Microsoft Word - Carvaja-Gamezl _30-34_.doc


 

  

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Publisher: Research Council of Science and Technology, Biratnagar, Nepal     p.30 
 
 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology and Mathematics 

ISSN 2091-0762 (print), 2382-5340 (online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

 

 Quaternions, 2x2 complex matrices and Lorentz transformations  

 
 

 

B. E. Carvajal-Gámez 
1
, I. J. Guerrero-Moreno 

2
, J. López-Bonilla 

2*
 

 

1 SEPI-ESCOM, Instituto Politécnico Nacional (IPN), Av. Bátiz, 07738 México DF, 
2 ESIME-Zacatenco, IPN, Edif. 5, Lindavista, CP 07738 México DF;    

*Email: jlopezb@ipn.mx 
Accepted for publication: December 9, 2014 

 

Abstract 

We show that the matrix method to construct Lorentz transformations  permit to deduce the 
corresponding quaternionic procedure. 
DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11671          © 2014 RCOST:  All rights reserved.         
 

Keywords:  Unitary quaternions; 2x2 complex matrices; Lorentz’s matrix. 

 

 

 

1. Introduction 
 

  In spacetime an event is represented by ( ) ( )zyxctx j ,,,= , ,3,...,0=j  with the metric 

( ) ( )1,1,1,1 Diag −−−=jrg . If it is necessary to employ another frame of reference, then the new 

coordinates rx~  are connected with jx  via the linear transformation: 
 

,~ r
r

jj xLx =                                                      (1) 
 

where the Lorentz’s matrix L~  verifies the restriction: 

 

,abb
r

rja
j gLgL =                                                      (2) 

 

because the Minkowskian line element must remain invariant under L~ , that is, r

r

r

r xxxx =~~  

   From (2) we see that L~  has six degrees of freedom, which permits to work with four complex 

numbers δγβα ,,,  such that 1=− βγαδ , then the components of homogeneous Lorentz 

transformation L~ can be written in the form [1-5] (i = √-1): 

 

( ) ( ) ,  c.c. 
2
1

,
2
1 **

1
0****

0
0 ++=+++= δγβαδδγγββαα LL  

 
 



 

  

Carvajal-Gámez et al. / BIBECHANA  (2015) 30-34: p. 31 
 

( ) ( ) ,
2
1

,c.c.
2

****
3

0**
2

0 δδγγββααδγβα −+−=++= L
i

L  

( ) ( ) ,c.c.  
2
1

,c.c. 
2
1 **

1
1**

0
1 ++=++= βγδαδβγα LL  

( ) ( ) (3)                         c.c. 
2
1

,c.c. 
2

**
3

1**
2

1 +−=++= δβγαβγδα L
i

L

( ) ( ) , c.c.
2

,c.c.
2

**
1

2**
0

2 ++=+−= βγαδδβαγ
i

L
i

L  

 

( ) ( ) ,c.c.
2

,c.c.
2
1 **

3
2**

2
2 ++=+−= δβαγγβδα

i
LL  

 

( ) ( ) ,
2
1

,c.c.
2

****
3

3**
2

3 δδγγββααδγβα +−−=+−= L
i

L  

 
where c.c. means the complex conjugate of all the previous terms. 
Therefore, any complex 2x2 matrix: 
 

,1~ Det,~ =−=







= βγαδ

δγ
βα

∪∪                                (4) 

 
generates one Lorentz matrix through (3), or via the expression [6-8]: 
 

=














+

−+
3~~2~1~

2~1~3~~

-      

       

xxixx

xixxx
o

o

 ,~       

       
~

†

321

213

∪∪ 










−+

−+

xxixx

xixxx
o

o

               (5) 

where: 

        ( ) ,1~ Det~Det ,~
*†

**

**
† ==








= ∪∪∪

δβ
γα

                                (6) 

 
thus the determinant of (5) implies the invariance of the line element: 
 
 

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .
232221223~22~21~2~

xxxxxxxx
oo

−−−=−−−                      (7) 
 

 
   In the next Section, from (5) we shall deduce a quaternionic expression to produce Lorentz 
transformations. 
 
 
2. Lorentz’s matrix via quaternions 
 

   The matrix (4) can be written in the form [9, 10]: 
 
 
 

( ) ( ) ,c.c. 
2

1
,

2

1 **
1

3****
0

3 +−=−−+= δγβαδδγγββαα LL



 

  

Carvajal-Gámez et al. / BIBECHANA  (2015) 30-34:  p.32 
 
 

,~I~ 3322110 σσσ iaiaiaa −−−=∪                                          (8) 

being jσ  the Cayley [11]-Sylvester [12]-Pauli matrices [13]: 

 

                          ,
1-0

01
 ,

0i

i-0
 ,

01

10
 321 








=








=








= σσσ                          (9) 

 
with the Euler-Olinde Rodrigues parameters [14]: 
   

