The matrices representation of division algebras, new properties, the Cartan-Weyl basis of SO(n) and the 2 n -dimensional harmonic oscillator basis Mehdi Hage-Hassan
To cite this version:
Mehdi Hage-Hassan. The matrices representation of division algebras, new properties, the Cartan- Weyl basis of SO(n) and the 2 n -dimensional harmonic oscillator basis. 2015. hal-01222005
HAL Id: hal-01222005 https://hal.archives-ouvertes.fr/hal-01222005 Preprint submitted on 8 Nov 2015
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés.
The matrices representation of division algebras, new properties, the Cartan-Weyl basis of SO(n) and the 2n-dimensional harmonic oscillator basis
M. Hage-Hassan* Université Libanaise, Faculté des Sciences Section (1) Hadath-Beyrouth
Abstract
We derive by a simple recurrence method the matrices representation of division algebras or Hurwitz matrices (Hn, n = 2,4,8,16). We introduce a generalization of the polar angles and coordinates and we show the connection between Hn and the polypolar matrices transformations. We find new properties of Hn matrices. We also find that the matrices Hn are generating functions of Cartan-Weyl basis of SO (n). We define the poly- cylindrical coordinates and we find the 2n-dimensional harmonic oscillator basis for n≥1.
1. Introduction
The real division algebras are: real numbers, complex numbers, quaternion, and Octonion. These algebras and the orthogonal groups SO(n) are very important in mathematics and physics [1-10]. This real division algebra is originated of the famous problem solving "the sums of squares" [10-11]. So we try to determine the values of n so that there is a solution of the identity:
2 2 2 2 2 2 2 2 2 x1 x2 ... xn x1 x2 ... xn z1 z2 ... zn (1.1)
With zi are bilinear forms xi, yi, i, j = 1... n. This problem was solved by Euler and Fermat for n = 2, by Hamilton for n = 4 and by Grave and Cayley for n = 8. Hurwitz shown that the equation (1.1) has solutions only for n = 1, 2, 4, and 8. And every solution corresponds to an algebra which is isomorphic to anti-symmetric matrices Hn (u1... un), representing the division algebras [11-12]. In this work we determine Hn (u) by a simple recurrence method. Then, by developing ii ii z j re , z j re ,( j 1k),k 1,2,4. and by introducing a new parameterization of{ j } polypolar angles, we find that the expansion of the polypolar coordinates are expressed in terms of matrices Mn with Mn = Hn (1... 1). By calculating the generating function of the cylindrical oscillator basis [13-14], we have observed a property of H2 (u) and after a simple calculation we found that this new property may be generalized to Hn (u) n = 4, 8, 16. The matrices representing the infinitesimal generators [15-20] of SO (n) are the anti- symmetric matrices which require the study of the connection between these matrices
1
and Hn (u). We find that Hn are generating matrices of Cartan-Weyl basis [15] of SO(2) ... SO(6) . The theory of N-dimensional harmonic oscillator is very useful in the study of Molecular vibration [9] and quantum physics [21-23]. It is natural to extend the polypolar coordinates, for that we generalize the cylindrical basis of the harmonic oscillator [18-20] by introducing a new parameterization of cylindrical coordinates [14, 23]. And thus we determine the eigenstates of the harmonic oscillator in N-dimensional, N= 2n with n> 1. Finally we give a simple application of division algebras.
In this paper we expose the derivation of the representation matrices of division algebras in section two. In section three we introduce the polypolars coordinates and its relation with the representation matrix of division algebra. The new properties of Hurwitz matrices or the representation matrices of division algebras are given in section four. In section five we give the representation matrix of division algebras and the Cartan-Weyl basis of . The generalized cylindrical coordinates and the N- dimensional radial function of the oscillators will be given in section six. In section seven we expose the derivation the wave function of the N-dimensional harmonic oscillator. In the appendix we give the polycylindric coordinates for n = 4 and the new simple application of division algebras.
