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August, 1995 hep-th/95mmnnn
Quaternions and Sp ecial Relativity
a)
Stefano De Leo
UniversitadiLecce, Dipartimento di Fisica
Instituto di Fisica Nucleare, Sezione di Lecce
Lecce, 73100, ITALY
Abstract
We reformulate Sp ecial Relativityby a quaternionic algebra on reals. Using real
linear quaternions,we show that previous diculties, concerning the appropriate
transformations on the 3 + 1 space-time, maybeovercome. This implies that a
complexi ed quaternionic version of Sp ecial Relativityisachoice and not a necessity.
a) e-mail: [email protected]
1 Intro duction
\The most remarkable formula in mathematics is:
i
e = cos + i sin : (1)
This is our jewel. Wemay relate the geometry to the algebra by representing complex
numb ers in a plane
i
x + iy = re :
This is the uni cation of algebra and geometry."-Feynman [1].
i
We know that a rotation of -angle around the z -axis, can b e represented by e ,
in fact
i i(+ )
e (x + iy )=re :
In 1843, Hamilton in the attempt to generalize the complex eld in order to describ e
1
the rotation in the three-dimensional space, discovered quaternions .
Today a rotation ab out an axis passing trough the origin and parallel to a given uni-
tary vector ~u (u ;u;u)by an angle , can b e obtained taking the transformation
x y z
(iu +ju +ku ) (iu +ju +ku )
x y z x y z
2 2
e (ix + jy + kz)e : (2)
Therefore if we wish to represent rotations in the three-dimensional space and com-
plete \the uni cation of algebra and geometry"we need quaternions.
The quaternionic algebra has b een exp ounded in a series of pap ers [2] and b o oks [3]
with particular reference to quantum mechanics, the reader may refer to these for
further details. For convenience we rep eat and develop the relevant p oints in the
following section, where the terminology is also de ned.
Nothing that U (1;q) is algebraically isomorphic to SU (2;c), the imaginary units
i; j; k can b e realized by means of the 2 2Pauli matrices through
(i; j; k ) $ (i ; i ; i ) :
3 2 1
(this particular representation of the imaginary units i; j; k has b een intro duced in ref. [4]).
So a quaternion q can b e represented bya 22 complex matrix
!
z z
1
2
q $ Q = ; (3)
z z
2
1
1
Quaternions, as used in this pap er, will always mean \real quaternions"
q = a + ib + jc + kd ; a; b; c; d 2R: 1
where
z = a + ib ; z = c id 2 C (1;i) :
1 2
It follows that a quaternion with unitary norm is identi ed by a unitary 2 2 matrix
2
with unit determinant, this gives the corresp ondence b etween unitary quaternions
U (1;q) and SU (2;c).
Let us consider the transformation law of a spinor (two-dimensional representations
of the rotation group)
0
= U ; (4)
where
!
z
1
= ; U2SU (2;c) :
z
2
We can immediately verify that
!
z
2
~
=
z
1
transforms as follows
0
~ ~
= U ; (5)
so
! !
0
z z z z
1 1
2 2
= U
z z z z
2 2
1 1
represents again the transformation law of a spinor.
Thanks to the identi cation (3) we can write the previous transformations by real
quaternions as follows
0
q = U q;
+
with q = z + jz and U quaternion with unitary norm (N (U )=U U =1). Note that
1 2
we don't need right op erators to indicate the transformation law of a spinor.
Nowwe can obtain the transformation law of a three-dimensional vector ~r
(x; y ; z )by pro duct of spinors, in fact if we consider the purely imaginary quaternion
+
! = qiq = ix + jy + kz
or the corresp onding traceless 2 2 complex matrix
!
ix y iz
+
= i = ;
y iz ix
3
a rotation in the three-dimensional space can b e written as follows
2
In a recent pap er [5] the representation theory of the group U (1;q) has b een discussed in detail.
3
\No such `trick' works to relate the full four-vector (ct; x; y ; z ) with real quaternions." -
Penrose [6]. 2
0 +
! = U ! U (quaternions) ,
0 +
= U U (2 2 complex matrices) .
