Prog. Theor. Phys. 1953 Murai 147 686D Ver

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147Progress of Theoretical Physics, Vol. 9, No.2, February 1953

On the roup of Transformations in Six dimensional SpaceYasuhisa MURAl

Saz tama Unz versity(Received December 7, 1952)

The possible irreducible representations of Lie algebra of the transformation group which leavesthe fundamental quadratic form,

(x1)2+ x2)2+ x ~ 2 - ( ,0)2+ .x5) 2_ (",6)2,invariant, are investigated. Three invariants which commute with all the elements of the algebra aregiven and every irreducible representation is characterised by the values of these invariants.

1. IntroductionSeveral years ago Snyder ) has considered the quantization of space-time on the basis

of the infinitesimal transformation group in five-dimensional space and given one directionto the theory of non-local field. A little later Yang l has proposed the similar operatorsrelated to the transformation group in six-dimensional ~ p a t e In this case the operatorsconstitute a Lie algebra, while Snyder's were not. Then Tokuoka3) has examined the equations given y Snyder ftom the view point of reciprocity, and also taken the infinitesimaltransformations in six-dimensional space, of which the. fundamental quadratic form wasdifferent from Yang's in the sign of one term. This type of operators was, then, takenup by SegaI4) , in his work on A class of operator algebras which are determined bygroups Here he emphasizes the following four points: 1) An analogue to the squareof rest-mass is automatically obtained. 2) The analogue to the angular momentum automatically includ.esthe spin angular mOmentum. 3) The space associated with the particleis discrete. 4) The Lie algebra of the group is not a limiting case of any non-isomorphicLie algebra. We shaH addalittlea:bout tpe last statement, as the others are shown inthe course of constructing representations. The fact that one Lie algebra is the limitingcase of another is visualized by th following example, which he gave in his paper. We

Iitake X1=z'p, X 2:::;;;z q and Xs=il ([p, q]=iI) . then the Lie algebra generated by Xihas the commu.tation xelations [Xl X 2]=nXS [X l X:i]=[X2 X : ~ J = o Now the Liealgebra of the groLJPof .transformations which leave invariant kt12+t22-ta2 has as bases(Y Y2, Va), among which hold the commutation reiations[YJ , Y2]=IiY3 [Y Y1]=likY2,[Y2 Y11=n-1kY;. These Lie algebras are not . isomorphic and when k--o the latter tendsto the former. In this sense the Lie algebra under consideration is not a limiting case ofany nop-isomorphic Lie algebra. (About the proof, see the last paragraph of his paper.)

147Progress of Theoretical Physics, Vol. 9, No.2, February 1953

On the roup of Transformations in Six dimensional SpaceYasuhisa MURAl

Saz tama Unz versity(Received December 7, 1952)

The possible irreducible representations of Lie algebra of the transformation group which leavesthe fundamental quadratic form,

(x1)2+ x2)2+ x ~ 2 - ( ,0)2+ .x5) 2_ (",6)2,invariant, are investigated. Three invariants which commute with all the elements of the algebra aregiven and every irreducible representation is characterised by the values of these invariants.

1. IntroductionSeveral years ago Snyder ) has considered the quantization of space-time on the basis

of the infinitesimal transformation group in five-dimensional space and given one directionto the theory of non-local field. A little later Yang l has proposed the similar operatorsrelated to the transformation group in six-dimensional ~ p a t e In this case the operatorsconstitute a Lie algebra, while Snyder's were not. Then Tokuoka3) has examined the equations given y Snyder ftom the view point of reciprocity, and also taken the infinitesimaltransformations in six-dimensional space, of which the. fundamental quadratic form wasdifferent from Yang's in the sign of one term. This type of operators was, then, takenup by SegaI4) , in his work on A class of operator algebras which are determined bygroups Here he emphasizes the following four points: 1) An analogue to the squareof rest-mass is automatically obtained. 2) The analogue to the angular momentum automatically includ.esthe spin angular mOmentum. 3) The space associated with the particleis discrete. 4) The Lie algebra of the group is not a limiting case of any non-isomorphicLie algebra. We shaH addalittlea:bout tpe last statement, as the others are shown inthe course of constructing representations. The fact that one Lie algebra is the limitingcase of another is visualized by th following example, which he gave in his paper. We

Iitake X1=z'p, X 2:::;;;z q and Xs=il ([p, q]=iI) . then the Lie algebra generated by Xihas the commu.tation xelations [Xl X 2]=nXS [X l X:i]=[X2 X : ~ J = o Now the Liealgebra of the groLJPof .transformations which leave invariant kt12+t22-ta2 has as bases(Y Y2, Va), among which hold the commutation reiations[YJ , Y2]=IiY3 [Y Y1]=likY2,[Y2 Y11=n-1kY;. These Lie algebras are not . isomorphic and when k--o the latter tendsto the former. In this sense the Lie algebra under consideration is not a limiting case ofany nop-isomorphic Lie algebra. (About the proof, see the last paragraph of his paper.)

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148 Y.MuraiTherefore, if we can construct the theory of elementary particle on the basis of this Liealgebra, we may expect this to be the most fundamental theory, provided that the relativisticwave equations are intimately related to some group such as the inhomogeneous Lorentzgroup.

On the other hand Schouten l has called attention of physicists to the conformalgeometry and thereby stated at the outset of his argument that the Maxwell equations areinvariant not only under the Lorentz transformation but also under the conformal transformation. Now, the. group under consideration, if the value of the fundamental quadraticform is kept equal to zero, has the conformal group in Minkowski space as a coveringgroup6l. Therefore, they have the same Lie algebra. The conformal group in Minkowskispace contains as a special case the inhomogeneous Lorentz transformation, of which Wigner7lclssified the irreducible reprsentations and then Wigner and Bargmann8l have applied themto the theory of relativistic wave equations.

Our intension is to form a theory of elementary particle on the basis of the conformalgroup in Minkowski space or the orthogonal group in six-dimensional pseudo-Euclidean space.Of course, the wave equations will appear in non-localizable form. Here, for the timebeing, we publish the result of classification of the irreducible representations of the abovementioned Lie a gebra, since it may have some merit by itself. And in the next paper weshall interprete the result in a rather simple way and obtain a mass spectrum tentatively.

2. tructure of the Lie algebra9lWe take as the fundamental quadratic form,

2.1)The Lie algebra of the group of transformations which leave (2. 1) invariant is of order15. Let denote an element of the Lie algebra

where a denotes parameters contained in the group element, i.e., iny =zeij(a)zj.

Corresponding to each X we shall introduce the differential operator

Then,

As X F) =0 we have

We define L l by

2.2)

2.3)

2.4)

2.5)

2.6)

148 Y.MuraiTherefore, if we can construct the theory of elementary particle on the basis of this Liealgebra, we may expect this to be the most fundamental theory, provided that the relativisticwave equations are intimately related to some group such as the inhomogeneous Lorentzgroup.

On the other hand Schouten l has called attention of physicists to the conformalgeometry and thereby stated at the outset of his argument that the Maxwell equations areinvariant not only under the Lorentz transformation but also under the conformal transformation. Now, the. group under consideration, if the value of the fundamental quadraticform is kept equal to zero, has the conformal group in Minkowski space as a coveringgroup6l. Therefore, they have the same Lie algebra. The conformal group in Minkowskispace contains as a special case the inhomogeneous Lorentz transformation, of which Wigner7lclssified the irreducible reprsentations and then Wigner and Bargmann8l have applied themto the theory of relativistic wave equations.

Our intension is to form a theory of elementary particle on the basis of the conformalgroup in Minkowski space or the orthogonal group in six-dimensional pseudo-Euclidean space.Of course, the wave equations will appear in non-localizable form. Here, for the timebeing, we publish the result of classification of the irreducible representations of the abovementioned Lie a gebra, since it may have some merit by itself. And in the next paper weshall interprete the result in a rather simple way and obtain a mass spectrum tentatively.

2. tructure of the Lie algebra9lWe take as the fundamental quadratic form,

2.1)The Lie algebra of the group of transformations which leave (2. 1) invariant is of order15. Let denote an element of the Lie algebra

where a denotes parameters contained in the group element, i.e., iny =zeij(a)zj.

Corresponding to each X we shall introduce the differential operator

Then,

As X F) =0 we have

We define L l by

2.2)

2.3)

2.4)

2.5)

2.6)



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On the Group of Transformations in Siz-dimensiollal Space 1 4 ~Fifteen Xkl which correspond to Lkl> can be taken as the bases of .The commutation relations for Lkl are, 2.7)

Next we shall consider the related spinor group. Following Cartan s procedure wefirst represent six-dimensional vector (Zi) by the matrixX.