2. The matrices representation of division algebras
We determine the matrices representation of division algebras [11], using a recurrence method and taking as a starting point trigonometry formulas.
The matrix representations of the real is H1 (u) u1I .
2.1 The matrix representation of complex algebra We begin with the well-known trigonometric formulas:
2 2 cos cos (/ 2) sin (/ 2) sin 2cos(/ 2)sin(/ 2) (2.1)
Multiplying these expressions by r: 2 2 (2.2) x1 u1 u2 , x2 2u1u2
With u1 r cos( / 2), u2 r sin( / 2) 2 2 2 2 2 2 2 2 And x1 x2 (u1 u2 ) or r (u ) (2.3)
We write (2.1) in matrix form: x u u u 1 1 2 2 t (H 2 (u)) (U 2 ) (2.4) x2 u2 u1 u1
H 2 (u1,u2 ) is an orthogonal matrix and is anti-symmetric with:
2 t 2 2 H 2 (u) u1I Ju 2 , and J 1, H 2 (u)H 2 (u) (u1 u2 )I (2.5)
2
t And H 2 (u) is the transpose of H 2 (u) .
2.2 The matrix representation of Quaternion The generalization of the transformation (2.2) and (2.3) is obtained by setting:
x u u u 1 2 1 2 3 (2.7) x u u u 2 2 1 4 2 2 2 2 2 2 2 We add x3 (u1 u2 ) (u3 u4 ) for the relationship r (u ) .
We find the Octonion quadratic transformation, [7-8], R4 R3
x u u u u u 1 3 1 2 3 4 x2 u4 u2 u1 u4 u3 t H (u) , H (u) (2.8) x 4 u 4 u u u u 3 1 3 4 1 2 0 u2 u4 u3 u2 u1
In quaternion’s notations we write: e1 I,e2 J, e3 K, e4 L . 2 2 2 Q= H 4 (u) u1e1 u2e2 u3e3 u4e4 , With J K L 1 (2.9)
2.3 The matrix representation of Octonion We obtain the orthogonal matrix H8(u) by generalization of the transformation (2.6). We find the Octonion quadratic transformation R8 R5 : x u 1 8 x2 u7 t H (u) , x 4 u 3 6 x u 4 5 We add x (u 2 u 2 u 2 u 2 ) (u 2 u 2 u 2 u 2 ) so that r 2 (u 2 )2 , we find: 5 1 2 3 4 5 6 7 8
u u u u u u u u 1 2 3 4 5 6 7 8 u2 u1 u4 u3 u6 u5 u8 u7 u u u u u u u u 3 4 1 2 7 8 5 6 u u u u u u u u H (u) 4 3 2 1 8 7 6 5 (2.10) 8 u u u u u u u u 5 6 7 8 1 2 3 4
u6 u5 u8 u7 u2 u1 u4 u3 u7 u8 u5 u6 u3 u4 u1 u2 u u u u u u u u 8 7 6 5 4 3 2 1
3
8 with H (u) u I u e , e e 1, 8 1 i2 i i i i
8 8 2 And t H (u) u I u e , t H (u) H (u) u (2.11) 8 1 i2 i i 8 8 i1 i a- From the orthogonality Hn, n = 1, 2, 4 it is simple to prove the identity (1.1) b- We also derive the non-orthogonal matrix H16 by the same method of recurrence.
2.4 The Hurwitz theorem: There are only anti-symmetric matrices and orthogonal Hn only if:
n=1, 2, 4,8 or “R, C, H=Q and O” (2.12)
3. The polypolar coordinates and the matrix representation of division algebra
We will define first the polypolar coordinates and then we determine the connection between these coordinates and the Hurwitz matrices of division algebras.
3.1 The polypolar coordinates
i j We consider the n-couples of complex numbers z j r e , z j , ( j 1k),k 1,2,4
and {i } are functions of angular variables ,, ,...etc.
For n=2, 1 , 2 ( ) , ( ) , For n=3 1 2 3 ( ) , 4 ( )
We will show that the developments z j , z j , ( j 1k),k 1,2,4 are expressed simply with the help of Hn (1).
