Conformal Newton–Hooke algebras, Niederer's transformation and Pais–Uhlenbeck oscillator

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Physics

Letters

B

www.elsevier.com/locate/physletb

Conformal

Newton–Hooke

algebras,

Niederer’s

transformation

and

Pais–Uhlenbeck

oscillator

Krzysztof Andrzejewski

DepartmentofComputerScience,UniversityofŁód´z,Pomorska149/153,90-236Łód´z,Poland

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received20September2014 Receivedinrevisedform30September 2014

Accepted3October2014 Availableonline8October2014 Editor: M.Cvetiˇc

Dynamicalsystemsinvariantundertheactionofthel-conformalNewton–Hookealgebrasareconstructed by the method of nonlinear realizations. The relevant first order Lagrangians together with the correspondingHamiltoniansarefound.TherelationtotheGalajinskyandMasterov[24]approachaswell asthehigherderivativesformulationisdiscussed.ThegeneralizedNiederer’stransformationis presented whichrelates thesystemsunder considerationtothose invariant underthe actionofthel-conformal Galileialgebra[25].Asaniceapplicationoftheseresults ananalogueofNiederer’stransformation,on theHamiltonianlevel,forthePais–Uhlenbeckoscillatorisconstructed.

©2014TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3.

1. Introduction

TheNewton–HookealgebraisageneralizationoftheGalileione tothe caseofnonvanishing cosmologicalconstant leading to the universal cosmologicalrepulsion orattraction (see, e.g., [1–3]). It is derived from the (anti) de Sitter algebra by the nonrelativis-ticcontractionina similar wayasthe Galilei algebraisobtained fromthe Poincaré one. The main difference betweenGalilei and theNewton–Hookealgebras isthatinthelattercasethestructure relations involving the generators of time andspace translations yieldtheGalileiboosts:

[

H

,

Pi

]

= ±

_R12Ki.The positiveconstant R is called the characteristic time (and is related to the radius of theparent (anti) de Sitter space).The upper/lower signabove is realizedinnonrelativisticspacetimewiththenegative/positive cos-mologicalconstant

Λ

= ∓

_R12.

ConformalextensionsoftheGalileiandNewton–Hookealgebras haverecentlyattractedconsiderableinterest,mostlyinthecontext ofthe nonrelativisticAdS/CFT correspondence (see [4–6] andthe references therein). Finite-dimensional extensions are parameter-izedbyapositivehalf-integerl [7–10],whichjustifiestheirname: l-conformalalgebras.Thedynamicalrealizationsofthel-conformal Galilei and Newton–Hooke algebras involve, in general, higher derivativesterms (see, e.g., [11–18]). However, it isalso possible (usingthe methodofnonlinear realizations[19–21]) toconstruct invariantdynamicsinvolvingonlysecondderivatives

[22–24]

.The

E-mailaddress:k-andrzejewski@uni.lodz.pl.

methodproposedinRefs. [22–24]allowsforelegantand algorith-micconstructionofinvariantdynamicalequations.However,there remains an open problemifthey admit Lagrangian and Hamilto-nianformalism.InRef.[25]ithasbeenshownthatthisispossible forthecaseofthel-conformalGalileialgebra.

Inthe presentpaper,first,we apply themethod developedin

[25]tothecaseofthel-conformalNewton–Hookealgebraandwe construct the invariant dynamics in terms of the first order La-grangianandHamiltonianformalism(Sections2and

3

).Moreover, we compare our approach with the one reported in Ref. [24] as wellaswiththePais–Uhlenbecktheory(Section4).

The second part of the paper is devoted to the problem of Niederer’s-typetransformations.InSection5weconstructan anal-ogy of the celebrated Niederer’s transformation [26] for our ap-proach,andweshow thatit leads totheresultsin

[25]

obtained for the l-conformal Galilei algebra; the relevance of Niederer’s transformation in the context of the both algebras, for the case l

=

1₂,wasalsodiscussedinRef.[27].Ontheotherhand,onthe La-grangianlevel,thegeneralizationofNiederer’stransformationhas beenalsoextensivelystudiedforthePais–Uhlenbecksystemwith odd frequencies (i.e., frequencies proportional to the consecutive odd integers); see, e.g., [28–30]. However, its Hamiltonian coun-terpart seems to be more involved dueto the lack of the direct transitionto theHamiltonianformalism fora theory withhigher derivatives. We solve this problemand give (see, Section 6) the explicit form of the canonical transformation which relates the Pais–UhlenbeckHamiltonian(withoddfrequencies)totheonefor thefreehigherderivativestheory.

http://dx.doi.org/10.1016/j.physletb.2014.10.008

0370-2693/©2014TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP3_.

