ISSN 2393-9257 Lorentz transformations via Pauli matrices Z. Ahsan 1, J. López-Bonilla 2, B. Man Tuladhar 3, 1 Dept. of Mathematics, Aligarh Muslim University, Aligarh 202 002, Aligarh, India 2 ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 5, 1er. Piso, Col. Lindavista 07738, México DF; jlopezb@ipn.mx 3 Kathmandu University, Dhulikhel, Kavre, Nepal Abstract- We exhibit expressions, in terms of Pauli matrices, which directly generate Lorentz transformations in Minkowski space. Key words- Pauli matrices, Lorentz transformations, Infeld-van der Waerden symbols Council for Innovative Research Peer Review Research Publishing System Journal of Advances in Natural Sciences Vol. 2, No. 1 editorjansonline@gmail.com www.cirworld.com 49 | P a g e September 25, 2014 ISSN 2393-9257 x   ct , x, y, z  , j  0,...,3, g   Diag 1,1,1,1 . If it is necessary to employ another frame of reference, then the new coordinates In space time an event is represented j by with jr connected with metric ~ xr where the Lorentz matrix L~ (1) , verifies the restriction : L j a g rj Lr b  g ab (2) , because the Minkowskian line element must remain invariant under L~ L~ , that is, ~ xr~ xr  x r xr .  ,  ,  ,     1, then the components of homogeneous Lorentz transformation L~ can be written in the form [1-4]: From (2) we see that  has six degrees of freedom, which permits to work with four complex numbers  1  *   *   *   * , 2 i L0 2   *    *  c.c. , 2 1 L10   *   *  c.c. , 2 i L12   *     c.c. , 2 i L2 0   *   *  c.c. , 2 1 L2 2   *   *  c.c. , 2 1 L3 0   *   *   *   * , 2 i L3 2   *    *  c.c. , 2 L0 0         are x j via the linear transformation: ~ x j  Lj r x r that the            1 *     *  c.c. , 2 1 L0 3   *   *   *   * , 2 1 L11   *   *  c.c. , 2 1 L13        c.c. , 2 i L21   *   *  c.c. , 2 i L2 3   *   *  c.c. , 2 1 L31   *    *  c.c. , 2 1 L3 3   *   *   *   * , 2 L01           (3)   such  where c.c. means the complex conjugate of all the previous terms. Therefore, any complex 2x2 matrix [4-7]:     ~     , Det       1 , ~   (4) generates a Lorentz matrix via (3). The following relations, which are not explicitly in the literature, give us directly all the components (3): 50 | P a g e September 25, 2014 ISSN 2393-9257 † 1 L     ar   a j  b r  b j 2 1 L 0    j r Q j r 2 ,   0,...,3 , ,  ,  1,2,3 1 L0     j k R j k 2 ,   1, 2,3 (5) such that:  *  *  † †     , R ~  ~  * * , Q ~  ~ , ~ ~~   † (6) with the Infeld-van der Waerden symbols [8-11]:       ~I  1 0 0 1   ,  ab    - 1ab    x   0 1 1 0 i  2ab    - 2ab   - y   -i 0  ,  3ab    - 3ab    z    0 ab where 0 ab  j , j  x, y, z 1  0 1 0     0 -1 (7) are the known Cayley-Sylvester-Pauli matrices [4, 6, 12-14]. The expressions (5) show explicitly a direct relationship between L~ and U ~ , which may be useful in applications of spinorial calculus [11] in Minkowski spacetime. 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 (1965) 3. J. López-Bonilla, J. Morales, G. Ovando, On the homogeneous Lorentz transformation, Bull. Allahabad Math. Soc. 17 (2002) 53-58 4. J. López-Bonilla, J. Morales, G. Ovando, Quaternions, 3-rotations and Lorentz transformations, Indian J. Theor. Phys. 52, No. 2 (2004) 91-96 5. J. L. Synge, Quaternions, Lorentz transformations, and the Conway-Dirac- Eddington matrices, Comm. Dublin Inst. Adv. Stud. A 21 (1972) 1-67 6. L. H. Ryder, Quantum field theory, Cambridge University Press (1985) 7. I. Guerrero, J. López-Bonilla, L. Rosales, Rotations in three and four dimensions via 2x2 complex matrices and quaternions, The Icfai Univ. J. Phys. 1, No. 2 (2008) 7-13 8. B. L. van der Waerden, Spinoranalyse, Nachr. Ges. Wiss. Göttingen Math.-Phys. (1929) 100-109 9. L. Infeld, Die verallgemeinerte spinorenrechnung und die Diracschen gleichungen, Phys. Z. 33 (1932) 475-483 10. L. Infeld, B. L. van der Waerden, Die wellengleichung des elektrons in der allgemeinen relativitätstheorie, Sitz. Ber. Preuss. Akad. Wiss. Physik-Math. Kl 9 (1933) 380-401 11. P. O’Donnell, Introduction to 2-spinors in general relativity, World Scientific, Singapore (2003) 12. A. Cayley, A memoir on the theory of matrices, London Phil. Trans. 148 (1858) 17-37 13. J. Sylvester, On quaternions, nonions and sedenions, Johns Hopkins Circ. 3 (1884) 7-9 14. W. Pauli, Zur quantenmechanik des magnetischen elektrons, Zeits f. Physik 43 (1927) 601-623 51 | P a g e September 25, 2014