X=(r iz2 Z3+Z0 Z5+ 6 0

OS); S=( Z3_ Z0 _zl+iz2 0 Z5+ Z6 ).S 0 Z5_ Z6 0 _zl+iz2 - zS-zoo Z5_ Z6 _Z3+ Z0 zl+iz22.8)

The following relations hold,XX= {(Zl)2+ (Z2?+ (zS)2_ (z)2+ (Z5) _ (x6nE,Det. S = {(Zl)2+(Z2)2+ (Z3) _ (ZO)2+ (Z5) _ (Z6)2}. 2.9)

The rotation in the six-dimensional space can be expressed by the equation,(2 10)

or S =wsw*-\ (2,11)where W is a certain unimodular matrix of 15 parameters. We shall decompose W inthe following way:

W=R/J?06IR10R06R4' (2.12)Here R4, Res and R10 are rotations in (zl, r Z3, z5)-space, ZO_%6 plane and Zl_ Z0 planerespectively. The desired forms of matrices can be easily obtained if we consider a rotation,for example ROB as a result of double reflections by two planes, whose normals lie in theZO': Z6. plane. The result is

(a 1 r a ) ( o s ~ S i n ~ COSh r; sinh J )-.(1* a* a -r* c o s ~ s i n ~ sinhr; cosh7J

R4= , R06= , R10=-r*-a* a* 1* s i n ~ c o s ~ cosh7j sinh7j-a - 1 a s i n ~ c o s ~ sinh7J cosh7j(aa*+(1(1*+rr*+ aa*= 1 213)

Fur.ther the matrix R4 can be decomposed as follows:

R4=( & ~ , , ,8)( ~ i ~-sm{} cos{} c o s ~

-sin{} cos{} i s i n ~

On the Group of Transformations in Siz-dimensiollal Space 1 4 ~Fifteen Xkl which correspond to Lkl> can be taken as the bases of .The commutation relations for Lkl are, 2.7)

Next we shall consider the related spinor group. Following Cartan s procedure wefirst represent six-dimensional vector (Zi) by the matrixX.

X=(r iz2 Z3+Z0 Z5+ 6 0

OS); S=( Z3_ Z0 _zl+iz2 0 Z5+ Z6 ).S 0 Z5_ Z6 0 _zl+iz2 - zS-zoo Z5_ Z6 _Z3+ Z0 zl+iz22.8)

The following relations hold,XX= {(Zl)2+ (Z2?+ (zS)2_ (z)2+ (Z5) _ (x6nE,Det. S = {(Zl)2+(Z2)2+ (Z3) _ (ZO)2+ (Z5) _ (Z6)2}. 2.9)

The rotation in the six-dimensional space can be expressed by the equation,(2 10)

or S =wsw*-\ (2,11)where W is a certain unimodular matrix of 15 parameters. We shall decompose W inthe following way:

W=R/J?06IR10R06R4' (2.12)Here R4, Res and R10 are rotations in (zl, r Z3, z5)-space, ZO_%6 plane and Zl_ Z0 planerespectively. The desired forms of matrices can be easily obtained if we consider a rotation,for example ROB as a result of double reflections by two planes, whose normals lie in theZO': Z6. plane. The result is

(a 1 r a ) ( o s ~ S i n ~ COSh r; sinh J )-.(1* a* a -r* c o s ~ s i n ~ sinhr; cosh7J

R4= , R06= , R10=-r*-a* a* 1* s i n ~ c o s ~ cosh7j sinh7j-a - 1 a s i n ~ c o s ~ sinh7J cosh7j(aa*+(1(1*+rr*+ aa*= 1 213)

Fur.ther the matrix R4 can be decomposed as follows:

R4=( & ~ , , ,8)( ~ i ~-sm{} cos{} c o s ~

-sin{} cos{} i s i n ~



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150 Y Murai

COS cosSin S i n ) i : : ~ ~ i ~ ~ COS X cosX SinxSinx\)s in cos cosa i s ina -sinX cosX . 2.14)sin cos i s ina cosa s inx OS X

The group composed of above Wand our original rotation group are locally isomorph, andso t h ~ y have the same Lie algebra. If we apply an infinitesimal transformation X to bothforms of transformations

y=w(a)x and H= WE VV- 1*,we have

Xy=LxY and XW=MxW,where Mx IS a matrix to be determined:

XH=(XW)EW-l*+ WE(XW-1*).To compute XW-1*, we use the decomposition 2.12), i.e.,

W,-U1 R U2 ,then from 2.13),

Therefore,

2.15)

XW- 1*=X(U/R- I 0; ) = {X (U1R I U2) },= (U1R-1U2 I ~ U;:l*R 1U11* ~=W - 1

where X UtR- t U2) = M ~ ( U I R U 2 ) And we getX H - M x H + H M ~ ,

of which Mx can be obtained and found to be1Mx 2

_xot_ i x02 _xSt + iX 23 XIS + iX 25 + xst+ i x62_ ~ _ ~ ~ 2 _ ~ ~ ~ ~_ x t 5 _XBO _ i i5+ XB3 X 3 _XS5+ X12

X15+ x 25 _ xG1 _ iX 62 _X l_ x O ~ + X3l _ iX 23

2.16)

217)Corresponding to Xk we have id and with the aid _of these matrices we can write asfollows cf. 2.13), 2.14) and 2.17:

150 Y Murai

COS cosSin S i n ) i : : ~ ~ i ~ ~ COS X cosX SinxSinx\)s in cos cosa i s ina -sinX cosX . 2.14)sin cos i s ina cosa s inx OS X

The group composed of above Wand our original rotation group are locally isomorph, andso t h ~ y have the same Lie algebra. If we apply an infinitesimal transformation X to bothforms of transformations

y=w(a)x and H= WE VV- 1*,we have

Xy=LxY and XW=MxW,where Mx IS a matrix to be determined:

XH=(XW)EW-l*+ WE(XW-1*).To compute XW-1*, we use the decomposition 2.12), i.e.,

W,-U1 R U2 ,then from 2.13),

Therefore,

2.15)

XW- 1*=X(U/R- I 0; ) = {X (U1R I U2) },= (U1R-1U2 I ~ U;:l*R 1U11* ~=W - 1

where X UtR- t U2) = M ~ ( U I R U 2 ) And we getX H - M x H + H M ~ ,

of which Mx can be obtained and found to be1Mx 2

_xot_ i x02 _xSt + iX 23 XIS + iX 25 + xst+ i x62_ ~ _ ~ ~ 2 _ ~ ~ ~ ~_ x t 5 _XBO _ i i5+ XB3 X 3 _XS5+ X12

X15+ x 25 _ xG1 _ iX 62 _X l_ x O ~ + X3l _ iX 23

2.16)

217)Corresponding to Xk we have id and with the aid _of these matrices we can write asfollows cf. 2.13), 2.14) and 2.17:



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On the Group of I'ransfOrmtltions in Six-dimensional Space 151and the corresponding elements of the original algebra are the same forms M.A. being reoplaced by XA'One parameter subgroup is written as a (t) =exp (tX) and its representative matrixUe=U(a(t is according to Stone s theorem exp (-itHx). with a selfadjoint operatorHx' As is easily seen the commutation relations among Hare

(2.19)if [XA' XR]=C%AXC' Therefore the structure constants are determined from [L.A.' L R ] = C A ~Lc (2.7). W r i t i n g ~ . Li .M,A. Xo.Po' N(i=1,2,3) for Efts. Hjk' HOi. Efto. Has. Hoo.Hoo. the commutation relations are as follows:

[Xi' L3J=iXk [X Lk ]= iXj. [Xi. L,]=O, [Xi. Po]=O. [Xi. N]= -ipi>[Pi' Lj]=iPk, [Pt L,.]=-ih, [Pi' L,]=O, [p.,poJ=iM, [p N] iXi,[M. Ljl=iM. [M. Lk]= -iMj. lM L,]=O. [M. Po] = -ip [M N]=O,[Xi. Xol=-iM [LuPoJ=[Li' N]=[Lj,Xo]=O, (220)[p Xo]=O, [ ~ Xj]=[Lj, L3]=iL .[M o l = - i ~ [pi,pj]=[M, Mj]=-iLk ,

[X Pj]=iJijN,[Xo Pol -iN,

[p,. lid,;F iJ.,tPo,[Po, N] iXo,

[Xi' Mj]=iJ.,jXo,[Xo. NJ= -ipo

This algebra is of rank 3. We take Ls. Xs and Po as the elements of Cartan subalgebraand in accordance with notation used by Weyl we form the following bases besides theabove ones ehis is because the following bases are convenient in constructing representations as we shall see later).

