119897(120579 120601) Positive and Negative Integer
Representations of su(1 1) for 119897 minus 119898 and 119897 + 119898
H Fakhri
Department of Theoretical Physics and Astrophysics Faculty of Physics University of Tabriz Tabriz 51666-16471 Iran
Correspondence should be addressed to H Fakhri hfakhritabrizuacir
Received 26 November 2015 Accepted 15 February 2016
Academic Editor Andrea Coccaro
Copyright copy 2016 H Fakhri This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3
The azimuthal and magnetic quantum numbers of spherical harmonics 119884119898119897(120579 120601) describe quantization corresponding to the
magnitude and 119911-component of angular momentum operator in the framework of realization of su(2) Lie algebra symmetry Theazimuthal quantum number 119897 allocates to itself an additional ladder symmetry by the operators which are written in terms of 119897Here it is shown that simultaneous realization of both symmetries inherits the positive and negative (119897 minus 119898)- and (119897 + 119898)-integerdiscrete irreducible representations for su(1 1) Lie algebra via the spherical harmonics on the sphere as a compact manifold So inaddition to realizing the unitary irreducible representation of su(2) compact Lie algebra via the 119884119898
119897(120579 120601)rsquos for a given 119897 we can also
represent su(1 1) noncompact Lie algebra by spherical harmonics for given values of 119897 minus 119898 and 119897 + 119898
1 Introduction
The set of principal azimuthal magnetic and spin quantumnumbers describe the unique quantum state of a singleelectron for any system in which the potential depends onlyon the radial coordinate The labels 119897 and 119898 of the usualcomplex spherical harmonics 119884119898
119897(120579 120601) are the second and
the third numbers of this set Spherical surface harmonicsare an orthonormal set of vibration solutions for eigenvalueequation of the Laplace-Beltrami operator on the sphere 1198782as a compact Riemannian manifoldThey also form the wavefunctions which represent the orbital angular momentumoperator L = r times p = minus119894r times nabla (ℏ = 1) and have a wide rangeof applications in theoretical and applied physics [1ndash4] Thethree components of the angular momentum operator thatis 119871119909 119871119910 and 119871
119911 are Killing vector fields that generate the
rotations about 119909- 119910- and 119911-axes respectively All sphericalharmonics with the given quantum number 119897 form a unitaryirreducible representation of su(2) cong so(3) Lie algebraIn fact they can be seen as representations of the SO(3)symmetry group of rotations about a point and its double-cover SU(2) It can also be noted that the spherical harmonics119884119898
119897(120579 120601) are just the independent components of symmetric
traceless tensors of rank 119897 The properties of the sphericalharmonics are well known and may be found in many textsand papers (eg see [5ndash9])
In spite of the fact that the problem of quantization ofparticle motion on a sphere is 80 years old there still existsome open questions concerning the symmetry properties ofthe bound states The aim of this work is to introduce newsymmetries based on the quantization of both azimuthal andmagnetic numbers 119897 and 119898 of the usual spherical harmonics119884119898
119897(120579 120601) In order to provide the necessary background and
also to attribute a quantization relation for azimuthal quan-tum number 119897 here we present some basic facts about thespherical harmonics [2 3] In Section 2 with the applicationof angular momentum operator we review realization of theunitary irreducible representations of su(2) Lie algebra on thesphere in terms of spherical harmonics by shifting119898 only InSection 3 the representations of the ladder symmetry withrespect to azimuthal quantum number 119897 are constructed interms of a pair of ladder operators and its correspondingquantization relation is also expressed as an operator identityoriginated from solubility in the framework of supersymme-try and shape invariance theories In Section 4 these resultsare applied to show that su(1 1) Lie algebra can also be
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 3732657 7 pageshttpdxdoiorg10115520163732657
2 Advances in High Energy Physics
represented irreducibly by using spherical harmonics Finallyin Section 5 we discuss the results and make some finalcomments
2 The Unitary IrreducibleRepresentations of su(2) Lie Algebra viaOrbital Angular Momentum Operator
This section covers the standard and the well-known formal-ism of su(2) commutation relations in order to encounterspherical harmonics In what follows we describe pointson 1198782 using the parametrization (119909 = 119903 sin 120579 cos120601 119910 =
119903 sin 120579 sin120601 119909 = 119903 cos 120579) where 0 le 120579 lt 120587 is the polar(or colatitude) angle and 0 le 120601 lt 2120587 is the azimuthal (orlongitude) angle For a given 119897with the lower bound 119897 ge 0 wedefine the (2119897 + 1)-dimensional Hilbert space
H119897fl span 119884119898
119897(120579 120601)
minus119897le119898le119897 (1)
with the spherical harmonics as bases
119884119898
119897(120579 120601) =
(minus1)119898
2119897Γ (119897 + 1)
sdot radic(2119897 + 1) Γ (119897 + 119898 + 1)
4120587Γ (119897 minus 119898 + 1)
(
119890119894120601
sin 120579)
119898
sdot (
1
sin 120579119889
119889120579
)
119897minus119898
(sin 120579)119897
(2)
Also the infinite dimensional Hilbert space H = 1198712
(1198782
119889Ω(120579 120601)) is defined as a direct sum of finite dimensionalsubspacesH = oplus
+infin
119897=0H119897Wemust emphasize that the bases of
H are independent spherical harmonics with different valuesfor both indices 119897 and 119898 The spherical harmonics as thebases ofH constitute an orthonormal set with respect to thefollowing inner product over the sphere 1198782
Therefore similar to the Fourier expansion they can be usedto expand any arbitrary square integrable function of latitudeand longitude angles 119889Ω(120579 120601) = 119889 cos 120579119889120601 is the naturalinvariant measure (area) on the sphere 1198782 The followingproposition is an immediate consequence of the raising andlowering relations of the index 119898 of the associated Legendrefunctions [2 4 10]
Proposition 1 Let one introduce three differential generators119871+ 119871minus and 119871
119911on the sphere 1198782 corresponding to the orbital
angular momentum operator L as
119871plusmn= 119890plusmn119894120601
(plusmn
120597
120597120579
+ 119894 cot 120579 120597120597120601
)
119871119911= minus119894
120597
120597120601
(4)
They satisfy the commutation relations of 119904119906(2) Lie algebra asfollows
[119871+ 119871minus] = 2119871
119911
[119871119911 119871plusmn] = plusmn119871
plusmn
(5)
119871119911is a self-adjoint operator and two operators 119871
+and 119871
minusare
Hermitian conjugate of each other with respect to the innerproduct (3) Each of the Hilbert subspaces H
119897realizes an 119897-
integer unitary irreducible representation of 119904119906(2) Lie algebraas
119897(120579 120601) minus 119897 le 119898 le 119897 (10)
Obviously the representation of the su(2) Lie algebra inthe Hilbert spaceH via (6a)ndash(6c) is reducible
A given unitary irreducible representation is character-ized by the index 119897 The spherical harmonics 119884119898
119897(120579 120601) via
their 119898 index describe quantization corresponding to com-mutation relations of the three components of orbital angularmomentumoperator 119871
119911= minus119894(120597120597120601) is always a Killing vector
field which corresponds to an angular momentum about thebody-fixed 119911-axis The Casimir operator L2su(2) along with theCartan subalgebra generator 119871
119911describes the Hamiltonian of
a free particle on the sphere with dynamical symmetry groupSU(2) and (2119897+1)-fold degeneracy for the energy spectrum Itmust be emphasized that the spherical harmonics and their
Advances in High Energy Physics 3
mathematical structure as given by Proposition 1 are playinga more visible and important role in different branchesof physics Proposition 1 implies that the spherical har-monics are created by orbital angular momentum operatorSchwinger has developed the realization of this proposition inthe framework of creation and annihilation operators of two-dimensional isotropic oscillator [11]
3 Ladder Symmetry for the AzimuthalQuantum Number 119897
It is evident that simultaneous realization of laddering rela-tions with respect to two different parameters 119897 and 119898 ofthe associated Legendre functions gives us the possibility torepresent laddering relations with respect to the azimuthalquantum number 119897 of spherical harmonics Representation ofsuch ladder symmetry by the spherical harmonics 119884119898
119897(120579 120601)
with the same119898 but different 119897 induces a new splitting on theHilbert spaceH
H =
+infin
⨁
119898=minusinfin
H119898
with H119898fl span 119884119898
119897(120579 120601)
119897ge|119898| (11)
The following proposition provides an alternative characteri-zation of the mathematical structure of spherical harmonics
Proposition 2 Let one define two first-order differentialoperators on the sphere 1198782
119869plusmn(119897) = plusmn sin 120579 120597
120597120579
+ 119897 cos 120579 (12)
They satisfy the following operator identity in the framework ofshape invariance theory
119869minus(119897 + 1) 119869
+(119897 + 1) minus 119869
+(119897) 119869minus(119897) = 2119897 + 1 (13)
119869plusmn(119897 plusmn 2) are the adjoint of the operators 119869
∓(119897) with respect to
the inner product (3) that is one has 119869dagger∓(119897) = 119869
plusmn(119897 plusmn 2) Each of
the Hilbert subspacesH119898realizes the semi-infinite raising and
lowering relations with respect to 119897 as
119869+(119897) 119884119898
119897minus1(120579 120601) = radic
2119897 minus 1
2119897 + 1
(119897 minus 119898) (119897 + 119898)119884119898
119897(120579 120601)
119897 ge |119898| + 1
(14a)
119869minus(119897) 119884119898
119897(120579 120601) = radic
2119897 + 1
2119897 minus 1
(119897 minus 119898) (119897 + 119898)119884119898
119897minus1(120579 120601)
119897 ge |119898|
(14b)
The lowest bases that is
119884119898
plusmn119898(120579 120601)
=
(minus1)minus1198982∓1198982
2plusmn119898Γ (1 plusmn 119898)
radicΓ (2 plusmn 2119898)
4120587
119890119894119898120601
(sin 120579)plusmn119898 (15)
belonging to the Hilbert subspaces H119898
with 119898 ge 0 and119898 le 0 are respectively annihilated by 119869
minus(119898) and 119869
minus(minus119898) as
119869minus(119898)119884119898
119898(120579 120601) = 0 and 119869
minus(minus119898)119884
119898
minus119898(120579 120601) = 0 Meanwhile
an arbitrary basis belonging to each of the Hilbert subspacesH119898with 119898 ge 0 and 119898 le 0 can be calculated by the algebraic
sdot 119869+(119897 minus 1) sdot sdot sdot 119869
+(1 plusmn 119898)119884
119898
plusmn119898(120579 120601)
(16)
Proof The proof follows immediately from the raising andlowering relations of the index 119897 of the associated Legendrefunctions [2]
According to the minus119897 le 119898 le +119897 limitation obtainedfrom the commutation relations of su(2) 2119897 + 1 must be anodd and even nonnegative integer for the orbital and spinangular momenta respectively Although the relation (13) isidentically satisfied for any constant number 119897 however it isrepresented only via the nonnegative integers 119897 (odd positiveinteger values for 2119897 + 1) of spherical harmonics 119884119898
119897(120579 120601)
This is an essential difference with respect to the spin angularmomentum In fact the relation (13) distinguishes the orbitalangularmomentum from the spin one It also implies that thenumber of independent components of spherical harmonicsof a given irreducible representation 119897of su(2)Lie algebra thatis 2119897+1 is derived by the shift operators corresponding to theazimuthal quantum number 119897 If we take the adjoint of (13)we obtain 119869
minus(119897 minus 1)119869
+(119897 + 3) minus 119869
+(119897 + 2)119869
minus(119897 minus 2) = 2119897 + 1
which is identically satisfied Thus Proposition 2 presentsa symmetry structure called ladder symmetry with respectto the azimuthal quantum number 119897 of spherical harmonicsNote that indeed the identical equality (13) has been orig-inated from a brilliant theory in connection with geometryand physics named supersymmetry In other words althoughcontrary to 119871
+and 119871
minusthe two operators 119869
+(119897) and 119869
minus(119897) do
not contribute in a set of closed commutation relationshowever the operator identity (13) for them can be inter-preted as a quantization relation in the framework of shapeinvariance symmetry (for reviews about supersymmetricquantummechanics and shape invariance see [12ndash17])Thusthe operators 119869
+(119897) and 119869
minus(119897) describe quantization of the
azimuthal quantum number 119897 which in turn lead to the pre-sentation of a different algebraic technique from (8) in orderto create the spherical harmonics 119884119898
119897(120579 120601) according to (16)
Furthermore spherical harmonics belonging to the Hilbertsubspaces H
119897have parity (minus1)119897 since L commutes with the
parity operator Thus the operators 119869+(119897) and 119869
minus(119897) can be
interpreted as the interchange operators of parity 119869+(119897)
H119897minus1
rarrH119897and 119869minus(119897) H
119897rarrH119897minus1
4 Positive and Negative Integer IrreducibleRepresentations of u(1 1) for 119897 ∓119898
The laddering equations (6a) and (6b) as well as (14a) and(14b) which describe shifting the indices 119898 and 119897 separately
4 Advances in High Energy Physics
lead to the derivation of twonew types of simultaneous laddersymmetries with respect to both azimuthal and magneticquantum numbers of spherical harmonics Our proposedladder operators for simultaneous shift of 119897 and 119898 are offirst-order differential type contrary to [2] They lead to anew perspective on the two quantum numbers 119897 and 119898 inconnection with realization of u(1 1) (consequently su(1 1))Lie algebra which in turn is accomplished by all sphericalharmonics 119884119898
119897(120579 120601) with constant values for 119897 minus 119898 and 119897 + 119898
separately First it should be pointed out that the Hilbertspace H can be split into the infinite direct sums of infinitedimensional Hilbert subspaces in two different ways asfollows
H = (
infin
⨁
119895=0
H+
119889=2119895+1) oplus (
infin
⨁
119896=1
H+
119889=2119896)
with
H+119889=2119895+1
= span 119884119898119898+2119895
(120579 120601)119898geminus119895
H+119889=2119896
= span 119884119898119898+2119896minus1