3.2 The polar coordinates and Hurwitz matrix of the complex variable We consider the polar transformation z r ei r(cos sin) x iy 1 1 1 (3.1) i z1 r e r(cos sin) x1 iy1
This polar transformation is written in the matrix form by: z x 1 1 1 1 M 2 , With M 2 (1,i) H 2 (1) (3.2) z1 iy1 1 1
3.3 The polar coordinates and Hurwitz matrix of the quaternion We consider the case of two complex variables z1 and z2 with:
i( ) z1 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 , i( ) z2 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 ,
4
i( ) z1 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 , i( ) z2 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 (3.3)
We write the polar transformation by: z x 1 1 1 1 1 1 z2 iy1 1 1 1 1 M , M (1,i,i,1) H (1) (3.4) z 4 ix 4 4 1 1 1 1 1 2 z2 y2 1 1 1 1
3.4 The polar coordinates of the Hurwitz matrix of Octonion and Sedenion 3.4.1 The polar coordinates of the Hurwitz matrix of the Octonion
We consider the case of four complex variables (z1,..., z4 , z1,..., z4 ) whose angular variables are functions of three variables, , with:
1 ( ) , 2 ( ) , 3 ( ) , 4 ( ) (3.5)
The transformation in this case becomes:
i1 z1 re r(cos i sin)(cos isin )(cos isin ) x1 iy1 ... iy 4 (3.6)
i4 z4 re r(cos i sin)(cos isin )(cos isin ) x1 iy1 ... iy 4 a-We get a matrix of the form 1 i i 1 i 1 1 i . b-We arrange the columns: the first column is followed by real-4 columns imaginary and 3-columns real 1 i i i i 1 1 1 M 8 H8 (1) and so we get the expression that generalizes (3.4).
3.4.2 The polar coordinates of the Hurwitz matrix of the Sedenion We consider the case of four variables:
1 (( ) ) , 2 (( ) ) , ,8 (( ) ) We follow the method above and we get the orthogonal matrix M16.
4. The new properties of Hurwitz matrices
In performing the calculation of the generating function of the cylindrical oscillator basis [8-13] we observe the existence of interesting properties of matrices H2 (u) which is simply generalized to Hn (u).
4.1 The cylindrical basis of the harmonic oscillator and the complex variables We know that the Cartesian basis of harmonic oscillator [21] in Dirac notations is:
5
(a ) n un (q) q n , And n 0 (4.1) n! 2 2 with a (x ip), a (x ip), and p id / dx 2 2
The generating function of the oscillator basis is well known: n 1 2 2 z q z G(z,q) u (q) q e za 0 4 exp{ 2qz } (4.2) n0 n n! 2 2 and the cylindrical basis is: A jm A jm jm j m, j m 1 2 0,0 . (4.3) ( j m)!( j m)! 2 2 with A (a ia ), A (a ia ) (4.4) 1 2 x y 2 2 x y In term of Cartesian coordinates r xi yj we derive the generating function [13]:
( jm) ( jm) u u F(u,r) 1 2 x, y jm x, y e(u1 u2 )ax i(u1 u2 )ay 00 jm ( j m)!( j m)! (4.5) 2 2 2 2 1 x y 2 2 H1 H 2 exp H1 x iH 2 y 2 2 with 2 2 (4.6) u1 (x iy) u2 (x iy) H1 x iH 2 y and 2 2 2 2 2 2 H1 H 2 4u1u2 ,, H1 H 2 2u1 u2 (4.7)
The properties of H2 (u) 2 2 2 2 H1 (u1 u2 ) , H 2 (u1 u2 )
4.2 The new properties of Quaternion matrix th n We denote the sum of elements of the i column of Hn by H i . n H n (H ) (4.8) i n i, j j1 After elementary calculation we find that:
(H 4 )2 (H 4 )2 (H 4 )2 (H 4 )2 8(u u u u ) (4.9) 1 2 3 4 1 3 2 4 4 4 2 4 2 4 2 4 2 2 (H1 ) (H 2 ) (H 3 ) (H 4 ) 4ui (4.10) i1 We deduce that 4 2 4 2 2 2 (H1 ) (H 4 ) 2((u1 u3 ) (u2 u4 ) ), (4.11)
6
4 2 4 2 2 2 (H 2 ) (H3 ) 2((u1 u3 ) (u2 u4 ) )
4.3 The new properties of Octonion and Sedenion matrices It is easy to verify the following relationships:
4.3.1 The new properties of Octonion matrix n4 8 2 8 2 8 2 8 2 8 2 8 H1 H 2 H 5 H 6 H 7 H8 16 uiui4 (4.12) i1 8 8 2 8 2 8 2 8 2 8 2 8 2 8 2 8 2 (4.13) H1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 8(ui ) i1 We deduce that 8 (H 8 ) 2 (H 8 )2 4((u u )2 (u u ) 2 (u u ) 2 (u u ) 2 ), 1 i 1 5 2 6 3 7 4 8 i6 (4.14) 5 8 2 2 2 2 2 (H i ) 4((u1 u5 ) (u2 u6 ) (u3 u7 ) (u4 u8 ) ), i2
4.3.2 The new properties of Sedenion matrix n8 16 2 8 2 16 2 16 2 16 H1 H 2 H 9 H10 .... H16 32 uiui8 (4.15) i1 16 2 2 16 H 16 H 8 .... H 16 16 u 2 (4.16) 1 1 2 16 1 i
The expressions (4.14) may be generalized to H16(u) .
t 4.3.3 The properties of Hn and Mn A- Exchanging columns by lines we find the same result with a change of sign of the first term of the expressions (4.10), (4.12) and (4.15). B- Let: n n u1 un M n M1 M n n n Replace H1 by M1 to find the results of equations (4.7-16).
5. Representation matrix of division algebra and Cartan-Weyl basis of SO(n)
The generators of SO (n), the group of rotations in n-dimensions [15-21], may be taken as
Lij L ji ;i j;i, j 1,,5. Lij what generates rotations in the ij plane.
The number of generators is therefore Ng = n (n+1)/2. The generators of rotation are p 1 i ˆ Lij (xi x j ) xi p j x j pi x1 xn Lij (5.1) x x j i pn ˆ with Lij ij ji (5.2)
7
In the following we put a hat to the adjoin representations.
V5.1 The generating matrices of Cartan-Weyl basis of SO(3) For SO(2) we have 0 u p u S x y 2 x 2 3 u 0 p 2 y (5.3) and u u H u I u Sˆ 1 2 2 1 2 3 u u 2 1 (5.4) The usual generators of SO(3) in term of angular momentum L r pˆ are :
