A non-Abelian SO(8) monopole as generalization of Dirac-Yang monopoles for a 9-dimensional space

JOURNAL OF MATHEMATICAL PHYSICS 52, 032105 (2011)

A non-Abelian SO(8) monopole as generalization ofDirac-Yang monopoles for a 9-dimensional space

Van-Hoang Lea) and Thanh-Son NguyenDepartment of Physics, HCMC University of Pedagogy, 280 An Duong Vuong, Dist. 5, HoChi Minh City, Vietnam

(Received 21 August 2010; accepted 17 February 2011; published online 18 March 2011)

We establish an explicit form of a non-Abelian SO(8) monopole in a 9-dimensionalspace and show that it is indeed a direct generalization of Dirac and Yang monopoles.Using the generalized Hurwitz transformation, we have found a connection betweena 16-dimensional harmonic oscillator and a 9-dimensional hydrogenlike atom in thefield of the SO(8) monopole (MICZ-Kepler problem). Using the built connection thegroup of dynamical symmetry of the 9-dimensional MICZ-Kepler problem is foundas SO(10, 2). C© 2011 American Institute of Physics. [doi:10.1063/1.3567422]

I. INTRODUCTION

In 1978, Yang generalized the Dirac monopole1 for a 5-dimensional space using the modelof interaction of SU(2) gauge field with an isospin particle.2 Two basic properties of the SU(2)monopole field are (i) flux into a Gaussian surface containing a monopole is not equal to zero; and(ii) the field has the O(5) symmetry. The SU(2) Yang monopole, which simultaneously has “electriccharge,” is called a Yang–Coulomb monopole that has been of great interests.3–10

A generalization of Yang results for higher dimensional space has become a natural need andthere have been some results in this direction.11–14 However, until now there has not been anydiscussion about the generalization of the monopole within the Yang logic while generalizing from3-dimensional to 5-dimensional space. It is a very interesting property of the Dirac monopole aswell as of the Yang monopole that their appearance does not break the symmetry of the Coulombproblem. In other words, the presence of the Dirac monopole does not change the O(4) symmetry,but in opposite, it even saves the invariance of the Runge–Lenz vector as well as the SO(4, 2)dynamical symmetry of the Coulomb problem.15 It is analogical that the 5-dimensional Coulombproblem with the presence of Yang monopole still has the O(6) symmetry and the SO(6, 2) dynamicalsymmetry.5, 16 In our point of view, a higher dimensional monopole if exists should have the similarproperty to be a direct generalization of Dirac and Yang monopoles.

In a previous paper,17 we have expanded the Hurwitz transformation, and then based on thisto establish a connection between the Schrodinger equation of a 16-dimensional isotropic harmonicoscillator and that of a 9-dimensional hydrogenlike atom in the field of a monopole described bya septet of potential vectors in a non-Abelian model of 28 generators. These results encourage usto research for a new way to generalize Yang monopole for a 9-dimensional space based on theconnection between an n-dimensional harmonic oscillator and an N-dimensional hydrogenlike atom.Until now, these connections have been established only for a few cases of dimension n → N suchas 2 → 2, 3 → 4, 5 → 8, and 9 → 16.18–21 It is interesting that when adding the field of a monopoleto the Coulomb problem (MICZ-Kepler problem), the connection with the harmonic oscillator stillexists. In the case of 3 → 4 we have a Dirac monopole19 whereas in the case of 5 → 8 it is a Yangmonopole.3–5 The question is what is the monopole for the case of 9 → 16?