(120579 120601)119898ge1minus119896
(17a)
H = (
infin
⨁
119895=0
Hminus
119904=2119895+1) oplus (
infin
⨁
119896=1
Hminus
119904=2119896)
with
Hminus119904=2119895+1
= span 119884119898minus119898+2119895
(120579 120601)119898le119895
Hminus119904=2119896
= span 119884119898minus119898+2119896minus1
(120579 120601)119898le119896minus1
(17b)
The constant values for the expressions 119897 minus 119898 and 119897 + 119898 ofspherical harmonics have been labeled by 119889 minus 1 and 119904 minus 1respectively
Proposition 3 Let one define two new first-order differentialoperators on the sphere 1198782
119870119889
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
minus (119889 minus
1
2
plusmn
1
2
) sin 120579) (18)
They together with the generators 119870119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra
[119870119889
+ 119870119889
minus] = minus8119870
119911minus 4119889 + 2
[119870119911 119870119889
plusmn] = plusmn119870
119889
plusmn
(19)
119870119889plusmn2
plusmnare the adjoint of the operators 119870119889
∓with respect to the
inner product (3) that is one has 119870119889∓
dagger
= 119870119889plusmn2
plusmn Each of the
Hilbert subspaces H+119889realizes separately (119889 minus 1)-integer irre-
ducible positive representations of 119906(1 1) Lie algebra as (Itmustbe pointed out that by defining 119878119889
119911fl 119871119911+ 1198892 minus 14 and 119878119889
plusmnfl
119870119889
plusmn2 the 119906(1 1) Lie algebra (19) can be considered as com-
mutation relations corresponding to the 119904119906(1 1) Lie algebra[119878119889
+ 119878119889
minus] = minus2119878
119889
119911and [119878119889
119911 119878119889
plusmn] = plusmn119878
119889
plusmn This means that 1 is a triv-
ial center for the semisimple Lie algebra 119906(1 1) In [18] a shortreview on the three different real forms ℎ
4 119906(2) and 119906(1 1) of
119892119897(2 119888) Lie algebra has been presented There their differencesin connection with the structure constants and their represen-tation spaces have also been pointed out)
119870119889
+119884119898minus1
119898+119889minus2(120579 120601)
= radic2119898 + 2119889 minus 3
2119898 + 2119889 minus 1
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898
119898+119889minus1(120579
120601)
(20a)
119870119889
minus119884119898
119898+119889minus1(120579 120601)
= radic2119898 + 2119889 minus 1
2119898 + 2119889 minus 3
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898minus1
119898+119889minus2(120579
120601)
(20b)
119870119911119884119898
119898+119889minus1(120579 120601) = 119898119884
119898
119898+119889minus1(120579 120601) (20c)
Also the Casimir operator corresponding to the generators119870119889+
119870119889
minus and 119870
119911
K1198892
119906(11)= 119870119889
+119870119889
minusminus 41198702
119911minus 2 (2119889 minus 3)119870
119911 (21)
has an infinite-fold degeneracy on the Hilbert subspaceH+119889as
K1198892
119906(11)119884119898
119898+119889minus1(120579 120601) = (119889 minus 1) (119889 minus 2) 119884
119898
119898+119889minus1(120579 120601) (22)
The Hilbert subspaces H+119889=2119895+1
= D+(minus119895) and H+119889=2119896
=
D+(1 minus 119896) with 119895 and 119896 as nonnegative and positive integerscontain respectively the following lowest bases
119884minus119895
119895(120579 120601) =
1
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890minus119894119895120601
(sin 120579)119895 (23a)
1198841minus119896
119896(120579 120601) =
1
2119896+12
Γ (119896 + 1)
sdot radic119896Γ (2119896 + 2)
120587
119890119894(1minus119896)120601
(sin 120579)119896minus1 cos 120579
(23b)
They are annihilated as 1198702119895+1minus
119884minus119895
119895(120579 120601) = 0 and 1198702119896
minus1198841minus119896
119896(120579
120601) = 0 and also have the lowest weights minus119895 and 1 minus
119896 Meanwhile the arbitrary bases of the Hilbert subspaces
Advances in High Energy Physics 5
H+119889=2119895+1
and H+119889=2119896
can be respectively calculated by thealgebraic methods as
Proof The relations (18) (20a) and (20b) can be followedfrom the realization of laddering relations with respect toboth azimuthal and magnetic quantum numbers 119897 and 119898simultaneously and agreeably It is sufficient to consider thattwo new differential operators
satisfy the simultaneous laddering relations with respect to 119897and119898 as
119860++
(119897) 119884119898minus1
119897minus1(120579 120601)
= radic2119897 minus 1
2119897 + 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898
119897(120579 120601)
(26a)
119860minusminus
(119897) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 minus 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898minus1
119897minus1(120579 120601)
(26b)
The relations (26a) and (26b) are obtained from (6a) (6b)(14a) and (14b) The relations (19) and (20c) are directlyfollowed The adjoint relation between the operators canbe easily checked by means of the inner product (3) Thecommutativity of operators 119870119889
+ 119870119889minus and 119870
119911with K1198892u(11)
results from (19)The eigenequation (22) follows immediatelyfrom the representation relations (20a)ndash(20c) The relation(20b) implies that 119884minus119895
119895(120579 120601) and 119884
1minus119896
119896(120579 120601) are the lowest
bases for the Hilbert subspacesH+2119895+1
andH+2119896 respectively
Then with repeated application of the raising relation (20a)one may obtain the arbitrary representation bases of u(1 1)Lie algebra as (24a) and (24b)
Although the commutation relations (19) are not closedwith respect to taking the adjoint however their adjointrelations [119870119889+2
+ 119870119889minus2
minus] = minus8119870
119911minus 4119889 + 2 and [119870
119911 119870119889∓2
∓] =
∓119870119889∓2
∓are identically satisfied
Proposition 4 Let one define two new first-order differentialoperators on the sphere 1198782 as
119868119904
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
+ (119904 minus
1
2
∓
1
2
) sin 120579) (27)
They together with the generators 119868119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra as
[119868119904
+ 119868119904
minus] = minus8119868
119911+ 4119904 minus 2
[119868119911 119868119904
plusmn] = plusmn119868
119904
plusmn
(28)
119868119904∓2
plusmnare the adjoint of the operators 119868119904
1) with 119895 and 119896 as nonnegative and positive integers containrespectively the following highest bases
119884119895
119895(120579 120601) =
(minus1)119895
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890119894119895120601
(sin 120579)119895 (32a)
119884119896minus1
119896(120579 120601) =
(minus1)119896minus1
2119896minus12
Γ (119896 + 1)
sdot radic(2119896 + 1) Γ (2119896)
2120587
119890119894(119896minus1)120601
(sin 120579)119896minus1 cos 120579
(32b)
They are annihilated as 1198682119895+1+
119884119895
119895(120579 120601) = 0 and 1198682119896
+119884119896minus1
119896(120579 120601) =
0 and also have the highest weights 119895 and 119896 minus 1 Meanwhile
6 Advances in High Energy Physics
the arbitrary bases of the Hilbert subspacesHminus119904=2119895+1
andHminus119904=2119896
can be respectively calculated by the algebraic methods as
119884119898
2119895minus119898(120579 120601)
=
(1198682119895+1
minus)
119895minus119898
119884119895
119895(120579 120601)
radic(2119895 + 1) Γ (2119895 minus 2119898 + 1) (4119895 minus 2119898 + 1)
119898 le 119895
(33a)
119884119898
2119896minus119898minus1(120579 120601)
=
(1198682119896
minus)
119896minus119898minus1
119884119896minus1
119896(120579 120601)
radic(2119896 + 1) Γ (2119896 minus 2119898) (4119896 minus 2119898 minus 1)
119898 le 119896 minus 1
(33b)
Proof Theproof is quite similar to the proof of Proposition 3So we have to take into account that the two new differentialoperators
119860∓plusmn
(119897) fl ∓ [119871plusmn 119869∓(119897)]
= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+
119894
sin 120579120597
120597120601
+ 119897 sin 120579)(34)
are represented by spherical harmonics whose correspondingladdering equations shift both the azimuthal and magneticquantum numbers 119897 and119898 simultaneously and inversely
119860minus+
(119897 + 1) 119884119898minus1
119897+1(120579 120601)
= radic2119897 + 3
2119897 + 1
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898
119897(120579 120601)
(35a)
119860+minus
(119897 + 1) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 + 3
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898minus1
119897+1(120579 120601)
(35b)
Here again the adjoint of commutation relations (28)becomes [119868119904minus2
+ 119868119904+2
minus] = minus8119868
119911+ 4119904 minus 2 and [119868
119911 119868119904plusmn2
∓] = ∓119868
119904plusmn2
∓
which are identically satisfiedThus all unitary and irreducible representations of su(2)
of dimensions 2119897+1with the nonnegative integers 119897 can carrythe new kind of irreducible representations for u(1 1) Thenew symmetry structures presented in the two recent propo-sitions the so-called positive and negative discrete represen-tations of u(1 1) in turn describe the simultaneous quan-tization of the azimuthal and magnetic quantum numbersTherefore the Hilbert spaces of all spherical harmonics notonly represent compact Lie algebra su(2) by ladder operatorsshifting 119898 for a given 119897 but also represent the noncompactLie algebra u(1 1) by simultaneous shift operators of bothquantum labels 119897 and119898 for given values 119897 minus 119898 and 119897 + 119898
5 Concluding Remarks
For a given azimuthal quantum number 119897 quantization of themagnetic number 119898 is customarily accomplished by repre-senting the operators 119871
+ 119871minus and 119871
3on the sphere with the
commutation relations su(2) compact Lie algebra in a (2119897+1)-dimensional Hilbert subspace H
119897 Furthermore for a given
magnetic quantum number119898 quantization of the azimuthalnumber 119897 is accomplished by representing the operators119869+(119897) and 119869
minus(119897) on the sphere 1198782 with the identity relation
(13) in an infinite-dimensional Hilbert subspaceH119898
Dealing with these issues together simultaneous quan-tization of both azimuthal and magnetic numbers 119897 and 119898is accomplished by representing two bunches of operators119870119889
+ 119870119889
minus 1198703 1 and 119868
119904
+ 119868119904
minus 1198683 1 on the sphere with their
corresponding commutation relations of u(1 1) noncompactLie algebra in the infinite-dimensionalHilbert subspacesH+
119889
and Hminus119904 respectively For given values 119889 = 119897 minus 119898 + 1 and
119904 = 119897 +119898+ 1 they are independent of each other the so-calledpositive and negative (119897 minus 119898)- and (119897 + 119898)-integer irreduciblerepresentations respectively As the spherical harmonics aregenerated from 119884
∓119897
119897(120579 120601) by the operators 119871
plusmn they are also
generated from 119884minus119895
119895(120579 120601) and 1198841minus119896
119896(120579 120601) by1198702119895+1
+and1198702119896
+ as
well as from 119884119895
119895(120579 120601) and 119884119896minus1
119896(120579 120601) by 1198682119895+1
minusand 1198682119896minus respec-
tivelyTherefore not only 119884119898119897(120579 120601)rsquos with the given value for 119897
represent su(2) Lie algebra but also 119884119898119897(120579 120601)rsquos with the given
values for subtraction and summation of the both quantumnumbers 119897 and119898 represent separately u(1 1) (hence su(1 1))Lie algebra as well In other words two different real forms ofsl(2 119888) Lie algebra that is su(2) and su(1 1) are representedby the space of all spherical harmonics119884119898
119897(120579 120601)This happens
because the quantization of both quantum numbers 119897 and 119898are considered jointly Indeed we have
Propositions 1 and 2 lArrrArr Propositions 3 and 4 (36)
We point out that the idea of this paper may find interest-ing applications in quantum devices For instance coherentstates of the SU(1 1) noncompact Lie group have beendefined by Barut and Girardello as eigenstates of the ladderoperators [19] and by Perelomov as the action of the displace-ment operator on the lowest and highest bases [20 21] So ourapproach to the representation of su(1 1) noncompact Liealgebra provides the possibility of constructing two differenttypes of coherent states of su(1 1) on compact manifold 1198782[22] Also realization of the additional symmetry namedsu(2) Lie algebra for Landau levels and bound states of a freeparticle on noncompact manifold 119860119889119878
2can be found based
on the above considerations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Advances in High Energy Physics 7
References
[1] T M Macrobert Spherical Harmonics An Elementary Treatiseon Harmonic Functions with Applications Methuen amp CoLondon UK 1947
[2] L Infeld and T E Hull ldquoThe factorization methodrdquo Reviews ofModern Physics vol 23 no 1 pp 21ndash68 1951
[3] Y Munakata ldquoA generalization of the spherical harmonic addi-tion theoremrdquo Communications in Mathematical Physics vol 9no 1 pp 18ndash37 1968
[4] D A Varshalovich A N Moskalev and V K KhersonskyQuantum Theory of Angular Momentum Irreducible TensorsSpherical Harmonics Vector Coupling Coefficients 3nj SymbolsWorld Scientific Singapore 1989
[5] M E Rose Elementary Theory of Angular Momentum WileyNew York 1957
[6] E Merzbacher Quantum Mechanics John Wiley amp Sons NewYork NY USA 1970
[7] B L Beers and A J Dragt ldquoNew theorems about spherical har-monic expansions and SU(2)rdquo Journal of Mathematical Physicsvol 11 no 8 pp 2313ndash2328 1970
[8] J M Dixon and R Lacroix ldquoSome useful relations using spher-ical harmonics and Legendre polynomialsrdquo Journal of PhysicsA General Physics vol 6 no 8 pp 1119ndash1128 1973
[9] R Beig ldquoA remarkable property of spherical harmonicsrdquoJournal ofMathematical Physics vol 26 no 4 pp 769ndash770 1985
[10] G B Arfken Mathematical Methods for Physicists AcademicPress New York NY USA 3rd edition 1985
[11] J Schwinger Quantum Theory of Angular Momentum Aca-demic Press New York NY USA 1952
[12] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3ndash5 pp 513ndash554 1981
[13] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982
[14] E Witten ldquoSupersymmetry and Morse theoryrdquo Journal ofDifferential Geometry vol 17 no 4 pp 661ndash692 1982
[15] L Alvarez-Gaume ldquoSupersymmetry and the Atiyah-Singerindex theoremrdquo Communications in Mathematical Physics vol90 no 2 pp 161ndash173 1983
[16] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[17] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[18] H Fakhri and A Chenaghlou ldquoQuantum solvable models withgl(2 c) Lie algebra symmetry embedded into the extension ofunitary parasupersymmetryrdquo Journal of Physics A Mathemati-cal and Theoretical vol 40 no 21 pp 5511ndash5523 2007