E(o) =_i_(Xo-iMa+P:i+iN)1 4 v2 E( o)=_i-eXo+iMa+Ps-iN),4 v2E( O)=-Z-(Xo-iMa-Ps-iN), E( o)=_i_(Xo+iMs-Ps+iN).1 4 v -i 4 .v2

Further we put

(221)

(2.22)

On the Group of I'ransfOrmtltions in Six-dimensional Space 151and the corresponding elements of the original algebra are the same forms M.A. being reoplaced by XA'One parameter subgroup is written as a (t) =exp (tX) and its representative matrixUe=U(a(t is according to Stone s theorem exp (-itHx). with a selfadjoint operatorHx' As is easily seen the commutation relations among Hare

(2.19)if [XA' XR]=C%AXC' Therefore the structure constants are determined from [L.A.' L R ] = C A ~Lc (2.7). W r i t i n g ~ . Li .M,A. Xo.Po' N(i=1,2,3) for Efts. Hjk' HOi. Efto. Has. Hoo.Hoo. the commutation relations are as follows:

[Xi' L3J=iXk [X Lk ]= iXj. [Xi. L,]=O, [Xi. Po]=O. [Xi. N]= -ipi>[Pi' Lj]=iPk, [Pt L,.]=-ih, [Pi' L,]=O, [p.,poJ=iM, [p N] iXi,[M. Ljl=iM. [M. Lk]= -iMj. lM L,]=O. [M. Po] = -ip [M N]=O,[Xi. Xol=-iM [LuPoJ=[Li' N]=[Lj,Xo]=O, (220)[p Xo]=O, [ ~ Xj]=[Lj, L3]=iL .[M o l = - i ~ [pi,pj]=[M, Mj]=-iLk ,

[X Pj]=iJijN,[Xo Pol -iN,

[p,. lid,;F iJ.,tPo,[Po, N] iXo,

[Xi' Mj]=iJ.,jXo,[Xo. NJ= -ipo

This algebra is of rank 3. We take Ls. Xs and Po as the elements of Cartan subalgebraand in accordance with notation used by Weyl we form the following bases besides theabove ones ehis is because the following bases are convenient in constructing representations as we shall see later).

E(o) =_i_(Xo-iMa+P:i+iN)1 4 v2 E( o)=_i-eXo+iMa+Ps-iN),4 v2E( O)=-Z-(Xo-iMa-Ps-iN), E( o)=_i_(Xo+iMs-Ps+iN).1 4 v -i 4 .v2

Further we put

(221)

(2.22)



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151 Y. MuraiThen the commutation relations become

Here the subscript /. is a root and means : ~ ) . The last commutation relations are, ex-a3

pliddy written,[ E ; ) , E - ~ ) J = _ i E( [ E l ) . E - ~ ) l = - i E ( ~ ) ,o -1 2.v2 -1 0 1 2.v2 1

[E(I).E( O ) J = ~ E 1),[E(1.),E( O ) l = ~ E l ) ,:1 2.v2 l 2 v'2[E( 1) E(O)J=_iE(I),[E( o),E( o)J=_iE( 1 ,, 1 2 v'2 -l \-l 2 v'2

[E( l ) , E - I ) J = ~ E O),[E r ) , E - I ) l = ~ E 0),2 v'2 - ~ - ~ . J 2 v'2 =i[ E ~ ) , E - ~ ) ] = _ i E(-i) E ~ ) , E 1)1=---=i-E(l)1 1 2.v2 1 1 2 v ' 2 , 0

[E(I) ,E( O)J=_iE(I),[E( 1),E( O ) J = ~ E 1).2 v'2 , _ ~ 2 v'2(2.24)

N G ~ = N _ _ 3 and N G , ~ is zero if a+f3 is not a root.This algebra has three invariants, of which one is the so-called Casimir operator:

(2.25)and the others are

R= Lp)Xo- ( L . j 1 f ) N - X ; ( L t P o + ~ P , , - M , . P i ) ' (2 26)W= L.X)2_ L.p)2_ L . .JI)2+ ( d o + M d , , - M , . h ) 2

M.N POXi - Pi XO2- LiN p;X,,-p .X; 2- (LiXo 1I1.;X,,- 1/1;.X;) 2. (2.27) 3. General forms of the rltpresentative matrices

In the following argument, the first procedure is almost the same as what Thomas10)has done in his On the Unitary Representations of the Group of De Sitter Space .So we shall state briefly.

Two subalgebras composed of H;. +H 2 E i) ,E =l) and lIt - H 2, E _D ECD respectively are commutative with each other. Among them hold the very familiar commutation relations, for example,

151 Y. MuraiThen the commutation relations become

Here the subscript /. is a root and means : ~ ) . The last commutation relations are, ex-a3

pliddy written,[ E ; ) , E - ~ ) J = _ i E( [ E l ) . E - ~ ) l = - i E ( ~ ) ,o -1 2.v2 -1 0 1 2.v2 1

[E(I).E( O ) J = ~ E 1),[E(1.),E( O ) l = ~ E l ) ,:1 2.v2 l 2 v'2[E( 1) E(O)J=_iE(I),[E( o),E( o)J=_iE( 1 ,, 1 2 v'2 -l \-l 2 v'2

[E( l ) , E - I ) J = ~ E O),[E r ) , E - I ) l = ~ E 0),2 v'2 - ~ - ~ . J 2 v'2 =i[ E ~ ) , E - ~ ) ] = _ i E(-i) E ~ ) , E 1)1=---=i-E(l)1 1 2.v2 1 1 2 v ' 2 , 0

[E(I) ,E( O)J=_iE(I),[E( 1),E( O ) J = ~ E 1).2 v'2 , _ ~ 2 v'2(2.24)

N G ~ = N _ _ 3 and N G , ~ is zero if a+f3 is not a root.This algebra has three invariants, of which one is the so-called Casimir operator:

(2.25)and the others are

R= Lp)Xo- ( L . j 1 f ) N - X ; ( L t P o + ~ P , , - M , . P i ) ' (2 26)W= L.X)2_ L.p)2_ L . .JI)2+ ( d o + M d , , - M , . h ) 2

M.N POXi - Pi XO2- LiN p;X,,-p .X; 2- (LiXo 1I1.;X,,- 1/1;.X;) 2. (2.27) 3. General forms of the rltpresentative matrices

In the following argument, the first procedure is almost the same as what Thomas10)has done in his On the Unitary Representations of the Group of De Sitter Space .So we shall state briefly.

Two subalgebras composed of H;. +H 2 E i) ,E =l) and lIt - H 2, E _D ECD respectively are commutative with each other. Among them hold the very familiar commutation relations, for example,



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On the Croup o Transformations in Six-dimmsiollal Space 153

Therefore the following representation matrices are obtained

m1 IE })lm 1- 1 = ~ 2 v j;- ?n1) (jl-m1+-i f,.0

(m2IE( lm2-1 = 1/ V ( j 2 + m 2 ) ( j 2 ~ m 2 + 1 ) .1 2 V 2o

J ~ = O , ~ . 1.... m = J ~ . J ~ - I , - J ~ I . - k2Then from the commutation relations

[E l), EG)l=O, i[E(I) ,E( O)J=-E(l)_1 2v2 0o 1 1[E(=i) E G ) ] = 2 ~ 2 E ( - D

we get the following relations:

where A and B are certain matrices. In this way using all the commutation relationsexcept such ones as [H3E,,]=* and [E . ~ , J = 1l3+0, [1'1+0), we find the similarrelations s above, and combining these results we have finally:

On the Croup o Transformations in Six-dimmsiollal Space 153

Therefore the following representation matrices are obtained

m1 IE })lm 1- 1 = ~ 2 v j;- ?n1) (jl-m1+-i f,.0

(m2IE( lm2-1 = 1/ V ( j 2 + m 2 ) ( j 2 ~ m 2 + 1 ) .1 2 V 2o

J ~ = O , ~ . 1.... m = J ~ . J ~ - I , - J ~ I . - k2Then from the commutation relations

[E l), EG)l=O, i[E(I) ,E( O)J=-E(l)_1 2v2 0o 1 1[E(=i) E G ) ] = 2 ~ 2 E ( - D

we get the following relations:

where A and B are certain matrices. In this way using all the commutation relationsexcept such ones as [H3E,,]=* and [E . ~ , J = 1l3+0, [1'1+0), we find the similarrelations s above, and combining these results we have finally:



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154

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Y. Murai

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I I )= v jl+m1) j ~ - m 2 ) f4 j;,j2) 1

IE I 1 . 1 1 1 )(-1 j l+- , }2+- , m1+ - , m2+o 2 2 2 21= i v t j l+m1+ 1) j2+m2+ 1)fl jl,/2)

I I 1 . 1}J + - , j 2 - - 12 2 )

)

)= v j l - m1) j2 -m 2).ft A, j2) 1

IE I 1 1 1 1 )(I l + - , f 2+ - m1- - m2- o 2 2 2 21=iv C];-m l + 1) j2-m2+ ) / l j ~ , j ~ ) .