In this paper, we seek for a complete answer to the given question by establishing a monopolein a 9-dimensional space in the model of SO(8) algebra in a form that the Coulomb problem with

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032105-2 V. H. Le and T. S. Nguyen J. Math. Phys. 52, 032105 (2011)

the presence of this monopole can be transformed into a 16-dimensional harmonic oscillator via thegeneralized Hurwitz transformation. Here, from a 9-dimensional to a 16-dimensional space, thereappear seven extra variables in the explicit form of 28 generators of SO(8) algebra. So the foundSO(8) monopole we consider as a direct generalization of Dirac and Yang monopoles in the scheme:

Dimension : 3 = 21 + 1 → Dirac monopole

: 5 = 22 + 1 → SU (2) Yang monopole

: 9 = 23 + 1 → SO(8) monopole

II. THE GENERALIZED HURWITZ TRANSFORMATION

In this section, we will review the generalized Hurwitz transformation given in the paper17 andderive some new formulae useful for further calculations in the paper.

The bilinear transformation for connecting a 9-dimensional space (x1, x2, . . . , x9) and a16-dimensional space (u1, u2, . . . , u8, v1, v2, . . . , v8) was first established21 in order to satisfy theEuler condition:

r = √xλxλ = usus + vsvs . (1)

Recently, in the paper17 the generalized Hurwitz transformation was given in the explicit formas follows:

x j = 2(� j )st usvt ,

x9 = usus − vsvs .(2)

Here, the matrices � j are defined as follows:

�1 =[

β

0

0

β

], �2 =

[βα1α3

0

0

βα1α3

], �3 =

[α3

0

0

α3

], �4 =

[α1

0

0

−α1

],

�5 =[

0

iα1α2

−iα1α2

0

], �6 =

[0

iβα2α3

−iβα2α3

0

], �7 =

[0

βα3

−βα3

0

], �8 =

[0

α1

α1

0

],

(3)

in which β = [ I0

0−I

], αk = [ 0

σk

σk

0

]are the Dirac matrices; σk are the Pauli matrices. In

Eq. (2) as well as further in the paper, repeating indices mean summation over them in wholeregion s, t = 1, 2, . . . , 8. From here on, Greek symbols are used for indicating the coordinates xλ

with λ = 1 → 9 but wherever necessary, we use the Latin symbols for denoting index of x j withvalue j = 1 → 8.

Matrices � j are either symmetric or antisymmetric. Indeed, �Tk = −�k for k = 2, 5, 6 and �T

k= �k for k = 1, 3, 4, 7, 8. Notation T as upper index means the transpose operation of matrix.Besides, they satisfy the following property:

�Ts �t + �T

t �s = 2δst I, (4)

with δst is Kronecker delta symbol. This means the product of any couple of different matrices (3)is an antisymmetric matrix.

We can add seven new variables φ1, φ2, φ3, α1, α2, α3, α4 in the generalized Hurwitz transfor-mation (2) as shown in Ref. 17. We will examine the physical meaning of these extra variables ina later work as it has been done for the cases of Kustaanheimo–Stiefel transformation22 (3 → 4)and of Hurwitz transformation16 5 → 8. We now show that the inverse transformation is established


032105-3 A non-Abelian SO(8) monopole J. Math. Phys. 52, 032105 (2011)

using these extra variables:

us =√

r + x9

2bs(φα),

vs = x j√2(r + x9)

Hs j (φα).(5)

Here, the functions bs(φα) depend on the extra variables only:

b1 = cos(φ1/2) cos(φ2/2) cos α1, b2 = cos(φ1/2) cos(φ2/2) sin α1,

b3 = cos(φ1/2) sin(φ2/2) cos α2, b4 = cos(φ1/2) sin(φ2/2) sin α2,

b5 = sin(φ1/2) cos(φ3/2) cos α3, b6 = sin(φ1/2) cos(φ3/2) sin α3,

b7 = sin(φ1/2) sin(φ3/2) cos α4, b8 = sin(φ1/2) sin(φ3/2) sin α4.

