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Abstract

We construct a nonunitary transformation that relates a given “asymptotically free” conformal quantum mechanical system Hf with its confined, harmonically trapped version Hc. In our construction, Jordan states corresponding to the zero eigenvalue of Hf, as well as its eigenstates and Gaussian packets, are mapped into the eigenstates, coherent states, and squeezed states of Hc, respectively. The transformation is an automorphism of the conformal sl(2,R) algebra of the nature of the fourth-order root of the identity transformation, to which a complex canonical transformation corresponds on the classical level being the fourth-order root of the spatial reflection. We investigate the one- and two-dimensional examples that reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.
Conformal bridge between asymptotic freedom and confinement
Luis Inzunza,1,* Mikhail S. Plyushchay,1,and Andreas Wipf2,
1Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 9170124, Chile
2Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena,
Max-Wien-Platz 1, D-07743 Jena, Germany
(Received 29 January 2020; accepted 12 May 2020; published 28 May 2020)
We construct a nonunitary transformation that relates a given asymptotically freeconformal quantum
mechanical system Hfwith its confined, harmonically trapped version Hc. In our construction, Jordan
states corresponding to the zero eigenvalue of Hf, as well as its eigenstates and Gaussian packets, are
mapped into the eigenstates, coherent states, and squeezed states of Hc, respectively. The transformation is
an automorphism of the conformal slð2;RÞalgebra of the nature of the fourth-order root of the identity
transformation, to which a complex canonical transformation corresponds on the classical level being the
fourth-order root of the spatial reflection. We investigate the one- and two-dimensional examples that
reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.
DOI: 10.1103/PhysRevD.101.105019
I. INTRODUCTION
A well-known deficiency of the conformal quantum
mechanics is that it has no invariant ground state that is
annihilated by the generators of the slð2;RÞalgebra, and in
this sense its conformal symmetry is spontaneously broken.
In the context of the AdS=CFT correspondence, this
phenomenon can be related with a peculiar nature of a
usual evolution variable which is not a good global
coordinate on AdS2, whose isometry is conformal sym-
metry [13]. The problem emerges from the fact that the
generators of the time translation H, the dilatation D, and
the special conformal transformations Kare noncompact
generators of the conformal slð2;RÞalgebra, and the
spectrum of the Hamiltonian operator Hof the system is
the open interval ð0;Þ. The deficiency, however, can be
cured from the perspective of Diracs different forms of
dynamics [4], by considering as a new Hamiltonian a linear
combination of these generators to be of a compact
topological nature. For example, one can take the operator
Hþmω2K, which has an equidistant spectrum bounded
from below, where mand ωare the mass and frequency
parameters. This regularizationwas first considered by
de Alfaro, Fubini, and Furlan (AFF) in their seminal
work [5], where by means of a canonical transformation
of the spatial and time coordinates, the conformal mechan-
ics action describing the asymptotically free trajectories is
transformed into a modified action principle with an
additional confining term in the form of a harmonic trap.
The AFF conformal mechanics model finds diverse inter-
esting applications including the black hole physics [6,7],
cosmology [8,9], and holographic QCD [10].
A similar transformation was considered earlier by
Niederer [11] as the canonical transformation by which
the conformal-invariant classical dynamics of a free non-
relativistic particle on the whole real line can be related to
the dynamics of the harmonic oscillator. A generalization
of the construction allows one to transform the time-
dependent Schrödinger equation of the free particle into
that for the quantum harmonic oscillator. However, a
relation between stationary states of the corresponding
pairs of the asymptotically freeand confinedquantum
systems remains to be unclear. To establish such a relation,
we construct here a nonunitary similarity transformation in
the form of a conformal bridge between the freedom and
confinement.
Before we pass over to the discussion of our construc-
tions, it is instructive to recall some aspects of the Darboux
transformations [12], a comparison that will be useful in
what follows.
The Darboux transformation and its generalizations
allow one to generate new quantum systems, sometimes
called superpartners, from a given one. Any generated
superpartner is completely or almost completely isospectral
to the initial system, and the energy eigenstates of both
systems are related to each other. The extended quantum
system composed from two superpartners is described by a
*luis.inzunza@usach.cl
mikhail.plyushchay@usach.cl
wipf@tpi.uni-jena.de
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PHYSICAL REVIEW D 101, 105019 (2020)
2470-0010=2020=101(10)=105019(14) 105019-1 Published by the American Physical Society
linear or nonlinear supersymmetry. An important class of
exactly solvable quantum systems to which Darboux trans-
formations are applied to produce new nontrivial systems
includes, in particular,
(i) the free particle,
(ii) the harmonic oscillator,
(iii) the conformal mechanics model without a confining
term (two-particle Calogero model with omitted
center of mass coordinate [13]),
(iv) the AFF conformal mechanics model with a con-
fining harmonic trap,
(v) the particle in the infinite potential well.
By applying Darboux transformations to the free particle
(i), reflectionless quantum systems can be obtained. The
covariance of the Lax representation of integrable systems
then allows one to promote reflectionless potentials to
soliton solutions of the Kortewegde Vries (KdV) equation.
The construction admits a further generalization to the case
of finite-gap solutions to the KdV equation [14].
Proceeding from the systems (ii) and (iv), an interesting
class of rational extensions of the harmonic oscillator and
conformal mechanics is constructed, which have a discrete
finite-gap type spectral structure described by exceptional
orthogonal polynomials, and are characterized by an ex-
tended deformed conformal symmetry [1517]. Exceptional
orthogonal polynomials also are generated by applying
Darboux transformations to the system (v) [18].
Darboux transformations are usually constructed by
using physical or formal, nonphysical eigenstates of the
original system. Via the appropriate confluent limit of
Darboux transformations, Jordan states of the original
system enter into the construction [12,19,20]. Such states
are used, particularly, in rational deformations of the
conformal mechanics with confining trap (iv) [21], and
in the construction of the extreme type wave solutions to
the complexified KdV equation based on the system (i) and
PT-regularized conformal mechanics (iii) [22].
Some pairs of the systems (i)(v) can be related among
themselves by using appropriate singular Darboux trans-
formations. In this way, conformal mechanics models
(iii) with certain values of the coupling constant and model
(v) can be generated from the system (i), while the AFF
model (iv) can be obtained from the system (ii). The
systems (i) and (ii), however, are not related by Darboux
transformations. The same is true for the pair of the systems
(iii) and (iv). One may ask whether the quantum systems in
the two indicated pairs can be related by an alternative
differential transformation.
In the present paper we show how the indicated systems
can be connected by employing conformal symmetry.
Aprioriit is obvious that, as for the Darboux transformations,
the sought for transformations have to be nonunitary. We shall
see, however, that being similarity transformations, they can
be related to the unitary Bargmann-Segal transformation in
the case of the pair of systems (i) and (ii), where a nonunitary
Weierstrass transformation plays an essential role. At the
same time, in correspondence with the above indicated
modification of conformal symmetry, the transformations
effectively relate Diracs different forms of dynamics with
respect to the conformal slð2;RÞsoð2;1Þsymmetry.
Also, we will observe the essential role played by the
Jordan as well as coherent and squeezed states in our
constructions. Our conformal bridge transformationcan
be generalized to higher dimensions, and as an example we
consider a relation of the two-dimensional (2D) free particle
system with the planar isotropic harmonic oscillator and the
Landau problem.
The paper is organized as follows. In Sec. II,we
investigate a relation between the quantum free particle
and the harmonic oscillator by exploiting the structure of the
inverse Weierstrass transformation and the generating func-
tion for Hermite polynomials in the light of conformal
symmetry. In Sec. III we generalize the obtained results in
the context of Diracs different forms of dynamics. In
Sec. IV we explore the meaning of our conformal bridge
transformation on the classical level. In Sec. Vwe apply the
transformation to establish a relation between the conformal
mechanics model and the AFF model. This way we generate
the Schrödinger odd cat states from eigenstates of the two-
particle Calogero system. In Sec. VI the conformal bridge
transformations are applied to two-dimensional systems to
establish a relation of certain quantum states of the 2D free
particle system with eigenstates and coherent states of the
planar isotropic harmonic oscillator as well as the Landau
problem. Section VII is devoted to the discussion and
outlook. Two Appendixes include some technical details.