~
~
For in nitesimal transformations, U =1+Q,we nd
0
~ ~
~ ~ ~ ~ ~ ~
Q ~r = Q ~r + Q Q ~r Q ~r Q ;
where
~
~
Q (i; j; k ) ; ( ; ; ) :
4
If we rewrite the mentioned ab ove transformation in the following form
0
~
~ ~ ~ ~
Q ~r =[1+(Q 1 j Q)] Q ~r; (6)
we identify
i 1 j i j 1 j j k 1 j k
; ; ;
2 2 2
5
as the generators for rotations in the three-dimensional space .
Up till now, wehave considered only particular op erations on quaternions. A
quaternion q can also b e multiplied by unitary quaternions V from the right. A
p ossible transformation which preserves the norm is given by
0
q = U q V ; (7)
+ +
( U U = V V =1) :
Since left and rightmultiplications commute, the group is lo cally isomorphic to
SU (2) SU (2), and so to O (4), the four-dimensional Euclidean rotation group.
As far as here we can recognize only particular real linear quaternions, namely
1 ; i; j; k; 1ji; 1jj; 1jk:
Wehaven't hop e to describ e the Lorentz group if we use only previous ob jects. An-
alyzing the most general transformation on quaternions (see section 4), weintro duce
new real linear quaternions which allow ustoovercome the ab ove diculty and so
obtain a quaternionic version of the Lorentz group, without the use of complexi ed
4
Barred op erators Ojq act on quaternionic ob jects as in
(Ojq) = O q:
1
5
The factor guarantees that our generators satisfy the usual algebra:
2
[A ;A ]= A m; n; p =1; 2; 3 :
m n mnp p 3
quaternions. This result app ears, to the b est of our knowledge, for the rst time in
print.
First of that we brie y recall the standard way to rewrite sp ecial relativitybya
quaternionic algebra on complex (see section 3).
In section 5, we present a quaternionic version of the sp ecial group SL(2;c), which
is as well-known collected to the Lorentz group. Our conclusions are drawn in the
nal section.
2 Quaternionic algebras
A quaternionic algebra over a eld F is a set
H = f + i + j + k j ; ; ; 2Fg
with multiplication op erations de ned by following rules for imaginary units i; j; k
2 2 2
i = j = k = 1 ;
jk = kj = i;
ki = ik = j;
ij = ji = k:
In our pap er we will work with quaternionic algebras de ned on reals and complex,
so in this section we give a panoramic review of such algebras.
We start with a quaternionic algebra on reals
H = f + i + j + k j ; ; ; 2Rg:
R
+
Weintro duce the quaternion conjugation denoted by and de ned by
+
q = i j k :
The previous de nition implies
+ + +
( ') = ' ;
for , ' quaternionic functions. A conjugation op eration which do es not reverse the
order of , ' factors is given, for example, by
q~ = i + j k :
An imp ortant di erence b etween quaternions and complexi ed quaternions is based
on the concept of division algebra, which is a nite dimensional algebra for which
a 6=0,b6= 0 implies ab 6= 0, in others words, which has not nonzero divisors of
zero. A classical theorem [7] states that the only division algebras over the reals are 4
algebras of dimension 1, 2, 4 and 8; the only asso ciative algebras over the reals are R,
C and H (Frob enius [8]); the nonasso ciative division algebras include the o ctonions
R
O (but there are others as well; see Okub o [9]).
A simple example of a nondivision algebra is provided by the algebra of complexi ed
quaternions
H = f + i + j + k j ; ; ; 2C(1; I )g ;
C
[I ;i]= [I;j]= [I;k]=0 :
In fact since
(1 + iI )(1 iI )=0 ;
there are nonzero divisors of zero.
For complexi ed quaternions wehave di erent opp ortunities to de ne conjugation
op erations, we shall use the following terminology:
1. The complex conjugate of q is
C
q = + i + j + k :
C
Under this op eration
(I ;i;j;k) ! ( I ;i;j;k) ;
and
(q p ) = q p :
C C
C C
2. The quaternion conjugate of q is
C
?
q = i j k :
C
Here
(I ;i;j;k) ! (I; i; j; k ) ;
and
? ? ?