( ) ( )

( ) ( )           ,
2

,
2
1

 ,
2

,
2

1

32

10

δαβγ

βγδα

−=−=

+=+=

i
aa

i
aa

                                  (10) 

 
 

If we employ the following formal association with the quaternionic units [15]: 
 

,    iσ , iσ, iσ,~I KJI →−→−→−→ 3211                  (11) 

then the properties of  the jσ  give us the rules of multiplication : 

 

 ,,   ,1 ,JI KIKJ KJ IKJI 222 ===−===                  (12) 
 
besides from (8) it appears the relationship: 
  

,      ~ 3210 KJIA aaaa +++=→∪                                     (13) 

where A is a unitary quaternion because the condition  1- =βγαδ and (10) imply: 
 

 ,       12
3

2
2

2
1

2
0 =+++= aaaaAA                                    (14) 

with  
.  3210 KJIA aaaa −−−=                                             (15) 

 
Therefore, any 2x2 complex matrix (4) has associated a unitary quaternion via (10) and (13). Now we 
shall apply this procedure to each matrix appearing in (5) to obtain its quaternionic version: 

 

R→








−+

−+
321

213

  

 

xxixx

xixxx
o

o

(16)                            ,      ct

,      3210

KJI

KJI

iziyix

ixixixx

+++=

+++=     

,        ~  ,    ~ 321
*

0
†

KJIAA **** aaaa −−−=→→ ∪∪  

 
thus (5) implies [16-21]: 
 



 

  

Carvajal-Gámez et al. / BIBECHANA  (2015) 30-34:  p.33 
 

,~ *AARR =                                                        (17) 
which generates the linear mapping (1) and reproduces all the components (3) because (10) gives the 
link: 

;,

,,

3012

1230

iaaiaa

iaaiaa

+=−=

−−=−=

δγ

βα
                                   (18) 

 
Then (17) is a quaternionic factory to elaborate Lorentz transformations. 
 

 

 

References 
 
[1] J. Aharoni, The special theory of relativity, Clarendon Press, Oxford (1959). 

[2]  J. L. Synge, Relativity: the special theory, North-Holland Pub., Amsterdam,  Chap. 4, Sec. 11. 
(1965) 

[3] P. J. Greenberg and J. P. Knauer, Stud. Appl. Math. 53 (1974) 165. 

[4] J. López-Bonilla, J. Morales and G. Ovando, Bull. Allahabad Math. Soc. 17 (2002) 53-58. 

[5] Z. Ahsan, J. López-Bonilla and B. Man Tuladhar, J. of Advances in Natural Sciences 2, No. 1 
(2014) 49-51. 

[6] E. Cartan, Bull. Soc. Math. de France 41 (1913) 53-96. 

[7] O. Veblen, Proc. Nat. Acad. Sci. USA 19 (1933) 462-474. 

[8] W. Rindler, Am. J. Phys. 34, No. 10 (1966) 937-942. 

http://dx.doi.org/10.1119/1.1972301 

[9] E. Piña G., Am. J. Phys. 51, No. 4 (1983) 375-379. 

http://dx.doi.org/10.1119/1.13253 

[10] G. F. Torres del Castillo, 3-D spinors, spin-weighted functions and their applications, 
Birkhäuser, Boston (2003) p. 10. 

http://dx.doi.org/10.1007/978-0-8176-8146-3 

[11]  A. Cayley, London Phil. Trans. 148 (1858) 17-37. 

[12]  J. Sylvester, Johns Hopkins Circ. 3 (1884) 7-9. 

[13]   W. Pauli, Zeits. für Physik 43 (1927) 601-623. 

 http://dx.doi.org/10.1007/BF01397326 

[14] R. E. Roberson, AIAA Journal 6 (1968) 5-71. 

[15] J. Kronsbein, Am. J. Phys. 35, No. 4 (1967) 335-342. 

 http://dx.doi.org/10.1119/1.1974074 

[16]  L. Silberstein, Phil. Mag. 23, No. 137 (1912) 790-809. 

 http://dx.doi.org/10.1080/14786440508637276 



 

  

Carvajal-Gámez et al. / BIBECHANA  (2015) 30-34:  p.34 
 

 

[17]  C. Lanczos, Zeits. für Physik 57 (1929) 447-473. 

 http://dx.doi.org/10.1007/BF01340274 

[18] C. Lanczos, The variational principles of mechanics, University of Toronto Press,  Chap. 9, Sec. 
4 (1970). 

[19]  J. L. Synge, Comm. Dublin Inst. Adv. Stud. Ser. A, No. 21 (1972) p. 19. 

[20]  R. Penrose and W. Rindler, Spinors and space-time, vol. 1, Cambridge University Press,  Chap. 
1, Sec. 2 (1984). 

[21]  R. Penrose, The road to reality, Ed. Jonathan Cape, London,  Chap. 24, Sec. 6 (2004).