I I 1 . 1jl+- , j 2 - -2 2 )

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)

. )

I I )= v jl+m1) j ~ - m 2 ) f4 j;,j2) 1

IE I 1 . 1 1 1 )(-1 j l+- , }2+- , m1+ - , m2+o 2 2 2 21= i v t j l+m1+ 1) j2+m2+ 1)fl jl,/2)

I I 1 . 1}J + - , j 2 - - 12 2 )

)

)= v j l - m1) j2 -m 2).ft A, j2) 1

IE I 1 1 1 1 )(I l + - , f 2+ - m1- - m2- o 2 2 2 21=iv C];-m l + 1) j2-m2+ ) / l j ~ , j ~ ) .

I I 1 . 1jl+- , j 2 - -2 2 )



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On the Group of Transfomations in Six-dimensional Space 55( I I 1 , 1) 1 - - , 1 2 - - ,2 2 )

= -iv j; +m]) ( J ~ + m 2 ) f . t ( j ] . j ; ) .( I I 1 . 1 1 1 )'( 0 j1+- .}2 ,+- , m1 -.m21 2 2 2 21

= - v. jl+ml+ 1) j2-m2+ 1) l ( j ~ . j 2 )( I I 1 . 1 )1+- , j2 - -2

= v +ml + 1) ( j ~ +m2) 12 j ~ , j2) ,( I I 1 . 1 )- - , j 2+ -2

= V (J l -m]) j2- m2+ 1 ) f ~ ( j 1 , j 2 )( I I i 1 . 1 )- - , j 2 - -2

= - v j ~ - m ; ) (j2+m2)f.t(jl.j2),( I I 1 . 1 1 1 )( 0 h+ - , j 2+ - m] - - , m2 +-] 2 2 2 2

-1

= v j ~ - m 1 + 1) j2-m2+ 1) g4(j]Jj2) 3 2)( I I .+ 1 . 1 )l -,j2--2

= v j ; -ml+1) j ~ - m 2 ) g 3 ( j ~ . j 2 )( I I 1 . + 1 )1- - , j2 .2

= 11'(;; +m]) j2+m2+ 1)g2(j1J}2),( I 1.1 1 )l--,j2--2

= v j; +m]) ( j ~ - ~ ) g 1 jp j2) ,( I I 1 . 1 1 1 ). ( 1) h+- . } 2+- . ml-- m22 2 2 2, - 1

=i v -m l+ 1) j 2 - m2+ 1) g4 (j;, j2)( I I 1 . 1 )1+- , j 2 - -2

= - i V U,-ml+ 1) (j2+ 1n2)gs (j1,j2) ,

On the Group of Transfomations in Six-dimensional Space 55( I I 1 , 1) 1 - - , 1 2 - - ,2 2 )

= -iv j; +m]) ( J ~ + m 2 ) f . t ( j ] . j ; ) .( I I 1 . 1 1 1 )'( 0 j1+- .}2 ,+- , m1 -.m21 2 2 2 21

= - v. jl+ml+ 1) j2-m2+ 1) l ( j ~ . j 2 )( I I 1 . 1 )1+- , j2 - -2

= v +ml + 1) ( j ~ +m2) 12 j ~ , j2) ,( I I 1 . 1 )- - , j 2+ -2

= V (J l -m]) j2- m2+ 1 ) f ~ ( j 1 , j 2 )( I I i 1 . 1 )- - , j 2 - -2

= - v j ~ - m ; ) (j2+m2)f.t(jl.j2),( I I 1 . 1 1 1 )( 0 h+ - , j 2+ - m] - - , m2 +-] 2 2 2 2

-1

= v j ~ - m 1 + 1) j2-m2+ 1) g4(j]Jj2) 3 2)( I I .+ 1 . 1 )l -,j2--2

= v j ; -ml+1) j ~ - m 2 ) g 3 ( j ~ . j 2 )( I I 1 . + 1 )1- - , j2 .2

= 11'(;; +m]) j2+m2+ 1)g2(j1J}2),( I 1.1 1 )l--,j2--2

= v j; +m]) ( j ~ - ~ ) g 1 jp j2) ,( I I 1 . 1 1 1 ). ( 1) h+- . } 2+- . ml-- m22 2 2 2, - 1

=i v -m l+ 1) j 2 - m2+ 1) g4 (j;, j2)( I I 1 . 1 )1+- , j 2 - -2

= - i V U,-ml+ 1) (j2+ 1n2)gs (j1,j2) ,



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156 Y. Murai

( I . 1 . 1j l - - , j 2+ -2 2 )=iV Ci +m1) U2-m2+ 1 )g2(jl j2)

( )=- i V (j l +m1) j ~ + m 2 ) g l ( j ; , j 2 )

IE I 1 . 1 1 + 1 )- ) j l+ - , }2+- , m1+ - , m2 -o 2 2 2 2-1

= -iv +m1+ 1) j ~ + m 2 + 1)g4(Jl>i2)'( 1 1. 111+- ,12- - ,2 2 )

= i V J ~ +m1+ ) ( . f 2 - m 2 ) g 3 ( j ~ . j 2 )( I 1 . 1j l - - , j 9+ - ,2 - 2 )

= i v {Jl-m 1) (J2+ m2 1)g2(Jl,j2),( I 1 . 1jl--,h--,2 2 )

=i v j ~ - m l ) (j2-m2)gl(; 1.j2)( I I 1 . 1 1 1 )) j l+ - , j 2+- m1+- , m2

1 2 2 2 2I

= - v(;;+ml + 1) (J2-1n2+ 1) 4 ( j ~ . j 2 ) 'I 1 . 1 ).I+- , j 2 - -2

= V j1+ml+ 1) (J2+m2)g3(j;, j2)'I 1 . 1 )j l - - , j 2+ -2

= V (j1-ml) (J2- m2 1 g 2 ( j ; j ~ ) ,( I 1 . 1jl--,j2--2 2 )

where ft, gi are certain matrices.Next w shall take up Hs Being commutative with ~ . and 2 , it may be diago

nalized. We put

156 Y. Murai

( I . 1 . 1j l - - , j 2+ -2 2 )=iV Ci +m1) U2-m2+ 1 )g2(jl j2)

( )=- i V (j l +m1) j ~ + m 2 ) g l ( j ; , j 2 )

IE I 1 . 1 1 + 1 )- ) j l+ - , }2+- , m1+ - , m2 -o 2 2 2 2-1

= -iv +m1+ 1) j ~ + m 2 + 1)g4(Jl>i2)'( 1 1. 111+- ,12- - ,2 2 )

= i V J ~ +m1+ ) ( . f 2 - m 2 ) g 3 ( j ~ . j 2 )( I 1 . 1j l - - , j 9+ - ,2 - 2 )

= i v {Jl-m 1) (J2+ m2 1)g2(Jl,j2),( I 1 . 1jl--,h--,2 2 )

=i v j ~ - m l ) (j2-m2)gl(; 1.j2)( I I 1 . 1 1 1 )) j l+ - , j 2+- m1+- , m2

1 2 2 2 2I

= - v(;;+ml + 1) (J2-1n2+ 1) 4 ( j ~ . j 2 ) 'I 1 . 1 ).I+- , j 2 - -2

= V j1+ml+ 1) (J2+m2)g3(j;, j2)'I 1 . 1 )j l - - , j 2+ -2

= V (j1-ml) (J2- m2 1 g 2 ( j ; j ~ ) ,( I 1 . 1jl--,j2--2 2 )

where ft, gi are certain matrices.Next w shall take up Hs Being commutative with ~ . and 2 , it may be diago

nalized. We put



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On the Group of Transformations i1l Six-dimmsiollal Space 157(33)

then the non-vanishing elements are ( A I . t . ( J ~ . J ~ ) I A - 1 ) and ( J . l g l j l o j ~ ) I A + 1 ) , whichwe shall write as f,(};, 72 A) and g;(};.j2' A) hereafter.