(q p ) = p q :
C C
C C
3. In the absence of standard terminology,we call that formed by combining these
op erations, the full conjugate
+
= i j k : q
C
Under this op eration
(I ;i;j;k) ! (I;i;j;k) ;
and
+ + +
(q p ) = p q :
C C
C C 5
3 Complexi ed Quaternions and Sp ecial Relativi-
ty
We b egin this section by recalling a sentence of Anderson and Joshi [10] ab out the
quaternionic reformulation of sp ecial relativity:
\There has b een a long tradition of using quaternions for Sp ecial Relativity ... The
use of quaternions in sp ecial relativity,however, is not entirely straightforward. Since
the eld of quaternions is a four-dimensional Euclidean space, complex comp onents
for the quaternions are required for the 3+1 space-time of sp ecial relativity."
In the following section, we will demonstrate that a reformulation of sp ecial rela-
tivityby a quaternionic algebra on reals is p ossible.
In the present section, we use complexi ed quaternions to reformulate sp ecial
relativity, for further details the reader may consult the pap ers of Edmonds [11],
Gough [12], Ab onyi [13], Gursey [14] and the b o ok of Synge [15 ].
A space-time p oint can b e represented by complexi ed quaternions as follows
X = I ct + ix + jy + kz : (8)
The Lorentz invariant in this formalism is given by
2 2 2 2
X X =(ct) x y z : (9)
If we consider the standard Lorentz transformation (b o ost ct - x)
0
ct = (ct x) ;
0
x = (x ct) ;
0
y = y;
0
z = z
and note that the rst two equations may b e rewritten as
0
ct = ct cosh x sinh ;
0
x = xcosh ct sinh ;
where cosh = ; sinh = ;
we can represent an in nitesimal transformation by
0
X = I (ct x )+i(x ct )+jy + kz =
i +1 j i
= X + I X :
2
Wethus recognize, in the previous transformation, the generator
i +1 j i
I :
2 6
It is nowvery simple to complete the translation, the set of generators of the Lorentz
group is provided with
i+1 j i
boost (ct; x) I ;
2
j +1 j j
boost (ct; y ) I ;
2
k +1 j k
boost (ct; z ) I ;
2
i 1 j i
r otation ar ound x ;
2
j 1 j j
r otation ar ound y ;
2
k 1 j k
r otation ar ound z :
2
And so a general nite Lorentz transformation is given by
I (i +j +k )+i +j +k I(i +j +k ) i j k
r r r r r r
b b b b b b
e (I ct + ix + jy + kz)e :
The previous results can b e elegantly summarize by the relation
0 + ?
X =X ; = 1 ; (10)
where is obviously a complexi ed quaternion. In this or similar way a lot of authors
have reformulated sp ecial relativity with complex quaternions.
In the following section we will intro duce real linear quaternions and reformulate
sp ecial relativity using a quaternionic algebra on reals. We remark that complex
comp onent for the quaternions representachoice and not a necessity.
4 A new p ossibility
We think that quaternions are the natural candidates to describ e sp ecial relativity.
It is simple to understand why, quaternions are characterized by four real numb ers
(whereas complexi ed quaternions by eight), thus we can collect these four real quan-
tities with a p oint(ct; x; y ; z ) in the space-time. In quaternionic notation wehave
X =ct + ix + jy + kz : (11)
In the rst section wehaveintro duced particular real linear quaternions, namely
~ ~
1 ; Q; 1jQ;
where
~
Q (i; j; k ) :
In order to write the most general real linear quaternions wemust consider the fol-
lowing quantities
~ ~ ~
Q j i; Qjj; Qjk: 7
In fact the most general transformation on quaternions is represented by
q + p j i + r j j + s j k; (12)
with
q ; p; r;s 2H :
R
New ob jects like
k j j; jjk; ijk; kji; jji; ijj
will b e essential to reformulate sp ecial relativity with real quaternions. They represent
the wedge which p ermit to overcome the diculties which in past did not allow a (real)
quaternionic version of sp ecial relativity.