Matrix elements H js(φα) also depend only on the extra variables. Their explicit form can beexpressed in terms of bs(φα) as follows:

H (βα) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

b1 −b2 b3 b4 −b5 −b6 b7 b8

b2 b1 −b4 b3 b6 −b5 −b8 b7

−b3 b4 b1 b2 −b7 b8 −b5 b6

−b4 −b3 −b2 b1 b8 b7 b6 b5

b5 −b6 b7 −b8 b1 b2 −b3 b4

b6 b5 −b8 −b7 −b2 b1 b4 b3

−b7 b8 b5 −b6 b3 −b4 b1 b2

−b8 −b7 −b6 −b5 −b4 −b3 −b2 b1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

This is the Hurwitz matrix23 whose properties: det H (φα) = 1, H T = H−1 which, accompaniedwith formulae (5), are very useful for our calculations in the next sections.

III. THE POTENTIAL VECTOR OF SO(8) MONOPOLE FOR A 9-DIMENSIONAL SPACE

Using the formulae (5), we can prove the following correlations:

−1

2i(� j

)st

(vt

∂

∂us+ us

∂

∂vt

)= r

(−i

∂

∂x j+ Ak(r) Qk j (φα)

), (6)

−1

2i

(us

∂

∂us− vs

∂

∂vs

)= −ir

∂

∂x9,

in which

Ak = xk

r (r + x9), k = 1, 2, . . . , 8. (7)

Operators Qk j , depending only on the extra variables (ϕφ), have the explicit form given in theAppendix. These operators are antisymmetric by the index ( jk) that is why there exist only 28different operators in total. They set up a closed SO(8) algebra by the following commutators:[

Q jk, Qlm] = iδ jm Qlk − iδkm Ql j + iδ jl Qkm − iδkl Q jm . (8)

Now we return to the formula (6), the operators in the right side of which without the factor (r ) arecomponents of an impulse operator in the 9-dimensional space:

π j = −i∂

∂x j+ Ak(r) Qk j (φα), (9)

π9 = −i∂

∂x9. (10)



Investigating the interaction term Ak(r) Qk j (φα) in (9), we see that there are seven operators ofSO(8) in total for every fixed index j . This suggests us to define a septet of potential vectors asfollows:

A1λ = 1

2(−A2, +A1, +A4, −A3, +A6, −A5, +A8, −A7, 0) ,

A2λ = 1

2(+A3, +A4, −A1, −A2, +A7, −A8, −A5, +A6, 0) ,

A3λ = 1

2(+A4, −A3, +A2, −A1, +A8, +A7, −A6, −A5, 0) ,

A4λ = 1

2(−A5, −A6, −A7, +A8, +A1, +A2, +A3, −A4, 0) ,

A5λ = 1

2(+A6, −A5, +A8, +A7, +A2, −A1, −A4, −A3, 0) ,

A6λ = 1

2(+A7, −A8, −A5, −A6, +A3, +A4, −A1, +A2, 0) ,

A7λ = 1

2(−A8, −A7, +A6, +A5, −A4, −A3, +A2, +A1, 0) .

(11)

Then, components of the impulse operators in a 9-dimensional space can be rewritten in theform:

π j = −i∂

∂x j+ 2 I ja(φα)Aaj (r). (12)

Here in (12) we put the sign ∼ on the index j to denote that the repeating index does not indicatethe summation over them; the potential vectors are defined in (11); operators I ja(φα) with indicesa = 1, 2, . . . , 7 and j = 1, 2, . . . , 8 are the same as Qk j with difference only in the sign ± asfollows:

I11 = −Q21, I12 = Q31, I13 = Q41, I14 = −Q51, I15 = Q61, I16 = Q71, I17 = −Q81,

I21 = Q12, I22 = Q42, I23 = −Q32, I24 = −Q62, I25 = −Q52, I26 = −Q82, I27 = −Q72,

I31 = Q43, I32 = −Q13, I33 = Q23, I34 = −Q73, I35 = Q83, I36 = −Q53, I37 = Q63,

I41 = −Q34, I42 = −Q24, I43 = −Q14, I44 = Q84, I45 = Q74, I46 = −Q64, I47 = Q54,

I51 = Q65, I52 = Q75, I53 = Q85, I54 = Q15, I55 = Q25, I56 = Q35, I57 = −Q45,

I61 = −Q56, I62 = −Q86, I63 = Q76, I64 = Q26, I65 = −Q16, I66 = Q46, I67 = −Q36,

I71 = Q87, I72 = −Q57, I73 = −Q67, I74 = Q37, I75 = −Q47, I76 = −Q17, I77 = Q27,

I81 = −Q78, I82 = Q68, I83 = −Q58, I84 = −Q48, I85 = −Q38, I86 = Q28, I87 = Q18.