Of the remaining commutation relations, we have to consider only

[E(I) E( O)J=_iE( 1)' [E( 0) E( o ) J = ~ ( H 2 - l f ~ )o I 2.v2 - 1 I 81 I 0 I 1and [E G) E c n J = [ E c ~ ) ' E(J)J=O,

the others being deduced from them. These commutation relations give the following relations among I. and f i :};(.t. ga+glh) = +1)(f2g1+ga.t.), 14 g ~ g d 2 = - f 2 g 1 + g 3 ~

} ~ U 4 g 2 + g d o ~ ) = j ~ + 1) (/Qgl+g2.t.), fit g2 g l /a= -fagl+g2./4,}; (fsg4 +g2.iJ) = J; +1) u g2+g4/Q)' j ; g4-g2 t= --j;g2+gJS'j ~ ( ; ; g 4 + g a . f r ) = (; 2+ 1) (fl gS+gd2)' fdf4-gst;= .iT ga+gJ2'

r; g4=g4h. j;ga=gs/2' fsg2= dq / tgl=gd4'j J d l= (jl+ l)t;/q, (Jl+ l)gS I=jl gl %a,

} ~ A j ; = ( } ~ + 1 ) f d 4 'j dd : = U2+ 1 )fl,h.jd4is= ) ~ + 1)/s14,

( } ~ + 1 ) g 4 g2=Ag2g4'(/2+ 1)g2g1=j2g1 g2'(; 2+ 1)g4 % 3 = j ~ g S % 4 '

{m 1(J2+ 1) +m2(jl +1)}j; t+ {m2(j;+ 1) - m l } ~ } f d ; '

(3.4)

+ { m l ( j ~ + 1) mdl} fd2 {md2+m2};}fij;= 0,{ml (; 2+ 1) +m 2(jl + 1 }g4 gl + {m2(};+ 1) m l j ~ } gS%2

+ {m 1(J2+ 1) -mdl}g2ga- {m1 / 2+mdl}gl 4=0.(jl +m1+1) ( j2- m2+ 1)j; %1- (1; +m1+ 1) (j2+ m2)f2g2- ( r-m I ) ( j ~ - m 2 + l).Ags

+ U l m1) ( J ~ + m 2 ) t g 4- CiI + m1) ( J ~ - m 2 k J l + (j; + m1) h +m2+ 1)gd2+ (; 1- m1+ 1) j ~ - m 2 ) % 3 / q

- (j ' t-m I + 1) (;'2+ m2+ 1 ) g J 4 = ~ ( m l - m 2 - A ) / .8In these relations Ii or gi which stands at the head of a term, has always the arguments(};.j2' A). and then those of the following matrix are automatically determined, if weconsider (3.2): for example, figs means .ftUl,j2, A ) g s C J ; - ~ , j 2 - ~ ' A-I) .2 2

By (31), (3.2), (3.3) and (3.4), three invariants Q, R, and Wcanbeexpres.sed as follows:

On the Group of Transformations i1l Six-dimmsiollal Space 157(33)

then the non-vanishing elements are ( A I . t . ( J ~ . J ~ ) I A - 1 ) and ( J . l g l j l o j ~ ) I A + 1 ) , whichwe shall write as f,(};, 72 A) and g;(};.j2' A) hereafter.

Of the remaining commutation relations, we have to consider only

[E(I) E( O)J=_iE( 1)' [E( 0) E( o ) J = ~ ( H 2 - l f ~ )o I 2.v2 - 1 I 81 I 0 I 1and [E G) E c n J = [ E c ~ ) ' E(J)J=O,

the others being deduced from them. These commutation relations give the following relations among I. and f i :};(.t. ga+glh) = +1)(f2g1+ga.t.), 14 g ~ g d 2 = - f 2 g 1 + g 3 ~

} ~ U 4 g 2 + g d o ~ ) = j ~ + 1) (/Qgl+g2.t.), fit g2 g l /a= -fagl+g2./4,}; (fsg4 +g2.iJ) = J; +1) u g2+g4/Q)' j ; g4-g2 t= --j;g2+gJS'j ~ ( ; ; g 4 + g a . f r ) = (; 2+ 1) (fl gS+gd2)' fdf4-gst;= .iT ga+gJ2'

r; g4=g4h. j;ga=gs/2' fsg2= dq / tgl=gd4'j J d l= (jl+ l)t;/q, (Jl+ l)gS I=jl gl %a,

} ~ A j ; = ( } ~ + 1 ) f d 4 'j dd : = U2+ 1 )fl,h.jd4is= ) ~ + 1)/s14,

( } ~ + 1 ) g 4 g2=Ag2g4'(/2+ 1)g2g1=j2g1 g2'(; 2+ 1)g4 % 3 = j ~ g S % 4 '

{m 1(J2+ 1) +m2(jl +1)}j; t+ {m2(j;+ 1) - m l } ~ } f d ; '

(3.4)

+ { m l ( j ~ + 1) mdl} fd2 {md2+m2};}fij;= 0,{ml (; 2+ 1) +m 2(jl + 1 }g4 gl + {m2(};+ 1) m l j ~ } gS%2

+ {m 1(J2+ 1) -mdl}g2ga- {m1 / 2+mdl}gl 4=0.(jl +m1+1) ( j2- m2+ 1)j; %1- (1; +m1+ 1) (j2+ m2)f2g2- ( r-m I ) ( j ~ - m 2 + l).Ags

+ U l m1) ( J ~ + m 2 ) t g 4- CiI + m1) ( J ~ - m 2 k J l + (j; + m1) h +m2+ 1)gd2+ (; 1- m1+ 1) j ~ - m 2 ) % 3 / q

- (j ' t-m I + 1) (;'2+ m2+ 1 ) g J 4 = ~ ( m l - m 2 - A ) / .8In these relations Ii or gi which stands at the head of a term, has always the arguments(};.j2' A). and then those of the following matrix are automatically determined, if weconsider (3.2): for example, figs means .ftUl,j2, A ) g s C J ; - ~ , j 2 - ~ ' A-I) .2 2

By (31), (3.2), (3.3) and (3.4), three invariants Q, R, and Wcanbeexpres.sed as follows:



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On the Group of Transformations in Six-dimensional Space 159property can be found, in the finite dimensional representation, out of consideration of thebehavior of weights under Wey 1 reflection.From now on w use P = j l + J ~ and q=jl-j2 instead of i1 and i2' Then

Q=p(P+2) + l + A 2 + 4 ~ - 2 4 [ p 2 _ l ) g d l - P + q ) (p-q+2)gd 2- (p-q) (P+q+2)gs/3+ (p+q+2) (P-q+2)gd4] (4.1)

R = - P + 1 ) q ~ + 2 ) +23q[(p+q+2) (P-q+2)gd4- ( f - l )gd lJ23(p+ 1)[(p+q) (p -q+2 g 2 /2 - (p-q) (p+q+2)gd,J. (4.2)

and from the last equation of (3. 4) we get(p+q+ 1 (p -q+ 1)/4 g4=gdl p - q + 2 ) g d ~ + (P+q+2)gd,

+ p + q + 2 ) p - q + 2 ) g d ~ - ~ P + ~ + 2 ) . (4.3)8These relations are to be understood as those of operators which are applied to I(P. q. ~ ) .Making use of them w can express g2/2' g3/1' and g4fi by gIl l as follows:

_2 5p(P+q+ 1)(p-q+2)gd2=P {p2+ ~ - q + 1)2_5_Q} - 2R_2 (p_q)(p+q+1) {(P+q-1)gdl+ (p-q+1)j;g4}, (44)

_2 5p(p_q+ 1 (p+q+ 2)gs/s=p{p2+ U+q+ 1)2_5-Q} +2R- 24 (p+q) (p -q+ 1) { (p-q-1)g , .k (p+ q+ 1 .ft g4}' (4.5)

24p(P+q+ 1 (P+q+2) (p -q+ 1 (p-q+2)gd4=P(P+ 1 {(P+2) (3P+2A)+ ~ 2 - l - Q } 2qR+24(P+q) (p+q+ 1 (p-q) (p -q+ 1) {(p+ 1V4g4-gJ/l}

(4.6)As p>O, in every representation there exists the minimum P say Po Then 11/(Po. q.and g4 (Pil. q. A are zero, if q is equal neither to Po nor to-po. Therefore w findfrom (44), (4.5) and (4.6)

-gdd(Po q . ~ = Po {Po: (..l-q+1)-5-Q} - 2R (Po. q. (-Po


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160 Y. Murai

tions of operators ~ H2 and 9., and belong to p, q), starting from a definite vector,we have to multiply an appropriate factor shown in (3.2) on 12 or hf, and this factoris zero for q= -Po in the former case and for q=Po in. the latter case. The correspondingexpressions for 11 gl 12 g2 and 13 gs can be obtained by changing the signs of A and Rin % 4 ~ ' g s/s and %2/2 respectively, for example,f 0 (p q A ) _ P O ( P o + 1 ) { ( P o + 2 ) ( 3 P o - 2 . i . ) + A ~ - l - Q } - 2 q R I(P q J.