2. Conformalmechanics

The prototype of all conformal groups is the one acting in

(

1

+

0

)

-dimensional spacetime, locally isomorphic to SL

(

2

,

R)

; for the recent developments in conformal mechanics see, e.g., Refs. [31–33].Inordertoconstructtheconformal Newton–Hooke dynamics,i.e.,the dynamicsoftheconformal particleinthe har-monic trap,we mustmodifythe Hamiltonianbyaddingthe con-formalgenerator.Thus wechoosethebasisofthesl

(

2

,

R)

algebra asfollows

[

H

,

D

] =

i

(

H

∓

2K

),

[

D

,

K

] =

i K

,

[

H

,

K

] =

2i D

.

(1)

Itisworthtonotethat,although weonlychangethebasis (H

→

H

±

K ) of sl

(

2

,

R)

algebra, this alters the dynamics and, conse-quently,thedynamicalrealizationsofthealgebra.

Letusconsider thedecompositionbasedon D as thestability subgroupgenerator. Then the coset space isparametrized as fol-lows

w

=

eit HeizK

,

(2)

andtheactionoftheSL

(

2

,

R)

groupisdefinedby

geit HeizK

=

eitHeizKeiuD

,

(3)

whichcanbeexplicitlyfoundbytakingtherepresentationspanned by H

=

i

(

−

σ

₊

±

σ

₋

),

K

=

i

σ

₋

,

D

= −

i 2

σ

3

.

(4) Itreads, t

=

arctan



α

tan t

+ β

γ

tan t

+ δ



,

z

=



(

α

sin t

+ β

cos t

)

2

+ (

γ

sin t

+ δ

cos t

)

2



z

,

+

1

2



β

2

+ δ

2

−

α

2

−

γ

2



sin 2t

− (

γ

δ

+

α

β)

cos 2t

,

u

= −

ln



(

α

sin t

+ β

cos t

)

2

+ (

γ

sin t

+ δ

cos t

)

2



,

(5)

intheoscillatorycase(

+

),and

t

=

arctanh



α

tanh t

+ β

γ

tanh t

+ δ



,

z

=



(

γ

sinh t

+ δ

cosh t

)

2

− (

α

sinh t

+ β

cosh t

)

2



z

,

+

1

2



β

2

− δ

2

+

α

2

−

γ

2



sinh 2t

+ (

α

β

−

γ

δ)

cosh 2t

,

u

= −

ln



(

γ

sinh t

+ δ

cosh t

)

2

− (

α

sinh t

+ β

cosh t

)

2



,

(6)

in the hyperbolic one (

−

); here, g

=



_γαβ_δ



∈

SL

(

2

,

R)

. Due to Eq.(1) the Cartan forms w−1dw

≡

i

(

ω

HH

+

ω

KK

+

ω

DD

)

coin-cidewiththoseintheoldbasisandread

ω

H

=

dt

ω

K

=

dz

+

z2dt

,

ω

D

= −

2zdt

.

(7) However,thetransformationruleschangeandtaketheform

ω

_H

=

eu

ω

H

,

(8)

ω

_K

=

e−u

ω

K

+



±

e−u

∓

eu



ω

H

,

ω

_D

=

ω

D

−

du

.

(9)

The covariantderivative ofz is definedastheratioofthe Cartan forms

∇

z

=

ω

K

ω

H

= ˙

z

+

z2

;

(10)

onecaneasilyobtains

∇

z

=

e−2u

∇

z

±

e−2u

∓

1

.

(11)

Inordertoconstructtheinvariant dynamicsitissufficienttofind the action integral invariant under the dilatation subgroup. This canbeeasilydonebytakingtheLagrangian

L0

=



˙

z

+

z2

_±

₁

_,

₍₁₂₎

orthecorrespondingHamiltonian

H0

=

−

1 4pz

−

pzz2

∓

pz

,

(13) with

{

z

,

pz

}

=

1. The Lagrangian (12) (or the Hamiltonian (13)) leadstothefollowingequationofmotion

¨

z

+

6zz

˙

+

4z3

±

4z

=

0

.