Now let us consider some formulae related to the septet of potential vectors A ja(r). It is easy toverify that the septet of potential vectors (11) satisfies the following formulae:

xλ Aλk = 0 , Aλ j Aλk = δ jkr − x9

4r2(r + x9), (13)

that is completely analogous to the properties of the Dirac monopole potential vector in a3-dimensional space:

Aλ = 1

2r (r + x3)(−x2, x1, 0), xλ Aλ = 0 , Aλ Aλ = r − x3

4r2(r + x3), (14)



as well as the properties of a triplet of potential vectors of a SU(2) Yang monopole for 5-dimensionalspace:4

A1,λ = 1

2r (r − x5)(−x2 , x1 , x4 ,−x3, 0)

A2,λ = 1

2r (r − x5)( x3 , x4 ,−x1 ,−x2, 0), (15)

A3,λ = 1

2r (r − x5)(x4 ,−x3 , x2 ,−x1, 0),

xλ Aλk = 0 A jλ Akλ = δ jkr − x5

4r2(r + x5). (16)

Consequently, we have constructed a septet of potential vectors for a 9-dimensional space inthe SO(8) model that is a direct generalization of a Dirac monopole in a 3-dimensional space and ofa Yang monopole in the SU(2)model in a 5-dimensional space.

IV. CONNECTION BETWEEN A HARMONIC OSCILLATOR AND A 9-DIMENSIONALHYDROGENLIKE ATOM WITH THE PRESENCE OF A SO(8) MONOPOLE

Let us consider a harmonic oscillator in a 16-dimensional real space (uv), whose Schrodingerequation can be written as follows:

{−1

8

(∂2

∂us∂us+ ∂2

∂vs∂vs

)+ 1

2ω2(usus + vsvs)

} (u, v) = Z (u, v). (17)

Here the real positive numbers ω, Z are the angular frequency and energy, respectively. Now wewill transform this equation into a 9-dimensional space x1, x2, . . . , x9 by the generalized Hurwitztransformation (2), in which the seven extra variables are denoted by φ1, φ2, φ3, α1, α2, α3, α4. Theequation is obtained in the new form:

{1

2πλ πλ + 1

8r2I 2(φϕ) − Z

r

}ψ(r, φα) = E ψ(r, φα), (18)

where πλ (λ = 1, 2, . . . , 9) are operators of the impulse components in the forms (10) and (12);operator I 2(φϕ) = I ja I ja commutes with all the operators I ja ; E = − 1

2ω2.The physical system described by (18) is a 9-dimensional hydrogenlike atom in the potential

vectors of a SO(8) monopole, expressed by (11). Here, Z is the nuclear charge whereas E is thebound energy. Particularly, when the wavefunctions do not depend on the extra variables but onlyon nine partial variables x1, x2, . . . , x9 , Eq. (18) is the Schrodinger equation for a 9-dimensionalhydrogenlike atom. We see that the seven extra variables are used for describing internal properties ofa particle via 28 generators of SO(8) algebra. This problem is the so-called MICZ-Kepler problem12, 24

investigated mostly for the cases of 3-dimensional and 5-dimensional spaces. In this paper, we havealready given the MICZ-Kepler problem for a 9-dimensional space and will discuss the dynamicalsymmetry hereon.