[19] A O Barut and L Girardello ldquoNew lsquocoherentrsquo states associatedwith non-compact groupsrdquo Communications in MathematicalPhysics vol 21 no 1 pp 41ndash55 1971
[20] A M Perelomov ldquoCoherent states for arbitrary Lie grouprdquoCommunications in Mathematical Physics vol 26 no 3 pp222ndash236 1972
[21] A M Perelomov Generalized Coherent States and Their Appli-cations Texts and Monographs in Physics Springer BerlinGermany 1986
[22] H Fakhri and A Dehghani ldquoCoherency of su(11)-Barut-Girardello type and entanglement for spherical harmonicsrdquoJournal ofMathematical Physics vol 50 no 5 Article ID 0521042009
represented irreducibly by using spherical harmonics Finallyin Section 5 we discuss the results and make some finalcomments
2 The Unitary IrreducibleRepresentations of su(2) Lie Algebra viaOrbital Angular Momentum Operator
This section covers the standard and the well-known formal-ism of su(2) commutation relations in order to encounterspherical harmonics In what follows we describe pointson 1198782 using the parametrization (119909 = 119903 sin 120579 cos120601 119910 =
119903 sin 120579 sin120601 119909 = 119903 cos 120579) where 0 le 120579 lt 120587 is the polar(or colatitude) angle and 0 le 120601 lt 2120587 is the azimuthal (orlongitude) angle For a given 119897with the lower bound 119897 ge 0 wedefine the (2119897 + 1)-dimensional Hilbert space
H119897fl span 119884119898
119897(120579 120601)
minus119897le119898le119897 (1)
with the spherical harmonics as bases
119884119898
119897(120579 120601) =
(minus1)119898
2119897Γ (119897 + 1)
sdot radic(2119897 + 1) Γ (119897 + 119898 + 1)
4120587Γ (119897 minus 119898 + 1)
(
119890119894120601
sin 120579)
119898
sdot (
1
sin 120579119889
119889120579
)
119897minus119898
(sin 120579)119897
(2)
Also the infinite dimensional Hilbert space H = 1198712
(1198782
119889Ω(120579 120601)) is defined as a direct sum of finite dimensionalsubspacesH = oplus
+infin
119897=0H119897Wemust emphasize that the bases of
H are independent spherical harmonics with different valuesfor both indices 119897 and 119898 The spherical harmonics as thebases ofH constitute an orthonormal set with respect to thefollowing inner product over the sphere 1198782
Therefore similar to the Fourier expansion they can be usedto expand any arbitrary square integrable function of latitudeand longitude angles 119889Ω(120579 120601) = 119889 cos 120579119889120601 is the naturalinvariant measure (area) on the sphere 1198782 The followingproposition is an immediate consequence of the raising andlowering relations of the index 119898 of the associated Legendrefunctions [2 4 10]
Proposition 1 Let one introduce three differential generators119871+ 119871minus and 119871
119911on the sphere 1198782 corresponding to the orbital
angular momentum operator L as
119871plusmn= 119890plusmn119894120601
(plusmn
120597
120597120579
+ 119894 cot 120579 120597120597120601
)
119871119911= minus119894
120597
120597120601
(4)
They satisfy the commutation relations of 119904119906(2) Lie algebra asfollows
[119871+ 119871minus] = 2119871
119911
[119871119911 119871plusmn] = plusmn119871
plusmn
(5)
119871119911is a self-adjoint operator and two operators 119871
+and 119871
minusare
Hermitian conjugate of each other with respect to the innerproduct (3) Each of the Hilbert subspaces H
119897realizes an 119897-
integer unitary irreducible representation of 119904119906(2) Lie algebraas
119897(120579 120601) minus 119897 le 119898 le 119897 (10)
Obviously the representation of the su(2) Lie algebra inthe Hilbert spaceH via (6a)ndash(6c) is reducible
A given unitary irreducible representation is character-ized by the index 119897 The spherical harmonics 119884119898
119897(120579 120601) via
their 119898 index describe quantization corresponding to com-mutation relations of the three components of orbital angularmomentumoperator 119871
119911= minus119894(120597120597120601) is always a Killing vector
field which corresponds to an angular momentum about thebody-fixed 119911-axis The Casimir operator L2su(2) along with theCartan subalgebra generator 119871
119911describes the Hamiltonian of
a free particle on the sphere with dynamical symmetry groupSU(2) and (2119897+1)-fold degeneracy for the energy spectrum Itmust be emphasized that the spherical harmonics and their
Advances in High Energy Physics 3
mathematical structure as given by Proposition 1 are playinga more visible and important role in different branchesof physics Proposition 1 implies that the spherical har-monics are created by orbital angular momentum operatorSchwinger has developed the realization of this proposition inthe framework of creation and annihilation operators of two-dimensional isotropic oscillator [11]
3 Ladder Symmetry for the AzimuthalQuantum Number 119897
It is evident that simultaneous realization of laddering rela-tions with respect to two different parameters 119897 and 119898 ofthe associated Legendre functions gives us the possibility torepresent laddering relations with respect to the azimuthalquantum number 119897 of spherical harmonics Representation ofsuch ladder symmetry by the spherical harmonics 119884119898
119897(120579 120601)
with the same119898 but different 119897 induces a new splitting on theHilbert spaceH
H =
+infin
⨁
119898=minusinfin
H119898
with H119898fl span 119884119898
119897(120579 120601)
119897ge|119898| (11)
The following proposition provides an alternative characteri-zation of the mathematical structure of spherical harmonics
Proposition 2 Let one define two first-order differentialoperators on the sphere 1198782
119869plusmn(119897) = plusmn sin 120579 120597
120597120579
+ 119897 cos 120579 (12)
They satisfy the following operator identity in the framework ofshape invariance theory
119869minus(119897 + 1) 119869
+(119897 + 1) minus 119869
+(119897) 119869minus(119897) = 2119897 + 1 (13)
119869plusmn(119897 plusmn 2) are the adjoint of the operators 119869
∓(119897) with respect to
the inner product (3) that is one has 119869dagger∓(119897) = 119869
plusmn(119897 plusmn 2) Each of
the Hilbert subspacesH119898realizes the semi-infinite raising and
lowering relations with respect to 119897 as
119869+(119897) 119884119898
119897minus1(120579 120601) = radic
2119897 minus 1
2119897 + 1
(119897 minus 119898) (119897 + 119898)119884119898
119897(120579 120601)
119897 ge |119898| + 1
(14a)
119869minus(119897) 119884119898
119897(120579 120601) = radic
2119897 + 1
2119897 minus 1
(119897 minus 119898) (119897 + 119898)119884119898
119897minus1(120579 120601)
119897 ge |119898|
(14b)
The lowest bases that is
119884119898
plusmn119898(120579 120601)
=
(minus1)minus1198982∓1198982
2plusmn119898Γ (1 plusmn 119898)
radicΓ (2 plusmn 2119898)
4120587
119890119894119898120601
(sin 120579)plusmn119898 (15)
belonging to the Hilbert subspaces H119898
with 119898 ge 0 and119898 le 0 are respectively annihilated by 119869
minus(119898) and 119869
minus(minus119898) as
119869minus(119898)119884119898
119898(120579 120601) = 0 and 119869
minus(minus119898)119884
119898
minus119898(120579 120601) = 0 Meanwhile
an arbitrary basis belonging to each of the Hilbert subspacesH119898with 119898 ge 0 and 119898 le 0 can be calculated by the algebraic
sdot 119869+(119897 minus 1) sdot sdot sdot 119869
+(1 plusmn 119898)119884
119898
plusmn119898(120579 120601)
(16)
Proof The proof follows immediately from the raising andlowering relations of the index 119897 of the associated Legendrefunctions [2]
According to the minus119897 le 119898 le +119897 limitation obtainedfrom the commutation relations of su(2) 2119897 + 1 must be anodd and even nonnegative integer for the orbital and spinangular momenta respectively Although the relation (13) isidentically satisfied for any constant number 119897 however it isrepresented only via the nonnegative integers 119897 (odd positiveinteger values for 2119897 + 1) of spherical harmonics 119884119898
119897(120579 120601)
This is an essential difference with respect to the spin angularmomentum In fact the relation (13) distinguishes the orbitalangularmomentum from the spin one It also implies that thenumber of independent components of spherical harmonicsof a given irreducible representation 119897of su(2)Lie algebra thatis 2119897+1 is derived by the shift operators corresponding to theazimuthal quantum number 119897 If we take the adjoint of (13)we obtain 119869
minus(119897 minus 1)119869
+(119897 + 3) minus 119869
+(119897 + 2)119869
minus(119897 minus 2) = 2119897 + 1
which is identically satisfied Thus Proposition 2 presentsa symmetry structure called ladder symmetry with respectto the azimuthal quantum number 119897 of spherical harmonicsNote that indeed the identical equality (13) has been orig-inated from a brilliant theory in connection with geometryand physics named supersymmetry In other words althoughcontrary to 119871
+and 119871
minusthe two operators 119869
+(119897) and 119869
minus(119897) do
not contribute in a set of closed commutation relationshowever the operator identity (13) for them can be inter-preted as a quantization relation in the framework of shapeinvariance symmetry (for reviews about supersymmetricquantummechanics and shape invariance see [12ndash17])Thusthe operators 119869
+(119897) and 119869
minus(119897) describe quantization of the
azimuthal quantum number 119897 which in turn lead to the pre-sentation of a different algebraic technique from (8) in orderto create the spherical harmonics 119884119898
119897(120579 120601) according to (16)
Furthermore spherical harmonics belonging to the Hilbertsubspaces H
119897have parity (minus1)119897 since L commutes with the
parity operator Thus the operators 119869+(119897) and 119869
minus(119897) can be
interpreted as the interchange operators of parity 119869+(119897)
H119897minus1
rarrH119897and 119869minus(119897) H
119897rarrH119897minus1
4 Positive and Negative Integer IrreducibleRepresentations of u(1 1) for 119897 ∓119898
The laddering equations (6a) and (6b) as well as (14a) and(14b) which describe shifting the indices 119898 and 119897 separately
4 Advances in High Energy Physics
lead to the derivation of twonew types of simultaneous laddersymmetries with respect to both azimuthal and magneticquantum numbers of spherical harmonics Our proposedladder operators for simultaneous shift of 119897 and 119898 are offirst-order differential type contrary to [2] They lead to anew perspective on the two quantum numbers 119897 and 119898 inconnection with realization of u(1 1) (consequently su(1 1))Lie algebra which in turn is accomplished by all sphericalharmonics 119884119898
119897(120579 120601) with constant values for 119897 minus 119898 and 119897 + 119898
separately First it should be pointed out that the Hilbertspace H can be split into the infinite direct sums of infinitedimensional Hilbert subspaces in two different ways asfollows
H = (
infin
⨁
119895=0
H+
119889=2119895+1) oplus (
infin
⨁
119896=1
H+
119889=2119896)
with
H+119889=2119895+1
= span 119884119898119898+2119895
(120579 120601)119898geminus119895
H+119889=2119896
= span 119884119898119898+2119896minus1
(120579 120601)119898ge1minus119896
(17a)
H = (
infin
⨁
119895=0
Hminus
119904=2119895+1) oplus (
infin
⨁
119896=1
Hminus
119904=2119896)
with
Hminus119904=2119895+1
= span 119884119898minus119898+2119895
(120579 120601)119898le119895
Hminus119904=2119896
= span 119884119898minus119898+2119896minus1
(120579 120601)119898le119896minus1
(17b)
The constant values for the expressions 119897 minus 119898 and 119897 + 119898 ofspherical harmonics have been labeled by 119889 minus 1 and 119904 minus 1respectively
Proposition 3 Let one define two new first-order differentialoperators on the sphere 1198782
119870119889
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
minus (119889 minus
1
2
plusmn
1
2
) sin 120579) (18)
They together with the generators 119870119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra
[119870119889
+ 119870119889
minus] = minus8119870
119911minus 4119889 + 2
[119870119911 119870119889
plusmn] = plusmn119870
119889
plusmn
(19)
119870119889plusmn2
plusmnare the adjoint of the operators 119870119889
∓with respect to the
inner product (3) that is one has 119870119889∓
dagger
= 119870119889plusmn2
plusmn Each of the
Hilbert subspaces H+119889realizes separately (119889 minus 1)-integer irre-
ducible positive representations of 119906(1 1) Lie algebra as (Itmustbe pointed out that by defining 119878119889
119911fl 119871119911+ 1198892 minus 14 and 119878119889
plusmnfl
119870119889
plusmn2 the 119906(1 1) Lie algebra (19) can be considered as com-
mutation relations corresponding to the 119904119906(1 1) Lie algebra[119878119889
+ 119878119889
minus] = minus2119878
119889
119911and [119878119889
119911 119878119889
plusmn] = plusmn119878
119889
plusmn This means that 1 is a triv-
ial center for the semisimple Lie algebra 119906(1 1) In [18] a shortreview on the three different real forms ℎ
4 119906(2) and 119906(1 1) of
119892119897(2 119888) Lie algebra has been presented There their differencesin connection with the structure constants and their represen-tation spaces have also been pointed out)
119870119889
+119884119898minus1
119898+119889minus2(120579 120601)
= radic2119898 + 2119889 minus 3
2119898 + 2119889 minus 1
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898
119898+119889minus1(120579
120601)
(20a)
119870119889
minus119884119898
119898+119889minus1(120579 120601)
= radic2119898 + 2119889 minus 1
2119898 + 2119889 minus 3
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898minus1
119898+119889minus2(120579
120601)
(20b)
119870119911119884119898
119898+119889minus1(120579 120601) = 119898119884
119898
119898+119889minus1(120579 120601) (20c)
Also the Casimir operator corresponding to the generators119870119889+
119870119889
minus and 119870
119911
K1198892
119906(11)= 119870119889
+119870119889
minusminus 41198702
119911minus 2 (2119889 minus 3)119870
119911 (21)
has an infinite-fold degeneracy on the Hilbert subspaceH+119889as
K1198892
119906(11)119884119898
119898+119889minus1(120579 120601) = (119889 minus 1) (119889 minus 2) 119884
119898
119898+119889minus1(120579 120601) (22)
The Hilbert subspaces H+119889=2119895+1
= D+(minus119895) and H+119889=2119896
=
D+(1 minus 119896) with 119895 and 119896 as nonnegative and positive integerscontain respectively the following lowest bases
119884minus119895
119895(120579 120601) =
1
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890minus119894119895120601
(sin 120579)119895 (23a)
1198841minus119896
119896(120579 120601) =
1
2119896+12
Γ (119896 + 1)
sdot radic119896Γ (2119896 + 2)
120587
119890119894(1minus119896)120601
(sin 120579)119896minus1 cos 120579
(23b)