II. FREE PARTICLE AND HARMONIC
OSCILLATOR
In this section we investigate the transformation, based
on conformal symmetry, by which the free particle and
quantum harmonic oscillator systems can be related.
Consider the generating function for Hermite polyno-
mials HnðxÞ,
Gðx;tÞ¼expð2xt t2Þ¼ex2eðxtÞ2¼X
n¼0
tn
n!HnðxÞ:
ð2:1Þ
Using it, one can obtain a chain of various representations
for HnðxÞ:
HnðxÞ¼n
tnGðx;tÞjt¼0¼ð1Þnex2dn
dxnex2
¼ex2d
dx ex2n·1¼2xd
dxn·1
¼e1
2x2xd
dxn
e1
2x2:ð2:2Þ
INZUNZA, PLYUSHCHAY, and WIPF PHYS. REV. D 101, 105019 (2020)
105019-2
Normalized eigenfunctions of the quantum harmonic
oscillator of mass mand frequency ωdescribed by the
Hamiltonian operator ˆ
Hosc ¼1
2m
ˆ
p2þ1
2mω2q2are
ψnðqÞ¼Cne1
2x2HnðxÞ;C
n¼1
ffiffiffiffiffi
l0
p1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π1=22nn!
p;ð2:3Þ
where l0¼ffiffiffiffiffi
mω
qand x¼q=l0is a dimensionless vari-
able. Take the units with ¼m¼ω¼1, and introduce
the ladder operators aþ¼1ffiffi2
pðxd
dxÞ,a¼ðaþÞ¼
1ffiffi2
pðxþd
dxÞ,½a;a
þ¼1. Then the last equality in
Eq. (2.2) multiplied from the left by e1
2x2gives, up to a
normalization, the eigenfunctions (2.3), that correspond to
the coordinate representation of the Fock states jn
ðaþÞn
ffiffiffi
n!
pj0igenerated from the ground state of the quantum
harmonic oscillator.
Based on the relation ð1
4
d2
dx2Þe2xt ¼t2e2xt, one can
also represent the generating function (2.1) as follows:
Gðx; tÞ¼exp 1
4
d2
dx2e2xt ¼X
n¼0
2ntn
n!exp 1
4
d2
dx2xn:
ð2:4Þ
Comparison of (2.4) with (2.1) yields yet another repre-
sentation of Hermite polynomials via the formal inverse of
the Weierstrass transform [23,24],
HnðxÞ¼2nexp 1
4
d2
dx2xn:ð2:5Þ
Equation (2.5) allows us to generate the eigenfunctions
(2.3) of the harmonic oscillator by applying the operator
ˆ
S0¼e
ˆ
Ke1
2
ˆ
H0ð2:6Þ
to the monomials xn,n¼0;1;,
ψnðxÞ¼ 1
ffiffiffiffiffiffiffiffiffiffiffiffi
π1=2n!
pˆ
S0ðffiffi
2
pxÞn:ð2:7Þ
The operator (2.6) is constructed from the operators
ˆ
H0¼1
2
d2
dx2;
ˆ
K¼1
2x2;ð2:8Þ
which together with the dilatation operator
ˆ
D¼1
4ðxˆ
pþˆ
pxÞ¼i
2xd
dx þ1
2ð2:9Þ
generate the dynamical conformal symmetry of the quan-
tum free particle system,
½ˆ
H0;
ˆ
D¼i
ˆ
H0;½ˆ
H0;
ˆ
K¼2i
ˆ
D; ½ˆ
K;
ˆ
D¼i
ˆ
K:
ð2:10Þ
The operators ˆ
Kand ˆ
Dare the t¼0form of the explicitly
depending on time integrals of motion ˆ
K¼1
2
ˆ
X2and
ˆ
D¼1
4ðˆ
p
ˆ
Xþˆ
Xˆ
pÞ, where ˆ
X¼ðxˆ
ptÞis the generator
of Galileo transformations of the free particle, d
dt
ˆ
X¼
t
ˆ
Xi½ˆ
X;
ˆ
H0¼0. The set ˆ
H0,ˆ
D, and ˆ
Kgenerates the
same conformal algebra (2.10). Its extension by the
integrals ˆ
p,ˆ
Xand central element 1 (in the chosen units)
corresponds to the dynamical Schrödinger symmetry of the
free particle system [25].
In (2.7),χ0¼x0¼1is an eigenstate of the lowest, zero
eigenvalue of the free particle Hamiltonian ˆ
H0, while χ1¼
xis a nonphysical, unbounded at infinity, eigenstate of ˆ
H0
of the same zero eigenvalue. The states given by wave
functions χnðxÞ¼xnwith n2are the Jordan states [22]
of the free particle corresponding to the zero energy E¼0:
ðˆ
H0Þnχ2n¼1
2n
ð2nÞ!χ0;
ðˆ
H0Þnχ2nþ1¼1
2n
ð2nþ1Þ!χ1;ð2:11Þ
ðˆ
H0Þnþ1χ2n¼ðˆ
H0Þnþ1χ2nþ1¼0;n¼1;:ð2:12Þ
The χnðxÞare, at the same time, the formal eigenstates of
the operator 2i
ˆ
Dwith the same eigenvalues as the eigen-
functions ψnðxÞof the quantum harmonic oscillator
Hamiltonian,
2i
ˆ
Dχn¼nþ1
2χn:ð2:13Þ
Behind the last observation lies the fact that the nonunitary
operator ˆ
S0intertwines generators of the conformal
slð2;RÞsymmetry of the quantum free particle system
with generators of the Newton-Hooke symmetry of the
quantum harmonic oscillator by changing the form of
dynamics. In details, this can be traced out as follows.
Using the relations ˆ
aψnðxÞ¼ ffiffi
n
pψn1ðxÞand ˆ
aþψnðxÞ¼
ffiffiffiffiffiffiffiffiffiffiffi
nþ1
pψnþ1ðxÞ, we obtain from (2.7) that
ˆ
a¼
ˆ
S01ffiffiffi
2
pd
dxˆ
S1
0;ˆ
aþ¼
ˆ
S0ðffiffi
2
pxÞ
ˆ
S1
0:ð2:14Þ
Therefore, the nonunitary operator ˆ
S0intertwines the free
particle momentum and coordinate operators multiplied by
i= ffiffi
2
pand ffiffi
2
p, respectively, with the annihilation and
creation operators of the quantum harmonic oscillator,
CONFORMAL BRIDGE BETWEEN ASYMPTOTIC PHYS. REV. D 101, 105019 (2020)
105019-3
ˆ
S01ffiffi
2
pd
dx¼ˆ
aˆ
S0;
ˆ
S0ðffiffi
2
pxÞ¼ˆ
aþˆ
S0:ð2:15Þ
Then we find that the same operator intertwines the
generators of conformal symmetry of the free particle with
the generators of the Newton-Hooke symmetry [26] of the
quantum harmonic oscillator,
ˆ
S0
ˆ
H0¼ð2
ˆ
JÞ
ˆ
S0;
ˆ
S0
ˆ
K¼1
2
ˆ
Jþ
ˆ
S0;
ˆ
S0ðˆ
DÞ¼i
2
ˆ
Hosc
ˆ
S0;ð2:16Þ
where
ˆ
J¼1
2ðˆ
aÞ2;
ˆ
Jþ¼1
2ðˆ
aþÞ2;
ˆ
Hosc ¼ˆ
aþˆ
aþ1
2¼2
ˆ
J0:ð2:17Þ
In this picture, the zero energy eigenstates, χ0¼1and
χ1¼xof the free particle, together with the Jordan states
χn¼xn,n2, corresponding to the same zero energy
[being at the same time formal eigenstates of the non-
compact slð2;RÞgenerator ˆ
D¼ˆ
J2multiplied by 2i] are
transformed by ˆ
S0into eigenstates of the harmonic
oscillator Hamiltonian. As a consequence of the intertwin-
ing relations (2.15) and Eqs. (2.13) and (2.16), the
operators 1ffiffi2
pd
dx and ffiffi
2
pxact on the described states
χnðxÞ,n¼0;1;, of the free particle in the same way
as the ladder operators ˆ
aand ˆ
aþact on the energy
eigenstates of the harmonic oscillator.