S Lˆ , S Lˆ , S Lˆ 1 23 2 31 3 12 (5.5) For n= 3 we must add S1 and S2 to this case (5.3). u u u 1 2 3 H 3 u1I (u2 S3 u3S2 u4 S1 ) u2 u1 u4 (5.6) u3 u4 u1
5.2 The generating matrices of Cartan-Weyl basis of SO(4) For n= 4 and Ng=6 , we obtain two orthogonal matrices:
ˆ ˆ ˆ ˆ ˆ ˆ S1 L23 L14 / 2, S2 L31 L24 / 2, S3 L12 L34 / 2 ˆ ˆ ˆ ˆ ˆ ˆ T1 L23 L14 / 2, T2 L31 L24 / 2, T3 L12L34 / 2 (5.7)
u u u u 1 2 3 4 u2 u1 u4 u3 H 1 u I 2(u S u S u S ) (5.8) 4 1 2 3 3 2 4 1 u u u u 3 4 1 2 u4 u3 u2 u1 And u u u u 1 2 3 4 u2 u1 u4 u3 H 2 u I 2(u T u T u T ) (5.9) 4 1 2 3 3 2 4 1 u u u u 3 4 1 2 u4 u3 u2 u1
5.3 The generating matrices of Cartan-Weyl basis of SO(5) The generators SO(5), Ng=10, are:
ˆ ˆ ˆ ˆ ˆ ˆ S1 L23 L14 / 2, S2 L31 L24 / 2, S3 L12 L34 / 2 ˆ ˆ ˆ ˆ ˆ ˆ T1 L23 L14 / 2, T2 L31 L24 / 2, T3 L12L34 / 2
8
ˆ ˆ ˆ ˆ U1 L15, U 2 L25, V1 L35 , V2 L45, (5.10)
S and T are two commuting spins which generate SO(4) transformations. The generating matrices are:
1 H5 u1I 2(u2 S3 u3S2 u4 S1 u5U1 u6U 2 u7V1 u8V2 ) u u u u u 1 2 3 4 5 u2 u1 u4 u3 u6 u u u u u (5.11) 3 4 1 2 7 u4 u3 u2 u1 u8 u5 u6 u7 u8 u1 And 2 H5 u1I 2(u2T3 u3T2 u4T1 u5U1 u6U 2 u7V1 u8V2 ) u u u u u 1 2 3 4 5 u2 u1 u4 u3 u6 u u u u u (5.12) 3 4 1 2 7 u4 u3 u2 u1 u8 u5 u6 u7 u8 u1
5.3 The generating matrices of Cartan-Weyl basis of SO(6) We The generators of SO(6) [24],Ng=15, are:
ˆ ˆ ˆ ˆ ˆ ˆ S1 L23 L14 / 2, S2 L31 L24 / 2, S3 L12 L34 / 2 ˆ ˆ ˆ ˆ ˆ ˆ T1 L23 L14 / 2, T2 L31 L24 / 2, T3 L12L34 / 2 1 1 ˆ ˆ ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ U1 L15 L26 , U 2 L25 L16 , U 3 L26 L15 , U 4 L25 L16 , 2 2 2 2 1 ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ V1 L35 L46 , V2 L45 L36 , V3 L46 L35 , V4 L45 L36 (5.13) 2 2 2 2 In addition we also have L 56 The generating matrices of Cartan-Weyl basis are: 1 ˆ H8 u1I 2(u2 (S3 L56) u3S2 u4S1 u5U1 u6U2 u7V1 u8V2 ) u u u u u u 1 2 3 4 5 6 u2 u1 u4 u3 u6 u5 u u u u u u 3 4 1 2 7 8 (5.14) u4 u3 u2 u1 u8 u7 u u u u u u 5 6 7 8 1 2 u6 u5 u8 u7 u2 u1
9
And 2 ˆ H8 u1I 2(u2 (T3 L56) u3T2 u4T1 u5U3 u6U4 u7V3 u8V4 ) u u u u u u 1 2 3 4 5 6 u2 u1 u4 u3 u6 u5 u u u u u u 3 4 1 2 7 8 (5.15) u4 u3 u2 u1 u8 u7 u u u u u u 5 6 7 8 1 2 u6 u5 u8 u7 u2 u1
6. The generalized cylindrical coordinates and the N-dimensional harmonic oscillators
We introduce the polycylindrical coordinates to the resolution of the N-dimensions, N = 2n, harmonic oscillator which are natural extensions of the cylindrical [14, 23] and polypolar coordinates (3.1). We expose also the method of solving the Schrödinger equation and we give the radial function of N-dimensional Harmonic oscillator.