(14)

The aboveequationcontains thewholefamilyofconformal mod-els.Infact,withthesubstitution

z

=

ρ

˙

ρ

,

(15)

suggestedinRefs.[23]and

[34]

,Eq.(14)yields

d dt



¨

ρρ

3

±

ρ

4



=

0

,

(16) or

¨

ρ

=

γ

2

ρ

3

∓

ρ

,

(17)

i.e.,conformalparticleintheharmonictrap. Letusnotethatthereplacement

t

→

it

,

z

→ −

iz

,

(18)

performed in Eq.(14), relatesthe oscillatory case(

+

) to the hy-perbolicone (

−

).This,together withthetransformationrules de-scribed by thefirst equation of(5) and(6)impliesthe following changeoftheactionoftheSL

(

2

,

R)

group:

g

=



α

β

γ

δ



→

g

=



α

i

β

−

i

γ

δ



.

(19)

Note that both realizationsof SL

(

2

,

R)

are equivalent butnot re-latedbyaninnerautomorphism.

Togetridofthesquarerootintheactionintegralonecan fol-lowthestandardprocedurebywriting

L1

= −

γ

2

η

−

1 2

η



˙

z

+

z2

±

1



,

(20)

where

γ

isanarbitraryconstantwhile

η

isanadjointfield trans-formingaccordingto

η



=

e−u

_η

_.

Now, let usperform a simple canonical analysis. The primary constraintsread

χ

1

≡

pη

≈

0

,

χ

2

≡

pz

+

1

2

η

≈

0

,

(21)

whiletheHamiltonianiswrittenas

H1

=

γ

2

η

+

1 2

η



z2

±

1



+

uηpη

+

uz



pz

+

1 2

η



,

(22)

uη,uz beingtheappropriateLagrangemultipliers.Imposing d dtpη

≈

0

,

d dt



pz

+

1 2

η



≈

0

,

(23)

wefindnonewconstraintswhile

uz

=

2

γ

2

η

2

−



z2

±

1



,

uη

= −

2z

η

.

(24) So the constraints (21) are of the second kind. This allows us to eliminate pη and pz at the expense of introducing the Dirac bracketandfinallyweobtain

HD

=

γ

2

η

+

z2

±

1 2

η

,

{

z

,

η

}

D

=

2

η

2

_.

₍₂₅₎ Putting

η

=

1

ρ

2

,

z

=

p_ρ

ρ

,

(26)

onearrivesatthestandardform

HD

=

1 2p 2 ρ

+

γ

2

ρ

2

±

1 2

ρ

2

_,

_{

_ρ

_,

_p ρ

} =

1

.

(27) 3. Dynamicalrealizationsofthel-conformalNewton–Hooke algebras

The l-conformal Newton–Hookealgebra (in three-dimensional case)isspannedby thegenerators H

,

D

,

K satisfying

(1)

together withso

(

3

)

generators Jk and2l

+

1 additionalgeneratorsC

(n),n

=

0

,

1

,

. . . ,

2l obeying



H

,

C(n)



=

i



nC

(n−1)

± (

n

−

2l

)

C(n+1)



,



K

,

C(n)



=

i

(

n

−

2l

)

C(n+1)

,



D

,

C(n)



=

i

(

n

−

l

)

C(n)

,



Ji

,

C_k(n)



=

i

ε

ikmCm(n)

.

(28)

Consider the nonlinear action definedby selecting the subgroup generatedby

J andD.Withsuchachoicewearenotdealingwith thesymmetric decomposition.However, the generators H

,

K and

C(n)_{spanthelinearrepresentationundertheadjointactionofthe} stabilitysubgroup.Therefore,ourrealizationlinearizesonit.In or-der toconstructthe invariant dynamicsit is sufficientto respect theinvarianceunderrotationsanddilatation.

Letuschoosethefollowingparametrizationofthecoset mani-fold

w

=

eit Hei x(n)C (n)eizK

;

(29)

note the difference with respect to the parametrization used in[24].

TheCartanforms

w−1dw

=

i



ω

HH

+

ω

DD

+

ω

KK

+

ω

(n)C

(n)



,

(30)

aregivenbyEqs.(7)togetherwith

ω

(n)

=

n

p=0



2l

−

p 2l

−

n



(

−

z

)

n−p



d

x(p)

− (

p

+

1

)

x(p+1)dt

∓ (

p

−

1

−

2l

)

x(p−1)dt



.

(31)

Theforms

ω

(n)_{arevectorsunderSO}

₍

₃

₎

_{whileunderdilatation}

ω

 (n)

₌

_e(l−n)u

_ω

(n)

_.