The ladder operators connect the states from the two
irreducible slð2;RÞrepresentations of the discrete type
series Dþ
αwith α¼1=4and α¼3=4, in which the Casimir
operator (A5) takes the same value ˆ
C¼αðα1Þ¼
3=16, while the compact generator ˆ
J0¼1
2
ˆ
Hosc has the
values j0;n ¼αþn,n¼0;1;. These two irreducible
representations of slð2;RÞare realized on subspaces
spanned by even, fψ2nðxÞg, and odd, fψ2nþ1ðxÞg,
n¼0;1;, eigenstates of the quantum harmonic oscil-
lator. Together they constitute an irreducible representation
of the ospð1j2Þsuperconformal algebra with ˆ
aand ˆ
aþas
odd generators [27]. The indicated separation of eigenstates
of the harmonic oscillator corresponds to a separation of the
set of Jordan states in (2.11) with odd and even values of
the index.
The generating function (2.1) is related with a unitary
map from the Hilbert space L2ðRÞto the Fock-Bargmann
space [28,29],
Gðx; z= ffiffi
2
pÞ¼ðπÞ1=4e1
2x2Uðx; zÞ:ð2:18Þ
Here
Uðx; zÞ¼X
n¼0
ψnðxÞfnðzÞ;ð2:19Þ
with ψnðxÞbeing the energy eigenstates of the harmonic
oscillator in the coordinate representation, while
fnðzÞ¼ zn
ffiffiffi
n!
p,zC, describe the same orthonormal states
in the Fock-Bargmann representation with scalar product
ðf; gÞ¼1
πZR2
ejzj2fðzÞgðzÞd2z; d2z¼dðRezÞdðImzÞ:
ð2:20Þ
The explicit form of (2.19) is
Uðx; zÞ¼π1=4exp 1
2ðx22ffiffi
2
pzx þz2Þ¼hxjziS
ð2:21Þ
that corresponds to the Schrödinger non-normalized
coherent state in the coordinate representation [30].
Equations (2.4) and (2.18) yield then the relation
Uðx; zÞ¼
ˆ
S0ϕzðxÞ;where ϕzðxÞ¼π1=4effiffi2
pzx:ð2:22Þ
The wave function ϕzðxÞwith zCcorresponds to a
formal eigenstate of the operator ˆ
H0with eigenvalue z2,
which at pure imaginary values z¼ik= ffiffi
2
pof zis a plane
wave eigenstate eikx of the free particle.1Therefore, in
addition to relation (2.7) we find that the nonunitary
operator ˆ
S0maps (formal in the general case of zC)
plane wave type eigenstates of the free particle into the
Schrödinger coherent states of the quantum harmonic
oscillator. The standard, or canonical Schrödinger-
Klauder-Glauber coherent states of the quantum harmonic
oscillator in coordinate representation [29,3234],
hxjze1
2jzj2Uðx; zÞψzðxÞ;ð2:23Þ
are generated from the rescaled formal plane wave eigen-
states of the free particle :
ψzðxÞ¼
ˆ
S0e1
2jzj2ϕzðxÞ:ð2:24Þ
Thus, the eigenstates of the quantum harmonic oscillator
are obtained from the Jordan states and eigenstates of the
free particle corresponding to zero energy by applying to
1Note that linear combination of the states ϕzðxÞwith real zare
used as seed states for Darboux transformations to generate
soliton solutions to the KdV equation. These states with complex
zare employed in the construction of the KdV soliton Baker-
Akhiezer function [31].
INZUNZA, PLYUSHCHAY, and WIPF PHYS. REV. D 101, 105019 (2020)
105019-4
them the nonunitary operator ˆ
S0. These free particles
states are also formal eigenstates (with purely imaginary
eigenvalues) of the dilatation operator ˆ
D. On the other
hand, in correspondence with the first relation in (2.15), the
plane wave eigenstates of the free particle, being eigen-
functions of the momentum operator, are mapped by ˆ
S0
into coherent states of the quantum harmonic oscillator
(QHO) being eigenstates of its annihilation operator. The
free particle plane wave eigenstates are produced then by
action of the inverse operator ˆ
S1
0¼e1
2H0eKon the
coherent states of the QHO.
Another important class of the states of the quantum
harmonic oscillator corresponds to the squeezed states
[29,33]. The so-called single-mode squeezed states jr
ˆ
SðrÞj0iare obtained from the ground state j0iby acting
with the unitary operator ˆ
SðrÞ¼expðr
ˆ
Jr
ˆ
JþÞ¼
expð2ir
ˆ
J2Þon it; see Eqs. (2.17) and (A2). In the
coordinate representation it is an infinite linear combination
of eigenfunctions of the QHO,
ψrðxÞ¼ 1
ffiffiffiffiffiffiffiffiffiffiffi
cosh r
pX
n¼0ðtanh rÞnffiffiffiffiffiffiffiffiffiffi
ð2nÞ!
p2nn!ψ2nðxÞ:ð2:25Þ
Using relation (2.7), we find that the preimage of (2.25)
under the action of the nonunitary operator ˆ
S0is given by a
Gaussian wave packet. Namely,
ˆ
S01
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π1=2cosh r
pexpðtanh r·x2Þ¼ψrðxÞ:ð2:26Þ
Therefore, the Gaussian wave packets of the quantum free
particle system correspond to the single-mode squeezed
states of the QHO under the similarity transformation
generated by the nonunitary operator ˆ
S0.
III. GENERALIZATION FOR FURTHER
APPLICATIONS
The similarity transformation that allowed us to establish
abridgebetween the quantum free particle system and
the quantum harmonic oscillator is given by the nonunitary
operator (2.6) constructed from generators of the conformal
algebra. The generators of this algebra correspond to the
dynamical symmetry slð2;RÞsoð2;1Þof the quantum
free particle system. This transformation generates a
change of dynamics in the sense of Dirac [4]. The initial
system is described by a Hamiltonian operator being a
noncompact linear combination of the generators ˆ
Jμof the
Lorentz algebra soð2;1Þ,ˆ
H0¼ˆ
J0þˆ
J1. The dynamics of
the final system being the QHO is described by the compact
generator 2
ˆ
J0¼ˆ
Hosc ¼ˆ
H0þˆ
Kof the same algebra. The
similarity transformation produced by ˆ
S0does not change
the conformal algebra, but being a nonunitary operator, it
intertwines slð2;RÞgenerators of different topological
natures according to Eq. (2.16).
One can modify the similarity transformation by chang-
ing operator (2.6) to
ˆ
S¼e
ˆ
Ke1
2
ˆ
H0eiln 2·ˆ
D¼e
ˆ
Keiln 2·ˆ
De
ˆ
H0:ð3:1Þ
As a result of the inclusion of the additional unitary factor
eiln 2·ˆ
D, operator (3.1) acts on the coordinate and momen-
tum ˆ
p¼id
dx operators in a more symmetric way. Instead
of (2.14),wehave
ˆ
Sˆ
p
ˆ
S1¼iˆ
a;
ˆ
Sx
ˆ
S1¼ˆ
aþ:ð3:2Þ
This is a nonunitary canonical transformation that is
identified as the fourth order root of the space reflection
operator,
ˆ
Sðx; ˆ
p; ˆ
aþ;ˆ
aÞðˆ
aþ;iˆ
a;iˆ
p; xÞ;
ˆ
S2ðx; ˆ
p; ˆ
aþ;ˆ
aÞðiˆ
p; ix; ˆ
a;ˆ
aþÞ;
ˆ
S4ðx; ˆ
p; ˆ
aþ;ˆ
aÞðx; ˆ
p; ˆ
aþ;ˆ
aÞ:ð3:3Þ
Therefore, from the point of view of the quantum phase
space, transformation (3.2) is the eighth order root of the
identity transformation.