They are annihilated as 1198702119895+1minus
119884minus119895
119895(120579 120601) = 0 and 1198702119896
minus1198841minus119896
119896(120579
120601) = 0 and also have the lowest weights minus119895 and 1 minus
119896 Meanwhile the arbitrary bases of the Hilbert subspaces
Advances in High Energy Physics 5
H+119889=2119895+1
and H+119889=2119896
can be respectively calculated by thealgebraic methods as
Proof The relations (18) (20a) and (20b) can be followedfrom the realization of laddering relations with respect toboth azimuthal and magnetic quantum numbers 119897 and 119898simultaneously and agreeably It is sufficient to consider thattwo new differential operators
satisfy the simultaneous laddering relations with respect to 119897and119898 as
119860++
(119897) 119884119898minus1
119897minus1(120579 120601)
= radic2119897 minus 1
2119897 + 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898
119897(120579 120601)
(26a)
119860minusminus
(119897) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 minus 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898minus1
119897minus1(120579 120601)
(26b)
The relations (26a) and (26b) are obtained from (6a) (6b)(14a) and (14b) The relations (19) and (20c) are directlyfollowed The adjoint relation between the operators canbe easily checked by means of the inner product (3) Thecommutativity of operators 119870119889
+ 119870119889minus and 119870
119911with K1198892u(11)
results from (19)The eigenequation (22) follows immediatelyfrom the representation relations (20a)ndash(20c) The relation(20b) implies that 119884minus119895
119895(120579 120601) and 119884
1minus119896
119896(120579 120601) are the lowest
bases for the Hilbert subspacesH+2119895+1
andH+2119896 respectively
Then with repeated application of the raising relation (20a)one may obtain the arbitrary representation bases of u(1 1)Lie algebra as (24a) and (24b)
Although the commutation relations (19) are not closedwith respect to taking the adjoint however their adjointrelations [119870119889+2
+ 119870119889minus2
minus] = minus8119870
119911minus 4119889 + 2 and [119870
119911 119870119889∓2
∓] =
∓119870119889∓2
∓are identically satisfied
Proposition 4 Let one define two new first-order differentialoperators on the sphere 1198782 as
119868119904
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
+ (119904 minus
1
2
∓
1
2
) sin 120579) (27)
They together with the generators 119868119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra as
[119868119904
+ 119868119904
minus] = minus8119868
119911+ 4119904 minus 2
[119868119911 119868119904
plusmn] = plusmn119868
119904
plusmn
(28)
119868119904∓2
plusmnare the adjoint of the operators 119868119904
1) with 119895 and 119896 as nonnegative and positive integers containrespectively the following highest bases
119884119895
119895(120579 120601) =
(minus1)119895
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890119894119895120601
(sin 120579)119895 (32a)
119884119896minus1
119896(120579 120601) =
(minus1)119896minus1
2119896minus12
Γ (119896 + 1)
sdot radic(2119896 + 1) Γ (2119896)
2120587
119890119894(119896minus1)120601
(sin 120579)119896minus1 cos 120579
(32b)
They are annihilated as 1198682119895+1+
119884119895
119895(120579 120601) = 0 and 1198682119896
+119884119896minus1
119896(120579 120601) =
0 and also have the highest weights 119895 and 119896 minus 1 Meanwhile
6 Advances in High Energy Physics
the arbitrary bases of the Hilbert subspacesHminus119904=2119895+1
andHminus119904=2119896
can be respectively calculated by the algebraic methods as
119884119898
2119895minus119898(120579 120601)
=
(1198682119895+1
minus)
119895minus119898
119884119895
119895(120579 120601)
radic(2119895 + 1) Γ (2119895 minus 2119898 + 1) (4119895 minus 2119898 + 1)
119898 le 119895
(33a)
119884119898
2119896minus119898minus1(120579 120601)
=
(1198682119896
minus)
119896minus119898minus1
119884119896minus1
119896(120579 120601)
radic(2119896 + 1) Γ (2119896 minus 2119898) (4119896 minus 2119898 minus 1)
119898 le 119896 minus 1
(33b)
Proof Theproof is quite similar to the proof of Proposition 3So we have to take into account that the two new differentialoperators
119860∓plusmn
(119897) fl ∓ [119871plusmn 119869∓(119897)]
= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+
119894
sin 120579120597
120597120601
+ 119897 sin 120579)(34)
are represented by spherical harmonics whose correspondingladdering equations shift both the azimuthal and magneticquantum numbers 119897 and119898 simultaneously and inversely
119860minus+
(119897 + 1) 119884119898minus1
119897+1(120579 120601)
= radic2119897 + 3
2119897 + 1
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898
119897(120579 120601)
(35a)
119860+minus
(119897 + 1) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 + 3
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898minus1
119897+1(120579 120601)
(35b)
Here again the adjoint of commutation relations (28)becomes [119868119904minus2
+ 119868119904+2
minus] = minus8119868
119911+ 4119904 minus 2 and [119868
119911 119868119904plusmn2
∓] = ∓119868
119904plusmn2
∓
which are identically satisfiedThus all unitary and irreducible representations of su(2)
of dimensions 2119897+1with the nonnegative integers 119897 can carrythe new kind of irreducible representations for u(1 1) Thenew symmetry structures presented in the two recent propo-sitions the so-called positive and negative discrete represen-tations of u(1 1) in turn describe the simultaneous quan-tization of the azimuthal and magnetic quantum numbersTherefore the Hilbert spaces of all spherical harmonics notonly represent compact Lie algebra su(2) by ladder operatorsshifting 119898 for a given 119897 but also represent the noncompactLie algebra u(1 1) by simultaneous shift operators of bothquantum labels 119897 and119898 for given values 119897 minus 119898 and 119897 + 119898
5 Concluding Remarks
For a given azimuthal quantum number 119897 quantization of themagnetic number 119898 is customarily accomplished by repre-senting the operators 119871
+ 119871minus and 119871
3on the sphere with the
commutation relations su(2) compact Lie algebra in a (2119897+1)-dimensional Hilbert subspace H
119897 Furthermore for a given
magnetic quantum number119898 quantization of the azimuthalnumber 119897 is accomplished by representing the operators119869+(119897) and 119869
minus(119897) on the sphere 1198782 with the identity relation
(13) in an infinite-dimensional Hilbert subspaceH119898
Dealing with these issues together simultaneous quan-tization of both azimuthal and magnetic numbers 119897 and 119898is accomplished by representing two bunches of operators119870119889
+ 119870119889
minus 1198703 1 and 119868
119904
+ 119868119904
minus 1198683 1 on the sphere with their
corresponding commutation relations of u(1 1) noncompactLie algebra in the infinite-dimensionalHilbert subspacesH+
119889
and Hminus119904 respectively For given values 119889 = 119897 minus 119898 + 1 and
119904 = 119897 +119898+ 1 they are independent of each other the so-calledpositive and negative (119897 minus 119898)- and (119897 + 119898)-integer irreduciblerepresentations respectively As the spherical harmonics aregenerated from 119884
∓119897
119897(120579 120601) by the operators 119871
plusmn they are also
generated from 119884minus119895
119895(120579 120601) and 1198841minus119896
119896(120579 120601) by1198702119895+1
+and1198702119896
+ as
well as from 119884119895
119895(120579 120601) and 119884119896minus1
119896(120579 120601) by 1198682119895+1
minusand 1198682119896minus respec-
tivelyTherefore not only 119884119898119897(120579 120601)rsquos with the given value for 119897
represent su(2) Lie algebra but also 119884119898119897(120579 120601)rsquos with the given
values for subtraction and summation of the both quantumnumbers 119897 and119898 represent separately u(1 1) (hence su(1 1))Lie algebra as well In other words two different real forms ofsl(2 119888) Lie algebra that is su(2) and su(1 1) are representedby the space of all spherical harmonics119884119898
119897(120579 120601)This happens
because the quantization of both quantum numbers 119897 and 119898are considered jointly Indeed we have
Propositions 1 and 2 lArrrArr Propositions 3 and 4 (36)
We point out that the idea of this paper may find interest-ing applications in quantum devices For instance coherentstates of the SU(1 1) noncompact Lie group have beendefined by Barut and Girardello as eigenstates of the ladderoperators [19] and by Perelomov as the action of the displace-ment operator on the lowest and highest bases [20 21] So ourapproach to the representation of su(1 1) noncompact Liealgebra provides the possibility of constructing two differenttypes of coherent states of su(1 1) on compact manifold 1198782[22] Also realization of the additional symmetry namedsu(2) Lie algebra for Landau levels and bound states of a freeparticle on noncompact manifold 119860119889119878
2can be found based
on the above considerations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Advances in High Energy Physics 7
References
[1] T M Macrobert Spherical Harmonics An Elementary Treatiseon Harmonic Functions with Applications Methuen amp CoLondon UK 1947
[2] L Infeld and T E Hull ldquoThe factorization methodrdquo Reviews ofModern Physics vol 23 no 1 pp 21ndash68 1951
[3] Y Munakata ldquoA generalization of the spherical harmonic addi-tion theoremrdquo Communications in Mathematical Physics vol 9no 1 pp 18ndash37 1968
[4] D A Varshalovich A N Moskalev and V K KhersonskyQuantum Theory of Angular Momentum Irreducible TensorsSpherical Harmonics Vector Coupling Coefficients 3nj SymbolsWorld Scientific Singapore 1989
[5] M E Rose Elementary Theory of Angular Momentum WileyNew York 1957
[6] E Merzbacher Quantum Mechanics John Wiley amp Sons NewYork NY USA 1970
[7] B L Beers and A J Dragt ldquoNew theorems about spherical har-monic expansions and SU(2)rdquo Journal of Mathematical Physicsvol 11 no 8 pp 2313ndash2328 1970
[8] J M Dixon and R Lacroix ldquoSome useful relations using spher-ical harmonics and Legendre polynomialsrdquo Journal of PhysicsA General Physics vol 6 no 8 pp 1119ndash1128 1973
[9] R Beig ldquoA remarkable property of spherical harmonicsrdquoJournal ofMathematical Physics vol 26 no 4 pp 769ndash770 1985
[10] G B Arfken Mathematical Methods for Physicists AcademicPress New York NY USA 3rd edition 1985
[11] J Schwinger Quantum Theory of Angular Momentum Aca-demic Press New York NY USA 1952
[12] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3ndash5 pp 513ndash554 1981
[13] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982
[14] E Witten ldquoSupersymmetry and Morse theoryrdquo Journal ofDifferential Geometry vol 17 no 4 pp 661ndash692 1982
[15] L Alvarez-Gaume ldquoSupersymmetry and the Atiyah-Singerindex theoremrdquo Communications in Mathematical Physics vol90 no 2 pp 161ndash173 1983
[16] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[17] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[18] H Fakhri and A Chenaghlou ldquoQuantum solvable models withgl(2 c) Lie algebra symmetry embedded into the extension ofunitary parasupersymmetryrdquo Journal of Physics A Mathemati-cal and Theoretical vol 40 no 21 pp 5511ndash5523 2007
[19] A O Barut and L Girardello ldquoNew lsquocoherentrsquo states associatedwith non-compact groupsrdquo Communications in MathematicalPhysics vol 21 no 1 pp 41ndash55 1971
[20] A M Perelomov ldquoCoherent states for arbitrary Lie grouprdquoCommunications in Mathematical Physics vol 26 no 3 pp222ndash236 1972
[21] A M Perelomov Generalized Coherent States and Their Appli-cations Texts and Monographs in Physics Springer BerlinGermany 1986
[22] H Fakhri and A Dehghani ldquoCoherency of su(11)-Barut-Girardello type and entanglement for spherical harmonicsrdquoJournal ofMathematical Physics vol 50 no 5 Article ID 0521042009
mathematical structure as given by Proposition 1 are playinga more visible and important role in different branchesof physics Proposition 1 implies that the spherical har-monics are created by orbital angular momentum operatorSchwinger has developed the realization of this proposition inthe framework of creation and annihilation operators of two-dimensional isotropic oscillator [11]
3 Ladder Symmetry for the AzimuthalQuantum Number 119897
It is evident that simultaneous realization of laddering rela-tions with respect to two different parameters 119897 and 119898 ofthe associated Legendre functions gives us the possibility torepresent laddering relations with respect to the azimuthalquantum number 119897 of spherical harmonics Representation ofsuch ladder symmetry by the spherical harmonics 119884119898
119897(120579 120601)
with the same119898 but different 119897 induces a new splitting on theHilbert spaceH
H =
+infin
⨁
119898=minusinfin
H119898
with H119898fl span 119884119898
119897(120579 120601)
119897ge|119898| (11)
The following proposition provides an alternative characteri-zation of the mathematical structure of spherical harmonics
Proposition 2 Let one define two first-order differentialoperators on the sphere 1198782
119869plusmn(119897) = plusmn sin 120579 120597
120597120579
+ 119897 cos 120579 (12)
They satisfy the following operator identity in the framework ofshape invariance theory
119869minus(119897 + 1) 119869
+(119897 + 1) minus 119869
+(119897) 119869minus(119897) = 2119897 + 1 (13)
119869plusmn(119897 plusmn 2) are the adjoint of the operators 119869
∓(119897) with respect to
the inner product (3) that is one has 119869dagger∓(119897) = 119869
plusmn(119897 plusmn 2) Each of
the Hilbert subspacesH119898realizes the semi-infinite raising and
lowering relations with respect to 119897 as
119869+(119897) 119884119898
119897minus1(120579 120601) = radic
2119897 minus 1
2119897 + 1
(119897 minus 119898) (119897 + 119898)119884119898
119897(120579 120601)
119897 ge |119898| + 1
(14a)
119869minus(119897) 119884119898
119897(120579 120601) = radic
2119897 + 1
2119897 minus 1
(119897 minus 119898) (119897 + 119898)119884119898
119897minus1(120579 120601)
119897 ge |119898|
(14b)
The lowest bases that is
119884119898
plusmn119898(120579 120601)
=
(minus1)minus1198982∓1198982
2plusmn119898Γ (1 plusmn 119898)
radicΓ (2 plusmn 2119898)
4120587
119890119894119898120601
(sin 120579)plusmn119898 (15)
belonging to the Hilbert subspaces H119898
with 119898 ge 0 and119898 le 0 are respectively annihilated by 119869
minus(119898) and 119869
minus(minus119898) as
119869minus(119898)119884119898
119898(120579 120601) = 0 and 119869
minus(minus119898)119884
119898
minus119898(120579 120601) = 0 Meanwhile
an arbitrary basis belonging to each of the Hilbert subspacesH119898with 119898 ge 0 and 119898 le 0 can be calculated by the algebraic
sdot 119869+(119897 minus 1) sdot sdot sdot 119869