The connection built between Eqs. (17) and (18) provides us a simple way to investigatethe symmetry of MICZ-Kepler problem in a 9-dimensional space. Indeed, the simplicity of theHamiltonian of a harmonic oscillator in Eq. (17) allows us to build the closed algebra including theHamilton operator and other operators of system dynamics. We have 66 antisymmetric operators



�ab (a, b = 1, 2, . . . , 11, 12) as follows:

� jk = 1

2i

[(�k�

Tj

)st

us∂

∂ut+ (

�Tk � j

)st vs

∂

∂vt

],

� j,9 = 1

2i(� j

)st

(us

∂

∂vt− vt

∂

∂us

), �10, j = 1

4ω

(� j

)st

(− ∂2

∂us∂vt+ 4ω2usvt

),

� j,11 = −1

2i(� j

)st

(us

∂

∂vt+ vt

∂

∂us

), �12, j = 1

4ω

(� j

)st

(∂2

∂us∂vt+ 4ω2usvt

),

�10,9 = 1

8ω

(− ∂2

∂us∂us+ ∂2

∂vs∂vs+ 4ω2usus − 4ω2vsvs

), �9,11 = −1

2i

(us

∂

∂us− vs

∂

∂vs

),

�12,9 = 1

8ω

(∂2

∂us∂us− ∂2

∂vs∂vs+ 4ω2usus − 4ω2vsvs

), �10,12 = −1

2i

(us

∂

∂us+ vs

∂

∂vs+ 8

),

�11,10 = 1

8ω

(∂2

∂us∂us+ ∂2

∂vs∂vs+ 4ω2usus + 4ω2vsvs

),

�11,12 = 1

8ω

(− ∂2

∂us∂us− ∂2

∂vs∂vs+ 4ω2usus + 4ω2vsvs

), (19)

which satisfy the commutators of SO(10, 2)algebra:

[�ab, �cd

] = i(gbc�da − gda�bc + gdb�ac − gac�db

), (20)

with the metric (gab) = diag(1, 1, 1, 1, 1, 1, 1, 1, 1, 1,−1,−1).Thus, we have built the SO(10, 2) dynamical symmetry of Eq. (17). We can easily transform 66

operators (19) into the 9-dimensional space via the generalized Hurwitz transformation (2) and, as aresult, obtain the SO(10, 2) group of dynamical symmetry of Eq. (18), the MICZ-Kepler problem in9-dimensional space. Many physical operators, in particular the Runge–Lenz vector, can be obtainedfrom (19). Detailed investigation will be discussed in our next work.

V. SUMMARY AND OUTLOOK

We have developed the generalized Hurwitz transformation with some explicit formulae suitablefor analytical calculations. Basing on this development, we have first established a direct connectionbetween a 16-dimensional harmonic oscillator and a 9-dimensional hydrogenlike atom in the fieldof a SO(8) monopole. The explicit form of a septet of potential vectors of the SO(8) monopolehas been first given in the present paper too. Considering the Dirac and Yang monopoles in thecontext of similar connections between a harmonic oscillator and a hydrogenlike atom in the fieldof the (Dirac or Yang) monopole, we found that the SO(8) monopole is a direct generalization ofthe Dirac monopole as well as of the Yang monopole. We also show SO(10, 2) as the dynamicalsymmetry group of the 9-dimensional MICZ-Kepler problem and will investigate in detail in ournext work.

ACKNOWLEDGMENTS

This work is financially supported by the Foundation for Science and Technology Project ofVietnam Ministry of Education and Training.



APPENDIX: PRESENTATION OF SO(8) ALGEBRA VIA EXPLICIT FORMS OF Qkj

Because the operators are antisymmetric, Qk j = −Q jk , we have the following 28 independentoperators:

Q12 = − i

2

(J 0

1 + J 02

), Q13 = i

2

(J+

1 + J+2

), Q14 = − i

2

(J−

1 − J−2

), Q15 = i

2

(Q+

1 + Q+2

),

Q16 = − i

2

(Q−

1 − Q−2

), Q17 = i

2

(K +

1 + K +2

), Q18 = i

2

(K −

1 − K −2

), Q23 = i

2

(J−

1 + J−2

),

Q24 = i

2

(J+

1 − J+2

), Q25 = i

2

(Q−

1 + Q−2

), Q26 = i

2

(Q+

1 − Q+2

), Q27 = i

2

(K −

1 + K −2

),

Q28 = − i

2

(K +

1 − K +2

), Q34 = i

2

(J 0

1 − J 02

), Q35 = i

2

(K +

3 − K +4

), Q36 = − i

2

(K −

3 − K −4

),

Q37 = i

2

(Q+

3 + Q+4

), Q38 = i

2

(Q−

3 + Q−4

), Q45 = i

2

(K −

3 + K −4

), Q46 = i

2

(K +

3 + K +4

),

Q47 = i

2

(Q−

3 − Q−4

), Q48 = − i

2

(Q+

3 − Q+4

), Q56 = i

2

(J 0

3 + J 04

), Q57 = − i

2

(J+

3 + J+4

),

Q58 = i

2

(J−

3 − J−4

), Q67 = i

2

(J−

3 + J−4

), Q68 = i

2

(J+

3 − J+4

), Q78 = i

2

(J 0

3 − J 04

),

where we use the following notations:

J 01 = ∂

∂α1+ ∂

∂α2, J 0

2 = ∂

∂α3+ ∂

∂α4, J 0

3 = ∂

∂α1− ∂

∂α2, J 0

4 = ∂

∂α3− ∂

∂α4,

J+1 = tan(φ2/2) sin(α1 + α2)

∂

∂α1− cot(φ2/2) sin(α1 + α2)

∂

∂α2+ 2 cos(α1 + α2)

∂

∂φ2,

J−1 = tan(φ2/2) cos(α1 + α2)

∂

∂α1− cot(φ2/2) cos(α1 + α2)

∂

∂α2− 2 sin(α1 + α2)

∂

∂φ2,

J+2 = tan(φ3/2) sin(α3 + α4)

∂

∂α3− cot(φ3/2) sin(α3 + α4)

∂

∂α4+ 2 cos(α3 + α4)

∂

∂φ3,

J−2 = tan(φ3/2) cos(α3 + α4)

∂

∂α3− cot(φ3/2) cos(α3 + α4)

∂

∂α4− 2 sin(α3 + α4)

∂

∂φ3,

J+3 = tan(φ2/2) sin(α1 − α2)

∂

∂α1+ cot(φ2/2) sin(α1 − α2)

∂

∂α2+ 2 cos(α1 − α2)

∂

∂φ2,

J−3 = tan(φ2/2) cos(α1 − α2)

∂

∂α1+ cot(φ2/2) cos(α1 − α2)

∂

∂α2− 2 sin(α1 − α2)

∂

∂φ2,

J+4 = tan(φ3/2) sin(α3 − α4)

∂

∂α3+ cot(φ3/2) sin(α3 − α4)

∂

∂α4+ 2 cos(α3 − α4)

∂

∂φ3,

J−4 = tan(φ3/2) cos(α3 − α4)

∂

∂α3+ cot(φ3/2) cos(α3 − α4)

∂

∂α4− 2 sin(α3 − α4)

∂

∂φ3,

Q+1 = − tan(φ1/2) cos(φ3/2)

cos(φ2/2)sin(α1 + α3)

∂

∂α1+ cot(φ1/2) cos(φ2/2)


∂

∂α3

−2 cos(φ2/2) cos(φ3/2) cos(α1 + α3)∂

∂φ1

−2 sin(φ2/2) cos(φ3/2) cos(α1 + α3) tan(φ1/2)∂

∂φ2

+2 cos(φ2/2) sin(φ3/2) cos(α1 + α3) cot(φ1/2)∂

∂φ3,



Q−1 = − tan(φ1/2) cos(φ3/2)

cos(φ2/2)cos(α1 + α3)

∂

∂α1+ cot(φ1/2) cos(φ2/2)