₍₃₂₎

Definethecovariantderivatives

∇

x(n)

≡

ω

(n)

ω

H

,

(33)

with the dilatation dimension l

−

n

−

1. Let

λ

(n) _{be additional} (adjoint) variables with dilatation dimension n

−

l. Consider the followingfirstorderLagrangian

L

= −

γ

2

η

−

1 2

η



˙

z

+

z2

±

1



+

2l

n=0

λ

(n)

_∇

_x(n)

_.

₍₃₄₎ Bythe very construction it yields theinvariant action functional. Theequationsofmotionareoftheform

2

γ

2

η

2

−



z

˙

+

z2

±

1



=

0

,

˙

η

+

2z

η

=

0

,

˙

x(n)

− (

n

+

1

)

x(n+1)

∓

x(n−1)

(

n

−

1

−

2l

)

=

0

,

n

=

0

, . . . ,

2l

,

2l

−p n=0



2l

−

p n



d dt



(

−

z

)

n

λ

(n+p)



+

p 2l−p+

1 n=0



2l

−

p

+

1 n



(

−

z

)

n

λ

(n+p−1)

± (

p

−

2l

)

2l−

p−1 n=0



2l

−

p

−

1 n



(

−

z

)

n

λ

(n+p+1)

=

0

.

(35)

We see that they decouple.The first two describe theconformal mechanicsinthe harmonictrap. Then,thereisa setofequations for

x(n) _{describing higherderivatives system. Letus note that in} ourapproachwe donotneedtoperformtheredefinitionoftime asinRef. [24].Finally,once z

(

t

)

isdetermined one cansolve the last equationfor

λ

(n);they donotimpose anyfurther constraints onz.

Finally,letusnotethatextendingthetransformationrules

(18)

by

η

→ −

i

η

,

λ

p

→

ip

λ

p

,

xp

→ (−

i

)

p

xp

,

(36) onecan transformtheLagrangian

(34)

fromtheoscillatory tothe hyperboliccase.

OurLagrangian,beingofthefirstorder,providesanexampleof aconstrainedsystem.Following

[35]

,theHamiltoniandynamicsis givenby H

=

γ

2

η

+

z 2

_±

₁ 2

η

+

2l

n=0

λ

(n) n

p=0



2l

−

p 2l

−

n



(

−

z

)

n−p

×



(

p

+

1

)

x(p+1)

± (

p

−

1

−

2l

)

xp−1



,

(37) togetherwith

x(an)

, λ

_b(m)



D

=

zn−m



2l

−

m 2l

−

n



δ

ab

,

{

z

,

η

}

D

=

2

η

2

,

λ

(k)

,

η



_D

=

2

(

2l

−

k

)

η

2

λ

(k+1)

.

(38)

Again,itisstraightforward,althoughslightlytedious,tocheckthat Eqs.(37)and

(38)

yieldthecorrectdynamics.

4. Pais–Uhlenbeckoscillator

Ithasbeenshownin Ref.[30] that forl half-integer (i.e.2l is odd)thePais–Uhlenbeckoscillatoroforder2l

+

1[36]

L

=

(

−

1

)

l+1 2 2

x l+1 2



k=1



d2 dt2

+

ω

2 k



x

,

(39)

withoddfrequencies

ω

k

= (

2k

−

1

)

ω

= (

2k

−

1

)

(inwhat follows weput

ω

=

1

R

=

1),k

=

1

,

2

,

. . . ,

l

+

1

2 enjoysl-conformalNewton–

Hookesymmetry(infact,itisthemaximalsymmetrygroup). In orderto compare thisfinding withour results, let usnote thatforthel half-integer,wecanputtheoscillatorsystemdefined bythedecoupledequationsfor

x’s

˙

x(n)

− (

n

+

1

)

x(n+1)

∓

x(n−1)

(

n

−

1

−

2l

)

=

0

,

n

=

0

, . . . ,

2l

,

(40)

intotheunconstrainedHamiltonianform.Toseethiswedefine

H

=

l−3 2

k=0

pk

qk+1

+

1 2

p 2 l−1 2

±

−

l−3 2

k=0

(

k

+

1

)(

2l

−

k

)

q

k

pk+1

+

1 2



l

+

1 2



2

q2 l−1 2



,

(41)

whichcorresponds toourchangeof thebasis H

→

H

±

K inthe algebraofthefreetheory;thestandardPoissonbracketsread

{

qka

,

pjb

} = δ

kj

δ

ab

.