+(1 plusmn 119898)119884
119898
plusmn119898(120579 120601)
(16)
Proof The proof follows immediately from the raising andlowering relations of the index 119897 of the associated Legendrefunctions [2]
According to the minus119897 le 119898 le +119897 limitation obtainedfrom the commutation relations of su(2) 2119897 + 1 must be anodd and even nonnegative integer for the orbital and spinangular momenta respectively Although the relation (13) isidentically satisfied for any constant number 119897 however it isrepresented only via the nonnegative integers 119897 (odd positiveinteger values for 2119897 + 1) of spherical harmonics 119884119898
119897(120579 120601)
This is an essential difference with respect to the spin angularmomentum In fact the relation (13) distinguishes the orbitalangularmomentum from the spin one It also implies that thenumber of independent components of spherical harmonicsof a given irreducible representation 119897of su(2)Lie algebra thatis 2119897+1 is derived by the shift operators corresponding to theazimuthal quantum number 119897 If we take the adjoint of (13)we obtain 119869
minus(119897 minus 1)119869
+(119897 + 3) minus 119869
+(119897 + 2)119869
minus(119897 minus 2) = 2119897 + 1
which is identically satisfied Thus Proposition 2 presentsa symmetry structure called ladder symmetry with respectto the azimuthal quantum number 119897 of spherical harmonicsNote that indeed the identical equality (13) has been orig-inated from a brilliant theory in connection with geometryand physics named supersymmetry In other words althoughcontrary to 119871
+and 119871
minusthe two operators 119869
+(119897) and 119869
minus(119897) do
not contribute in a set of closed commutation relationshowever the operator identity (13) for them can be inter-preted as a quantization relation in the framework of shapeinvariance symmetry (for reviews about supersymmetricquantummechanics and shape invariance see [12ndash17])Thusthe operators 119869
+(119897) and 119869
minus(119897) describe quantization of the
azimuthal quantum number 119897 which in turn lead to the pre-sentation of a different algebraic technique from (8) in orderto create the spherical harmonics 119884119898
119897(120579 120601) according to (16)
Furthermore spherical harmonics belonging to the Hilbertsubspaces H
119897have parity (minus1)119897 since L commutes with the
parity operator Thus the operators 119869+(119897) and 119869
minus(119897) can be
interpreted as the interchange operators of parity 119869+(119897)
H119897minus1
rarrH119897and 119869minus(119897) H
119897rarrH119897minus1
4 Positive and Negative Integer IrreducibleRepresentations of u(1 1) for 119897 ∓119898
The laddering equations (6a) and (6b) as well as (14a) and(14b) which describe shifting the indices 119898 and 119897 separately
4 Advances in High Energy Physics
lead to the derivation of twonew types of simultaneous laddersymmetries with respect to both azimuthal and magneticquantum numbers of spherical harmonics Our proposedladder operators for simultaneous shift of 119897 and 119898 are offirst-order differential type contrary to [2] They lead to anew perspective on the two quantum numbers 119897 and 119898 inconnection with realization of u(1 1) (consequently su(1 1))Lie algebra which in turn is accomplished by all sphericalharmonics 119884119898
119897(120579 120601) with constant values for 119897 minus 119898 and 119897 + 119898
separately First it should be pointed out that the Hilbertspace H can be split into the infinite direct sums of infinitedimensional Hilbert subspaces in two different ways asfollows
H = (
infin
⨁
119895=0
H+
119889=2119895+1) oplus (
infin
⨁
119896=1
H+
119889=2119896)
with
H+119889=2119895+1
= span 119884119898119898+2119895
(120579 120601)119898geminus119895
H+119889=2119896
= span 119884119898119898+2119896minus1
(120579 120601)119898ge1minus119896
(17a)
H = (
infin
⨁
119895=0
Hminus
119904=2119895+1) oplus (
infin
⨁
119896=1
Hminus
119904=2119896)
with
Hminus119904=2119895+1
= span 119884119898minus119898+2119895
(120579 120601)119898le119895
Hminus119904=2119896
= span 119884119898minus119898+2119896minus1
(120579 120601)119898le119896minus1
(17b)
The constant values for the expressions 119897 minus 119898 and 119897 + 119898 ofspherical harmonics have been labeled by 119889 minus 1 and 119904 minus 1respectively
Proposition 3 Let one define two new first-order differentialoperators on the sphere 1198782
119870119889
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
minus (119889 minus
1
2
plusmn
1
2
) sin 120579) (18)
They together with the generators 119870119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra
[119870119889
+ 119870119889
minus] = minus8119870
119911minus 4119889 + 2
[119870119911 119870119889
plusmn] = plusmn119870
119889
plusmn
(19)
119870119889plusmn2
plusmnare the adjoint of the operators 119870119889
∓with respect to the
inner product (3) that is one has 119870119889∓
dagger
= 119870119889plusmn2
plusmn Each of the
Hilbert subspaces H+119889realizes separately (119889 minus 1)-integer irre-
ducible positive representations of 119906(1 1) Lie algebra as (Itmustbe pointed out that by defining 119878119889
119911fl 119871119911+ 1198892 minus 14 and 119878119889
plusmnfl
119870119889
plusmn2 the 119906(1 1) Lie algebra (19) can be considered as com-
mutation relations corresponding to the 119904119906(1 1) Lie algebra[119878119889
+ 119878119889
minus] = minus2119878
119889
119911and [119878119889
119911 119878119889
plusmn] = plusmn119878
119889
plusmn This means that 1 is a triv-
ial center for the semisimple Lie algebra 119906(1 1) In [18] a shortreview on the three different real forms ℎ
4 119906(2) and 119906(1 1) of
119892119897(2 119888) Lie algebra has been presented There their differencesin connection with the structure constants and their represen-tation spaces have also been pointed out)
119870119889
+119884119898minus1
119898+119889minus2(120579 120601)
= radic2119898 + 2119889 minus 3
2119898 + 2119889 minus 1
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898
119898+119889minus1(120579
120601)
(20a)
119870119889
minus119884119898
119898+119889minus1(120579 120601)
= radic2119898 + 2119889 minus 1
2119898 + 2119889 minus 3
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898minus1
119898+119889minus2(120579
120601)
(20b)
119870119911119884119898
119898+119889minus1(120579 120601) = 119898119884
119898
119898+119889minus1(120579 120601) (20c)
Also the Casimir operator corresponding to the generators119870119889+
119870119889
minus and 119870
119911
K1198892
119906(11)= 119870119889
+119870119889
minusminus 41198702
119911minus 2 (2119889 minus 3)119870
119911 (21)
has an infinite-fold degeneracy on the Hilbert subspaceH+119889as
K1198892
119906(11)119884119898
119898+119889minus1(120579 120601) = (119889 minus 1) (119889 minus 2) 119884
119898
119898+119889minus1(120579 120601) (22)
The Hilbert subspaces H+119889=2119895+1
= D+(minus119895) and H+119889=2119896
=
D+(1 minus 119896) with 119895 and 119896 as nonnegative and positive integerscontain respectively the following lowest bases
119884minus119895
119895(120579 120601) =
1
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890minus119894119895120601
(sin 120579)119895 (23a)
1198841minus119896
119896(120579 120601) =
1
2119896+12
Γ (119896 + 1)
sdot radic119896Γ (2119896 + 2)
120587
119890119894(1minus119896)120601
(sin 120579)119896minus1 cos 120579
(23b)
They are annihilated as 1198702119895+1minus
119884minus119895
119895(120579 120601) = 0 and 1198702119896
minus1198841minus119896
119896(120579
120601) = 0 and also have the lowest weights minus119895 and 1 minus
119896 Meanwhile the arbitrary bases of the Hilbert subspaces
Advances in High Energy Physics 5
H+119889=2119895+1
and H+119889=2119896
can be respectively calculated by thealgebraic methods as
Proof The relations (18) (20a) and (20b) can be followedfrom the realization of laddering relations with respect toboth azimuthal and magnetic quantum numbers 119897 and 119898simultaneously and agreeably It is sufficient to consider thattwo new differential operators
satisfy the simultaneous laddering relations with respect to 119897and119898 as
119860++
(119897) 119884119898minus1
119897minus1(120579 120601)
= radic2119897 minus 1
2119897 + 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898
119897(120579 120601)
(26a)
119860minusminus
(119897) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 minus 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898minus1
119897minus1(120579 120601)
(26b)
The relations (26a) and (26b) are obtained from (6a) (6b)(14a) and (14b) The relations (19) and (20c) are directlyfollowed The adjoint relation between the operators canbe easily checked by means of the inner product (3) Thecommutativity of operators 119870119889
+ 119870119889minus and 119870
119911with K1198892u(11)
results from (19)The eigenequation (22) follows immediatelyfrom the representation relations (20a)ndash(20c) The relation(20b) implies that 119884minus119895
119895(120579 120601) and 119884
1minus119896
119896(120579 120601) are the lowest
bases for the Hilbert subspacesH+2119895+1
andH+2119896 respectively
Then with repeated application of the raising relation (20a)one may obtain the arbitrary representation bases of u(1 1)Lie algebra as (24a) and (24b)
Although the commutation relations (19) are not closedwith respect to taking the adjoint however their adjointrelations [119870119889+2
+ 119870119889minus2
minus] = minus8119870
119911minus 4119889 + 2 and [119870
119911 119870119889∓2
∓] =
∓119870119889∓2
∓are identically satisfied
Proposition 4 Let one define two new first-order differentialoperators on the sphere 1198782 as
119868119904
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
+ (119904 minus
1
2
∓
1
2
) sin 120579) (27)
They together with the generators 119868119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra as
[119868119904
+ 119868119904
minus] = minus8119868
119911+ 4119904 minus 2
[119868119911 119868119904
plusmn] = plusmn119868
119904
plusmn
(28)
119868119904∓2
plusmnare the adjoint of the operators 119868119904
1) with 119895 and 119896 as nonnegative and positive integers containrespectively the following highest bases
119884119895
119895(120579 120601) =
(minus1)119895
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890119894119895120601
(sin 120579)119895 (32a)
119884119896minus1
119896(120579 120601) =
(minus1)119896minus1
2119896minus12
Γ (119896 + 1)
sdot radic(2119896 + 1) Γ (2119896)
2120587
119890119894(119896minus1)120601
(sin 120579)119896minus1 cos 120579
(32b)
They are annihilated as 1198682119895+1+
119884119895
119895(120579 120601) = 0 and 1198682119896
+119884119896minus1
119896(120579 120601) =
0 and also have the highest weights 119895 and 119896 minus 1 Meanwhile
6 Advances in High Energy Physics
the arbitrary bases of the Hilbert subspacesHminus119904=2119895+1
andHminus119904=2119896
can be respectively calculated by the algebraic methods as
119884119898
2119895minus119898(120579 120601)
=
(1198682119895+1
minus)
119895minus119898
119884119895
119895(120579 120601)
radic(2119895 + 1) Γ (2119895 minus 2119898 + 1) (4119895 minus 2119898 + 1)
119898 le 119895
(33a)
119884119898
2119896minus119898minus1(120579 120601)
=
(1198682119896
minus)
119896minus119898minus1
119884119896minus1
119896(120579 120601)
radic(2119896 + 1) Γ (2119896 minus 2119898) (4119896 minus 2119898 minus 1)
119898 le 119896 minus 1
(33b)
Proof Theproof is quite similar to the proof of Proposition 3So we have to take into account that the two new differentialoperators
119860∓plusmn
(119897) fl ∓ [119871plusmn 119869∓(119897)]
= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+
119894
sin 120579120597
120597120601
+ 119897 sin 120579)(34)
are represented by spherical harmonics whose correspondingladdering equations shift both the azimuthal and magneticquantum numbers 119897 and119898 simultaneously and inversely
119860minus+
(119897 + 1) 119884119898minus1
119897+1(120579 120601)
= radic2119897 + 3
2119897 + 1
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898
119897(120579 120601)
(35a)
119860+minus
(119897 + 1) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 + 3
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898minus1
119897+1(120579 120601)
(35b)
Here again the adjoint of commutation relations (28)becomes [119868119904minus2
+ 119868119904+2
minus] = minus8119868
119911+ 4119904 minus 2 and [119868
119911 119868119904plusmn2
∓] = ∓119868
119904plusmn2
∓
which are identically satisfiedThus all unitary and irreducible representations of su(2)
of dimensions 2119897+1with the nonnegative integers 119897 can carrythe new kind of irreducible representations for u(1 1) Thenew symmetry structures presented in the two recent propo-sitions the so-called positive and negative discrete represen-tations of u(1 1) in turn describe the simultaneous quan-tization of the azimuthal and magnetic quantum numbersTherefore the Hilbert spaces of all spherical harmonics notonly represent compact Lie algebra su(2) by ladder operatorsshifting 119898 for a given 119897 but also represent the noncompactLie algebra u(1 1) by simultaneous shift operators of bothquantum labels 119897 and119898 for given values 119897 minus 119898 and 119897 + 119898
5 Concluding Remarks
For a given azimuthal quantum number 119897 quantization of themagnetic number 119898 is customarily accomplished by repre-senting the operators 119871
+ 119871minus and 119871
3on the sphere with the
commutation relations su(2) compact Lie algebra in a (2119897+1)-dimensional Hilbert subspace H
119897 Furthermore for a given
magnetic quantum number119898 quantization of the azimuthalnumber 119897 is accomplished by representing the operators119869+(119897) and 119869
minus(119897) on the sphere 1198782 with the identity relation
(13) in an infinite-dimensional Hilbert subspaceH119898
Dealing with these issues together simultaneous quan-tization of both azimuthal and magnetic numbers 119897 and 119898is accomplished by representing two bunches of operators119870119889
+ 119870119889
minus 1198703 1 and 119868
119904
+ 119868119904
minus 1198683 1 on the sphere with their
corresponding commutation relations of u(1 1) noncompactLie algebra in the infinite-dimensionalHilbert subspacesH+
119889
and Hminus119904 respectively For given values 119889 = 119897 minus 119898 + 1 and
119904 = 119897 +119898+ 1 they are independent of each other the so-calledpositive and negative (119897 minus 119898)- and (119897 + 119898)-integer irreduciblerepresentations respectively As the spherical harmonics aregenerated from 119884
∓119897
119897(120579 120601) by the operators 119871
plusmn they are also
generated from 119884minus119895
119895(120579 120601) and 1198841minus119896