The operator (3.1) can also be factorized as
ˆ
S¼expðˆ
J1
ˆ
J0Þ· expðiln 2·ˆ
J2Þ· expðˆ
J0þˆ
J1Þð3:4Þ
involving the slð2;RÞgenerators ˆ
Jμ. It produces the
following nonunitary automorphism of slð2;RÞ:
ˆ
S
ˆ
J0
ˆ
S1¼i
ˆ
J2;
ˆ
S
ˆ
J1
ˆ
S1¼
ˆ
J1;
ˆ
S
ˆ
J2
ˆ
S1¼i
ˆ
J0:
ð3:5Þ
The action of the squared nonunitary similarity trans-
formation on the slð2;RÞgenerators is a rotation by π
about J1,ˆ
S2ðˆ
J0;
ˆ
J1;
ˆ
J2Þð
ˆ
J0;
ˆ
J1;
ˆ
J2Þ, such that the
action of ˆ
Son ˆ
Jμis identified as the fourth order root of
the identity.
Equations (3.4) and (3.5) can be used to relate other pairs
of the quantum systems described by different realizations
of the conformal slð2;RÞdynamics in the sense of Diracs
different forms of dynamics [4].
IV. CLASSICAL PICTURE
In this section, we consider the classical picture under-
lying the quantum transformation based on conformal
symmetry and relating the quantum mechanical systems
of the free particle and harmonic oscillator. This will allow
us, in particular, to obtain an interesting reinterpretation of
some aspect of the quantum conformal bridge transforma-
tion in relation with the unitary Bargmann-Segal trans-
formation and the Neumann-Stone theorem.
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The classical analogs of the free particles generators of
conformal symmetry, H0¼1
2p2,D¼1
2xp, and K¼1
2x2,
satisfy the Poisson bracket relations
fD; H0H0;fD; KK; fK; H02D;
ð4:1Þ
which correspond to the quantum slð2;RÞsoð2;1Þ
algebra (A1) under the identification
J0¼1
2ðH0þKÞ;J
1¼1
2ðH0KÞ;J
2¼D:
ð4:2Þ
The classical analog of the Casimir element (A5) is
presented here in the form C¼H0KþD2, and takes
zero value, C¼0. Since J00, the dynamics of both the
free particle and the harmonic oscillator takes place on the
upper cone surface in coordinates of the slð2;RÞconformal
algebra generators ðJ0;J
1;J
2Þ.
The phase space functions H0,K, and Dgenerate the
following canonical transformations of xand p; see
Appendix B:
TH0ðαÞðxÞ¼xαp; TH0ðαÞðpÞ¼p; ð4:3Þ
TKðβÞðxÞ¼x; TKðβÞðpÞ¼pþβx; ð4:4Þ
TDðγÞðxÞ¼e1
2γx; TDðγÞðpÞ¼e1
2γp: ð4:5Þ
Then the composition
Tβαγ TKðβÞTH0ðαÞTDðγÞ¼TKðβÞTDðγÞTH0ð2αÞ
ð4:6Þ
transforms the canonical variables as follows:
TβαγðxÞ¼e1
2γðxð1αβÞαpÞ˜
x;
TβαγðpÞ¼e1
2γðpþβxÞ˜
p: ð4:7Þ
The choice
α¼i
2;β¼i; γ¼ln 2ð4:8Þ
gives
˜
x¼aþ;˜
p¼ia;˜
aþ¼ip; ˜
a¼x;
ð4:9Þ
where a¼1ffiffi2
pðxþipÞand aþ¼1ffiffi2
pðxipÞ,
fa;a
þi, are the classical analogs of the crea-
tion and annihilation operators. This shows that the
transformation Tβαγ with the parameters fixed as in (4.8)
is the classical analog of the operator (3.1).
The same complex canonical transformation applied to the
generators of conformal symmetry gives ˜
H0¼1
2ðaÞ2,
˜
K¼1
2ðaþÞ2,˜
D¼i
2aþa.The2iD transforms into the
Hamiltonian of the classical harmonic oscillator, and we find
that it generates the complex flow in the phase space given by
x0fx; 2iDix,p0fp; 2iDip. Taking into
account relations (4.9) and 2i
˜
D¼Hosc, one concludes that
these relations correspond exactly to the classical equations of
motion for the harmonic oscillator, _
a¼fa;Hoscg¼ia.
The similarity transformation (3.2) preserves the com-
mutation relations. However, the operator (3.1) is not
unitary with respect to the scalar product
ðψ1;ψ2Þ¼Zþ
−∞
ψ1ðxÞψ2ðxÞdx: ð4:10Þ
This is the case since the parameters αand βin the
corresponding classical canonical transformation are purely
imaginary, as a result of which the transformed canonical
variables ˜
x¼aþand ˜
p¼iaare complex variables,
while their quantum analogs (3.2) are not Hermitian
operators with respect to the scalar product (4.10). This
deficiency can be curedand reinterpreted in correspon-
dence with the Stonevon Neumann theorem [28]. The
transformed classical coordinate takes complex values,
˜
x¼aþC, and its complex conjugation is
¯
˜
x¼a¼i˜
p: ð4:11Þ
In correspondence with these properties, at the quantum
level we pass over from the coordinate representation to
representation in which ˆ
˜
x¼ˆ
aþacts as the operator of
multiplication by a complex variable z,ˆ
aþψðzÞ¼zψðzÞ,
while the operator iˆ
˜
p¼ˆ
aacts as a differential operator,
ˆ
aψðzÞ¼d
dz ψðzÞ. Hence the quantum relation ½ˆ
a;ˆ
aþ¼
1is the analog of the classical relation f˜
x; ˜
p1.
Replacing finally the scalar product (4.10) by the scalar
product (2.20),
ðψ1;ψ2Þ¼1
πZR2
ψ1ðzÞψ2ðzÞe¯zzd2z;
d2z¼dðRezÞdðImzÞ;ð4:12Þ
we arrive at the Fock-Bargmann representation, in which
the operators ˆ
aþ¼zand ˆ
a¼d
dz satisfy the relation
ðˆ
aþÞ¼ˆ
athat corresponds to the classical identity
(4.11). In this representation
2i
ˆ
˜
D¼ˆ
Hosc ¼zd
dz þ1
2;
ˆ
˜
H0¼1
2
d2
dz2;
ˆ
˜
K¼1
2z2;
ð4:13Þ
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where
ˆ
˜
Dis constructed from the phase space function ˜
Dvia
an antisymmetrization. In other words, the transformed
operators in the Fock-Bargmann representation can be
obtained from the corresponding initial generators of
conformal symmetry of the quantum free particle by a
formal change of xto z. The change of the scalar product
from (4.10) to (4.12) transmutes then the nonunitary
similarity transformation (3.2) into the unitary transforma-
tion from the coordinate to the holomorphic representation
for the Heisenberg algebra in correspondence with the
Neumann-Stone theorem, to which the kernel (2.19)
corresponds.
V. CONFORMAL MECHANICS BRIDGE
In this section, we construct in a similar way a conformal
bridge between the quantum conformal mechanics systems
(iii) and (iv). The peculiarity in comparison with the
previous case of the pair of the systems (i) and (ii) is that
here the operators ˆ
xand ˆ
pare not observables anymore
since ˆ
pis not self-adjoint and the commutator of ˆ
xwith the
corresponding Hamiltonian ˆ
Hνgives ˆ
p. Thus we work in
terms of an operator analogous to (3.4) and its associated
relations (3.5), which are well-defined being quadratic in ˆ
x
and ˆ
p.
The Hamiltonian operator of the two-body Calogero
model with an omitted center of mass degree of freedom is
ˆ
Hν¼1
2
d2
dx2þΔν;Δν¼νðνþ1Þ
2x2;ð5:1Þ
where ν≥−1=2,xRþ, and we impose the Dirichlet
boundary condition ψð0Þ¼0for the wave functions. The
case ν¼0corresponds to the free particle on the half-line Rþ
with eigenstates ψκðxÞ¼sin κxof energies E¼1
2κ2>0.
The limit limκ0sin κx=κ¼xgives a nonphysical,
unbounded zero energy eigenstate, which simultaneously
is the eigenstate of the dilatation operator (2.9),2i
ˆ
Dx ¼3
2x.