6.1 The Polycylindrical coordinates
i j We consider the complex numbers z j r e , z j , j 1,..,n. with i are functions of the angular variables ,i 1,...,2n . i i A- For n=1 r, , z1 re , 0,2
B- For n=2 1 1 2 , 2 1 2
i1 i2 z1 r cos e , z2 r sin e , 0, / 2 Following the notations of Euler’s angles [25] we write:
1 , 2 . C- For n=3
r ( ) ( ), ( ) ( ), 1 1 2 3 4 2 1 2 3 4 cosθ1 sinθ1 3 (1 2 ) (3 4 ), 4 (1 2 ) (3 4 ) i1 i3 z1 r cos1 cos 2 e , z3 r sin1 cos3 e
z r cos sin ei2 , z r sin sin ei3 2 1 2 4 1 3 (6.1) Tree of coordinates N=2n
2 2 It is easy to verify that: r zi (6.2) i N / 2 2 The invariant metric in this case is: ds 2 dz (6.3) N i1 i n-1 We therefore conclude that the number of angles {i }is 2 , the number {i }
10
is 2n-1-1, and it follows that the total number of variables is 2n.
D- We emphasize that the transformation {1,i i2} {i }is a matrix MN / 2. 6.2 The Schrödinger equation in N-dimensional harmonic oscillator The time independent Schrödinger equation for a particle of mass M in N-dimensional space has the form [17, 21- 22] 2 1 ( M 2r 2 ) E (6.4) 2M N 2
Where N is the Laplacian operator: N 2 / x2 (6.5) N i1 i We write the wave function by:
(r,) R(r)Y(), With (1 ,..., N ,1 ,..., N (6.6) 1 2 2 R(r) And Y() are the radial and angular parts of the wave functions. To determine the eigenfunctions of the oscillator it is important to firstly determine the expression of Laplacian in terms of the polycylindrical coordinates and then we calculate R(r) and Y() by the separating of variables method.
6.2.1 Expression of Laplacian The invariant metric on the manifold is: N 2 i j dsN gij dx dx (6.7) i, j And the invariant Laplace-Beltrami operator is
1 ij ij 1 N g g , g det(gij ), g (g )ij (6.8) g xi x j We derive 1 N 1 S N r , (6.9) r N 1 r r 2r 2 ν S is the Laplace-Beltrami operator on ν-sphere S . Louck [16] define the total angular- momentum by L2 and find that is given by: k1 i1 2 2 L S Lij , k N 1 (6.10) i2 j1
The eigevalues of the Laplace operator on a ν-sphere Sν are –ℓ (ℓ + ν -1).
SY() l(l 1)Y() 0, (6.11)
6.2.2 Expression of radial Harmonic oscillator The wave function of radial part by:
11
2 d N 1 d ( N 2) 2 2 2 2 2 r R(r) 0 (6.12) dr r dr r
R (r) was determined from long time by several authors and by different methods [22, 23] and therefore we will give only the result:
r 2 N 2 p! ( ) R (r) 2 e 2 r 2 L( 1) (r 2 ) (6.13) N , p ( p) p N N With , p , E (2p N / 2) (6.14) 2 2
7. The cylindrical basis of the N-dimensional harmonic oscillator
We simply determine the proper functions of N-dimensional harmonic oscillator for N = 2, 4, 8, with the help of a formula of Vilenkin [20].
7.1 The cylindrical basis of harmonic oscillator The invariant metric is 2 2 2 ds2 z d (7.1) 2 and 2 Y() Y() m2Y() (7.2) S 2 eim The solution is: Y() (7.3) 2 The solution of the cylindrical basis is: im pm( ,) R2, p (r)e (7.4)
7.2 The basis of 4-dimensions of harmonic oscillator The invariant metric is given by 2 2 2 2 2 2 2 2 ds3 (12 ) dz1 dz2 d cos d1 sin d2 (7.5) 1 g sin cos sin(2 ) (7.6) 2 1 The Laplacian is then written: 2 2 S 1 1 1 3 (12 ) sin cos 2 2 2 2 (7.7) sin cos cos 1 sin 2 To determine the solution of this equation we use the formula [26]:
1 r(r p 1) s(s q 1) cos p sin q Y l(l 1) Y 0 p q 2 2 cos sin cos sin and p q 1 with l a positive integer number.