119896(120579 120601) by1198702119895+1
+and1198702119896
+ as
well as from 119884119895
119895(120579 120601) and 119884119896minus1
119896(120579 120601) by 1198682119895+1
minusand 1198682119896minus respec-
tivelyTherefore not only 119884119898119897(120579 120601)rsquos with the given value for 119897
represent su(2) Lie algebra but also 119884119898119897(120579 120601)rsquos with the given
values for subtraction and summation of the both quantumnumbers 119897 and119898 represent separately u(1 1) (hence su(1 1))Lie algebra as well In other words two different real forms ofsl(2 119888) Lie algebra that is su(2) and su(1 1) are representedby the space of all spherical harmonics119884119898
119897(120579 120601)This happens
because the quantization of both quantum numbers 119897 and 119898are considered jointly Indeed we have
Propositions 1 and 2 lArrrArr Propositions 3 and 4 (36)
We point out that the idea of this paper may find interest-ing applications in quantum devices For instance coherentstates of the SU(1 1) noncompact Lie group have beendefined by Barut and Girardello as eigenstates of the ladderoperators [19] and by Perelomov as the action of the displace-ment operator on the lowest and highest bases [20 21] So ourapproach to the representation of su(1 1) noncompact Liealgebra provides the possibility of constructing two differenttypes of coherent states of su(1 1) on compact manifold 1198782[22] Also realization of the additional symmetry namedsu(2) Lie algebra for Landau levels and bound states of a freeparticle on noncompact manifold 119860119889119878
2can be found based
on the above considerations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Advances in High Energy Physics 7
References
[1] T M Macrobert Spherical Harmonics An Elementary Treatiseon Harmonic Functions with Applications Methuen amp CoLondon UK 1947
[2] L Infeld and T E Hull ldquoThe factorization methodrdquo Reviews ofModern Physics vol 23 no 1 pp 21ndash68 1951
[3] Y Munakata ldquoA generalization of the spherical harmonic addi-tion theoremrdquo Communications in Mathematical Physics vol 9no 1 pp 18ndash37 1968
[4] D A Varshalovich A N Moskalev and V K KhersonskyQuantum Theory of Angular Momentum Irreducible TensorsSpherical Harmonics Vector Coupling Coefficients 3nj SymbolsWorld Scientific Singapore 1989
[5] M E Rose Elementary Theory of Angular Momentum WileyNew York 1957
[6] E Merzbacher Quantum Mechanics John Wiley amp Sons NewYork NY USA 1970
[7] B L Beers and A J Dragt ldquoNew theorems about spherical har-monic expansions and SU(2)rdquo Journal of Mathematical Physicsvol 11 no 8 pp 2313ndash2328 1970
[8] J M Dixon and R Lacroix ldquoSome useful relations using spher-ical harmonics and Legendre polynomialsrdquo Journal of PhysicsA General Physics vol 6 no 8 pp 1119ndash1128 1973
[9] R Beig ldquoA remarkable property of spherical harmonicsrdquoJournal ofMathematical Physics vol 26 no 4 pp 769ndash770 1985
[10] G B Arfken Mathematical Methods for Physicists AcademicPress New York NY USA 3rd edition 1985
[11] J Schwinger Quantum Theory of Angular Momentum Aca-demic Press New York NY USA 1952
[12] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3ndash5 pp 513ndash554 1981
[13] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982
[14] E Witten ldquoSupersymmetry and Morse theoryrdquo Journal ofDifferential Geometry vol 17 no 4 pp 661ndash692 1982
[15] L Alvarez-Gaume ldquoSupersymmetry and the Atiyah-Singerindex theoremrdquo Communications in Mathematical Physics vol90 no 2 pp 161ndash173 1983
[16] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[17] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[18] H Fakhri and A Chenaghlou ldquoQuantum solvable models withgl(2 c) Lie algebra symmetry embedded into the extension ofunitary parasupersymmetryrdquo Journal of Physics A Mathemati-cal and Theoretical vol 40 no 21 pp 5511ndash5523 2007
[19] A O Barut and L Girardello ldquoNew lsquocoherentrsquo states associatedwith non-compact groupsrdquo Communications in MathematicalPhysics vol 21 no 1 pp 41ndash55 1971
[20] A M Perelomov ldquoCoherent states for arbitrary Lie grouprdquoCommunications in Mathematical Physics vol 26 no 3 pp222ndash236 1972
[21] A M Perelomov Generalized Coherent States and Their Appli-cations Texts and Monographs in Physics Springer BerlinGermany 1986
[22] H Fakhri and A Dehghani ldquoCoherency of su(11)-Barut-Girardello type and entanglement for spherical harmonicsrdquoJournal ofMathematical Physics vol 50 no 5 Article ID 0521042009
lead to the derivation of twonew types of simultaneous laddersymmetries with respect to both azimuthal and magneticquantum numbers of spherical harmonics Our proposedladder operators for simultaneous shift of 119897 and 119898 are offirst-order differential type contrary to [2] They lead to anew perspective on the two quantum numbers 119897 and 119898 inconnection with realization of u(1 1) (consequently su(1 1))Lie algebra which in turn is accomplished by all sphericalharmonics 119884119898
119897(120579 120601) with constant values for 119897 minus 119898 and 119897 + 119898
separately First it should be pointed out that the Hilbertspace H can be split into the infinite direct sums of infinitedimensional Hilbert subspaces in two different ways asfollows
H = (
infin
⨁
119895=0
H+
119889=2119895+1) oplus (
infin
⨁
119896=1
H+
119889=2119896)
with
H+119889=2119895+1
= span 119884119898119898+2119895
(120579 120601)119898geminus119895
H+119889=2119896
= span 119884119898119898+2119896minus1
(120579 120601)119898ge1minus119896
(17a)
H = (
infin
⨁
119895=0
Hminus
119904=2119895+1) oplus (
infin
⨁
119896=1
Hminus
119904=2119896)
with
Hminus119904=2119895+1
= span 119884119898minus119898+2119895
(120579 120601)119898le119895
Hminus119904=2119896
= span 119884119898minus119898+2119896minus1
(120579 120601)119898le119896minus1
(17b)
The constant values for the expressions 119897 minus 119898 and 119897 + 119898 ofspherical harmonics have been labeled by 119889 minus 1 and 119904 minus 1respectively
Proposition 3 Let one define two new first-order differentialoperators on the sphere 1198782
119870119889
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
minus (119889 minus
1
2
plusmn
1
2
) sin 120579) (18)
They together with the generators 119870119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra
[119870119889
+ 119870119889
minus] = minus8119870
119911minus 4119889 + 2
[119870119911 119870119889
plusmn] = plusmn119870
119889
plusmn
(19)
119870119889plusmn2
plusmnare the adjoint of the operators 119870119889
∓with respect to the
inner product (3) that is one has 119870119889∓
dagger
= 119870119889plusmn2
plusmn Each of the
Hilbert subspaces H+119889realizes separately (119889 minus 1)-integer irre-
ducible positive representations of 119906(1 1) Lie algebra as (Itmustbe pointed out that by defining 119878119889
119911fl 119871119911+ 1198892 minus 14 and 119878119889
plusmnfl
119870119889
plusmn2 the 119906(1 1) Lie algebra (19) can be considered as com-
mutation relations corresponding to the 119904119906(1 1) Lie algebra[119878119889
+ 119878119889
minus] = minus2119878
119889
119911and [119878119889
119911 119878119889
plusmn] = plusmn119878
119889
plusmn This means that 1 is a triv-
ial center for the semisimple Lie algebra 119906(1 1) In [18] a shortreview on the three different real forms ℎ
4 119906(2) and 119906(1 1) of
119892119897(2 119888) Lie algebra has been presented There their differencesin connection with the structure constants and their represen-tation spaces have also been pointed out)
119870119889
+119884119898minus1
119898+119889minus2(120579 120601)
= radic2119898 + 2119889 minus 3
2119898 + 2119889 minus 1
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898
119898+119889minus1(120579
120601)
(20a)
119870119889
minus119884119898
119898+119889minus1(120579 120601)
= radic2119898 + 2119889 minus 1
2119898 + 2119889 minus 3
(2119898 + 119889 minus 2) (2119898 + 119889 minus 1)119884119898minus1
119898+119889minus2(120579
120601)
(20b)
119870119911119884119898
119898+119889minus1(120579 120601) = 119898119884
119898
119898+119889minus1(120579 120601) (20c)
Also the Casimir operator corresponding to the generators119870119889+
119870119889
minus and 119870
119911
K1198892
119906(11)= 119870119889
+119870119889
minusminus 41198702
119911minus 2 (2119889 minus 3)119870
119911 (21)
has an infinite-fold degeneracy on the Hilbert subspaceH+119889as
K1198892
119906(11)119884119898
119898+119889minus1(120579 120601) = (119889 minus 1) (119889 minus 2) 119884
119898
119898+119889minus1(120579 120601) (22)
The Hilbert subspaces H+119889=2119895+1
= D+(minus119895) and H+119889=2119896
=
D+(1 minus 119896) with 119895 and 119896 as nonnegative and positive integerscontain respectively the following lowest bases
119884minus119895
119895(120579 120601) =
1
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890minus119894119895120601
(sin 120579)119895 (23a)
1198841minus119896
119896(120579 120601) =
1
2119896+12
Γ (119896 + 1)
sdot radic119896Γ (2119896 + 2)
120587
119890119894(1minus119896)120601
(sin 120579)119896minus1 cos 120579
(23b)
They are annihilated as 1198702119895+1minus
119884minus119895
119895(120579 120601) = 0 and 1198702119896
minus1198841minus119896
119896(120579
120601) = 0 and also have the lowest weights minus119895 and 1 minus
119896 Meanwhile the arbitrary bases of the Hilbert subspaces
Advances in High Energy Physics 5
H+119889=2119895+1
and H+119889=2119896
can be respectively calculated by thealgebraic methods as
Proof The relations (18) (20a) and (20b) can be followedfrom the realization of laddering relations with respect toboth azimuthal and magnetic quantum numbers 119897 and 119898simultaneously and agreeably It is sufficient to consider thattwo new differential operators
satisfy the simultaneous laddering relations with respect to 119897and119898 as
119860++
(119897) 119884119898minus1
119897minus1(120579 120601)
= radic2119897 minus 1
2119897 + 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898
119897(120579 120601)
(26a)
119860minusminus
(119897) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 minus 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898minus1
119897minus1(120579 120601)
(26b)
The relations (26a) and (26b) are obtained from (6a) (6b)(14a) and (14b) The relations (19) and (20c) are directlyfollowed The adjoint relation between the operators canbe easily checked by means of the inner product (3) Thecommutativity of operators 119870119889
+ 119870119889minus and 119870
119911with K1198892u(11)
results from (19)The eigenequation (22) follows immediatelyfrom the representation relations (20a)ndash(20c) The relation(20b) implies that 119884minus119895
119895(120579 120601) and 119884
1minus119896
119896(120579 120601) are the lowest
bases for the Hilbert subspacesH+2119895+1
andH+2119896 respectively
Then with repeated application of the raising relation (20a)one may obtain the arbitrary representation bases of u(1 1)Lie algebra as (24a) and (24b)
Although the commutation relations (19) are not closedwith respect to taking the adjoint however their adjointrelations [119870119889+2
+ 119870119889minus2
minus] = minus8119870
119911minus 4119889 + 2 and [119870
119911 119870119889∓2
∓] =
∓119870119889∓2
∓are identically satisfied
Proposition 4 Let one define two new first-order differentialoperators on the sphere 1198782 as
119868119904
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
+ (119904 minus
1
2
∓
1
2
) sin 120579) (27)
They together with the generators 119868119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra as
[119868119904
+ 119868119904
minus] = minus8119868
119911+ 4119904 minus 2
[119868119911 119868119904
plusmn] = plusmn119868
119904
plusmn
(28)
119868119904∓2
plusmnare the adjoint of the operators 119868119904
1) with 119895 and 119896 as nonnegative and positive integers containrespectively the following highest bases
119884119895
119895(120579 120601) =
(minus1)119895
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890119894119895120601
(sin 120579)119895 (32a)
119884119896minus1
119896(120579 120601) =
(minus1)119896minus1
2119896minus12
Γ (119896 + 1)
sdot radic(2119896 + 1) Γ (2119896)
2120587
119890119894(119896minus1)120601
(sin 120579)119896minus1 cos 120579
(32b)
They are annihilated as 1198682119895+1+
119884119895
119895(120579 120601) = 0 and 1198682119896
+119884119896minus1
119896(120579 120601) =
0 and also have the highest weights 119895 and 119896 minus 1 Meanwhile
6 Advances in High Energy Physics
the arbitrary bases of the Hilbert subspacesHminus119904=2119895+1
andHminus119904=2119896
can be respectively calculated by the algebraic methods as
119884119898
2119895minus119898(120579 120601)
=
(1198682119895+1
minus)
119895minus119898
119884119895
119895(120579 120601)
radic(2119895 + 1) Γ (2119895 minus 2119898 + 1) (4119895 minus 2119898 + 1)
119898 le 119895
(33a)
119884119898
2119896minus119898minus1(120579 120601)
=
(1198682119896
minus)
119896minus119898minus1
119884119896minus1
119896(120579 120601)
radic(2119896 + 1) Γ (2119896 minus 2119898) (4119896 minus 2119898 minus 1)
119898 le 119896 minus 1
(33b)
Proof Theproof is quite similar to the proof of Proposition 3So we have to take into account that the two new differentialoperators
119860∓plusmn
(119897) fl ∓ [119871plusmn 119869∓(119897)]
= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+
119894
sin 120579120597
120597120601
+ 119897 sin 120579)(34)
are represented by spherical harmonics whose correspondingladdering equations shift both the azimuthal and magneticquantum numbers 119897 and119898 simultaneously and inversely
119860minus+
(119897 + 1) 119884119898minus1
119897+1(120579 120601)
= radic2119897 + 3
2119897 + 1
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898
119897(120579 120601)
(35a)
119860+minus
(119897 + 1) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 + 3
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898minus1
119897+1(120579 120601)
(35b)
Here again the adjoint of commutation relations (28)becomes [119868119904minus2
+ 119868119904+2
minus] = minus8119868
119911+ 4119904 minus 2 and [119868
119911 119868119904plusmn2
∓] = ∓119868
119904plusmn2
∓
which are identically satisfiedThus all unitary and irreducible representations of su(2)
of dimensions 2119897+1with the nonnegative integers 119897 can carrythe new kind of irreducible representations for u(1 1) Thenew symmetry structures presented in the two recent propo-sitions the so-called positive and negative discrete represen-tations of u(1 1) in turn describe the simultaneous quan-tization of the azimuthal and magnetic quantum numbersTherefore the Hilbert spaces of all spherical harmonics notonly represent compact Lie algebra su(2) by ladder operatorsshifting 119898 for a given 119897 but also represent the noncompactLie algebra u(1 1) by simultaneous shift operators of bothquantum labels 119897 and119898 for given values 119897 minus 119898 and 119897 + 119898