In the general case of ν≥−1=2, solutions of the stationary
equation ˆ
Hνψκ;ν¼EνðκÞψκ;νwith EνðκÞ¼1
2κ2,κ>0are
ψκ;νðxÞ¼ ffiffi
x
pJνþ1=2ðκxÞ;ð5:2Þ
where
JαðηÞ¼X
n¼0
ð1Þn
n!Γðnþαþ1Þη
22nþα
ð5:3Þ
is the Bessel function of the first kind. The solutions (5.2)
include ν¼0as a particular case. The Hamiltonian operator
ˆ
Hνis invariant under the change νðνþ1Þ,butthesame
transformation applied to (5.2) produces eigenfunctions of ˆ
Hν
not satisfying the boundary condition ψð0Þ¼0[21].At
ν¼1=2,wehaveν¼ðνþ1Þ, and the eigenstates (5.2)
correspond to a particular case of a family of self-adjoint
extensions of ˆ
H1=2[21,35].
Consider the set of wave functions xνþ1þ2n,n¼0;1;.
The function with n¼0represents a formal, diverging at
infinity, eigenstate of the differential operator ˆ
Hνwith ν
1=2of eigenvalue E¼0, which follows from solutions
(5.2) by applying to them the same limit as in the case of
ν¼0. The wave functions with n1are the Jordan states
of rank ncorresponding to the same eigenvalue E¼0of
ˆ
Hν. The functions xνþ1þ2nare at the same time eigenstates
of the operator 2i
ˆ
Dwith eigenvalues νþ2nþ3=2. The
Jordan states with n1satisfy the relations
ðˆ
HνÞjxνþ1þ2n
¼ð2ÞjΓðnþ1Þ
Γðnþ1jÞ
Γðnþνþ3=2Þ
Γðnþνþ3=2jÞxνþ1þ2ðnjÞ;
j¼0;1;;n; ð5:4Þ
which can be proved by induction. Equation (5.4) extends
to the case j¼nþ1giving ðˆ
HνÞnþ1xνþ1þ2n¼0due to the
appearance of the simple pole in the denominator.
The operator ˆ
Hνtogether with the operators ˆ
Kand ˆ
D
defined by Eqs. (2.8) and (2.9) satisfy the conformal
slð2;RÞalgebra (2.10) as in the case of the free particle
on the whole real line. Now we can define the direct analog
of the operator ˆ
Sin the form2
ˆ
Sνe
ˆ
Ke1
2
ˆ
Hνeiln 2·ˆ
D¼e
ˆ
Keiln 2·ˆ
De
ˆ
Hν:ð5:5Þ
A similarity transformation with this nonunitary operator
produces, analogously to (2.16), the relations
ˆ
Sν
ˆ
Hν
ˆ
S1
ν¼
ˆ
J;
ˆ
Sν
ˆ
K
ˆ
S1
ν¼ˆ
Jþ;
ˆ
Sνðˆ
Dވ
S1
ν¼i
2
ˆ
HAFF
ν;ð5:6Þ
where now instead of (2.17)
ˆ
J¼1
2ðˆ
aÞ2Δν;
ˆ
Jþ¼1
2ðˆ
aþÞ2Δν;
2
ˆ
J0¼ˆ
HAFF
ν¼ˆ
aþˆ
aþ1
2þΔν:ð5:7Þ
The dilatation operator multiplied by 2itransforms in this
case into the Hamiltonian ˆ
HAFF
νof the conformal mechan-
ics model of de Alfaro et al. [5]. At the same time, we note
that the operators xand d
dx are transformed by the operator
ˆ
Sνinto nonlocal operators corresponding to the square root
2One can work, instead, with the analog of the operator ˆ
S0, but
such a change is not essential.
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of the operators ˆ
Jþand ˆ
Jin (5.7), whose action violates
the boundary condition ψð0Þ¼0.
Application of ˆ
Sνto the states xνþ1þ2nrelated to the
system ˆ
Hνproduces the energy eigenstates of the AFF
model,
ˆ
Sνð1ffiffi
2
pxÞνþ1þ2n¼21
4ð1Þnn!ψAFF
ν;n ðxÞ;ð5:8Þ
where
ψAFF
ν;n ðxÞ¼xνþ1Lðνþ1=2Þ
nðx2Þex2=2;E
ν;n ¼2nþνþ3=2
ð5:9Þ
are the non-normalized eigenstates of the AFF model and
their corresponding energy values, and
LðαÞ
nðηÞ¼X
n
j¼0
Γðnþαþ1Þ
Γðjþαþ1ÞðηÞj
j!ðnjÞ!ð5:10Þ
are the generalized Laguerre polynomials.
On the other hand, application of the operator ˆ
Sνto the
eigenstates (5.2) of the system ˆ
Hνgives
ˆ
Sνψκ;ν1ffiffi
2
px¼21
4e1
2x2þ1
4κ2ffiffi
x
pJνþ1=2ðκxÞϕνðx; κÞ:
ð5:11Þ
According to the first relation in (5.6), these are the
coherent states of the AFF model [33],
ˆ
Jϕνðx; κÞ¼1
4κ2ϕνðx; κÞ:ð5:12Þ
By allowing the κ>0to become a complex parameter z,
coherent states can be constructed with complex eigenval-
ues of the operator ˆ
J. Application of the evolution
operator exp fit
ˆ
HAFF
νgto these states gives the time-
dependent coherent states
ϕνðx; z; tÞ¼21=4ffiffi
x
pJνþ1=2ðzðtÞxÞex2=2þz2ðtÞ=4it;ð5:13Þ
where zðtÞ¼zeit. In the case of ν¼0, these time-
dependent coherent states of the AFF model are the
odd Schrödinger cat states of the quantum harmonic
oscillator [36],
ϕ0ðx; z; tÞex2
2þz2ðtÞ
4it
2sinðzðtÞxÞ:ð5:14Þ
Similar to the analysis presented in the previous section,
one can consider the classical picture underlying the
quantum nonunitary similarity transformation associated
with the constructed conformal bridge between the systems
(iii) and (iv). We will not discuss this picture in detail here,
and only note that instead of (4.6), we will have the phase
space function of a similar form but with H0changed for
the Hamiltonian of the classical two-particle Calogero
system, Hν¼1
2p2þΔν. The phase space functions Hν,
D, and Ksatisfy the classical algebra of the form (4.1), and
the Casimir element Cν¼JμJμ¼HAFF
νKþD2takes
here the value defined by the coupling constant, Cν¼
1
4νðνþ1Þ. Analogous to the pair discussed in Sec. IV,
classical relations ðx2Þ0fx2;2iD2ix2,ðp2Þ0
fp2;2iD2ip2for the two-particle Calogero system
correspond here to the dynamics of the classical AFF model
given by the relations d
dt J¼fJ;H
AFF
νg¼2iJ, where
Jare the classical analogs of the operators ˆ
Jfrom (5.6).
On the Hilbert space of the AFF system, the infinite-
dimensional unitary irreducible representation of the
slð2;RÞalgebra of the discrete type series Dþ
αwith
α¼1
2νþ3=4is realized, in which the states ψAFF
ν;n from
(5.9) are eigenstates of the compact generator ˆ
J0with
eigenvalues j0;n ¼αþn,n¼0;1;, and the Casimir
operator takes the value ˆ
Cν¼ˆ
Jμˆ
Jμ¼αðα1Þ¼3
16
1
4νðνþ1Þ[3739].
VI. TWO-DIMENSIONAL EXAMPLES
When applying the transformation ˆ
Sto generators of the
conformal algebra, one (formally) need not care about the
concrete realization of the ˆ
Jμ, since only the algebraic
relations presented by Eqs. (3.4) and (3.5) are used. In
particular, one may consider higher dimensional examples,
where the range of physical systems of interest is greater. In
this section we generalize our construction to relate the
two-dimensional free particle system with a planar iso-
tropic harmonic oscillator and the Landau problem.