(42)

TheHamiltonian

(41)

togetherwiththePoissonbrackets

(42)

yield thefollowingequationsofmotion

˙

q_k

=

q_k+1

∓ (

2l

+

1

−

k

)

kq

k−1

,

˙

p_k

= ±(

k

+

1

)(

2l

−

k

)

pk+1

−

pk−1

,

˙

q_l−1 2

=

pl−12

∓



l

+

3 2



l

−

1 2



q_l−3 2

,

˙

p_l−1 2

= ∓



l

+

1 2



2

q_l−1 2

−

pl−32

,

(43)

fork

=

0

,

. . . ,

l

−

3₂;whichaftermakingthesubstitution

qk

=

k

!

x(k)

,

pk

= (−

1

)

l− 1 2−k

₍

_2l

−

_k

₎

!

_x(2l−k)

_,

₍₄₄₎

fork

=

0

,

. . . ,

l

−

1₂,becomeequivalenttoEqs.(40).

Ontheotherhand,letusobserve(see,

[37]

and

[38]

)thatthe Hamiltonian(41)isrelated (inthe (

+

) case)through acanonical transformationtotheoneforthePais–UhlenbeckLagrangian

(39)

, i.e., H

=

l+1 2

k=1

(

−

1

)

l+12−k 2



P 2 k

+ (

2k

−

1

)

2Q

k2



.

(45)

So,inthecaseofl half-integerthereexistsanalternative Hamilto-nianformalismwithnoadditionalvariables.Onthecontrary,forl integertheauxiliarydynamicalvariables

λ

’sarenecessary.

5. GeneralizedNiederer’stransformation

As it was mentioned before, the l-conformal Newton–Hooke algebra is a counterpart of the l-conformal Galilei one in the presenceofauniversalcosmologicalrepulsionorattraction.Since thesealgebrasare isomorphic we expectthat they dynamical re-alizations should be relatedto each other inanalogy tothe case ofl

=

1₂,wheretheir realizations(motionofthe freeparticleand ahalf-periodmotionoftheharmonicoscillator)arerelatedby fa-mous Niederer’stransformation

[26]

.Inthissectionwewillshow that the realizations obtained in the preceding sections are also related, byacounterpartofNiederer’stransformation,totheones obtainedin

[25]

forthel-conformalGalileialgebra.

It isworthtonoticethat thisfact holdsforbothl integerand half-integer. However, inthe second case(as we saw inthe pre-cedingsection)wehaveatourdisposalanalternativeHamiltonian formalism–thePais–UhlenbeckHamiltonianwithoddfrequencies. Inthenextsection,wewillapplytheresultsobtainedheretothat importantcase.

First,let usdenotewithtildethedynamicalvariablesentering therealizationsofthel-conformalGalileialgebra1anddefine

˜

κ

(˜

t

)

=

 

1

+ ˜

t2

₍

₊₎

_{oscillatory case}

_,



1

− ˜

t2

₍

₋₎

_{hyperbolic case}

_.

(46)

Then

κ

˜

satisfiesthefollowingusefulrelations

˙˜

κ

κ

˜

= ±˜

t

,

κ

˙˜

2

_{= ±}

₁

_∓

1

˜

κ

2

,

¨˜

κ

= ±

1

˜

κ

−

˙˜

κ

2

˜

κ

,

(47)

and, consequently, theequation ofmotion fortheconformal me-chanics

κ

¨˜

= ±

_κ˜13.

Now,we define acounterpartofNiederer’s transformationsas follows

t

=

arctant

˜

,

(

+)

case

;

t

=

arctanh

˜

t

,

(

−)

case

;

z

= ˜

κ

2_z

_˜

_{− ˙˜}

_κ

_κ

_˜

_.

₍₄₈₎

First, by the direct calculations, we can check that the action of the SL

(

2

,

R

)

groupon

(

t

,

z

)

(Eqs. (5)and(6))transforms intothe one for

(˜

t

,

˜

z

)

,(cf. Ref. [25]). Next, we verifythat the Lagrangian

(12) transforms exactly (no total time derivative is needed) into theoneobtainedin

[25]

,i.e.,

˜

L0

=



˙˜

z

+ ˜

z2

_.