∂

∂α3

+2 cos(φ2/2) cos(φ3/2) sin(α1 + α3)∂

∂φ1

+2 sin(φ2/2) cos(φ3/2) sin(α1 + α3) tan(φ1/2)∂

∂φ2

−2 cos(φ2/2) sin(φ3/2) sin(α1 + α3) cot(φ1/2)∂

∂φ3,

Q+2 = tan(φ1/2) sin(φ3/2)

sin(φ2/2)sin(α2 + α4)

∂

∂α2− cot(φ1/2) sin(φ2/2)


∂

∂α4

+2 sin(φ2/2) sin(φ3/2) cos(α2 + α4)∂

∂φ1

−2 tan(φ1/2) cos(φ2/2) sin(φ3/2) cos(α2 + α4)∂

∂φ2

+2 cot(φ1/2) sin(φ2/2) cos(φ3/2) cos(α2 + α4)∂

∂φ3,

Q−2 = tan(φ1/2) sin(φ3/2)

sin(φ2/2)cos(α2 + α4)

∂

∂α2− cot(φ1/2) sin(φ2/2)


∂

∂α4

−2 sin(φ2/2) sin(φ3/2) sin(α2 + α4)∂

∂φ1

+2 tan(φ1/2) cos(φ2/2) sin(φ3/2) sin(α2 + α4)∂

∂φ2

−2 cot(φ1/2) sin(φ2/2) cos(φ3/2) sin(α2 + α4)∂

∂φ3,

Q+3 = − tan(φ1/2) cos(φ3/2)

cos(φ2/2)sin(α1 − α3)

∂

∂α1− cot(φ1/2) cos(φ2/2)


∂

∂α3

−2 cos(φ2/2) cos(φ3/2) cos(α1 − α3)∂

∂φ1

−2 tan(φ1/2) sin(φ2/2) cos(φ3/2) cos(α1 − α3)∂

∂φ2

+2 cot(φ1/2) cos(φ2/2) sin(φ3/2) cos(α1 − α3)∂

∂φ3,

Q−3 = tan(φ1/2) cos(φ3/2)

cos(φ2/2)cos(α1 − α3)

∂

∂α1+ cot(φ1/2) cos(φ2/2)


∂

∂α3

−2 cos(φ2/2) cos(φ3/2) sin(α1 − α3)∂

∂φ1

−2 tan(φ1/2) sin(φ2/2) cos(φ3/2) sin(α1 − α3)∂

∂φ2

+2 cot(φ1/2) cos(φ2/2) sin(φ3/2) sin(α1 − α3)∂

∂φ3,



Q+4 = tan(φ1/2) sin(φ3/2)

sin(φ2/2)sin(α2 − α4)

∂

∂α2+ cot(φ1/2) sin(φ2/2)


∂

∂α4

+2 sin(φ2/2) sin(φ3/2) cos(α2 − α4)∂

∂φ1

−2 tan(φ1/2) cos(φ2/2) sin(φ3/2) cos(α2 − α4)∂

∂φ2

+2 cot(φ1/2) sin(φ2/2) cos(φ3/2) cos(α2 − α4)∂

∂φ3,

Q−4 = tan(φ1/2) sin(φ3/2)

sin(φ2/2)cos(α2 − α4)

∂

∂α2+ cot(φ1/2) sin(φ2/2)


∂

∂α4

−2 sin(φ2/2) sin(φ3/2) sin(α2 − α4)∂

∂φ1

+2 tan(φ1/2) cos(φ2/2) sin(φ3/2) sin(α2 − α4)∂

∂φ2

−2 cot(φ1/2) sin(φ2/2) cos(φ3/2) sin(α2 − α4)∂

∂φ3,

K +1 = tan(φ1/2) cos(φ3/2)


∂

∂α2− cot(φ1/2) sin(φ2/2)