To begin with, consider the nonunitary operator
ˆ
Sðx; yÞ¼
ˆ
SðxÞ
ˆ
SðyÞð6:1Þ
with ˆ
SðxÞand ˆ
SðyÞof the form (3.1). Via a similarity
transformation, it produces a map
ˆ
Sðx; yÞðx; ˆ
px;y; ˆ
pyÞðˆ
aþ
x;iˆ
a
x;ˆ
aþ
y;iˆ
a
yÞ:ð6:2Þ
Then the two-dimensional free particle characterized by the
dynamical conformal symmetry with generators
ˆ
H¼1
2ðˆ
p2
xþˆ
p2
yÞ;
ˆ
D¼1
2ðxˆ
pxþyˆ
pyþ1Þ;
ˆ
K¼1
2ðx2þy2Þð6:3Þ
is related with the planar isotropic harmonic oscillator and
generators of its Newton-Hooke symmetry as follows:
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ˆ
Sðx; yÞˆ
H
ˆ
S1ðx; yÞ¼1
2ððˆ
a
xÞ2þðˆ
a
yÞ2Þ¼
ˆ
J;ð6:4Þ
ˆ
Sðx; yÞ2i
ˆ
D
ˆ
S1ðx; yÞ¼ˆ
aþ
x
ˆ
a
xþˆ
aþ
y
ˆ
a
yþ1¼ˆ
Hiso ¼2
ˆ
J0;
ð6:5Þ
ˆ
Sðx; yÞˆ
K
ˆ
S1ðx; yÞ¼1
2ððˆ
aþ
xÞ2þðˆ
aþ
yÞ2Þ¼ ˆ
Jþ;ð6:6Þ
where the operators ˆ
J0and ˆ
Jsatisfy the slð2;RÞalgebra
(A3). Analogously to the one-dimensional case, the sta-
tionary and coherent states of ˆ
Hiso are produced by ˆ
Sðx; yÞ
from the corresponding states of the two-dimensional free
particle system,
ˆ
Sðx; yÞxffiffi
2
pnyffiffi
2
pm
¼2nþmþ1
2eðx2þy2Þ
2HnðxÞHmðyÞ;
ð6:7Þ
ˆ
Sðx; yÞeiffiffi2
pðkxxþkyyÞ¼ffiffi
2
peðx2þy2Þ
2e
k2
xþk2
y
4eiðkxxþkyyÞ:ð6:8Þ
The angular momentum operator,
ˆ
M¼xˆ
pyyˆ
px¼iðˆ
a
x
ˆ
a
yˆ
a
y
ˆ
a
xÞ;ð6:9Þ
is an integral of the planar free particle and of the isotropic
harmonic oscillator systems. It is invariant under the
similarity transformation produced by ˆ
S, or, equivalently,
ˆ
Sðx; yÞˆ
M¼ˆ
M
ˆ
Sðx; yÞ:ð6:10Þ
Consider now the Landau problem for a scalar particle
on R2. In the symmetric gauge
A¼1
2Bðq2;q
1Þ, the
Hamiltonian operator (in units c¼m¼¼1),
ˆ
HL¼1
2
ˆ
Π2;
ˆ
Πj¼i
qj
eAj;½ˆ
Π1;
ˆ
Π2¼ieB;
ð6:11Þ
has an explicitly rotational invariant form. Assuming
ωc¼eB > 0, it can be factorized,
ˆ
HL¼ωcˆ
Aþˆ
Aþ1
2;ð6:12Þ
in terms of the Hermitian conjugate operators
ˆ
A¼1
ffiffiffiffiffiffiffi
2ωc
pðˆ
Π1i
ˆ
Π2Þ;½
ˆ
A;
ˆ
Aþ¼1:ð6:13Þ
Setting ωc¼2, we can identify qiwith dimensionless
variables q1¼x,q2¼y. Then we present ˆ
Aas linear
combinations of the usually defined ladder operators ˆ
a
x
and ˆ
a
y, in terms of which we also define the operators ˆ
B,
ˆ
A¼1ffiffiffi
2
pðˆ
a
yiˆ
a
xÞ;
ˆ
B¼1ffiffi
2
pðˆ
a
yiˆ
a
xÞ:ð6:14Þ
The operators ˆ
Bsatisfy relation ½
ˆ
B;
ˆ
Bþ¼1and com-
mute with ˆ
A. They are integrals of motion, and their
noncommuting Hermitian linear combinations ˆ
Bþþ
ˆ
B
and ið
ˆ
Bþ
ˆ
BÞcorrespond to coordinates of the center of
the cyclotron motion. In terms of the ladder operators ˆ
a
x,
ˆ
a
ythe Hamiltonian ˆ
HLtakes the form of a linear
combination of the Hamiltonian of the isotropic oscillator
and angular momentum operator,
ˆ
HL¼ˆ
Hiso
ˆ
M: ð6:15Þ
On the other hand, the angular momentum operator and
isotropic oscillators Hamiltonian are presented in terms of
ˆ
Aand ˆ
Bas follows:
ˆ
M¼
ˆ
Bþˆ
B
ˆ
Aþˆ
A;
ˆ
Hiso ¼
ˆ
Bþˆ
Bþ
ˆ
Aþˆ
Aþ1;
ð6:16Þ
and we have the commutation relations ½ˆ
M;
ˆ
B¼
ˆ
B,
½ˆ
M;
ˆ
A¼
ˆ
A. By taking into account the invariance of
the angular momentum under similarity transformation, we
find that its linear combination with the dilatation operator
is transformed into the Hamiltonian of the Landau problem,
ˆ
Sðx; yÞð2i
ˆ
D
ˆ
MÞ
ˆ
S1ðx; yÞ¼ ˆ
HL:ð6:17Þ
Let us now introduce a complex coordinate in the plane,
w¼1ffiffi
2
pðyþixÞ;and ¯
w: ð6:18Þ
The elements of conformal algebra and angular momentum
operator take then the form
ˆ
H¼2
w¯
w;
ˆ
D¼1
2iw
wþ¯
w
¯
wþ1;
ˆ
K¼w¯
w;
ˆ
M¼¯
w
¯
ww
w;ð6:19Þ
and we find that the operator (6.1) generates the similarity
transformations
ˆ
Sðx;yÞw
ˆ
S1ðx;yÞ¼
ˆ
Aþ;
ˆ
Sðx;yÞ
wˆ
S1ðx;yÞ¼
ˆ
A;
ð6:20Þ
ˆ
Sðx;yÞ¯
w
ˆ
S1ðx;yÞ¼
ˆ
Bþ;
ˆ
Sðx;yÞ
¯
wˆ
S1ðx;yÞ¼
ˆ
B;
ð6:21Þ
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ˆ
Sðx; yÞw
wˆ
S1ðx; yÞ¼
ˆ
Aþˆ
A;
ˆ
Sðx; yÞ¯
w
¯
wˆ
S1ðx; yÞ¼
ˆ
Bþˆ
B:ð6:22Þ
Observe that each pair of relations in (6.20) and (6.21) has a
form similar to the one-dimensional transformation (3.2),
where, however, the coordinate and momentum are
Hermitian operators.
Simultaneous eigenstates of the operators w
wand ¯
w
¯
w,
which satisfy the relations w
wϕn;m ¼nϕn;m and ¯
w
¯
wϕn;m ¼
mϕn;m with n; m ¼0;1;, are
ϕn;mðx; yÞ¼wnð¯
wÞm
¼2ðnþmÞ=2X
n
k¼0X
m
l¼0
×n
km
lðiÞnmþlkykþlxnþmkl;ð6:23Þ
where the binomial theorem has been used. Employing
Eq. (6.19) we find that
ˆ
Mϕn;m ¼ðmnÞϕn;m;2i
ˆ
Dϕn;m ¼ðnþmþ1Þϕn;m;
ð6:24Þ
ˆ
Kϕn;m ¼ϕnþ1;mþ1;
ˆ
Hϕn;m ¼nmϕn1;m1:ð6:25Þ
The last equality shows that ϕ0;m and ϕn;0are the zero
energy eigenstates of the two-dimensional free particle,
while the ϕn;m with n; m > 0are the Jordan states corre-
sponding to the same zero energy value. Application of the
operator ˆ
Sðx; yÞto these functions yields
ˆ
Sðx; yÞϕn;mðx; yÞ¼22ðnþmÞþ1
2eðx2þy2Þ
2Hn;mðy; xÞ
¼ψn;mðx; yÞ;ð6:26Þ
where
Hn;mðy;xÞ
¼2ðnþmÞX
n
k¼0X
m
l¼0n
km
lðiÞnmþlkHkþlðyÞHnþmklðxÞ
ð6:27Þ
are the complex Hermite polynomials; see [40]. These
functions are eigenstates of the operators ˆ
HL,ˆ
M, and ˆ
Hiso,
ˆ
HLψn;m ¼nþ1
2ψn;m;
ˆ
Mψn;m ¼ðmnÞψn;m;
ð6:28Þ
ˆ
Hisoψn;m ¼ðnþmþ1Þψn;m ;ð6:29Þ
and we note that ψn;n is rotational invariant.