Returning to Lorentz transformations, let us start with the following in nitesimal
transformation (b o ost ct - x)
0
X = ct x + i(x ct )+ jy + kz =
k j j j j k
= X + X :
2
We can immediately note that the generator which substitutes
i +1 j i
I
2
is
k j j j j k
:
2
So wehave (for the rst time in print) the p ossibili ty to list the generators of the
Lorentz group without the need to work with complexi ed quaternions
k j j j j k
boost (ct; x) ;
2
i j k k j i
boost (ct; y ) ;
2
j j i i j j
boost (ct; z ) ;
2
i 1 j i
r otation ar ound x ;
2
j 1 j j
r otation ar ound y ;
2
k 1 j k
r otation ar ound z :
2
In app endix A we explicitly prove that the action of previous generators leaves
2 2 2 2 2
Re X =(ct) x y z (13)
invariant.
In app endix B we will give an alternative but equivalent presentation of sp ecial rel-
ativityby a quaternionic algebra on reals. There weintro duce a real linear quaternion
g which substitutes the metric tensor g . 8
5 A quaternionic version of the complex group
SL(2)
In analogy to the connection b etween the rotation group O (3) to the sp ecial uni-
tary group SU (2), there is a natural corresp ondence [16] b etween the Lorentz group
O (3; 1) and the sp ecial linear group SL(2). In fact SL(2) is the universal covering
group of O (3; 1) in the same way that SU (2) is of O (3).
The aim of this Section is to give, by extending the consideration which collect
the sp ecial unitary group SU (2) with unitary real quaternions (as shown in section
1), a quaternionic version of the sp ecial linear group SL(2). Once more the aim will
be achieved with help of real linear quaternions.
A Lorentz spinor is a complex ob ject which transform under Lorentz transformati-
ons as
0
= A ;
where A is a SL(2) matrix. When we restrict ourselves to the three-dimensional
space and to rotations, this de nition gives the usual Pauli spinors
0
= U ;
where U is a SU (2) matrix.
Nowwe shall derive the generators of rotations and Lorentz b o osts in the spinor
representation by using real linear quaternions.
The action of generators of the sp ecial group SL(2)
! ! !
0 -i 0 -1 i 0
; ; ;
-i 0 1 0 0 -i
! ! !
0 1 0 -i -1 0
; ; ;
1 0 i 0 0 1
on the spinor
!
; =
can b e represented by the action of real linear quaternions
i; j; k;
iji; j ji; k ji;
on the quaternion
q = + j :
In section 1 wehave obtained a three-dimensional vector (x; y ; z )by pro duct of
Pauli spinors q :
P
+
q iq = ix + jy + kz
P
P 9
0 +
( q = U q ; U U =1) ;
P
P
consequently wehave written its transformation law as follows
+ 0 + +
(q iq ) = Uq iq U :
P P
P P
Nowwe start with a Lorentz spinor q
L
0
q = Aq ;
L
L
and construct a four-vector (ct; x; y ; z )by pro duct of such spinors
+
q (1 + i) q = ct + ix + jy + kz :
L
L
The transformation law is then given by
+ 0 +
(q (1 + i) q ) =(Aq )(1+i)(Aq ) :
L L L
L
If we consider in nitesimal transformations
~
Q
~ ~
A =1+ (+ ji) ;
2
~ ~ ~
( ; ; ) and (~ ; ; ~) ; with
wehave
0
T = T + [i; T ]+ [j; T ]+ [k; T ]+
2 2 2
~
~ ~
~ ~ ~
+ fi; Tg+ fj; Tg+ fk; Tg
2 2 2
where
+
T = q (1 + i) q ;
L
L
and
+ +
~
T = q i(1 + i) q = T 2q q :
L L
L L
In order to simplify next considerations we p ose
T = ix + jy + kz + ct = T + T + T + T ;
i j k 1
~
T = ix + jy + kz ct = T + T + T T ;
i j k 1
so the standard Lorentz transformations are given by
~
T ! T +~ iT + jT +~ kT
1 1 i j k
T ! T i~ T + jT kT
i i 1 k j
~
T ! T jT iT + kT
j j 1 k i
T ! T k~ T + iT jT :
k k 1 j i
In this waywe obtain a quaternionic version of the sp ecial group SL(2) and
6
demonstrate that, if real linear quaternions app ear, a `trick' similar to that one of
rotations works to relate the full four-vector (ct; x; y ; z ) with real quaternions.