1 6 1 0 - 2 4po(po+q+1) (Po+q+2) (Po-q+l) (Po-q+2) 0( -Po< q po) . (4 . 10)

Now we apply 14(Po+1, q, A+1) to both sides of (49),~ ( P o + 1 , A+1)g4(PO' q, A ) ~ ( P o + 1 , q , A+1lJ(po, q, i.)

Po Po 1) {(Po"+2)(3po+2A) +A2_ ql_Q} +2qR /4(Po+1. q, A+1)f(po. q, A).24po(Po+q+ 1) (p+q+2) (Po-q+ 1) (p -q+2)We denote J,; Po 1. q, J 1\f, which, multiplied by an appropriate factor, is a propervector of Hp H2 and Ifl (of proper value A+ 1) and belongs to 1 0+ 1 and q, by

I(Po+ 1, q, A+ 1). Thus we have

Similarly from (4.10),gdJ/(Po+ 1. q, A)

Po(Po+ 1) {(Po 2)(3 Po 2A-2) A - l ) ~ - l - Q} 2qR24 Po(Po+ q+ 1) (Po+q+ 2) (Po-q+ 1}{Po-q+ 2)

I(Po+ 1, q. A).

Po(po+ 1) {(Po 2) (3po-2A-2) 0 1)2_ q2_Q} -2qR24 Po(Po+q+ 1) (Po+ q+ 2) (Po-q+ 1) (Po-q+2)

1(1'0+ 1, q, A).Using these results we can find, by m e ~ s o f (4. 4), (4 5) ,md (5 6) , the expressionsfor g2 2' %3/1 and g 4 ~ as operators applying to 1(1 0+ 1, q. A) Following in this waywe can get the similar expressions for .P=Po+2, Po+3,....... And from the first fewcases we can perceive the general forms, which are legitimated by inserting into (4.4);( 4 .5) and (4.6) ;gt/d(Po+s, q, A)

s(Po+1/2(s+ [- - - - - - - - - ' - - - ' ' - - ' - - - - - - ' - - - - - ' - - - ~ ' - - ' ' - - - - - - - - - - - : - - Po Po 1 )24 Po(Po+ 1) (Po+q+s) (Po+q+s+ 1) (Po-q+s) (Po-q+s+ 1). {3Po(Po+2) (s -1) (2po+s+3) 2 (A+ 1) (Po+s+3) (A+ 1 ) 2 - l -Q} -2qR] / ,-gdd(Po+s , q, A)

(Po+q) (Po-q+ 1)25po(Po+ 1) (Po+q+s) (Po+q+s+ 1) (Po-q+s+ 1) (Po-q+s+2)

X [Po Po 1) {P02-2spo-s(s+ 2) - 5 (A-q+ 1)2_ Q} -2{Po+s+ 1)Rlf .(4 11)

160 Y. Murai

tions of operators ~ H2 and 9., and belong to p, q), starting from a definite vector,we have to multiply an appropriate factor shown in (3.2) on 12 or hf, and this factoris zero for q= -Po in the former case and for q=Po in. the latter case. The correspondingexpressions for 11 gl 12 g2 and 13 gs can be obtained by changing the signs of A and Rin % 4 ~ ' g s/s and %2/2 respectively, for example,f 0 (p q A ) _ P O ( P o + 1 ) { ( P o + 2 ) ( 3 P o - 2 . i . ) + A ~ - l - Q } - 2 q R I(P q J.

1 6 1 0 - 2 4po(po+q+1) (Po+q+2) (Po-q+l) (Po-q+2) 0( -Po< q po) . (4 . 10)

Now we apply 14(Po+1, q, A+1) to both sides of (49),~ ( P o + 1 , A+1)g4(PO' q, A ) ~ ( P o + 1 , q , A+1lJ(po, q, i.)

Po Po 1) {(Po"+2)(3po+2A) +A2_ ql_Q} +2qR /4(Po+1. q, A+1)f(po. q, A).24po(Po+q+ 1) (p+q+2) (Po-q+ 1) (p -q+2)We denote J,; Po 1. q, J 1\f, which, multiplied by an appropriate factor, is a propervector of Hp H2 and Ifl (of proper value A+ 1) and belongs to 1 0+ 1 and q, by

I(Po+ 1, q, A+ 1). Thus we have

Similarly from (4.10),gdJ/(Po+ 1. q, A)

Po(Po+ 1) {(Po 2)(3 Po 2A-2) A - l ) ~ - l - Q} 2qR24 Po(Po+ q+ 1) (Po+q+ 2) (Po-q+ 1}{Po-q+ 2)

I(Po+ 1, q. A).

Po(po+ 1) {(Po 2) (3po-2A-2) 0 1)2_ q2_Q} -2qR24 Po(Po+q+ 1) (Po+ q+ 2) (Po-q+ 1) (Po-q+2)

1(1'0+ 1, q, A).Using these results we can find, by m e ~ s o f (4. 4), (4 5) ,md (5 6) , the expressionsfor g2 2' %3/1 and g 4 ~ as operators applying to 1(1 0+ 1, q. A) Following in this waywe can get the similar expressions for .P=Po+2, Po+3,....... And from the first fewcases we can perceive the general forms, which are legitimated by inserting into (4.4);( 4 .5) and (4.6) ;gt/d(Po+s, q, A)

s(Po+1/2(s+ [- - - - - - - - - ' - - - ' ' - - ' - - - - - - ' - - - - - ' - - - ~ ' - - ' ' - - - - - - - - - - - : - - Po Po 1 )24 Po(Po+ 1) (Po+q+s) (Po+q+s+ 1) (Po-q+s) (Po-q+s+ 1). {3Po(Po+2) (s -1) (2po+s+3) 2 (A+ 1) (Po+s+3) (A+ 1 ) 2 - l -Q} -2qR] / ,-gdd(Po+s , q, A)

(Po+q) (Po-q+ 1)25po(Po+ 1) (Po+q+s) (Po+q+s+ 1) (Po-q+s+ 1) (Po-q+s+2)

X [Po Po 1) {P02-2spo-s(s+ 2) - 5 (A-q+ 1)2_ Q} -2{Po+s+ 1)Rlf .(4 11)



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On the Group of Transformations in Six-dimensional Space 161

25poCPo+ 1) (Po-q+s)(Po-q+s+ 1) (Po + q+s+ 1) (Po+ q+s+2)x [Po (Po + 1) {P02-2sPo-s(s+2) 5 + (A+q+ 1)2-Q} + 2(Po+s+ I)R]f,

%4f4f(Po+s, q, A)(s+ 1) Po+ 1 + s/2)

24po(Po+ 1) (Po+q+s+ 1) (Po+q+s+2) (Po-q+s+ 1) (Po-q+s+ 2)x [Po (Po+ 1) {3Po(Po+2) +s(2po+s+4) +2A.(Po+s+2) +A2 Q} +2qR]f.

The corresponding expressions for 11 1' f2 2' 1 3 and i 4 are obtained by changing thesigns of A and R in %4/J. %3/1' K2 (2 and ii] j respectively.

From 221) and 3.2), we see thatf1(P" q. A) and %1(P+l, q, A-I ) ,12(P, q, A) and -%2(P, q+l , A-I) ,f3(P,q,A) and - 3 (p ,q - l ,A- l ) ,/J p, q, A) and %4(p-l , q, ).-1)

are Hermitian conjugate with each other. Therefore, forming scalar products of both sidesof 4.11) with /(Po+s, q, A), we have,

Ilfl(Po'+s-l, q, A+l)f(Po+s, q, A)W=O'l(S, q, A)lIf(Po+s, q, A)W,IIf2(Po+s, q - l A+l)f( ) 11 2=0'2( )I f( ) 1 2.lif3(Po+s, q+ 1, A+ l ) f ( . ) 1 2=0'3( ) flf( ) 21 ~ ( P o + s + I, q, A+ 1V( ) 2=0'4( )I f( ) 12,li%l(PO+s+l, q, A-1) f ( ) 1 2=Pl( ) I f( )1 2, 4.12)I K2(Po +s, q+ 1, A- l ) f ( )1 2=P2( )lIf( ) 1 2,II%g(Po+s, q - l , A- l ) f ( ) 1 2=P3( )llf( ) 21 %4(Po+s-l, q, A- l ) / ( ) 1 2=P4 ( )llf( ) 112

We simplify the expression for 0'. and Pi by introducing two constants A and B relatedto QandRby :

(413)Then 0',/ and Pi come to be as follows;

( 1) s(Po+ 1/2(s+ 1 [(Po+S-A)2- (q+A )(q+B)1~ ~ A ,24 (Po+q+s) (Po+q+s+l) (Po-q+s)(Po-q+s+ 1) (4 14-1)

On the Group of Transformations in Six-dimensional Space 161

25poCPo+ 1) (Po-q+s)(Po-q+s+ 1) (Po + q+s+ 1) (Po+ q+s+2)x [Po (Po + 1) {P02-2sPo-s(s+2) 5 + (A+q+ 1)2-Q} + 2(Po+s+ I)R]f,

%4f4f(Po+s, q, A)(s+ 1) Po+ 1 + s/2)

24po(Po+ 1) (Po+q+s+ 1) (Po+q+s+2) (Po-q+s+ 1) (Po-q+s+ 2)x [Po (Po+ 1) {3Po(Po+2) +s(2po+s+4) +2A.(Po+s+2) +A2 Q} +2qR]f.