12
Put Y tg s cosl v, tg 2 x We find the solution: s l r s l r p 1 q 1 u tg s cos l F , ;s ;tg 2 (7.8) 2 2 2 This expression is written using polynomial Jacobi [24]: n ( , ) (n 1) 1 z z 1 Pn (z) F(n,n ; 1; ) (7.9) n!( 1) 2 z 1 We find s r (, ) u sin cos Pn (cos(2)), n (s l r)/ 2 (7.10)
The solution is the matrix elements of finite rotation or D-Wigner function [25]: ( j m')!( j m')! d ( j) (2) (cos)m'm (sin)m'm P(m'm,m'm) (cos(2)) m'm ( j m)!( j m)! jm' We finally find: q1 p1 u (sin) (cos) d j (2) /(sin) 2 (cos) 2 (7.11) (m,m') q 1 p 1 s , r 2 2 1 p q j l 1, m , m' (7.12) 2 2 2 2 From expression (7.6) we deduce that p = q = 1, hence the solution is: [27]:
Y j (, ,) d j (2)ei(m11m22 ) m1m2 (m1m2 ) / 2,(m1m2 ) / 2
The wave function of 4-dimensional oscillator in terms of the cylindrical coordinates is:
(r,, ,) R (r)Y j (, ,) (7.13) p, jm1m2 4, pj m1m2
7.3 The basis of 8-dimensions of harmonic oscillator The invariant metric in this case is
ds 2 d 2 cos 2 (d 2 cos 2 d 2 sin 2 d 2 ) 7 1 1 2 2 1 2 2 (7.14) sin 2 (d 2 cos 2 d 2 sin 2 d 2 ) 1 3 3 3 3 4 1 g sin 3 (2 )sin(2 )sin(2 ) (7.15) 25 1 2 3
The Laplacian is then written
1 1 1 S sin 3 (2 ) S ( ) S ( ) 7 3 1 2 3 2 1 2 2 3 3 3 4 sin (21 ) 1 1 cos 1 sin 1
13
We have
r l1, s l2 , j1 l1 / 2, j2 l2 / 2 2 l l l l p q 3 7, s 1, r 1, j , m' 1 2 1, m 1 2 2 2 2 We get the angular function of (6.5):
( j) 4 d(m;m) (21 ) i( mii ) Y ( j, j1 , j2 ) (,) d j2 (2 )d j1 (2 ),e i1 (m,m,mi ) (m3 m4 ) / 2,(m3 m4 ) / 2 2 (m1 m2 ) / 2,(m1 m2 ) / 2 3 sin(1 )cos(1 )
The wave function of 8-dimensional oscillator in terms of the cylindrical coordinates is:
R (r)Y ( j, j1, j2 ) (,) (7.16) p, j, j1, j2 8, pj (m,m,mi )
8. Appendix
We give the expressions of polycylindric coordinates for n = 4 and we present a new application of division algebras to "change the ideas".
8.1 The Polycylindrical coordinates for n=4
i1 i5 z1 r cos1 cos2 cos4e , z5 r sin1 cos3 cos6e
i2 i6 z2 r cos1 cos2 sin4e , z6 r sin1 cos3 sin6e (8.1)
i3 i7 z3 r cos1 sin2 cos5e , z7 r sin1 sin3 cos7e
i4 i8 z4 r cos1 sin2 sin5e , z8 r sin1 sin3 sin7e With:
1 (1 2 ) (3 4 ) ((5 6 ) (7 8 )), 2 (1 2 ) (3 4 ) ((5 6 ) (7 8 )) …………………………………………………………………………………………. ( ) ( ) (( ) ( )), ( ) ( ) (( ) ( )) 7 1 2 3 4 5 6 7 8 8 1 2 3 4 5 6 7 8
The invariant metric in this case is
8 2 2 2 2 2 ds dzi dr r ds15, (8.2) i1 With: 2 2 2 2 2 2 2 2 ds15 d1 cos 1 d2 cos 2ds4 (412 ) sin 2ds4 (534 ) 2 2 2 2 2 2 sin 1 d3 cos 3ds4 (656 ) sin 3ds4 (778 ) (8.3)
The calculation of the Laplacian is simple and it is quite obvious that the angular j functions of the wave functions are products of functionsD(m,m') .