₍₄₉₎

ThesamesituationoccursontheHamiltonianlevel.Indeed, defin-ing

pz

=

˜

pz

˜

κ

2

,

(50)

we obtain the time dependent canonical transformation, which transformstheHamiltonian

(13)

intotheconformalone,i.e.,

H0 dt d

˜

t

+

∂

F

∂

t

=

−

1 4p

˜

z

− ˜

pz

˜

z2

= ˜

H0

,

(51) where

1 _{However,forsimplicity,thederivativeswithrespectto˜t arealsodenotedby} dots.

dt dt

˜

=

1

˜

κ

2

,

(52)

whileF0

(

z

,

p

˜

z

,

t

˜

)

= ˜

pz

(

z

κ

˜

−2

+ ˙˜

κ

κ

˜

−1

)

isthegeneratingfunctionfor thetransformation

(48)

and

(50)

.

Moreover,addingthefollowingtransformationruleforthe dy-namicalvariable

η

η

= ˜

κ

2

η

˜

,

(53)

we obtainthe generalization ofNiederer’s transformation forthe Lagrangian

(20)

.

Next,wesupplythetransformations

(48)

and

(53)

bytheones fortheremainingdynamicalvariables

x(p)

=

p

m=0



2l

−

m 2l

−

p



(

− ˙˜

κ

)

p−m

κ

˜

m+p−2l_x

˜

(m)

_,

λ

(p)

_{= ˜}

_κ

2l−2p

˜ λ

(p)

_,

₍₅₄₎

where p

=

0

,

. . . ,

2l. Now, making the substitution defined by Eqs.(48),

(53)

and

(54)

intheLagrangian

(34)

andusingEqs.(47)

togetherwiththefollowingidentities

0

= (

m

−

p

)



2l

−

m 2l

−

p



+ (

2l

−

p

+

1

)



2l

−

m 2l

−

p

+

1



,

0

=

m



2l

−

m

+

1 2l

−

p



− (

p

+

1

)



2l

−

m 2l

−

p

−

1



+ (

2l

−

m

−

p

)



2l

−

m 2l

−

p



,

(55)

wearrive,afterstraightforwardbutrathertediouscomputations,at theLagrangianinvariantundertheactionofthel-conformalGalilei algebra(see,

[25]

)

˜

L

= −

γ

2

η

˜

−

1 2

η

˜

˙˜

z

+ ˜

z 2



+

2l

n=0 n

p=0

˜ λ

(n)



2l

−

p 2l

−

n



(

−˜

z

)

n−p

×

˙˜

x(p)

− (

p

+

1

)˜

x(p+1)



.

(56)

Let us stress that there is no total time derivative entering the transformationrule.

6. Niederer’stransformationforPais–Uhlenbeckmodelonthe Hamiltonianlevel

Let us recall (see, Ref. [30]) that the Pais–Uhlenbeck oscilla-tordescribed bythe Lagrangian

(39)

isrelatedtothe freehigher derivativestheory,definedbytheLagrangian

˜

L

=

1 2



dl+12

˜

_x dl+12

˜

t



2

.

(57)

Therelevanttransformationreads

t

=

arctan

˜

t

,

x

= ˜

κ

−2lx

˜

.

(58)

However, passing to the Hamiltonian counterpart of this trans-formationwe encountersome difficulties; thereisno straightfor-wardtransitiontotheHamiltonianformalismforLagrangianswith higherderivatives (in general, we have to introduce some auxil-iary variables and next apply the Dirac’s method for constraint systems). We will fill this gap below. Namely, using the results fromtheprecedingsections,weconstructacanonical transforma-tionrelatingtheHamiltonian

(41)

totheonecorrespondingtothe

freetheory,i.e.,theOstrogradskiHamiltoniancorrespondingtothe Lagrangian

(57)

:

˜

H

=

l−3 2

k=0

˜

pkq

˜

k+1

+

1 2p

˜

2 l−1 2

.

(59)

Wewillworkintermsofthevariablesq’sandp’s and Hamil-tonian

(41)

sinceinthisapproachthePais–UhlenbeckHamiltonian (foroddfrequencies)isthe sumoftheHamiltonianandthe con-formalgenerator(attime zero)ofthefree theorywhichperfectly correspondswiththerelationbetweenthel-conformalGalileiand Newton–Hook algebra.An explicitformofthecanonical transfor-mation betweenq’s andp’s andthedecoupleharmonicvariables aswellasOstrogradskioneswillbegivenintheforthcomingpaper

[38];what enables tofind thistransformation inboth remaining approaches.