The corresponding expressions for 11 1' f2 2' 1 3 and i 4 are obtained by changing thesigns of A and R in %4/J. %3/1' K2 (2 and ii] j respectively.

From 221) and 3.2), we see thatf1(P" q. A) and %1(P+l, q, A-I ) ,12(P, q, A) and -%2(P, q+l , A-I) ,f3(P,q,A) and - 3 (p ,q - l ,A- l ) ,/J p, q, A) and %4(p-l , q, ).-1)

are Hermitian conjugate with each other. Therefore, forming scalar products of both sidesof 4.11) with /(Po+s, q, A), we have,

Ilfl(Po'+s-l, q, A+l)f(Po+s, q, A)W=O'l(S, q, A)lIf(Po+s, q, A)W,IIf2(Po+s, q - l A+l)f( ) 11 2=0'2( )I f( ) 1 2.lif3(Po+s, q+ 1, A+ l ) f ( . ) 1 2=0'3( ) flf( ) 21 ~ ( P o + s + I, q, A+ 1V( ) 2=0'4( )I f( ) 12,li%l(PO+s+l, q, A-1) f ( ) 1 2=Pl( ) I f( )1 2, 4.12)I K2(Po +s, q+ 1, A- l ) f ( )1 2=P2( )lIf( ) 1 2,II%g(Po+s, q - l , A- l ) f ( ) 1 2=P3( )llf( ) 21 %4(Po+s-l, q, A- l ) / ( ) 1 2=P4 ( )llf( ) 112

We simplify the expression for 0'. and Pi by introducing two constants A and B relatedto QandRby :

(413)Then 0',/ and Pi come to be as follows;

( 1) s(Po+ 1/2(s+ 1 [(Po+S-A)2- (q+A )(q+B)1~ ~ A ,24 (Po+q+s) (Po+q+s+l) (Po-q+s)(Po-q+s+ 1) (4 14-1)



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162

Ga( )

Y. Murai

(Po+q) (Po-q+ 1)[ (q_) _1)2_ (Po+s+l +A) (Po+s+ 1+B)l25 (Po+q+s) (Po+q+s+ 1) (Po-q+s+ 1) (Po-q+s+2)(414-2)

(Po-q) (Po+q+ l)[(q+A+ 1)2- (Po+s+ I -A (Po+s+ I -B ]25(Po-q+s) (Po-q+s+ I){Po+ q+s+ 1) (Po+q+s+2) ,

(4.14-3),, = (s+l){po+l+S/2)r(po+S+A+2)2-(q-A)(q-B)] ,

24 (Po+q+s+ 1) (Po+q+s+ 2) (Po-q+s+ 1) (PIJ-q+s+2)(4.14-4)

Pa( )

(s+ 1)[(1 0+ 1+s/2)[ (Po+S-A+2)2- (q+A) (q+B)]24 (Po+q+S+ 1) (Po+q+s+2)(Po-q+s+ 1) (Po-q+s+2) ,(4.15-1)

(Po-q) (Po q+ 1)[(q-A+ 1)2_ (Po+s+ 1+A) (Po+s+ 1 +B)]25 (Po-q+s) (Po-q+s+ 1) (Po+ q+s+ 1) (Po +q+s+ 2)

(4 15-2)(Po+q) (Po-q+ 1)[(q+A- 1)2- (Po+s+ A) (Po+s+ I -B ]

25 (Po+q+s) (Po+q+s+ 1) (Po-q+s+ 1) (Po-q+s+2)(4.15-3)(4.15-4)

At first we see that always (i.e., for any value of s) - P o ~ q < p o Then except GlandP4 for s=O, it depends upon the expression in the square brackets, whether G. or p. ispositive, negative or zero.

We first consider (4.14-2, 3) and (4 15-2, 3) for a fixed s. The existence ofI(Po+s, q, A) being assumed for some values q and A we may construct other vectors byapplying g2 ga, /2 or la on I(q, A). From the condition that G, and p. can not .benegative, there arise restrictions for the domain on which j(q, A) exists. We see thatthese domains are necessarily limited by a boundary-line q+A=const. or q-A=const .From the consideration of G 2 and P2 we have the following three different cases.

1) (Po+s+ 1 +A) (Po+s+ 1 +B) < 0 or (Po+s+ 1 +A)(po+s+ 1 +B) < 1(when qA is even). In this case G 2 and P2 are always positive and starting from anyvector, we can form other vectors of which q ranges from-po to Po. (Fig. 1)

2. (Po+s+ 1+A) (Po+s+ 1+B) =(t-l)2, t > oLet G 2 be zero for q-A=t and positive for q-A=t+2, then (Po+s+l+B) (Po+s+l+B) = (t-l)2 and t > O For q-A > t, P2 is positive and the allowable domain becomesas such one shown in Fig. 2a. Similarly let P2 be zero for q-A=t and positive for q A=t -2, then (Po+s+1+A)(Po+s+1+B) = (t +1)2 and 1 oLet G 2 be zero for q-A=t and positive for q-A=t+2, then (Po+s+l+B) (Po+s+l+B) = (t-l)2 and t > O For q-A > t, P2 is positive and the allowable domain becomesas such one shown in Fig. 2a. Similarly let P2 be zero for q-A=t and positive for q A=t -2, then (Po+s+1+A)(Po+s+1+B) = (t +1)2 and 1


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On the Group of Transformations in Siz-dimensiollat Space 1633. (Po+s+ 1+A) po+s+ l-j-B)=I, q-A=O.

In this case 0 2=P2=0 and f(q, A) can exist only for q-).=O. (Fig. 3)

Fig. 1 Fig.2a(

Fig.2b(

Fig. 3In the same way from (4.14-3) and (4 .15-3) we have three cases as to the behaviorof /3 and g3

1 . (Po+s+ l -A) po+s+ I -B) O.3 . (Po+s+ l -A) po+s+ I -B) =1 q+).=O.

Combining these results we have nine cases in all. (1-1 , 1-2 , etc.) Now we shall annulthe inappropriate combinations examining the behavior of vectors under the application ofA r iJ and g4

1) (Po+s+ 1 +A) po+s+ 1+B)


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164 Y. Murai

consider the former case where the domain is q A r; the latter case is treated exactlyin the same way. In this case 114 is always positive and if P is not zero on q+A=r,the domain comes to be limited by q+A=r-l for P=Po+s+1. But such a situationcan not continue without limit, since the domain q A k > 0). Butsuch an equation can not be satisfied as A and B are compelled to satisfy only one equation and an inequality. This case should be excluded too. This is the combination 1 - 2 ,and 1 - 2 is treated similarly.

3) (Po+s+l+A)(Po+s+l+B) Po 2. Therefore the combination2-3 , and similarly 2 -3, are excluded.

5) (Po+s+ I -A ) (Po+s+ I -B)=l , (Po+s+ 1+A)(po+s+ 1+B) =1,q=A=O.smust be zero and A+B=o and AB=I-(Po+1)2. But as in 4) A must be equalto B, and as Po=l=o this can not be satisfi.'ed. This is the combination 3-3 .

Thus we have seen that all the combinations except 2 - 2 are to be excluded. Inthe case of 2 - 2 we have the condition

t and are positive integers.(Po+s+I-A) (Po+s+ I -B) = (1'-1)2.

The domain of vectors is q+A > r, q-A::: t or q+..l -1 , q - . ~ t . Since this is theonly possibility remained here, the similar condition and the similar shape of the domainmust be maintained for any value of s. From the simple consideration of above equationswe see that A = B and when s changes by 1, t and l change also by 1. We take s tobe zero, then

Po+l+A= t - l ) , Po+I-A= r- l ) .Thus there are four possibilities for the choice of signs.