∂

∂α3

+2 sin(φ2/2) cos(φ3/2) cos(α2 + α3)∂

∂φ1

−2 cos(φ3/2) cos(φ2/2) cos(α2 + α3) tan(φ1/2)∂

∂φ2

−2 sin(φ3/2) sin(φ2/2) cos(α2 + α3) cot(φ1/2)∂

∂φ3,

K −1 = tan(φ1/2) cos(φ3/2)


∂

∂α2− cot(φ1/2) sin(φ2/2)


∂

∂α3

−2 sin(φ2/2) cos(φ3/2) sin(α3 + α2)∂

∂φ1

+2 cos(φ3/2) cos(φ2/2) sin(α2 + α3) tan(φ1/2)∂

∂φ2

+2 sin(φ3/2) sin(φ2/2) sin(α2 + α3) cot(φ1/2)∂

∂φ3,

K +2 = tan(φ1/2) sin(φ3/2)


∂

∂α1− cot(φ1/2) cos(φ2/2)


∂

∂α4

+2 cos(φ2/2) sin(φ3/2) cos(α1 + α4)∂

∂φ1

+2 tan(φ1/2) sin(φ2/2) sin(φ3/2) cos(α1 + α4)∂

∂φ2

+2 cot(φ1/2) cos(φ2/2) cos(φ3/2) cos(α1 + α4)∂

∂φ3,



K −2 = tan(φ1/2) sin(φ3/2)


∂

∂α1− cot(φ1/2) cos(φ2/2)


∂

∂α4

−2 cos(φ2/2) sin(φ3/2) sin(α1 + α4)∂

∂φ1

−2 tan(φ1/2) sin(φ2/2) sin(φ3/2) sin(α1 + α4)∂

∂φ2

−2 cot(φ1/2) cos(φ2/2) cos(φ3/2) sin(α1 + α4)∂

∂φ3,

K +3 = tan(φ1/2) cos(φ3/2)

sin(φ2/2)sin(−α2 + α3)

∂

∂α2+ cot(φ1/2) sin(φ2/2)

cos(φ3/2)sin(−α2 + α3)

∂

∂α3

−2 sin(φ2/2) cos(φ3/2) cos(α2 − α3)∂

∂φ1

+2 tan(φ1/2) cos(φ2/2) cos(φ3/2) cos(α2 − α3)∂

∂φ2

+2 cot(φ1/2) sin(φ2/2) sin(φ3/2) cos(α2 − α3)∂

∂φ3,

K −3 = tan(φ1/2) cos(φ3/2)


∂

∂α2+ cot(φ1/2) sin(φ2/2)


∂

∂α3

−2 sin(φ2/2) cos(φ3/2) sin(α2 − α3)∂

∂φ1

+2 tan(φ1/2) cos(φ2/2) cos(φ3/2) sin(α2 − α3)∂

∂φ2

+2 cot(φ1/2) sin(φ2/2) sin(φ3/2) sin(α2 − α3)∂

∂φ3,

K +4 = tan(φ1/2) sin(φ3/2)


∂

∂α1+ cot(φ1/2) cos(φ2/2)


∂

∂α4

+2 cos(φ2/2) sin(φ3/2) cos(α1 − α4)∂

∂φ1

+2 tan(φ1/2) sin(φ2/2) sin(φ3/2) cos(α1 − α4)∂

∂φ2

+2 cot(φ1/2) cos(φ2/2) cos(φ3/2) cos(α1 − α4)∂

∂φ3,

K −4 = tan(φ1/2) sin(φ3/2)


∂

∂α1+ cot(φ1/2) cos(φ2/2)


∂

∂α4

−2 cos(φ2/2) sin(φ3/2) sin(α1 − α4)∂

∂φ1

−2 tan(φ1/2) sin(φ2/2) sin(φ3/2) sin(α1 − α4)∂

∂φ2

−2 cot(φ1/2) cos(φ2/2) cos(φ3/2) sin(α1 − α4)∂

∂φ3.

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A non-Abelian SO(8) monopole as generalization of Dirac-Yang monopoles for a 9-dimensional space

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