Letusstartwiththecrucialobservationthattherelations

(44)

canbe usedalsointhecaseofthefree theoryandthatEqs.(54)

defineNiederer’s-typetransformationinourLagrangianformalism (withnototaltimederivativeentering).Followingthisideawe ob-tainthetransformation

qk

=

l−1 2

m=0 bkmq

˜

m

,

pk

=

l−l 2

m=0



b−1



_mkp

˜

_m

+

l−l 2

m=0 cmkq

˜

m

,

(60) where bkm

=

k

!

m

!



2l

−

m 2l

−

k



(

− ˙˜

κ

)

k−m

κ

˜

m+k−2l

_,

cmk

=

(

2l

−

k

)

!

m

!

(

−

1

)

l−1 2−k



2l

−

m k



(

− ˙˜

κ

)

2l−k−m

κ

˜

m−k

,



b−1



_mk

= (−

1

)

k+m

κ

˜

4l−2m−2kbmk (61) and, by definition,



_mk



=

0 ifk

<

m. We willcheck that Eqs.(60)

define,ontheHamiltonianlevel,ananalogue(totheclassicalcase l

=

1₂)ofNiederer’s transformationrelatingPais–Uhlenbeckmodel withoddfrequenciesandthefreehigherderivativestheory,i.e.,

Hdt d

˜

t

+



∂

F

∂ ˜

t



= ˜

H

;

(62)

whereF isthegeneratingfunctionforthetransformation

(60)

and bothsidesareexpressedintermsofq’s

˜

andp’s.

˜

2

First,by thestandard calculationswe checkthat Eqs.(60) de-fine a canonical transformation. Further, we find the generating function F

(

q

0

, . . . ,

q_l−1 2

, ˜

p0

, . . . , ˜

pl−12

, ˜

t

)

=

l−1 2

k=0

˜

p_kq

˜

_k

(

q

0

, . . . ,

q

l−1 2

, ˜

t

)

+

1 2 l−1 2

k,m=0 akmq

k

qm

,

(63) where

2 _{Eq. (62)isthe wellknow transformation rulefor the Hamiltonian,under a} canonicaltransformation,inthecasewhentimevariable.

akm

=

(

−

1

)

k+m

(

2l

−

k

)

!(

2l

−

m

)

!

k

!

m

!(

l

−

1₂

−

k

)

!(

l

−

1₂

−

m

)

!

(

− ˜

κ

κ

˙˜

)

2l−k−m

(

2l

−

k

−

m

)

=

amk

,

(64) and,byvirtueof

(60)

˜

q_m

(

q0

, . . . ,

q

_l−1 2

, ˜

t

)

=

l−1 2

k=0



b−1

(˜

t

)



_mk

qk

.

(65) Tothisendtheidentity

a

k=0

(

−

1

)

k



a

+

b k



= (−

1

)

a



a

+

b

−

1 a



,

0

≤

a

,

1

≤

b

,

(66)

appearstobeveryuseful.Next,weproveEq.(62):duetothefact that the Pais–Uhlenbeck model is traditionally considered in the oscillatoryregime(cf.Eqs.(39)and

(45)

),wewillfocusonthe(

+

) case;the

(

−)

casecanbetreatedinthesamewayorbyusingthe observationthatthetransformation

qk

→ (−

i

)

l+ 1

2+k_q

_k

_,

_pk

→ (−

₁

₎

l+21

₍

−

_i

₎

l−12−k_p

_k

_,

₍₆₇₎ relates(

+

)and(

−

)cases.

Using Eq. (47) as well as the known properties of the bino-mialcoefficientsweobtain,afterstraightforwardbutrathertedious computations,thederivativeofF withrespecttot –

˜

expressedin termsofq’s

˜

andp’s:

˜

∂

F

∂ ˜

t

(˜

q

0

, . . . , ˜

ql−12

, ˜

p0

, . . . , ˜

pl−12

, ˜

t

)

=

_˜

1

κ

2 l−3 2

m=0

(

2l

−

m

)(

m

+

1

) ˜

p

_m+1q

˜

_m

+

2

_˜

κ

˙˜

κ

l−1 2

m=0

(

l

−

m

) ˜

p

_mq

˜

_m

−

1 2

κ

˜

2



l

+

1 2



2

˜

q2 l−1 2

.

(68)

So,toprove Eq.(62)itremains toexpress H interms ofq’s

˜

and

˜

p’s. The explicit calculations are troublesome so we will sketch onlythemainsteps.