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On the Group of Transformations in Six-dimensional Space 165a) +, +). In this case t+r=2(po+2), 2A=t-r . As to the values of 14 and

Pl' we have the following result;on q+A=-r, 14=0, P1> 0,on q+A=r, 74) 0, Pl=O,on q-).=t, 74=0, PI> 0,on q- .= - t , 74) 0, P =0

(cf. Fig. 4). When s increases the similar situation continues as is seen from 4.14-4)and 4.15-4).

j l b) - , - ) . This should be excluded0 1 since 2 Po + 1) =2 - t - r 0,on q+).=r, 74) 0,

P l ~ O , = 0 if 1 =1),-t-----,orr---+ I on q-A=t, 74=0, PI> 0,

Fig. 4 on q-) .=-t , 74) 0, Pl=O.In this case therefore, 1 =1 and t=2po+3, i.e., A=Po+l is the only case to be takeninto account, but this is included in a).

d) -, +). Similarly as above the permissible values are 1=1 and r=2po+3,i.e., A = - Po 1) and this is included in a).

To find the value of W, e.g., in the case where the domain is q A ~ - r , q-).>tfor P=Po q+k::;;-r-l , q-A > t+ 1 for P=Po+ 1,, we apply 3.7) to the vectorI(P-Po= ( t+r-4) , q= ( t-r) , ) .= - ( t+r, (when It-rl.:5::2po). We getthe following result,

W = ~ ( t + r - 2 ) ( t + r - 4 ) [ ( t + r - 2 ) ( t+r-4) +2(t-r+2) ( t -r-2)] . 4.16.)16When It-rl>2Po, the values of t and l are forced to be t=2po+3, 1 =1 or t=l, 1 =2po+3. And if we apply Won I(Po, q=Po A=Po-t) or on I(Po q=-Po A=po-r),we see that the result is nothing else the special case of 4.16).

If vectors are confined in the domain q+A>r, q - . _ ~ - t for P=Po the value ofW is the same as above. Because this case is obtained from the former by changing thesigns of A and exchanging the role of t and l (this comes to be A ~ - A , and R ~ Rand so II l , f 2 g 2 / . ~ g3,h g4 are replaced by g4h, g31a, g2lz, gIll respectively and viceversa), and thereby the expression W 3.7) is seen to be left invariant.

On the Group of Transformations in Six-dimensional Space 165a) +, +). In this case t+r=2(po+2), 2A=t-r . As to the values of 14 and

Pl' we have the following result;on q+A=-r, 14=0, P1> 0,on q+A=r, 74) 0, Pl=O,on q-).=t, 74=0, PI> 0,on q- .= - t , 74) 0, P =0

(cf. Fig. 4). When s increases the similar situation continues as is seen from 4.14-4)and 4.15-4).

j l b) - , - ) . This should be excluded0 1 since 2 Po + 1) =2 - t - r 0,on q+).=r, 74) 0,

P l ~ O , = 0 if 1 =1),-t-----,orr---+ I on q-A=t, 74=0, PI> 0,

Fig. 4 on q-) .=-t , 74) 0, Pl=O.In this case therefore, 1 =1 and t=2po+3, i.e., A=Po+l is the only case to be takeninto account, but this is included in a).

d) -, +). Similarly as above the permissible values are 1=1 and r=2po+3,i.e., A = - Po 1) and this is included in a).

To find the value of W, e.g., in the case where the domain is q A ~ - r , q-).>tfor P=Po q+k::;;-r-l , q-A > t+ 1 for P=Po+ 1,, we apply 3.7) to the vectorI(P-Po= ( t+r-4) , q= ( t-r) , ) .= - ( t+r, (when It-rl.:5::2po). We getthe following result,

W = ~ ( t + r - 2 ) ( t + r - 4 ) [ ( t + r - 2 ) ( t+r-4) +2(t-r+2) ( t -r-2)] . 4.16.)16When It-rl>2Po, the values of t and l are forced to be t=2po+3, 1 =1 or t=l, 1 =2po+3. And if we apply Won I(Po, q=Po A=Po-t) or on I(Po q=-Po A=po-r),we see that the result is nothing else the special case of 4.16).

If vectors are confined in the domain q+A>r, q - . _ ~ - t for P=Po the value ofW is the same as above. Because this case is obtained from the former by changing thesigns of A and exchanging the role of t and l (this comes to be A ~ - A , and R ~ Rand so II l , f 2 g 2 / . ~ g3,h g4 are replaced by g4h, g31a, g2lz, gIll respectively and viceversa), and thereby the expression W 3.7) is seen to be left invariant.



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166 Y. MuraiNow we shall study the case Po=O. In the same way as before, starting from

Q=A2_4A-26}; 1=A2+4A-26 J4'Q being considered as an operator on /(0, 0, A we get.. F F, )_A2+2A(s+2) +s(s+4) -Q j_ (s+A+2)2+ Ar_ r4.14.1'S, JI - - - 14.1 ,25(s+ 1) (S+2) 22(S+ 1) (s+2)

( ) ).2-2A(s.+ 2) +s(s+4)-Q f (s-A+2)2+A2f= 1./j; d s 0, A 25(s+1)(s+2) 25(s+1)(s+2) P/2 2=/3 3= 2/2- 3/3=0, R=o and q is always zero. (We put Q= - 4 - A2 in thelast expression.) There are three cases as to the value of A.

In this case (14 and P is positive for any sand A2) >A2>_l .

Only when A is an odd integer, (14 and PI is positive for any s.

In this case the support of vectors, to which both /1 and 1 are applicable, is confined tos - A < - t + 2 ) or s-A>t, and similarly fo r /4 andg4 the support is s A ~ t or s+A> - (t+ 2). Combining them we have for the domain of vectors, to which any operatorcan be applied, the following:

a) s -A - (t+ 2). Combining them we have for the domain of vectors, to which any operatorcan be applied, the following:

a) s -A


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On the Group of Transformations in Si -dimensional Space 167III ' Q=t2-4, t=O, 1, 2, 3 ' , R= W-O.

j;+ 2=0 } 1 } 2 }A ~ - (t+2) -( t+3) - ( t+4) ...... j;-j2=0.IV. Q= -4 , R= W=O.

j;+j2: 0 } } 2 } , l - j ~ = O .A-O - 1 , 1 -2 0,

V. Q= [2(t+r-2) ( t+r-4) + (t-r+4) (t-1'-4) J.1R=s t - r ) t+r-2) t+r-4) ,

W - 116 ( t+r-2) ( t+1'-4)[( t+r-2)(t+r-4) +2(6-1'-2) t - r -2) J.t=1, 2, 3, 4,, ' , 1 = 1, 2, 3, 4 ,(t+r > 5).

1 . 1 5).

1 . 1


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168 Y. MuraiReferences

1) H. S. Snyder, Phys. Rev 7 1947), 38.2) C N. Yang, Phys. Rev 72 1947), 874.3) Z. Tokuoka, SoryushironKenkyu Mimeographed Circular in Japanese) Vol. 1 1949), No. 3-2,23.4) 1 E. Segal, Duke Math. Journ. 18 1951), 221.5) J A. Schouten, Rev Mod. Phys. 21 1949), 421.6) N. H. Kuiper, Ann. Math. 50 1949),916.7) E P. Wigner, Ann. Math. 40 1939), 149.8) V. Bargmann and E. P. Wigner, Proc. Nat. Acad. Sci. 34 1948) 211.9) V. Bargmann, Ann. Math. 48 1947), 568.

10) L. H. Thomas, Ann. Math. 42 1941), 113. Cf alsoT. D. Newton, Ann. Math. 51 1950), 730.

168 Y. MuraiReferences

1) H. S. Snyder, Phys. Rev 7 1947), 38.2) C N. Yang, Phys. Rev 72 1947), 874.3) Z. Tokuoka, SoryushironKenkyu Mimeographed Circular in Japanese) Vol. 1 1949), No. 3-2,23.4) 1 E. Segal, Duke Math. Journ. 18 1951), 221.5) J A. Schouten, Rev Mod. Phys. 21 1949), 421.6) N. H. Kuiper, Ann. Math. 50 1949),916.7) E P. Wigner, Ann. Math. 40 1939), 149.8) V. Bargmann and E. P. Wigner, Proc. Nat. Acad. Sci. 34 1948) 211.9) V. Bargmann, Ann. Math. 48 1947), 568.

10) L. H. Thomas, Ann. Math. 42 1941), 113. Cf alsoT. D. Newton, Ann. Math. 51 1950), 730.



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