5 Concluding Remarks
For a given azimuthal quantum number 119897 quantization of themagnetic number 119898 is customarily accomplished by repre-senting the operators 119871
+ 119871minus and 119871
3on the sphere with the
commutation relations su(2) compact Lie algebra in a (2119897+1)-dimensional Hilbert subspace H
119897 Furthermore for a given
magnetic quantum number119898 quantization of the azimuthalnumber 119897 is accomplished by representing the operators119869+(119897) and 119869
minus(119897) on the sphere 1198782 with the identity relation
(13) in an infinite-dimensional Hilbert subspaceH119898
Dealing with these issues together simultaneous quan-tization of both azimuthal and magnetic numbers 119897 and 119898is accomplished by representing two bunches of operators119870119889
+ 119870119889
minus 1198703 1 and 119868
119904
+ 119868119904
minus 1198683 1 on the sphere with their
corresponding commutation relations of u(1 1) noncompactLie algebra in the infinite-dimensionalHilbert subspacesH+
119889
and Hminus119904 respectively For given values 119889 = 119897 minus 119898 + 1 and
119904 = 119897 +119898+ 1 they are independent of each other the so-calledpositive and negative (119897 minus 119898)- and (119897 + 119898)-integer irreduciblerepresentations respectively As the spherical harmonics aregenerated from 119884
∓119897
119897(120579 120601) by the operators 119871
plusmn they are also
generated from 119884minus119895
119895(120579 120601) and 1198841minus119896
119896(120579 120601) by1198702119895+1
+and1198702119896
+ as
well as from 119884119895
119895(120579 120601) and 119884119896minus1
119896(120579 120601) by 1198682119895+1
minusand 1198682119896minus respec-
tivelyTherefore not only 119884119898119897(120579 120601)rsquos with the given value for 119897
represent su(2) Lie algebra but also 119884119898119897(120579 120601)rsquos with the given
values for subtraction and summation of the both quantumnumbers 119897 and119898 represent separately u(1 1) (hence su(1 1))Lie algebra as well In other words two different real forms ofsl(2 119888) Lie algebra that is su(2) and su(1 1) are representedby the space of all spherical harmonics119884119898
119897(120579 120601)This happens
because the quantization of both quantum numbers 119897 and 119898are considered jointly Indeed we have
Propositions 1 and 2 lArrrArr Propositions 3 and 4 (36)
We point out that the idea of this paper may find interest-ing applications in quantum devices For instance coherentstates of the SU(1 1) noncompact Lie group have beendefined by Barut and Girardello as eigenstates of the ladderoperators [19] and by Perelomov as the action of the displace-ment operator on the lowest and highest bases [20 21] So ourapproach to the representation of su(1 1) noncompact Liealgebra provides the possibility of constructing two differenttypes of coherent states of su(1 1) on compact manifold 1198782[22] Also realization of the additional symmetry namedsu(2) Lie algebra for Landau levels and bound states of a freeparticle on noncompact manifold 119860119889119878
2can be found based
on the above considerations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Advances in High Energy Physics 7
References
[1] T M Macrobert Spherical Harmonics An Elementary Treatiseon Harmonic Functions with Applications Methuen amp CoLondon UK 1947
[2] L Infeld and T E Hull ldquoThe factorization methodrdquo Reviews ofModern Physics vol 23 no 1 pp 21ndash68 1951
[3] Y Munakata ldquoA generalization of the spherical harmonic addi-tion theoremrdquo Communications in Mathematical Physics vol 9no 1 pp 18ndash37 1968
[4] D A Varshalovich A N Moskalev and V K KhersonskyQuantum Theory of Angular Momentum Irreducible TensorsSpherical Harmonics Vector Coupling Coefficients 3nj SymbolsWorld Scientific Singapore 1989
[5] M E Rose Elementary Theory of Angular Momentum WileyNew York 1957
[6] E Merzbacher Quantum Mechanics John Wiley amp Sons NewYork NY USA 1970
[7] B L Beers and A J Dragt ldquoNew theorems about spherical har-monic expansions and SU(2)rdquo Journal of Mathematical Physicsvol 11 no 8 pp 2313ndash2328 1970
[8] J M Dixon and R Lacroix ldquoSome useful relations using spher-ical harmonics and Legendre polynomialsrdquo Journal of PhysicsA General Physics vol 6 no 8 pp 1119ndash1128 1973
[9] R Beig ldquoA remarkable property of spherical harmonicsrdquoJournal ofMathematical Physics vol 26 no 4 pp 769ndash770 1985
[10] G B Arfken Mathematical Methods for Physicists AcademicPress New York NY USA 3rd edition 1985
[11] J Schwinger Quantum Theory of Angular Momentum Aca-demic Press New York NY USA 1952
[12] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3ndash5 pp 513ndash554 1981
[13] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982
[14] E Witten ldquoSupersymmetry and Morse theoryrdquo Journal ofDifferential Geometry vol 17 no 4 pp 661ndash692 1982
[15] L Alvarez-Gaume ldquoSupersymmetry and the Atiyah-Singerindex theoremrdquo Communications in Mathematical Physics vol90 no 2 pp 161ndash173 1983
[16] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[17] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[18] H Fakhri and A Chenaghlou ldquoQuantum solvable models withgl(2 c) Lie algebra symmetry embedded into the extension ofunitary parasupersymmetryrdquo Journal of Physics A Mathemati-cal and Theoretical vol 40 no 21 pp 5511ndash5523 2007
[19] A O Barut and L Girardello ldquoNew lsquocoherentrsquo states associatedwith non-compact groupsrdquo Communications in MathematicalPhysics vol 21 no 1 pp 41ndash55 1971
[20] A M Perelomov ldquoCoherent states for arbitrary Lie grouprdquoCommunications in Mathematical Physics vol 26 no 3 pp222ndash236 1972
[21] A M Perelomov Generalized Coherent States and Their Appli-cations Texts and Monographs in Physics Springer BerlinGermany 1986
[22] H Fakhri and A Dehghani ldquoCoherency of su(11)-Barut-Girardello type and entanglement for spherical harmonicsrdquoJournal ofMathematical Physics vol 50 no 5 Article ID 0521042009
Proof The relations (18) (20a) and (20b) can be followedfrom the realization of laddering relations with respect toboth azimuthal and magnetic quantum numbers 119897 and 119898simultaneously and agreeably It is sufficient to consider thattwo new differential operators
satisfy the simultaneous laddering relations with respect to 119897and119898 as
119860++
(119897) 119884119898minus1
119897minus1(120579 120601)
= radic2119897 minus 1
2119897 + 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898
119897(120579 120601)
(26a)
119860minusminus
(119897) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 minus 1
(119897 + 119898 minus 1) (119897 + 119898)119884119898minus1
119897minus1(120579 120601)
(26b)
The relations (26a) and (26b) are obtained from (6a) (6b)(14a) and (14b) The relations (19) and (20c) are directlyfollowed The adjoint relation between the operators canbe easily checked by means of the inner product (3) Thecommutativity of operators 119870119889
+ 119870119889minus and 119870
119911with K1198892u(11)
results from (19)The eigenequation (22) follows immediatelyfrom the representation relations (20a)ndash(20c) The relation(20b) implies that 119884minus119895
119895(120579 120601) and 119884
1minus119896
119896(120579 120601) are the lowest
bases for the Hilbert subspacesH+2119895+1
andH+2119896 respectively
Then with repeated application of the raising relation (20a)one may obtain the arbitrary representation bases of u(1 1)Lie algebra as (24a) and (24b)
Although the commutation relations (19) are not closedwith respect to taking the adjoint however their adjointrelations [119870119889+2
+ 119870119889minus2
minus] = minus8119870
119911minus 4119889 + 2 and [119870
119911 119870119889∓2
∓] =
∓119870119889∓2
∓are identically satisfied
Proposition 4 Let one define two new first-order differentialoperators on the sphere 1198782 as
119868119904
plusmn= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+ 119894 (
1
sin 120579+ sin 120579) 120597
120597120601
+ (119904 minus
1
2
∓
1
2
) sin 120579) (27)
They together with the generators 119868119911= 119871119911= minus119894(120597120597120601) and 1
satisfy the commutation relations of 119906(1 1) Lie algebra as
[119868119904
+ 119868119904
minus] = minus8119868
119911+ 4119904 minus 2
[119868119911 119868119904
plusmn] = plusmn119868
119904
plusmn
(28)
119868119904∓2
plusmnare the adjoint of the operators 119868119904
1) with 119895 and 119896 as nonnegative and positive integers containrespectively the following highest bases
119884119895
119895(120579 120601) =
(minus1)119895
2119895Γ (119895 + 1)
radicΓ (2119895 + 2)
4120587
119890119894119895120601
(sin 120579)119895 (32a)
119884119896minus1
119896(120579 120601) =
(minus1)119896minus1
2119896minus12
Γ (119896 + 1)
sdot radic(2119896 + 1) Γ (2119896)
2120587
119890119894(119896minus1)120601
(sin 120579)119896minus1 cos 120579
(32b)
They are annihilated as 1198682119895+1+
119884119895
119895(120579 120601) = 0 and 1198682119896
+119884119896minus1
119896(120579 120601) =
0 and also have the highest weights 119895 and 119896 minus 1 Meanwhile
6 Advances in High Energy Physics
the arbitrary bases of the Hilbert subspacesHminus119904=2119895+1
andHminus119904=2119896
can be respectively calculated by the algebraic methods as
119884119898
2119895minus119898(120579 120601)
=
(1198682119895+1
minus)
119895minus119898
119884119895
119895(120579 120601)
radic(2119895 + 1) Γ (2119895 minus 2119898 + 1) (4119895 minus 2119898 + 1)
119898 le 119895
(33a)
119884119898
2119896minus119898minus1(120579 120601)
=
(1198682119896
minus)
119896minus119898minus1
119884119896minus1
119896(120579 120601)
radic(2119896 + 1) Γ (2119896 minus 2119898) (4119896 minus 2119898 minus 1)
119898 le 119896 minus 1
(33b)
Proof Theproof is quite similar to the proof of Proposition 3So we have to take into account that the two new differentialoperators
119860∓plusmn
(119897) fl ∓ [119871plusmn 119869∓(119897)]
= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+
119894
sin 120579120597
120597120601
+ 119897 sin 120579)(34)
are represented by spherical harmonics whose correspondingladdering equations shift both the azimuthal and magneticquantum numbers 119897 and119898 simultaneously and inversely
119860minus+
(119897 + 1) 119884119898minus1
119897+1(120579 120601)
= radic2119897 + 3
2119897 + 1
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898
119897(120579 120601)
(35a)
119860+minus
(119897 + 1) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 + 3
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898minus1
119897+1(120579 120601)
(35b)
Here again the adjoint of commutation relations (28)becomes [119868119904minus2
+ 119868119904+2
minus] = minus8119868
119911+ 4119904 minus 2 and [119868
119911 119868119904plusmn2
∓] = ∓119868
119904plusmn2
∓
which are identically satisfiedThus all unitary and irreducible representations of su(2)
of dimensions 2119897+1with the nonnegative integers 119897 can carrythe new kind of irreducible representations for u(1 1) Thenew symmetry structures presented in the two recent propo-sitions the so-called positive and negative discrete represen-tations of u(1 1) in turn describe the simultaneous quan-tization of the azimuthal and magnetic quantum numbersTherefore the Hilbert spaces of all spherical harmonics notonly represent compact Lie algebra su(2) by ladder operatorsshifting 119898 for a given 119897 but also represent the noncompactLie algebra u(1 1) by simultaneous shift operators of bothquantum labels 119897 and119898 for given values 119897 minus 119898 and 119897 + 119898
5 Concluding Remarks
For a given azimuthal quantum number 119897 quantization of themagnetic number 119898 is customarily accomplished by repre-senting the operators 119871
+ 119871minus and 119871
3on the sphere with the
commutation relations su(2) compact Lie algebra in a (2119897+1)-dimensional Hilbert subspace H
119897 Furthermore for a given
magnetic quantum number119898 quantization of the azimuthalnumber 119897 is accomplished by representing the operators119869+(119897) and 119869
minus(119897) on the sphere 1198782 with the identity relation
(13) in an infinite-dimensional Hilbert subspaceH119898
Dealing with these issues together simultaneous quan-tization of both azimuthal and magnetic numbers 119897 and 119898is accomplished by representing two bunches of operators119870119889
+ 119870119889
minus 1198703 1 and 119868
119904
+ 119868119904
minus 1198683 1 on the sphere with their
corresponding commutation relations of u(1 1) noncompactLie algebra in the infinite-dimensionalHilbert subspacesH+
119889
and Hminus119904 respectively For given values 119889 = 119897 minus 119898 + 1 and
119904 = 119897 +119898+ 1 they are independent of each other the so-calledpositive and negative (119897 minus 119898)- and (119897 + 119898)-integer irreduciblerepresentations respectively As the spherical harmonics aregenerated from 119884
∓119897
119897(120579 120601) by the operators 119871
plusmn they are also
generated from 119884minus119895
119895(120579 120601) and 1198841minus119896
119896(120579 120601) by1198702119895+1
+and1198702119896
+ as
well as from 119884119895
119895(120579 120601) and 119884119896minus1
119896(120579 120601) by 1198682119895+1
minusand 1198682119896minus respec-
tivelyTherefore not only 119884119898119897(120579 120601)rsquos with the given value for 119897
represent su(2) Lie algebra but also 119884119898119897(120579 120601)rsquos with the given
values for subtraction and summation of the both quantumnumbers 119897 and119898 represent separately u(1 1) (hence su(1 1))Lie algebra as well In other words two different real forms ofsl(2 119888) Lie algebra that is su(2) and su(1 1) are representedby the space of all spherical harmonics119884119898
119897(120579 120601)This happens
because the quantization of both quantum numbers 119897 and 119898are considered jointly Indeed we have
Propositions 1 and 2 lArrrArr Propositions 3 and 4 (36)
We point out that the idea of this paper may find interest-ing applications in quantum devices For instance coherentstates of the SU(1 1) noncompact Lie group have beendefined by Barut and Girardello as eigenstates of the ladderoperators [19] and by Perelomov as the action of the displace-ment operator on the lowest and highest bases [20 21] So ourapproach to the representation of su(1 1) noncompact Liealgebra provides the possibility of constructing two differenttypes of coherent states of su(1 1) on compact manifold 1198782[22] Also realization of the additional symmetry namedsu(2) Lie algebra for Landau levels and bound states of a freeparticle on noncompact manifold 119860119889119878
