [23]. The transmission and re-
flection coefficients were evaluated in closed analytic
form exhibiting the coincidence of the bound-state en-
ergies with the poles of the transmission coefficients.
Thus, after a moderate modification a number of ob-
servations forming the analytic theory of S-matrix has
been found transferrable to the cryptohermitian quan-
tum theory.

5 Conclusions
OneofparadoxescharacterizingQuantumTheorymay
be seen in the contrast between its stable status in ex-
periments (where, typically, its first principles are ap-
preciated as unexpectedly robust [24]) and its fragile
status in themathematical context, where virtually all
of its rigorous formulations are steadily being found,
for this or that reason, not entirely satisfactory [25].
In fact, at least a part of this apparent conflict is just a
pseudoconflict. Its roots can be traced back to various
purely conceptual misunderstandings. In our present
review we emphasized that within the comparatively
narrow frameworkof quantumtheoryusing cryptoher-
mitian representations of observables the majority of
these misunderstandings can be clarified, mostly via a
careful use of an adequate notation.
The core of our present message can be seen in

the unified outline of the resolution of the internet-
mediateddebate (cf. [3] for references) inwhich thead-
missibility and consistent tractability of themanifestly
time-dependent cryptohermitian observables has been
questioned. It is now clear that the reduction of the
scope of the theory to the mere quasistationary sys-
tems as proposed by Mostafazadeh [15] is unfounded.
This bound-state-related message can be seen ac-

companied by the clarification of a return to unitarity
in the models of scattering mediated by cryptohermi-
tian interactions. The currently valid conclusion is
that itmakes sense to combine the complexification of
the short-range interactions with our making them at
least slightly nonlocal. We have seen that, in parallel,
also the metric can be required to exhibit a certain
limited degree of nonlocality.

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Acta Polytechnica Vol. 50 No. 3/2010

New questions emerge in this context. This means
that in spite of all the recent rapidprogress the current
intensive development of the cryptohermitian quan-
tum theory is still fairly far from its completion.

Appendix A: Hilbert space in
our present notation

In our review paper [3] we explained that one of the
most natural formulations of the abstract Quantum
Theory should follow the ideas of Scholtz et al [1] by
constructing the three parallel representatives of any
givenwave function living in the three separateHilbert
spaces. We argued that the use of the three-Hilbert-
space (3HS) formulation of Quantum Theory seems
best capable of clarifying a few paradoxes emerging in
connection with the concept of Hermiticity and en-
countered in the recent literature. We emphasized
in [3] thatmanyquantumHamiltonianswith real spec-
tra, characterized by their authors as manifestly non-
Hermitian, should and must be re-classified as Her-
mitian. In this sense we fully accepted the dictum
of standard textbooks on quantum theory and com-
plemented the corresponding postulates just by a few
explanatory comments.
In a brief summary of this argument let us recall

that the states ψ of a (say, one-dimensional) quan-
tum system are often assumed represented by nor-
malized elements of the simplest physical and com-
putation friendly concrete Hilbert space L2(R). This
is already just a specific assumption with restrictive
consequences. Thus, in a more ambitious picture of
a general quantum system each state ψ should only
be perceived as an element |ψ〉 of an abstract vector
space V. The equally abstract dual vector space V′
of linear functionals over V may be bigger, V′ ⊃ V.
In the most common selfdual case with V′ = V one
speaks about the Hilbert space H(F) := (V, V′) where
the superscript (F) stands, say, for (user-)friendly or
“feasible”.
In many standard formulations of the first princi-

ples of Quantum Theory the well known Dirac’s bra-
ket notation is used, with |ψ 〉 ∈ V and 〈ψ| ∈ V′ for
a fixed or “favored” Hilbert space H(F). At the same
time, this choice of the notation does not exclude a
transition (say, Ω) to some other vector and Hilbert
spaces denoting, e.g., Ω |ψ〉 := |ψ � ∈ W and using
here the slightly deformed, spiked ket symbols [3].