6
In contrast with the opinion of Penrose [6], cited in fo otnote 3. 10
6 Conclusions
The study of sp ecial relativity with a quaternionic algebra on reals has yielded a result
of interest. While we cannot demonstrate in this pap er that one numb er system
(quaternions) is preferable to another (complexi ed quaternions) wehave p ointed
out the advantages of using real linear quaternions which naturally app ear when we
work with a non commutativenumb er system, like the quaternionic eld. As seen in
this pap er these ob jects are very useful if we wish to rewrite sp ecial relativitybya
quaternionic algebra on reals. The complexi ed quaternionic reformulation of sp ecial
relativityisthusachoice and not a necessity. This armation is in contrast with the
standard folklore (see, for example, Anderson and Joshi [10 ]).
Our principal aim in this work is to underline the p otentialities of real linear
quaternions. We wish to rememb er that a lot of diculties have b een overcome
thanks to these ob jects (which in our colourful language wehave named generalized
ob jects [4]).
To remark their p otentialities let us list the situations whichhave requested their use.
The need of such ob jects naturally app ears, for example, in the construction
of quaternion group theory and tensor pro duct group representations [5]. Also
starting with only standard quaternions i; j; k in order to represent the gen-
erators of the group U (1;q), we nd generalized quaternions when we analyze
quaternionic tensor pro ducts.
1 i j k
S pin g ener ator s : ; ; :
2 2 2 2
S pin 1 0 g ener ator s :
! ! !
i+1 j i
k 1 0 j 1 j i
2
; : ;
i 1 j i
1 k 1 j i j
0
2
If we desire to extend the isomorphism of SU (2;c) with U (1;q) to the group
U (2;c), wemust intro duce the additional real linear quaternion `1 j i'. In
this way there exists at least one version of quaternionic quantum mechanics
in which a `partial' set of translations may b e de ned [4], in fact, tanks to real
linear op erators, a translation b etween 2n 2n complex and n n quaternionic
matrices is p ossible.
In the work of ref. [17] a quaternion version of the Dirac equation was derived
in the form
@ i = m ;
where the are twobytwo quaternionic matrices satisfying the Dirac condition
f ; g=2g :
11
In the Rotelli's formalism the momentum op erator must b e de ned as
p = @ j i
which is also a generalized ob ject.
In this pap er, contrary to the common opinion, wehave given a real quaternionic
formulation of sp ecial relativity. In order to obtain that wehaveintro duced the
following real linear quaternions
~ ~ ~
Q j i; Qjj; Qjk;
~
Q(i; j; k ) :
A quaternionic version of the sp ecial group SL(2) has also b een given.
We nally note that the pro cess of generalization can b e extend also to complexi-
ed quaternions. In a recent pap er [18] we give an elegant one-comp onent formulation
of the Dirac equation and, thanks to our generalization, weovercome previous dicul-
ties concerning the doubling of solutions [10 , 11 , 12 ] in the complexi ed quaternionic
Dirac equation.
In seeking a b etter understanding of the success of mathematical abstraction in
physics and in particular of the wide applicabili ty of quaternionic numb ers in the-
ories of physical phenomena, we found that generalized quaternions should b e not
undervalued. We think that there are go o d reasons to hop e that these generalized
structures provide new p ossibili ties concerning physical applications of quaternions.
\The most p owerful metho d of advance that can b e suggested at presentisto
employ all the resources of pure mathematics in attempts to p erfect and generalize
the mathematical formalism that forms the existing basis of theoretical physics, and
after each success in this direction, to try to interpret the new mathematical features
in terms of physical entities..." - Dirac [19 ].
App endix A
In this app endix we prove that the Lorentz invariantis
0 2 2
Re X = Re X ; (14)
where
X = ct + ix + jy + kz :
Under an in nitesimal transformation, wehave
k jj j jk i 1ji
0
X =(1+ + +:::)X
2 2 12
so, neglecting second order terms,
0 2 2
X = X + fX ;kXj jXkg+ fX ;iX Xig+:::
2 2
Equation (14) is then satis ed since
2
fX ;iX Xig = (i 1ji)X ;
fX ;kXj jXkg = (1 j j j )X k X +(k 1 j k)XjX ;
are purely imaginary quaternions.