2can be found based
on the above considerations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Advances in High Energy Physics 7
References
[1] T M Macrobert Spherical Harmonics An Elementary Treatiseon Harmonic Functions with Applications Methuen amp CoLondon UK 1947
[2] L Infeld and T E Hull ldquoThe factorization methodrdquo Reviews ofModern Physics vol 23 no 1 pp 21ndash68 1951
[3] Y Munakata ldquoA generalization of the spherical harmonic addi-tion theoremrdquo Communications in Mathematical Physics vol 9no 1 pp 18ndash37 1968
[4] D A Varshalovich A N Moskalev and V K KhersonskyQuantum Theory of Angular Momentum Irreducible TensorsSpherical Harmonics Vector Coupling Coefficients 3nj SymbolsWorld Scientific Singapore 1989
[5] M E Rose Elementary Theory of Angular Momentum WileyNew York 1957
[6] E Merzbacher Quantum Mechanics John Wiley amp Sons NewYork NY USA 1970
[7] B L Beers and A J Dragt ldquoNew theorems about spherical har-monic expansions and SU(2)rdquo Journal of Mathematical Physicsvol 11 no 8 pp 2313ndash2328 1970
[8] J M Dixon and R Lacroix ldquoSome useful relations using spher-ical harmonics and Legendre polynomialsrdquo Journal of PhysicsA General Physics vol 6 no 8 pp 1119ndash1128 1973
[9] R Beig ldquoA remarkable property of spherical harmonicsrdquoJournal ofMathematical Physics vol 26 no 4 pp 769ndash770 1985
[10] G B Arfken Mathematical Methods for Physicists AcademicPress New York NY USA 3rd edition 1985
[11] J Schwinger Quantum Theory of Angular Momentum Aca-demic Press New York NY USA 1952
[12] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3ndash5 pp 513ndash554 1981
[13] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982
[14] E Witten ldquoSupersymmetry and Morse theoryrdquo Journal ofDifferential Geometry vol 17 no 4 pp 661ndash692 1982
[15] L Alvarez-Gaume ldquoSupersymmetry and the Atiyah-Singerindex theoremrdquo Communications in Mathematical Physics vol90 no 2 pp 161ndash173 1983
[16] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[17] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[18] H Fakhri and A Chenaghlou ldquoQuantum solvable models withgl(2 c) Lie algebra symmetry embedded into the extension ofunitary parasupersymmetryrdquo Journal of Physics A Mathemati-cal and Theoretical vol 40 no 21 pp 5511ndash5523 2007
[19] A O Barut and L Girardello ldquoNew lsquocoherentrsquo states associatedwith non-compact groupsrdquo Communications in MathematicalPhysics vol 21 no 1 pp 41ndash55 1971
[20] A M Perelomov ldquoCoherent states for arbitrary Lie grouprdquoCommunications in Mathematical Physics vol 26 no 3 pp222ndash236 1972
[21] A M Perelomov Generalized Coherent States and Their Appli-cations Texts and Monographs in Physics Springer BerlinGermany 1986
[22] H Fakhri and A Dehghani ldquoCoherency of su(11)-Barut-Girardello type and entanglement for spherical harmonicsrdquoJournal ofMathematical Physics vol 50 no 5 Article ID 0521042009
the arbitrary bases of the Hilbert subspacesHminus119904=2119895+1
andHminus119904=2119896
can be respectively calculated by the algebraic methods as
119884119898
2119895minus119898(120579 120601)
=
(1198682119895+1
minus)
119895minus119898
119884119895
119895(120579 120601)
radic(2119895 + 1) Γ (2119895 minus 2119898 + 1) (4119895 minus 2119898 + 1)
119898 le 119895
(33a)
119884119898
2119896minus119898minus1(120579 120601)
=
(1198682119896
minus)
119896minus119898minus1
119884119896minus1
119896(120579 120601)
radic(2119896 + 1) Γ (2119896 minus 2119898) (4119896 minus 2119898 minus 1)
119898 le 119896 minus 1
(33b)
Proof Theproof is quite similar to the proof of Proposition 3So we have to take into account that the two new differentialoperators
119860∓plusmn
(119897) fl ∓ [119871plusmn 119869∓(119897)]
= 119890plusmn119894120601
(plusmn cos 120579 120597120597120579
+
119894
sin 120579120597
120597120601
+ 119897 sin 120579)(34)
are represented by spherical harmonics whose correspondingladdering equations shift both the azimuthal and magneticquantum numbers 119897 and119898 simultaneously and inversely
119860minus+
(119897 + 1) 119884119898minus1
119897+1(120579 120601)
= radic2119897 + 3
2119897 + 1
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898
119897(120579 120601)
(35a)
119860+minus
(119897 + 1) 119884119898
119897(120579 120601)
= radic2119897 + 1
2119897 + 3
(119897 minus 119898 + 1) (119897 minus 119898 + 2)119884119898minus1
119897+1(120579 120601)
(35b)
Here again the adjoint of commutation relations (28)becomes [119868119904minus2
+ 119868119904+2
minus] = minus8119868
119911+ 4119904 minus 2 and [119868
119911 119868119904plusmn2
∓] = ∓119868
119904plusmn2
∓
which are identically satisfiedThus all unitary and irreducible representations of su(2)
of dimensions 2119897+1with the nonnegative integers 119897 can carrythe new kind of irreducible representations for u(1 1) Thenew symmetry structures presented in the two recent propo-sitions the so-called positive and negative discrete represen-tations of u(1 1) in turn describe the simultaneous quan-tization of the azimuthal and magnetic quantum numbersTherefore the Hilbert spaces of all spherical harmonics notonly represent compact Lie algebra su(2) by ladder operatorsshifting 119898 for a given 119897 but also represent the noncompactLie algebra u(1 1) by simultaneous shift operators of bothquantum labels 119897 and119898 for given values 119897 minus 119898 and 119897 + 119898
5 Concluding Remarks
For a given azimuthal quantum number 119897 quantization of themagnetic number 119898 is customarily accomplished by repre-senting the operators 119871
+ 119871minus and 119871
3on the sphere with the
commutation relations su(2) compact Lie algebra in a (2119897+1)-dimensional Hilbert subspace H
119897 Furthermore for a given
magnetic quantum number119898 quantization of the azimuthalnumber 119897 is accomplished by representing the operators119869+(119897) and 119869
minus(119897) on the sphere 1198782 with the identity relation
(13) in an infinite-dimensional Hilbert subspaceH119898
Dealing with these issues together simultaneous quan-tization of both azimuthal and magnetic numbers 119897 and 119898is accomplished by representing two bunches of operators119870119889
+ 119870119889
minus 1198703 1 and 119868
119904
+ 119868119904
minus 1198683 1 on the sphere with their
corresponding commutation relations of u(1 1) noncompactLie algebra in the infinite-dimensionalHilbert subspacesH+
119889
and Hminus119904 respectively For given values 119889 = 119897 minus 119898 + 1 and
119904 = 119897 +119898+ 1 they are independent of each other the so-calledpositive and negative (119897 minus 119898)- and (119897 + 119898)-integer irreduciblerepresentations respectively As the spherical harmonics aregenerated from 119884
∓119897
119897(120579 120601) by the operators 119871
plusmn they are also
generated from 119884minus119895
119895(120579 120601) and 1198841minus119896
119896(120579 120601) by1198702119895+1
+and1198702119896
+ as
well as from 119884119895
119895(120579 120601) and 119884119896minus1
119896(120579 120601) by 1198682119895+1
minusand 1198682119896minus respec-
tivelyTherefore not only 119884119898119897(120579 120601)rsquos with the given value for 119897
represent su(2) Lie algebra but also 119884119898119897(120579 120601)rsquos with the given
values for subtraction and summation of the both quantumnumbers 119897 and119898 represent separately u(1 1) (hence su(1 1))Lie algebra as well In other words two different real forms ofsl(2 119888) Lie algebra that is su(2) and su(1 1) are representedby the space of all spherical harmonics119884119898
119897(120579 120601)This happens
because the quantization of both quantum numbers 119897 and 119898are considered jointly Indeed we have
Propositions 1 and 2 lArrrArr Propositions 3 and 4 (36)
We point out that the idea of this paper may find interest-ing applications in quantum devices For instance coherentstates of the SU(1 1) noncompact Lie group have beendefined by Barut and Girardello as eigenstates of the ladderoperators [19] and by Perelomov as the action of the displace-ment operator on the lowest and highest bases [20 21] So ourapproach to the representation of su(1 1) noncompact Liealgebra provides the possibility of constructing two differenttypes of coherent states of su(1 1) on compact manifold 1198782[22] Also realization of the additional symmetry namedsu(2) Lie algebra for Landau levels and bound states of a freeparticle on noncompact manifold 119860119889119878
2can be found based
on the above considerations
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Advances in High Energy Physics 7
References
[1] T M Macrobert Spherical Harmonics An Elementary Treatiseon Harmonic Functions with Applications Methuen amp CoLondon UK 1947
[2] L Infeld and T E Hull ldquoThe factorization methodrdquo Reviews ofModern Physics vol 23 no 1 pp 21ndash68 1951
[3] Y Munakata ldquoA generalization of the spherical harmonic addi-tion theoremrdquo Communications in Mathematical Physics vol 9no 1 pp 18ndash37 1968
[4] D A Varshalovich A N Moskalev and V K KhersonskyQuantum Theory of Angular Momentum Irreducible TensorsSpherical Harmonics Vector Coupling Coefficients 3nj SymbolsWorld Scientific Singapore 1989
[5] M E Rose Elementary Theory of Angular Momentum WileyNew York 1957
[6] E Merzbacher Quantum Mechanics John Wiley amp Sons NewYork NY USA 1970
[7] B L Beers and A J Dragt ldquoNew theorems about spherical har-monic expansions and SU(2)rdquo Journal of Mathematical Physicsvol 11 no 8 pp 2313ndash2328 1970
[8] J M Dixon and R Lacroix ldquoSome useful relations using spher-ical harmonics and Legendre polynomialsrdquo Journal of PhysicsA General Physics vol 6 no 8 pp 1119ndash1128 1973
[9] R Beig ldquoA remarkable property of spherical harmonicsrdquoJournal ofMathematical Physics vol 26 no 4 pp 769ndash770 1985
[10] G B Arfken Mathematical Methods for Physicists AcademicPress New York NY USA 3rd edition 1985
[11] J Schwinger Quantum Theory of Angular Momentum Aca-demic Press New York NY USA 1952
[12] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3ndash5 pp 513ndash554 1981
[13] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982
[14] E Witten ldquoSupersymmetry and Morse theoryrdquo Journal ofDifferential Geometry vol 17 no 4 pp 661ndash692 1982
[15] L Alvarez-Gaume ldquoSupersymmetry and the Atiyah-Singerindex theoremrdquo Communications in Mathematical Physics vol90 no 2 pp 161ndash173 1983
[16] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[17] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[18] H Fakhri and A Chenaghlou ldquoQuantum solvable models withgl(2 c) Lie algebra symmetry embedded into the extension ofunitary parasupersymmetryrdquo Journal of Physics A Mathemati-cal and Theoretical vol 40 no 21 pp 5511ndash5523 2007
[19] A O Barut and L Girardello ldquoNew lsquocoherentrsquo states associatedwith non-compact groupsrdquo Communications in MathematicalPhysics vol 21 no 1 pp 41ndash55 1971
[20] A M Perelomov ldquoCoherent states for arbitrary Lie grouprdquoCommunications in Mathematical Physics vol 26 no 3 pp222ndash236 1972
[21] A M Perelomov Generalized Coherent States and Their Appli-cations Texts and Monographs in Physics Springer BerlinGermany 1986
[22] H Fakhri and A Dehghani ldquoCoherency of su(11)-Barut-Girardello type and entanglement for spherical harmonicsrdquoJournal ofMathematical Physics vol 50 no 5 Article ID 0521042009
[1] T M Macrobert Spherical Harmonics An Elementary Treatiseon Harmonic Functions with Applications Methuen amp CoLondon UK 1947
[2] L Infeld and T E Hull ldquoThe factorization methodrdquo Reviews ofModern Physics vol 23 no 1 pp 21ndash68 1951
[3] Y Munakata ldquoA generalization of the spherical harmonic addi-tion theoremrdquo Communications in Mathematical Physics vol 9no 1 pp 18ndash37 1968
[4] D A Varshalovich A N Moskalev and V K KhersonskyQuantum Theory of Angular Momentum Irreducible TensorsSpherical Harmonics Vector Coupling Coefficients 3nj SymbolsWorld Scientific Singapore 1989
[5] M E Rose Elementary Theory of Angular Momentum WileyNew York 1957
[6] E Merzbacher Quantum Mechanics John Wiley amp Sons NewYork NY USA 1970
[7] B L Beers and A J Dragt ldquoNew theorems about spherical har-monic expansions and SU(2)rdquo Journal of Mathematical Physicsvol 11 no 8 pp 2313ndash2328 1970
[8] J M Dixon and R Lacroix ldquoSome useful relations using spher-ical harmonics and Legendre polynomialsrdquo Journal of PhysicsA General Physics vol 6 no 8 pp 1119ndash1128 1973
[9] R Beig ldquoA remarkable property of spherical harmonicsrdquoJournal ofMathematical Physics vol 26 no 4 pp 769ndash770 1985
[10] G B Arfken Mathematical Methods for Physicists AcademicPress New York NY USA 3rd edition 1985
[11] J Schwinger Quantum Theory of Angular Momentum Aca-demic Press New York NY USA 1952
[12] E Witten ldquoDynamical breaking of supersymmetryrdquo NuclearPhysics B vol 188 no 3ndash5 pp 513ndash554 1981
[13] E Witten ldquoConstraints on supersymmetry breakingrdquo NuclearPhysics B vol 202 no 2 pp 253ndash316 1982
[14] E Witten ldquoSupersymmetry and Morse theoryrdquo Journal ofDifferential Geometry vol 17 no 4 pp 661ndash692 1982
[15] L Alvarez-Gaume ldquoSupersymmetry and the Atiyah-Singerindex theoremrdquo Communications in Mathematical Physics vol90 no 2 pp 161ndash173 1983
[16] A V Turbiner ldquoQuasi-exactly-solvable problems and sl(2)algebrardquo Communications in Mathematical Physics vol 118 no3 pp 467ndash474 1988
[17] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[18] H Fakhri and A Chenaghlou ldquoQuantum solvable models withgl(2 c) Lie algebra symmetry embedded into the extension ofunitary parasupersymmetryrdquo Journal of Physics A Mathemati-cal and Theoretical vol 40 no 21 pp 5511ndash5523 2007
[19] A O Barut and L Girardello ldquoNew lsquocoherentrsquo states associatedwith non-compact groupsrdquo Communications in MathematicalPhysics vol 21 no 1 pp 41ndash55 1971
[20] A M Perelomov ldquoCoherent states for arbitrary Lie grouprdquoCommunications in Mathematical Physics vol 26 no 3 pp222ndash236 1972
[21] A M Perelomov Generalized Coherent States and Their Appli-cations Texts and Monographs in Physics Springer BerlinGermany 1986
[22] H Fakhri and A Dehghani ldquoCoherency of su(11)-Barut-Girardello type and entanglement for spherical harmonicsrdquoJournal ofMathematical Physics vol 50 no 5 Article ID 0521042009
![Notes on Spherical Harmonics and Linear …cis610/sharmonics.pdfChapter 1 Spherical Harmonics and Linear ... The reader will nd the above formulae in Fourier’s famous book [12] in](https://static.documents.pub/doc/80x56/5af1620f7f8b9ad0618f5829/notes-on-spherical-harmonics-and-linear-cis610-1-spherical-harmonics-and-linear.jpg)