Firstwefindthecoefficientsinfrontofthetermsq

˜

mq

˜

m¯.Using Eq.(47)andtheidentities

(55)

wederivethefollowingrelations

(

k

+

1

)(

2l

−

k

)

cm,k+1

−

cm,k−1

= ˜

κ

2_c

m−1,k

−

2

κ

˜

κ

˙˜

(

l

−

m

)

cmk

− (

2l

−

m

)(

m

+

1

)

cm+1,k

,

(69) fork

,

m

=

0

,

. . . ,

l

−

1₂.Next,applyingtheidentity

(66)

,wecompute the expressions ofthe type



l−

1 2

k=mcmk¯ bkm. Due to the symmetry

m

↔ ¯

m thefinalresultisoftheform₂1

(

l

+

1₂

)

2_q

˜

2 l−1

2,andbyEq.(52) itcancelsagainstthelasttermofEq.(68).

To compute the coefficients in front of the terms bilinear in

˜

q’s and p’s,

˜

we first derive, by virtue of Eqs. (47) and (55) the followingidentity

bk+1,m

−

k

(

2l

−

k

+

1

)

bk−1,m

= ˜

κ

2bk,m−1

−

2

κ

˜

κ

˙˜

(

l

−

m

)

bkm

− (

2l

−

m

)(

m

+

1

)

bk,m+1

,

(70)

form

=

0

,

. . . ,

l

−

1₂ andk

=

0

. . . ,

l

+

1₂. Using (70)and (52)we concludethat the final result contains three terms: two ofthem cancelagainst the first two terms of Eq.(68) andthere only re-mainsthesum



l−

3 2

m=0p

˜

mq

˜

m+1.Finally,itisquiteeasytocheckthat

theonlynonvanishingtermbilinearin p’s

˜

is 1₂p

˜

2 l−1

2

.Insummary, weobtaintheHamiltonian H (cf.

˜

Eq.(59)) and,consequently,the relation

(62)

.

7. Conclusions

We have used the method of the nonlinear realizations to construct dynamical systems invariant under the action of the l-conformal Newton–Hooke algebra for both integer and half-integervaluesofl.WeputemphasisontheLagrangianand Hamil-tonian formulation. Therefore,instead of imposing invariant con-straints on the Cartan forms we enlarged the stability subgroup (in order to abandon one constraint) and added new variables whichallowustoconstructasimpleinvariant Lagrangianinsuch a way that these new degrees of freedom do not enter the dy-namics oftheoriginalones. Theresultingdynamical equationsof motion are described by Eqs. (35). The characteristicproperty of Eqs. (35) is that they decouple. We have achieved this by the appropriate choice of the subgroup, on which the action of the l-conformalgrouplinearizes(rotationsanddilatation)andthe spe-cificparametrizationofthecosetmanifold(cf.Eq.(29)).

We haveshownthat thisdescriptionis universal inthesense that it works whetherl is half-integer orinteger. The difference betweenthecaseofl integerorhalf-integeristhat thelatter ad-mits, besidesthe Hamiltonianformalism presentedhere, an alter-nativeonewherenoadditionalvariablesarenecessary,namely,the Hamiltonianformalism of thePais–Uhlenbeckoscillator withodd frequencies. Notethat,when

λ

(n) _{variables arepresent,the group} action isno longer transitiveandthe phase spaceisnot a coad-jointorbitandcannotbedirectlyobtainedbytheorbitmethod.

Next, we constructed an analogy of Niederer’s transformation relating the dynamics described in Sections 2 and 3 to the one constructed inRef.[25] forthel-conformalGalilei algebra. More-over, we use this transformation as well as the relations be-tween our Lagrangian formalism and the Pais–Uhlenbeck theory to findthecounterpart ofNiederer’s transformationforthePais– UhlenbeckoscillatorontheHamiltonianlevel.Thisisaccomplished by the canonical transformation

(60)

. Webelieve that this trans-formation can be useful to extend Niederer’s transformation to the quantumversion ofthePais–Uhlenbeckmodelaswell asthe study of its quantum symmetries. It is also tempting (especially in the context of the recent results [39]) to extend the present considerations to the supersymmetric case: in particular, to find supersymmetricextensionsofNiederer’stransformations.

Acknowledgements

Special thanks are to Piotr Kosi ´nski for valuable comments and suggestions. The discussions with Joanna Gonera and Paweł Ma´slanka are gratefully acknowledged.The work issupported by thegrantofNationalResearchCenternumberDEC-2013/09/B/ST2/ 02205.

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