Obviously we can derive the generators of Lorentz group by starting from the
in nitesimal transformation
0
X = X + AX
and imp osing that they satisfy the relation
Re fX ; AX g =0 (15)
0 2 2
( Re X = Re X ) Re fX ; AX g =0) .
With straightforward mathematical calculus we can nd the generators requested. In
order to simplify following considerations let us p ose
X = a + ib + jc + kd ; A = q + q j i + q j j + q j k
0 1 2 3
where q = + i + j + k (m =0;1;2;3) are real quaternions.
m m m m m
The only quantities whichwemust calculate are
Re fX ; Xg ; Re fX ;iXig ; Re fX ;iXg ; Re fX ;kXjg ;
in fact the other quantities can b e obtained from previous ones, by simple manipulati-
ons.
2 2 2 2 2 2 2 2
Re fX ; Xg = 2(+a b c d ) ; Re fX ;iXig =2( a +b c d )
2 2 2 2 2 2 2 2
Re fX ;jXjg=2( a b +c d ) ; Re fX ;kXkg=2( a b c +d )
Re fX ;iXg = Re fX ; X ig = 4ab ; Re fX ;kXjg=Re fX ;jXkg=4cd
Re fX ;jXg = Re fX ; X j g = 4ac ; Re fX ;jXig =Re fX ;iXjg =4bc
Re fX ;kXg = Re fX ; X k g = 4ad ; Re fX ;iXkg =Re fX ;kXig =4bd :
The previous relations imply the following conditions on the real parameters of the
generator A
=0 ; =0
0 1
=0 ; =0
2 3
= = ; = =
0 1 0 2
= = ; = =
0 3 2 3
= =' ; = =:
1 2 3 1
We can immediately recognize the Lorentz generators given in section 4. 13
App endix B
Weintro duce the usual four-vector x by the following quaternion
0 1 2 3
X = x + ix + jx + kx ;
and de ne a scalar pro duct of twovectors X , Y by
+
(X ;gY) =Re (X g Y )=x g y ; (16)
R
where g is the generalized quaternion
1
(1 + i j i + j j j + k j k ) :
2
We can de ne a real norm (or metric)
+
(X ;gX) =Re (X g X )= x g x :
R
The vectors which transform under a Lorentz transformation L, will b e denoted by
0
X = LX ;
with L real linear op erators (see eq. (12)).
From the p ostulated invariance of the norm we can deduce the generators of Lorentz
group.
If we consider in nitesimal transformations
L =1+A ;
wehave
0+ 0 + + + +
Re (X g X )=Re (X g X + X (A g + g A)X )= Re (X g X ) ;
and therefore
+
A g + g A =0 : (17)
Using real scalar pro ducts, given an op erator
A = q + p j i + r j j + s j k;
q ; p; r;s 2H ;
R
we can write its hermitian conjugate as follows
+ + + + +
A = q p j i r j j s j k:
Then the equation (17) can b e rewritten as
g A + h:c: =0 : 14
If we p ose
g A = B =~q+~pji+~r j j +~sj k;
we obtain the following conditions on the op erator B
Re q~ = Vec p~= Vec r~= Vec s~=0 :
Noting that A = gB we can quickly write the generators of Lorentz group. We give
explicitly an example
1
A = g (1 j i)= ( i+1 j i+j j k k j j) ;
1
2
1
A = gi= (i 1ji+j jk kjj) ;
2
2
i 1 j i
A=A A = ;
1 2
2
k j j j j k
~
: A = A +A =
1 2
2
References
[1] Feynman RP - The Feynman Lectures on Physics,vol. I - part 1.
Inter Europ ean Editions, Amsterdam (1975).
[2] Finkelstein D, Jauch JM, Schiminovich S, Speiser D -
J. Math. Phys. 3, 207 (1962); 4, 788 (1963).
Finkelstein D, Jauch JM, Speiser D - J. Math. Phys. 4, 136 (1963).
Ad ler SL -Phys. Rev. D21, 550 (1980), D21, 2903 (1980);
Ad ler SL -Phys. Rev. Letts 55, 783 (1985);
Ad ler SL -Phys. Rev. D34, 1871 (1986);
Ad ler SL - Comm. Math. Phys. 104, 611 (1986);
Ad ler SL -Phys. Rev. D37, 3654 (1988);
Ad ler SL - Nuc. Phys. B415, 195 (1994).