14
8.2 The new simple application of division algebra Ten years of work after retirement, (2004), [28], I wanted to be "illiterate". The deception pushed me in the street one morning to relax and I found a lot of women who go to the market. This made me think of the movie "God created woman” and: 1- Why god is masculine in all languages! "Break the left- right symmetry" 2- The definition of God in the Quran: "Who has never begotten and has not been begotten, and that no one is able to match!" a- So God is not composed of matter and is Pure Imaginary "As time“ b- The imaginary in division algebras are: Complex, Quaternions and Octonion. This is reflected in the Christian religion by: the father, the son and saint- spirit (3 in1).
9. References [1] S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge, 1995. [2] G. M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Dordrecht, Netherlands: Kluwer, 1994. [3] J.C. Baez, Division Algebras and quantum theory, arXiv:1101.5690[quant-ph [4] J.C. Baez, “The Octonion”, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 145 [5] T. Levi-Civita opera Mathematiche (Bolognia), Vol.2 (1956) [6] P.Kutaanheimo and E. Steifel; J. Reine Angew. Math. 218, 204 (1965) [7] M. Hage-Hassan, “Inertia tensor and cross product in n-dimensions space” ArXiv: math-ph/0604051(2006) [8] M. Hage-Hassan “Generating function method and its applications to Quantum, Nuclear and the Classical Groups”, arXiv: 1203.2892 (10/03/2012) [9] W.H. Shaffer, Rev. Mod. Phys.16, 245 (1944) [10] K. Conrad, The Hurwitz theorem on sums of squares, Internet. [11] Y. Tian , matrix Representations of Octonion and their applications, ArXiv: math.RA/0003166 (2000) [12] M. Hage-Hassan, “Hurwitz’s matrices, Cayley transformation and the Cartan-Weyl basis for the orthogonal groups”, arXiv: math-ph/0610021 (2006) [13] M. Hage-Hassan, Fock Bargmann space and Feynman propagator of charged Harmonic oscillator in a constant magnetic field, HAL : hal-00512123 (2010) [14] E. Karimi and al. , Radial quantum number of Laguerre-Gauss modes, ArXiv: 1401.4985 [math.phys.] 2014 [15] B.G. Wybourne “Classical Groups for Physicists” Wiley& Sons (1974)” [16] J.D. Louck, J. Mol.Spectrosc. 4,334 (1960) [17] K.D. Granzow,N-dimensional orbital angular momentum operator, J. Math. Phys. 4,7(1963) 897 [18] C. K. Chew and R. T. Sharp, Can. J. Phys., 44 (1966) 2789 [19] R. Sharp and S. Pieper; “O(5) Polynomials Bases”, J. Math. Phys. 9,5(1968) 663 [20] N. J. Vilenkin, “Fonctions spéciales et théorie de la représentation des groupes” Dunod (1969) [21] A. Messiah, Mécanique Quantique Tomes I et II E. Dunod, Paris (1965) [22] T. Das and A. Arda, arXiv:1308.5295 [math.phys.] 2014 [23] C. Grosche and F. Steiner, Handbook of Feynman path Integrals, Springer (1998). [24] R. Piepenbring and al., J.Physique 48 (1987) 577
15
[25] A. R. Edmonds, Angular Momentum in Quantum Mechanics Princeton, U.P., Princeton, N.J., (1957) [26] I.S. Gradshteyn and I.M. Ryzhik,”table of integrals series and products” Academic Press New-York (1965) [27] M.A.B. Beg and H. Ruegg, J. Math. Phys. 6, 5(1965) 677 [28] M. Hage-Hassan, A note on the normalization of the momentum eigenfunctions and Dirac delta function, arXiv math-ph/ : 1404.3926 (12/04/2014)
------*Email: [email protected]
16