Appendix B: Dyson mapping Ω
as a nonunitary generalization
of the Fourier transformation F
In the context of nuclear physics the use of the sin-
gle, favored Hilbert space H(F) is rather restrictive.
For example, in the context of the so called inter-

acting boson model and in the way inspired by the
well known advantages of the use of the usual unitary

Fourier transformation F =
[
F†

]−1
, nuclear physi-

cists discovered that their constructive purposes may
bemuchbetter servedby a suitable generalized,mani-
festly non-unitary (often calledDyson) invertiblemap-
ping Ω.
More details may be found in paper [1], where the

operators Ω were described as mediating the transi-
tion from a friendly bosonic vector space V into an-
other, fermionic and “physical” vector space W. The
deepenedmathematical differences between “bosonic”
(i.e., simpler) V and fermionic (i.e., complicated, com-
putationallymuch less accessible)W weakens the par-
allelism between Ω and F since the latter operator
merely switches between the so called coordinate- and
momentum-representations of ψs lying in the same
Hilbert space L2(R).
This encouraged us to propose, in [3], visual iden-

tification of the bras and kets in one-to-one correspon-
dence to the space in which they live, with |ψ 〉 ∈ V
whileΩ |ψ〉 := |ψ � ∈ W. For duals (i.e., bra-vectors)
we recommended the same notation, with 〈ψ| ∈ V′
while 〈ψ|Ω† :=≺ ψ| ∈ W′.

Appendix C: The connection
between Dyson map Ω and
metric Θ
In the notation of Appendix B one represents the
same state ψ in two non-equivalent Hilbert spaces,
viz., in the friendly F-space H(F) := (V, V′) and
in the physical P-space H(P) := (W, W′) (character-
ized by the “spiked” kets and bras). The latter space
is, by construction, manifestly non-equivalent to the
former one since, by definition, we have, for overlaps,
≺ ψa|ψb � = 〈ψa|Ω†Ω |ψb〉 �= 〈ψa|ψb〉.

final

initial

friendly
S-space:

P-space:

F-space:
constructive,

usual,

auxiliary,
predictive

inaccessible

unphysical

(unitary) equivalence

(spiritual role)

(paternal role)

(filial role)

 change of
 metric

Dyson map

Fig. 2: The samephysics is predicted in H(P) and in H(S)

while, presumably, the calculations are all performed in
H(F)

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Acta Polytechnica Vol. 50 No. 3/2010

According to our review [3], the demonstration of
unitary non-equivalence between H(F) and H(P) can
easily be converted into a proof of unitary equivalence
betweenH(P) andanother, third, standardizedHilbert
space H(S) := (V, V′′). Indeed, we are free to intro-
duce a redefined vector space of linear functionals V′′
such that the equivalence will be achieved. For the
latter purpose it is sufficient to introduce the special
duals 〈〈ψ| ∈ V′′ denoted by the new, “brabra” Dirac-
inspired symbol. In terms of a givenDysonoperatorΩ
we may define these brabras, for the sake of definite-
ness, by the formula 〈〈ψa| = 〈ψa|Θ of ref. [6], where
we abbreviated Θ=Ω†Ω.
In [1] the new operator Θ has been called met-

ric. It defines the inner products in the “second aux-
iliary” (i.e., in its nuclear-physics exemplification, sec-
ond bosonic)Hilbert spaceH(S) which is, by construc-
tion, unitarily equivalent to the originalphysicalH(P).
The whole 3HS scheme is given a compact graphical
form in figure 2.

Acknowledgement

This work has been supported by MŠMT Doppler
Institute project LC06002, by Institutional Re-
search Plan AV0Z10480505 and by GAČR grant
202/07/1307.

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Miloslav Znojil, DrSc.
Phone: +420 266 173 286
E-mail: znojil@ujf.cas.cz
Nuclear Physics Institute
Academy of Science of the Czech Republic
250 68 Řež, Czech Republic

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