De LeoS,Rotel li P -Phys. Rev. D45, 575, (1992);
De LeoS,Rotel li P - Quaternions Higgs and Electroweak Gauge Group - to app ear
in Int. J. of Mo d. Phys. A.
De LeoS -Quaternions for GUTs - submitted for publication in J. Math. Phys.
Dimitirc R, Goldsmith B - Math. Intell. 11, 29 (1989).
Horwitz LP, Biedenharn LC - Ann. of Phys. 157, 432 (1984).
Horwitz LP -J. Math. Phys. 34, 3405 (1993).
Razon A, Horwitz LP - Acta Appl. Math. 24, 141 (1991); 179 (1991);
Razon A, Horwitz LP - J. Math. Phys. 33, 3098 (1992). 15
Rembielinski J -J.Phys. A11, 2323 (1978), A13, 15 (1980);
Rembielinski J -J.Phys. A13, 23 (1980), A14, 2609 (1981).
Nash CC and Joshi GC - J. Math. Phys. 28, 2883, 2886 (1987);
Nash CC and Joshi GC -Int. J. Theor. Phys. 27, 409 (1988);
Nash CC and Joshi GC -Int. J. Theor. Phys. 31, 965 (1992).
[3] Finkelstein D, Jauch JM, Speiser D - Notes on quaternion quantum mechanics
in Logico-Algebraic Approach to Quantum Mechanics I I,Hooker (Reidel, Dordrecht
1979), 367-421.
Hamilton WR - Elements of Quaternions - Chelsea Publishing Co., N.Y., 1969.
GilmoreR-Lie Groups, Lie Algebras and Some of their Applications - John
Wiley & Sons, 1974.
Altmann SL - Rotations, Quaternions, and Double Groups - Claredon, Oxford,
1986.
Ad ler SL - Quaternion quantum mechanics and quantum elds - Oxford UP,
Oxford, 1995.
[4] De LeoS,Rotel li P - Prog. Theor. Phys. 92, 917 (1994).
[5] De LeoS,Rotel li P - Nuovo Cim. B110, 33, (1995).
[6] Penrose R, Rind ler W - Spinors and space-time - Cambridge UP, Cambridge,
1984 ( vol. 1, pag. 23).
[7] Bott R, Milnor J - Bull. Amer. Math. So c. 64, 87 (1958).
KervaireM- Pro c. Nat. Acad. Sci. 44, 280 (1958).
[8] Frobenius G - J. Reine Angew. Nath. 84, 59 (1878).
[9] OkuboS-Intro duction to Octonion and Other Non-Asso ciative Algebras in
Physics - unpublished.
[10] Anderson R, Joshi GC -Phys. Essays 6, 308 (1993).
[11] Edmonds JD -Int. J. Theor. Phys. 6, 205 (1972).
Edmonds JD -Found. of Phys. 3, 313 (1973).
Edmonds JD - Am. J. Phys. 42, 220 (1974).
[12] Gough W - Eur. J. Phys. 7, 35, (1986); 8, 164, (1987); 10, 188, (1989).
[13] Abonyi I, Bito JF, Tar JK -J.Phys. A 24, 3245 (1991).
st
[14] Gursey F - Symmetries in Physics (1600-1980 ): Pro ceedings of the 1 Inter-
national Meeting on the History of Scienti c Ideas - Seminari d' Historia de les
Ciences, Barcellona, Spain, 557 (1983). 16
[15] Synge JL - Quaternions, Lorentz Transformation, and the Conway Dirac Ed-
dington Matrices - Dublin Institute for Advanced Studies, Dublin, 1972.
[16] Wu-Ki Tung - Group Theory in Physics -World Scienti c, Singap ore, 1985.
[17] Rotel li P -Mod.Phys. Lett. A 4, 933 (1989).
[18] De LeoS-One-comp onent Dirac equation - submitted for publication in Int. J. of
Phys. A.
[19] Dirac PAM - Pro c. R. So c. London A133, 60 (1931). 17