Equations (6.20),(6.21), and (6.25) show that the
operators ˆ
Aand ˆ
Bact as the ladder operators for the
indexes nand m, respectively, while the operators ˆ
J
given by Eqs. (6.4) and (6.6) increase or decrease simulta-
neously nand mby one.
Application of the operator ˆ
Sðx; yÞto exponential
functions of the most general form eαwþβ¯
wwith α;βC
gives here, similar to the one-dimensional case, the
coherent states of the Landau problem as well of the
isotropic harmonic oscillator,
ψLðx; y; α;βÞ¼
ˆ
Sðx; yÞe1ffiffi2
pððαþβÞyþiðαβÞxÞ¼ffiffi
2
peðx2þy2Þ
2þðαþβÞyþiðαβÞxαβ
¼X
n¼0X
n
l¼0
1
n!n
lαlβnlψl;nlðx; yÞ:ð6:30Þ
Applying to them, in particular, the evolution operator
eit ˆ
HL, we obtain the time dependent solution to the Landau
problem,
ψLðx; y; α;β;tÞ¼eit
2ψLðx; y; αeit;βÞ;ð6:31Þ
whereas under rotations these states transform as
eiφ
ˆ
MψLðx; y; α;βÞ¼ψLðx; y; αeiφ;βeiφÞ:ð6:32Þ
As the function eαwþβ¯
wis a common eigenstate of the
differential operators
wand
¯
wwith eigenvalues αand β,
respectively, then our transformation yields
ˆ
AψLðx; y; α;βÞ¼αψLðx; y; α;βÞ;
ˆ
BψLðx; y; α;βÞ¼βψLðx; y; α;βÞ;ð6:33Þ
which provides another explanation why the wave func-
tions (6.30) are the coherent states for the planar harmonic
oscillator as well as for the Landau problem.
VII. DISCUSSION AND OUTLOOK
The intertwining operators of the Darboux transformations
allow one to construct eigenstates of a Hamiltonian of one
system in terms of eigenstates of a Hamiltonian of another,
related system. There, the intertwining operators are differ-
ential operators of finite order, and so the relation between
INZUNZA, PLYUSHCHAY, and WIPF PHYS. REV. D 101, 105019 (2020)
105019-10
corresponding eigenstates is local. Those operators also
factorize the corresponding Hamiltonians or polynomials
thereof. In our conformal bridgeconstructions, the generator
of the similarity transformation is a nonlocal operator, formally
given by an infinite series in the momentum operator. It has a
nature of the fourth order root of the identity operator from the
point of view of its action on generators of the conformal
algebra, where it acts as a nonunitary automorphism. Its
peculiarity in comparison with generators of the Darboux
transformations is that it intertwines the generators of the
slð2;RÞhaving a different topological nature corresponding
to a change of the dynamics in the sense of Dirac. In fact, our
transformation changes the conformally invariant asymptoti-
cally free dynamics described by the Hamiltonian being a
noncompact generator of the slð2;RÞto the conformal
Newton-Hooke dynamics generated by a compact generator.
In comparison with the well-known Niederers canonical
transformation [11,41],
ζ¼x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þðωtÞ2
p;dτ¼dt
1þðωtÞ2;ð7:1Þ
employed to transform the one-dimensional conformal
mechanics model into the regularized AFF model, our
quantum nonunitary similarity transformations and corre-
sponding complex classical canonical transformations are
time independent: we work within the stationary frame-
work. As a result, instead of the map between solutions of
the time-dependent Schrödinger equation of the free
particle and harmonic oscillator, or of the conformal
mechanics corresponding to the two-particle Calogero
model and the AFF model, our transformation acts in an
essentially different way. It maps formal eigenstates of the
dilatation operator with imaginary eigenvalues being also
Jordan states corresponding to the zero energy value of the
conformal mechanics model Hamiltonian ˆ
Hν(or of the free
particle Hamiltonian ˆ
H0) into physical bound eigenstates of
the regularized AFF model ˆ
HAFF
ν(or of the harmonic
oscillator ˆ
Hosc) of the real energy values. Besides, the
stationary eigenstates of ˆ
Hν(or of ˆ
H0) are transformed into
the coherent states of ˆ
HAFF
ν(ˆ
Hosc).
The interesting question is then what is a possible geomet-
rical interpretation of our complex canonical transformation,
being an eighth order root of the identity transformation. One
could try to characterize it within the framework of the
Eisenhart lift [42] or by using the Dirac trick which corre-
sponds to a generation of the time-dependent Schrödinger
equation via the presentation of the classical action in the
reparametrization invariant form. This way one may expect to
establish a possible relation (if it exists) of our conformal
bridgeconstructions with the transformations based on the
Niederers time-dependent canonical transformation (7.1).
Investigation of the indicated problem may be of interest in
the light of the AdS=CFT correspondence [1,2,43].
Our canonical transformations correspond to the
Hamiltonian vector flows produced by generators of the
conformal symmetry with particular complex values of
the parameters. The interesting question is whether such
transformations having the nature of the fourth order root of
the spatial reflection have any interpretation in the context
of PT symmetry [4446]. This question is intriguing
especially having in mind that perfectly invisible zero-
gap systems and the related extreme wave solutions (multi-
solitons) to the complexified KdVequation can be obtained
from the free particle system by application to it of the
PT -regularized Darboux transformations intimately
related to conformal symmetry [19,22,47].
It is not difficult to find the analog of our canonical
transformation that maps the dilatation generator of the free
particle into the Hamiltonian of the inverted (repulsive)
harmonic oscillator having a continuous spectrum at the
quantum level [48]. In this way, Gamow states [49] enter
naturally into the construction. However, at the quantum
level there appears a problem because the transformation,
which now will be unitary, acting on the eigenstates of the
dilatation operator with real eigenvalues will produce
divergent series. This implies the necessity of a deeper
analysis of the corresponding unitary transformation.
Yet another interesting question is whether by using a
conformal bridge transformation, new solutions to some
integrable systems in partial derivatives can be constructed.
We have here in mind the generation of the multisoliton
solutions to the KdV equation, including rational solutions
of the Calogero type, by application of the Darboux trans-
formations to the quantum free particle system on the one
hand, and the conformal bridge relation of the free particle
with the quantum harmonic oscillator that we have discussed.
Such a hypothetical possibility still remains a mystery for us.
As our transformation does not depend on a concrete
realization of the generators of the conformal algebra, as we
showed, one can study higher dimensional systems. In this
way we found an interesting relation between the 2D free
particle system and the Landau problem, where the two-
dimensional plane waves were mapped into the coherent
states of the planar harmonic oscillator and of the Landau
problem. It would be interesting to investigate what happens
in the case of the higher-dimensional nonseparable conformal
invariant systems such as the multiparticle Calogero model,3
or in the case of the Dirac magnetic monopole system.
3Based on a spectrum similarity, Calogero conjectured [13]
that there should exist a nontrivial relation between his model and
the system of decoupled oscillators. This conjecture was proved,
including the case without quadratic interactions, by constructing
some (non)unitary transformations that mutually map Hamilto-
nians of the corresponding systems of the coupled and decoupled
particles [5053] and give rise to a nonlocal slð2;RÞgenerator
[53]. The conformal transformations we discussed here, in spite
of being of a somewhat similar form to those in [5153], are
considered from an essentially different perspective of the change
of the form of dynamics álaDirac [1,2,4,5,10].
CONFORMAL BRIDGE BETWEEN ASYMPTOTIC PHYS. REV. D 101, 105019 (2020)
105019-11
Conformal bridge transformations that we considered are
based on the global dynamical slð2;RÞsymmetry. This
global symmetry is promoted to the local conformal
[symplectic, spð2;RÞslð2;RÞ] gauge symmetry in 2T
(two-time) physics; see [5459] and further references
therein. The 2T physics provides a holographicunifica-
tion of different 1T systems, which appear there under
different choices of gauge fixing and are treated as different
shadowsof the extended system with gauged conformal
symmetry. It would be very interesting to consider the
conformal bridge transformations in the light of the 2T
physics approach and somehow related to it contact
geometry and Reebs dynamics [60,61]. The particular
interesting question in this context is how the quantum
Jordan states corresponding to zero energy would reveal
themselves in the 2T physics and contact quantization, and
in the associated duality relations. Another interesting point
to clarify is how the complex nature of the generalized
canonical transformations, which underlie the quantum
conformal bridge transformations, will manifest itself in
2T physics. The study of the conformal bridge trans-
formations in the context of 2T physics also is promising
having in mind that the latter approach finds applications to
a very broad class of the physical systems.
As a final interesting problem that seems to be rather
natural for further investigation we indicate a generalization
of our constructions to the superconformal case [62,63],
bearing, particularly, in mind the presence of the hidden
superconformal symmetry in the quantum harmonic oscil-
lator [27]. It is worthwhile to note that the topic of
supersymmetry was also addressed within the framework
of 2T physics [55,59] and contact geometry [64].
Some results on the listed problems are presented in our
paper [65].
ACKNOWLEDGMENTS
The work was partially supported by the CONICYT
scholarship 21170053 (L. I.), FONDECYT Project
No. 1190842 (M. S. P.), by the Project USA 1899 (M. S.
P. and L. I.), and by DICYT, USACH (M.S.P.). L. I. and M.
S. P. are grateful to Jena University for the warm hospitality
where a part of this work was realized. We thank O.
Lechtenfeld for drawing our attention to [53] and related
Refs. [5052].
APPENDIX A: CONFORMAL SYMMETRY
The soð2;1Þslð2;RÞsuð1;1Þslð2;RÞspð2;RÞ
algebra is given by the commutation relations
½Jμ;J
ν¼iϵμνλJλ;ðA1Þ
where ϵμνλ is a totally antisymmetric tensor, ϵ012 ¼1, and
the metric is ημν ¼diagð1;1;1Þ. The algebra (A1) can be
generated, in particular, by Jμ¼ϵμνλxνpλrealized in
terms of the (2þ1)-dimensional coordinate and momenta
operators, ½xμ;p
ν¼iημν.
With
JJ1iJ2;ðA2Þ
the algebra (A1) takes the form of the conformal slð2;RÞ
algebra
½J;J
þ¼2J0;½J0;J
¼J:ðA3Þ
The transformation
J0J0;J
JðA4Þ
defines an outer automorphism of (A3), and the Casimir
element of the algebra is
C¼JμJμ¼ðJ0Þ2þðJ1Þ2þðJ2Þ2
¼ðJ0Þ2þ1
2ðJþJþJJþÞ:ðA5Þ
The realization
J¼
t;J
0¼t
t;J
þ¼t2
tðA6Þ
generates the conformal transformations
ttþα;teβt; t
t
1γtðA7Þ
of a time variable. The transformations (A7) together with
xe1
2βx; x
x
1γtðA8Þ
are generated by
J¼
t;J
0¼t
tþ1
2x
x;J
þ¼t2
tþxt
x;ðA9Þ
and represent the dynamical conformal symmetry of the
free particle action A¼m
2R_
x2dt.
The formal change tz,zC,in(A6) accompanied
by the automorphism (A4) with subsequent identification
L0¼J0,Lþ1¼J,L1¼Jþgives generators of the
suð1;1Þsubalgebra of the Virasoro algebra,
L0¼z
z;L
þ1¼z2
z;L
1¼
z:ðA10Þ
This can be realized as
J0¼aþa;J
þ¼ðaþÞ2a;J
¼aðA11Þ
with ladder operators a¼1ffiffi2
pðxd
dxÞof the quantum
harmonic oscillator. Another realization of (A3) is given by
INZUNZA, PLYUSHCHAY, and WIPF PHYS. REV. D 101, 105019 (2020)
105019-12
J0¼1
4ðaþaþaaþÞ;J
¼1
2ðaÞ2;ðA12Þ
which corresponds to the conformal Newton-Hooke
symmetry of the quantum harmonic oscillator.
APPENDIX B: HAMILTONIAN VECTOR FLOWS
AS CANONICAL TRANSFORMATIONS
A Hamiltonian vector flow generated by a function Fon
a phase space Mis given by
fðαÞ¼expðαFÞfðq; pÞ
fðq; pÞþX
n¼1
αn
n!fF; f;fF; fggg
|fflffl{zfflffl}
n
TFðαÞðfÞ:
ðB1Þ
The parameter αis usually assumed to be real, but we allow
for complex values. Transformations (B1) correspond to
the action of a one-parametric Lie group on M,
TFðαÞTFðβÞ¼TFðαþβÞ;T
Fð0Þ¼I;
ðTFðαÞÞ1¼TFðαÞ:
The composition of the Hamiltonian flows generated by
functions Fand Gwith fF; Gg0is noncommutative, and
ðTFðαÞTGðβÞÞ1¼TGðβÞTFðαÞ:
For functions fand gon phase space, the following relation
holds:
TFðαÞTGðβÞðf·gÞ
¼ðTFðαÞTGðβÞðfÞÞ ·ðTFðαÞTGðβÞðgÞÞ:ðB2Þ
A flow of a Hamiltonian vector field is a canonical trans-
formation : ffðαÞ;gðαÞg ¼ ff; gg. In the general case of
αC, the transformation (B1) corresponds to the quantum
similarity transformation ˆ
fðαÞ¼expðiα
ˆ
FÞ
ˆ
fexpðiα
ˆ
FÞ.
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105019-14
... Our goal here is to study the influence of the geometrical properties of this background (encoded in a "geometrical parameter" α given in terms of the linear mass density of the string) on the dynamics of the systems from the perspective of well-defined integrals of motion in the phase space when considering classical cases, and well-defined symmetry operators for the corresponding quantum versions of the systems. We are interested in the case of the free particle in the cone, which has the so(2, 1) conformal symmetry, and in the harmonic oscillators system in the same geometry, which is characterized by the sl(2, R) ∼ = so(2, 1) conformal Newton-Hooke symmetry [51][52][53][54][55]. The key construction in our investigation is the so-called conformal bridge transformation [55], which in general is a mapping that allows us to transform the complete set of symmetry generators from an so(2, 1) invariant asymptotically free system, to those of a harmonically confined model with the sl(2, R) conformal symmetry. ...
... We are interested in the case of the free particle in the cone, which has the so(2, 1) conformal symmetry, and in the harmonic oscillators system in the same geometry, which is characterized by the sl(2, R) ∼ = so(2, 1) conformal Newton-Hooke symmetry [51][52][53][54][55]. The key construction in our investigation is the so-called conformal bridge transformation [55], which in general is a mapping that allows us to transform the complete set of symmetry generators from an so(2, 1) invariant asymptotically free system, to those of a harmonically confined model with the sl(2, R) conformal symmetry. The plan is to first characterize the free case by identifying the corresponding classical and quantum integrals of motion for different values of α, and then to obtain the complete information on the harmonic oscillator system through the conformal bridge transformation. ...
... can be reinterpreted as the Landau problem's Hamiltonians in the symmetric gauge, with positive/negative magnetic field of the magnitude B = 2cmω q , where q and c correspond to the electric charge of the particle and the speed of light [55]. ...
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A bstract Hidden symmetries of non-relativistic $$ \mathfrak{so}\left(2,1\right)\cong \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$ so 2 1 ≅ sl 2 ℝ invariant systems in a cosmic string background are studied using the conformal bridge transformation. Geometric properties of this background are analogous to those of a conical surface with a deficiency/excess angle encoded in the “geometrical parameter” α , determined by the linear positive/negative mass density of the string. The free particle and the harmonic oscillator on this background are shown to be related by the conformal bridge transformation. To identify the integrals of the free system, we employ a local canonical transformation that relates the model with its planar version. The conformal bridge transformation is then used to map the obtained integrals to those of the harmonic oscillator on the cone. Well-defined classical integrals in both models exist only at α = q / k with q, k = 1 , 2 , . . . , which for q > 1 are higher-order generators of finite nonlinear algebras. The systems are quantized for arbitrary values of α ; however, the well-defined hidden symmetry operators associated with spectral degeneracies only exist when α is an integer, that reveals a quantum anomaly.
... See Refs. [68][69][70] for details, Refs. [71,72] for recent applications to cosmological systems and Ref. [41] for black hole mechanics. ...